Chapter 2 Coalition Formation: A Game-Theoretic Approach

COALITION FORMATION Henk A.M. Wilke (ed.) 0 Elsevier Science Publishers B.V. (North-Holland), 1985

29

Chapter 2 COALITION FORMATION: A GAME-THEORETICAPPROACH

Wim J. van der Linden

Twente University of Technology Enschede The Netherlands

&

Albert Verbeek University of Utrecht Utrecht The Netherlands

Game t h e o r y i s n o t h i n g e l s e t h a n t h e r e s u l t of r i g o r o u s mathematics a p p l i e d t o a s s u m p t i o n s about t h e b e h a v i o r of a c t o r s i n s i t u a t i o n s of p a r t i a l conf l i c t . From g a m e - t h e o r e t i c models it i s p o s s i b l e t o p r e d i c t what t h e i r s t r a t e g i c c h o i c e s w i l l b e , how t h e y w i l l u n i t e i n t o c o a l i t i o n s a g a i n s t each o t h e r , and what d i s t r i b u t i o n s of p a y o f f s w i l l o c c u r . The aim of t h i s c h a p t e r i s t o p r e s e n t t h e v a r i o u s g a m e - t h e o r e t i c models of c o a l i t i o n f o r ming b e h a v i o r a s w e l l as r e s u l t s from e x p e r i m e n t s t o t e s t t h e i r p r e d i c t i o n s e m p i r i c a l l y . S e c t i o n s 2.2 - 2.5 d e a l w i t h t h e o r e t i c a l work on game t h e o r y and c u l m i n a t e i n S e c t i o n 2.5 where t h e major g a m e - t h e o r e t i c models o f coal i t i o n forming a r e g i v e n . T h i s s e c t i o n c o n t a i n s some new m a t e r i a l . P a r t i c u l a r l y i n 2.5.4 and 2.5.6 - 2.5.8 i t is shown how gametheory i s a p p l i e d t o c o a l i t i o n f o r m a t i o n by i n t r o d u c i n g t h e r e a d e r t o t h e b a s i c a s s u m p t i o n s and c o n c e p t s of t h i s b r a n c h of a p p l i e d m a t h e m a t i c s . These s e c t i o n s a l s o p r o v i d e a taxonomy of models i n game t h e o r y and p r e s e n t a n a l y s e s of a few games t h a t have become p o p u l a r among game t h e o r i s t s b e c a u s e t h e y f o r m a l i z e w e l l known t y p e s of s o c i a l c o n f l i c t s . A s u r v e y of r e s u l t s from e x p e r i m e n t s t h a t have been run t o test g a m e - t h e o r e t i c p r e d i c t i o n s of c o a l i t i o n forming p r o cesses i s p r e s e n t e d i n S e c t i o n 2.6. I n t h i s s e c t i o n s p e c i a l emphasis i s given t o t h e m e t h o d o l o g i c a l s t a t u s of t h e s e p r e d i c t i o n s , t o t h e e x p e r i m e n t a l methods i n u s e , and t o t h e t r e n d s i n t h e outcomes of t h e s e e x p e r i m e n t s . T h i s s e c t i o n e n d s w i t h a p l e a f o r a more r i g o r o u s e x p e r i m e n t a l c o n t r o l of subj e c t ' s u t i l i t y f u n c t i o n s . F i n a l l y , some f u r t h e r t e c h n i c a l d e t a i l s and s p e c i f i c examples a r e g i v e n i n t h e Appendices c o n c l u d i n g t h i s c h a p t e r . Contents

2.1 2.2 2.3 2.4 2.5 2.6

Introduction B a s i c a s s u m p t i o n s and a b s t r a c t i o n s Some c h a r a c t e r i s t i c s of games S o l u t i o n c o n c e p t s f o r normal form game S o l u t i o n c o n c e p t s f o r c h a r a c t e r i s t i c f u n c t i o n games A s u r v e y of e x p e r i m e n t a l game t h e o r y Appendices

30 33 45 55 64 86 107

30

W.J. van der Linden and A. Verbeek

2.1 INTRODUCTION

2.1.1 GAME THEORY Game theory deals with safe or optimal decisions in social, economic, or political situations where each of the participants has only partial control of the outcomes. In most of these situations the participants (or "actors") have conflicting interests, either as direct opponents or in more collateral way. The presence of conflicting interests is an important factor in explaining such phenomena as negotiation, cooperation, and coalition formation. By its formalizations, game theory elucidates an impressive (but non-exhaustive) number of mechanisms at work in human decision making under conditions of uncertainty, in bargaining processes, group decision making, and social arbitration. The power of formal theory is the precision required in its formulation. Although game theory has yielded many valuable results, often with a strong intuitive appeal, we feel that its main achievement has been that several assumptions and definitions have become highly explicit and precise. An example is the Shapley value which measures an operationalization of power of, for instance, each member of the U.N. Security Council (for details, see section2.5.5E.On the other hand, many results have been obtained that are rather counterintuitive or even paradoxical. The prisoner's dilemma is perhaps the best known example; more will be given in section 5. Here game theory culminates (some may say: breaks down) into a number of precisely and frugally described conflicts successfully resisting any intelligent, rational attempt at solution or arbitration. Nevertheless, such conflicts abound in real life, so that it is not surprising that real life solutions are seldom cheered with enthusiasm.

Chapter 2 1 Coalition formation: a game-theoretic approach 2.1.2

31

A I M O F THlS CHAPTER

The a i m o f t h i s c h a p t e r i s t w o f o l d . F i r s t , it p r o v i d e s an i n f o r m a l i n t r o d u c t i o n t o g a m e - t h e o r e t i c c o n c e p t s and r e s u l t s r e l e v a n t t o c o a l i t i o n formation ( s e c t s . 2 . 2 - 2 . 5 ) .

W e s h a l l s t a r t , however, w i t h a d i s c u s s i o n of

t h e b a s i c a s s u m p t i o n s , c o n c e p t s , and a taxonomy of models t h a t u n d e r l i e t h e a p p l i c a t i o n s of game t h e o r y t o n e g o t i a t i o n s and c o a l i t i o n f o r m a t i o n . Hence, t h e theme o f c o a l i t i o n f o r m a t i o n w i l l n o t be dominant u n t i l s e c t i o n 2 . 5 . S e c t i o n 2 . 5 a l s o c o n t a i n s some new m a t e r i a l , p a r t i c u l a r l y i n 2 . 5 . 4 . 2.5.6-2.5.8.

and

S e c o n d l y , t h e c h a p t e r o f f e r s a s u r v e y o f t h e l i t e r a t u r e on

e x p e r i m e n t s on c o a l i t i o n forming b e h a v i o r i n g a m e - t h e o r e t i c

settings

( s e c t . 2 . 6 ) . A g r e a t v a r i e t y of e x p e r i m e n t s have been run t o t e s t gamet h e o r e t i c " s o l u t i o n s " o f c o a l i t i o n forming p r o c e s s e s . S p e c i a l emphasis

w i l l b e g i v e n t o t h e i s s u e whether g a m e - t h e o r e t i c

resultslendthemselves

t o e x p e r i m e n t a l t e s t i n g , t o a d e s c r i p t i o n of t h e t y p i c a l e x p e r i m e n t a l methods i n u s e , and t o t h e most i m p o r t a n t c o n c l u s i o n s t h a t c a n b e drawn from t h e s e e x p e r i m e n t s .

F o r t h e d e s c r i p t i o n of c o n f l i c t s o f i n t e r e s t game t h e o r y h a s produced two d i f f e r e n t c l a s s e s of models: c h a r a c t e r i s t i c f u n c t i o n games and normal form games. H i s t o r i c a l l y , c h a r a c t e r i s t i c f u n c t i o n games were a t f i r s t i n t r o d u c e d as a n a t u r a l r e p r e s e n t a t i o n of c e r t a i n normal form games ( c f . Appendix 4 ) b u t g r a d u a l l y i t h a s become c l e a r t h a t c h a r a c t e r i s t i c f u n c t i o n games are much more i m p o r t a n t f o r d e s c r i b i n g c o a l i t i o n f o r m a t i o n t h a n normal form games. I n t h i s c h a p t e r t h e two models are developed s i d e by s i d e s u p p o s i n g thal; t h e r e a d e r w i l l be i n t e r e s t e d i n comparing t h e i r p r o p e r t i e s . Those i n t e r e s t e d o n l y i n c h a r a c t e r i s t i c f u n c t i o n games, however, c a n r e a d s e c t i o n s 2 . 2 . 1 ,

2.3.1-2.3.3,

2.3.7-2.3.9,

2 . 5 , and 2.6

and o m i t t h e o t h e r s e c t i o n s w i t h o u t l o s s of c o n t i n u i t y . Both t h e t h e o r e t i c a l and e x p e r i m e n t a l l i t e r a t u r e on game t h e o r y are f a r t o o e x t e n s i v e t o o f f e r a c o m p l e t e l i s t o f r e f e r e n c e s , l e t alone t o g i v e

a c o m p l e t e r e v i e w o f t h e work done. O u t s t a n d i n g t e x t b o o k s f o r s o c i a l s c i e n c e a p p l i c a t i o n s o f game t h e o r y s t i l l a r e Luce and R a i f f a (1957) and von Neumann and Morgenstern ( 1 9 4 4 ) ; o t h e r r e f e r e n c e s are A r r o w and Hahn ( 1 9 7 1 ) , B r a i t h w a i t e ( 1 9 5 5 ) , Davis (19701, Hamburger ( 1 9 7 9 ) , J o n e s ( 1 9 6 0 ) , Rapoport (1966, 1 9 7 0 ) , S c h e l l i n g ( 1 9 6 0 ) , Shubik (1964), and Vorob'ev (1977). J o u r n a l s w i t h many c o n t r i b u t i o n s on game t h e o r y a r e American

32

W.J. van der Linden and A. Verbeek

P o l i t i c a l Science Review, Behavioral S c i e n c e , Comparative P o l i t i c a l S t u d i e s , J o u r n a l of C o n f l i c t R e s o l u t i o n , J o u r n a l o f Experimental S o c i a l Psychology, J o u r n a l of Mathematical Psychology, J o u r n a l of Mathematical Sociology, J o u r n a l of P e r s o n a l i t y and S o c i a l Psychology, J u r i m e t r i c s J o u r n a l , and S i m u l a t i o n and G a m e s .

2.1.3

AN INTRODUCTORY EXAMPLE

W e conclude t h i s i n t r o d u c t i o n w i t h an example of a t y p i c a l c o a l i t i o n -

formation problem i n a game-theoretic f o r m u l a t i o n . Only t h e r u l e s of t h e game a r e e x p l a i n e d h e r e , and i t i s l e f t t o t h e r e a d e r t o ponder on r e a s o n a b l e s o l u t i o n s . We s h a l l r e t u r n t o t h i s example s e v e r a l times. EXAMPLE: "Me and my aunt'' (quoted from Davis and Maschler, 1965, p . 236). "My a u n t ( a c t o r A ) and I ( a c t o r I ) can e n t e r a p a r t n e r s h i p i n

which w e s h a l l both win 100 u n i t s . "In p r i n c i p l e " , w e a g r e e t o form t h e p a r t n e r s h i p , provided t h a t w e r e a c h an agreement on t h e s p l i t . Each of us have o t h e r a l t e r n a t i v e s which a r e shown i n D i s p l a y 1: One can see t h a t my aunt need convince one, and o n l y

Display 1.

The 5 - a c t o r game "Me and my Aunt" (Me

I , and

Aunt = A). Only one of f i v e c o a l i t i o n s i n d i c a t e d c a n

form and win $ 100: A I , AP, AQ, AR or IPQR. How w i l l t h e p r i z e be divided?

33

Chapter 2 J Coalition formation: a game-theoretic approach o n e , of t h r e e a c t o r s , P , Q , and R , w h i l e I need t h e agreement of a l l t h e s e a c t o r s a s my o n l y a l t e r n a t i v e . I n t u i t i v e l y , my a u n t

seems s t r o n g e r t h a n I and i t seems t h a t s h e s h o u l d g e t more t h a n 50, i f w e b o t h form a c o a l i t i o n . I f so, how much?" 2.2

BASIC ASSUMPTIONS AND ABSTRACTIONS

2.2.1

CHARACTERISTIC FUNCTION GAMES

Two d i f f e r e n t models are w i d e l y used t o d e s c r i b e c o n f l i c t s o f i n t e r e s t : c h a r a c t e r i s t i c f u n c t i o n games and normal form games. A t y p i c a l example o f a c h a r a c t e r i s t i c f u n c t i o n game i s "Me and my a u n t " d e s c r i b e d i n t h e i n t r o d u c t i o n . G e n e r a l l y , a c h a r a c t e r i s t i c f u n c t i o n game c o n s i s t s of t h e following elements: There i s a set of n a c t o r s u s u a l l y d e n o t e d b y 1, 2 , 3,

...,

n. The game

i s s u p e r v i s e d by a n umpire who c a n make p a y o f f s . For e a c h s u b s e t of a c t o r s , c a l l e d a c o a l i t i o n , t h e r e i s a payoff c a l l e d t h e

value of

the

c o a l i t i o n . A c o a l i t i o n d o e s n o t o b t a i n i t s v a l u e from t h e umpire u n l e s s

i t s a c t o r s unanimously a g r e e on two p o i n t s , namely, (1) t o form t h i s c o a l i t i o n and ( 2 ) t h e d i s t r i b u t i o n of t h e v a l u e o v e r t h e actors i n t h i s c o a l i t i o n . A f t e r r e c e i v i n g t h e i r i n d i v i d u a l p a y o f f s , t h e s e a c t o r s are removed from t h e game. For " M e and my a u n t " t h e v a l u e s c o u l d be d e f i n e d a s i n D i s p l a y 2. The f u n c t i o n a s s i g n i n g a v a l u e t o e a c h c o a l i t i o n i s

co a1i t i on

characteristic function

value

D i s p l a y 2 . The c h a r a c t e r i s t i c f u n c t i o n of t h e game

"Me and my Aunt" from D i s p l a y 1.

34

W.J. van der Linden and A. Verbeek

c a l l e d t h e c h a r a c t e r i s t i c f u n c t i o n . Formally, i t i s a f u n c t i o n mapping t h e s e t of a l l p o s s i b l e c o a l i t i o n s , C , i n t o t h e set of r e a l numbers, R :

v: C +R. I t is common t o r e q u i r e t h a t v s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n :

(1)

I f c o a l i t i o n s S1 and S v(S

us ) 1 2

2 V(S1)

2

are d i s j o i n t , t h e n

+ V(S2)

(superadditivity)

By d e f i n i t i o n , t h e v a l u e of t h e empty c o a l i t i o n i s e q u a l t o z e r o : v ( @ ) = 0 . Moreover, it assumed t h a t f o r at least one p a i r S1, S t h e s t r i c t g r e a t e r 2 t h a n s i g n i n (1) h o l d s . Such games are c a l l e d e s s e n t i a l . Other games, n o t meeting t h i s assumption, are c a l l e d i n e s s e n t i a l . With i n e s s e n t i a l games t h e r e i s no i n c e n t i v e at a l l t o any form of co-operation o r c o a l i t i o n formation; it i s e a s y t o prove t h a t

Thus it h o l d s f o r i n e s s e n t i a l games t h a t any c o a l i t i o n o b t a i n s j u s t as much by co-operation as i t s members would o b t a i n when o p e r a t i n g on t h e i r own. The f o l l o w i n g remarks can be made: 1. The s u p e r a d d i t i v i t y c o n d i t i o n (1) c a n be j u s t i f i e d by o b s e r v i n g t h a t

S1 and S2 c a n always go t o t h e umpire s e p a r a t e l y and c o l l e c t v(S1)

and

v(S ) , r e s p e c t i v e l y . Note, however, t h a t when S1 a n d S j o i n i n t o S U S and 2 2 1 2 j o i n t l y c o l l e c t p r e c i s e l y v ( S ) + v ( S ) , more d i s t r i b u t i o n s o f t h e 1 2 v a l u e are p o s s i b l e t h e n when S1 r e c e i v e s v ( S ) and S r e c e i v e s v ( S 2 ) . 1 2 2 . Condition (1) can be d i s p e n s e d w i t h without much harm. In f a c t , w e d i d so i n Display 1. I f (1) d o e s n o t hold f o r c e r t a i n d i s j o i n t S1 and S

then r e a s o n a b l y speaking t h i s p r e c l u d e s t h e f o r m a t i o n o f S1uS2.

2' In a

s e n s e , c h a r a c t e r i s t i c f u n c t i o n games n o t s a t i s f y i n g (1) a r e e q u i v a l e n t t o games t h a t do s a t i s f y (1) (see Appendix 2 . 3 ) . 3. Furthermore, c h a r a c t e r i s t i c f u n c t i o n games s a t i s f y i n g (1) c a n be

c o n s i d e r e d a s p e c i a l b u t very important r e p r e s e n t a t i o n of c o - o p e r a t i v e normal form games with exchangeable u t i l i t y ( s e e Appendix 4 ) .

Chapter 2 f Coalition formation: a game-theoretic approach

35

Typically, in characteristic function games the possibility of negotiations is assumed. It is also assumed that each actor desires to maximize his part of the value paid off to the coalition. The latter assumption implies that no other interests are at stake, that is, that the game is not played for want of social contact, for instance, or to enjoy the thrill of taking a risk. In more technical terms, each actor's preferences for the various payoffs are one-dimensional, and strictly increasing with the payoff. Moreover, the payoffs are expressed in means of payment exchangeable between the actors and in an interpersonally comparable unit. The payoffs within a coalition are assumed to be mutually restrictive in an additive manner: if one actor gets more, the others must get equally much less. Finally, it is assumed that the value of each coalition is known to all actors. The assumptions needed to model situations of conflicting interests into characteristic function games impose certain limitations on their applicability. The most important limitations seem to be: (2)

The one-dimensionality and interpersonal comparability of the preferences.

(3)

The perspective of a limited duration of the coalition once it has formed.

(4)

The assumed perfect knowledge and constancy of the payoffs v(S) during the stage of negotiations.

As to (21, in for example government coalitions, marriages, business

mergings, and war alliance, a great variety of goals are usually involved, either short-term or long-term. Usually, these goals do not combine into one dimension, let alone into a dimension with the same scale unit for all parties involved. All these examples are no one-shot affairs but coalitions typically agreed on for an extended period, often of uncertain length. Nevertheless, in characteristic function games (and in the experimental settings generally used to verify their results) this perspective of durability is lacking and coalitions are assumed to break up immediately after reaching agreement and collecting the payoffs. As to ( 4 1 , it should be observed that the material and immaterial costs of

extensive negotiations (which are not known in advance) are not included

36

W.J. van der Linden and A. Verbeek

in the model. Moreover, the frustrations or the pleasure of real negotiations may change the actors' preferences, for instance into the direction of certain coalitions or for a fast settlement.

As an aside, we note that in real life the reputation of each actor is often based on the other actors' experience of his/her previous behavior and may play a critical role in the present game. Hence, present behavior will also affect one's chances in the future. This point will be further expanded in section 2.2.7. The reader is urged to spend some time on extending the characteristic function model beyond the above limitations. This is not so difficult. Finding generalizations leading to a powerful theory with interesting, non-trivial theorems, on the other hand, is a difficult job. A s will be seen in section 5, characteristic function game theory mainly

concentrates on predicting reasonable distributions of payoffs among the actors. 2.2.2

NORMAL FORM GAMES

We now return to the second class of models known as normal form games. These games consist of the following elements: For each actor a precise and exhaustive list of options open to him/her is available. In game theory these options are called strategies. They are prescriptions of how to act under every possible circumstance in a situation of conflicting interests. After an actor has decided on such a strategy, the prescription should be so precise that its execution can be delegated to a caretaker (e.g., a computer). For each combination of strategies (one for each actor), there is a precise description of the consequences or outcome of this combination for each actor. It is important to note that each actor contributes only one strategy to the combination of strategies but that his/her outcome dependson the full combination and is in this way also a function of the choice of the other actors. This formalizes the notion that each actor has only partial control of his/her situation. Further, it is assumed that for each actor there is a preference scheme, i.e., a list of all possible outcomes (linearly) ordered

Chapter 2 1 Coalition f o m t i o n : a game-theoretic approach

37

Unless s t a t e d o t h e r w i s e , i t is assumed t h a t t h e p r e f e r e n c e s

by p r e f e r e n c e .

a r e measured on an i n t e r v a l s c a l e i n which c a s e t h e y a r e called u t i l i t i e s or p a y o f f s . W e s h a l l u s e t h e two words a s synonyms. Note t h a t t h i s i n t e r v a l s c a l e assumption i s a l s o i m p l i c i t i n t h e f o r m u l a t i o n of c h a r a c t e r i s t i c f u n c t i o n games. F i n a l l y , each a c t o r is assumed t o have complete knowledge n o t o n l y about h i s / h e r own s t r a t e g i e s , outcomes, and p r e f e r e n c e s b u t a l s o ahout t h o s e of a l l o t h e r a c t o r s .

Normal form games a r e o f t e n p l a y e d i n s u c h a way t h a t no one a c t o r h a s any knowledge o f t h e a c t u a l c h o i c e s of any o t h e r u n t i l a l l t h e a c t o r s have s u b m i t t e d t h e i r c h o i c e o f s t r a t e g y t o t h e umpire. For two a c t o r s , l e t us s a y A and B , w e can r e p r e s e n t t h e above games a s a b i - m a t r i x game ( s e e D i s p l a y 3). I n a b i - m a t r i x game t h e rows c o r r e s p o n d t o t h e s t r a t e g i e s open t o A

,

t h e columns t o t h o s e open t o B , and e a c h

strategies for actor B B1

...

B2

Bc

strategies for actor A ‘41

D i s p l a y 3.

(Ull’Wll)

(Ul2,Wl2)

...

I

wlc)

The “ b i - m a t r i x r e p r e s e n t a t i o n “ of an a r b i t r a r y t w o - a c t o r normal form game. The two a c t o r s a r e : A ( t h e r o w c h o o s e r ) and B ( t h e column c h o o s e r ) . I f A c h o o s e s t h e i - t h row ( = o p t i o n = s t r a t e g y ) , and B c h o o s e s t h e j - t h

column, t h e consequences f o r A a r e

symbolized by u . . and t h o s e f o r B by w . . . Each a c t o r is assumed 1J

1J

t o have complete knowledge of t h i s b i - m a t r i x , b u t n o t n e c e s s a r i l y of t h e i n t e n t i o n s o f t h e o t h e r a c t o r . I f F-u t h e w may b e dropped from t h e b i - m a t r i x

everywhere,

and w e g e t a m a t r i x

game, a s i n D i s p l a y 4 . F o r o t h e r b i - m a t r i x games see D i s p l a y s 7 , 11, 12, 13, and 15.

38

W .J. van der Linden and A. Verbee k

cell contains two outcomes, the first for A and the second for B. If the preference of the two actors is on the same one-dimensional utility scale (see section 2.4. for particulars about utility measurement) and the game is strictly competitive in the sense that what A wins has to be paid for by B, it holds that w = -U everywhere. In such cases the notation is simplified if we drop the outcome for B from the matrix. This type of game is called a matrix game, An example of a matrix game is the game in which A and B each have a coin and each havetwo options: "show heads" or "show tails". If both choose the same option than A pays $2.00 to B. If A shows heads and B shows tails then

B pays $1.00 to A , whereas B pays $3.00 to A if the reverse occurs. The outcomes of the game can be set out as in Display 4. (Incidentally, would options open to actor E show heads

show tails

options open to actor A show heads

- 2

show tails

3

1 - 2

Display 4 . Example of a matrix game, that is a two-actor zero-sum normal form game, here with only two strategies for each actor. The payoff is in dollars. If it is positive, B pays A: if it

is negative A pays B. you rather be A or B? If you have not read section 2.4.1 yet and don't care much either way, we would like to invite you to a long series of plays.) The theory of normal games can be applied in a variety of situations: 1. Conflicts of interest in which the actors are opponents. This includes

many parlor games, war situations, and problems of the distribution of scarce commodities. 2. To optimize decisions under uncertainty. In a matrix game the actors are

usually uncertain about the actions planned by the others. Decision making under uncertainty is a special class of matrix games in which the opponent ("Nature") is indifferent to the outcome. An important branch

Chapter 2 f Coalition formation: a game-theoretic approach

39

of game theory is the decision-theoretic approach to statistics (see, e.g. Ferguson, 1967). 3. Co-operation and coalition formation in situations with more than two actors and only partial conflicts of interests. 4.

To measure the power of each actor in negotiation situations. An example is the Shapley value which will be discussed in section 2.5.5.

5 . To establish arbitration schemes for conflicts. A typical example is the

Nash solution of a normal form game discussed extensively in for instance Luce and Raiffa (1957, chap. 6). Of course, many real-life games and types of conflict cannot be modeled into a matrix game. Chess and bridge, for example, have the form and structure of matrix games.(See Appendix 1 for a more extensive commentary.) But they are far too complex to satisfy the assumption that all possible strategies are known beforehand. Yet it is enlightening to think of chess or bridge as matrix games and to theorize about their properties. In many negotiations the rules of the game and the strategies available are not explicitely known at the start. Also, many "psychological tricks" that are sometimes used in negotiation situations, such as well-timed assertiveness or deafness, a poker face, etc., are difficult to formalize. Often preferences only show up when people are forced to choose. Also, it can be very hard to obtain precise measurements of preferences for immaterial outcomes as, for example, friendship, trust, safety, o r the thrill of gambling. 2.2.3 ASSUMPTIONS OF RATIONALITY IN GAME THEORY It is often contended that the basic assumption underlying game theory is the rationality of the actors. Occasionally, this contention is not further explicated, and in that case it seems to be too vague to be refutable. Usually, however, more explicit notions are implied, a few of which will be reviewed in the following paragraphs. For authoritative reviews, see Howard (1971), Luce and Raiffa (1957),

Rapoport (19701, and

Riker and Ordeshook (1973, pp.8-44). The first notion is that game theory supposes actors to have only one goal in mind--maximization of their payoff. This is, however, an observation to be interpreted with care: it is in fact a tautology due to the way in which the entries in the game matrix are obtained. As will be seen in the

W.J. van der Linden and A . Verbeek

40

next section, game theory assumes that the payoffs are made in utilities-an abstract psychological concept introduced to describe the preferences of the actors. What actors prefer most receives the highest utility. So actors behave

they are maximizing their utilities, not because they

are rational but because their strongest preferences have been assigned the highest utility. "Explaining" the behavior of actors by referring to utilities rests on a circular argument: actors are assumed t o do what they prefer to do because they prefer to do so. Luce and Raiffa (1957, chap. 7) were among the first to point to the tautological character of this notion of rationality and to give it a thorough discussion.

In many experiments with and applications of game theory, it is assumed that actors strive to maximize their monetary profit. Experiments can be designed in which the monetary aspect of the outcome is predominant, and then subjects are often asked to try for a maximum outcome in money. Also, in many economic situations all outcomes are measured in terms of that outstanding medium of exchange: money. This notion of rationality, which equates utility to monetary value, is an extra-game-theoretic assumption. It cannot be defined with reference to game theory but must be verified separately. It may have been unfortunate that this assumption was already introduced on page 8 of the first treatise on game theory (von Neumann & Morgenstern, 1944). Luce and Raiffa's (1957) authoritative treatise on game theory, however, strongly de-emphasizes the assumption. To us, it does not seem proper to call this assumption "rationality", nor to suggest that it is typcial of game theory. As mentioned earlier, in normal and characteristic function games it is assumed that the order of preference for each actor is constant during negotiation and decision making. The actors are supposed not be get frustrated or carried away to the extent of changing their preferences. Occasionally, this stability is also called "rationality", but "constancy of preferences" describes much better what is assumed.

On the other hand, to suppose that game theory does not expect anything like

"rational behavior" from the actors would certainly go too far. When

actors are asked to state their preferences for all pairs of possible outcomes, utility theory expects them to be fully consistent. For example, if

Chapter 2 J Coalition formation: a game-theoretic approach

41

an a c t o r p r e f e r s outcome O1 o v e r O2 and O2 o v e r 03, h e / s h e i s assumed t o have a p r e f e r e n c e f o r O1 o v e r 03. F o r o t h e r e x a m p l e s , s e e t h e a x i o m a t i c t r e a t m e n t o f s t a n d a r d u t i l i t y t h e o r y i n Luce and R a i f f a ( 1 9 5 7 , c h a p . 2 ) . I n t h e case of minor or random i n c o n s i s t e n c i e s , some o f t h e p r o b a b i l i s t i c models of p r e f e r e n c e measurement ( e . g . B r a d l e y , 1 9 7 6 ; B r a d l e y 81 T e r r e y ,

1952; Mokken, 1971; Rasch, 1960) c o u l d be u s e d t o s c a l e t h e p r e f e r e n c e s , b u t when t h e i n c o n s i s t e n c i e s a r e p e r s i s t e n t even t h e s e models f a i l . Game t h e o r y , or more p r e c i s e l y u t i l i t y t h e o r y (which i s a t t h e b a s i s o f game t h e o r y ) , cannot s a t i s f a c t o r i l y d e a l w i t h such " i r r a t i o n a l i t i e s " .

I n summary,

game t h e o r y assumes a f u l l U n d e r s t a n d i n g o f t h e s t r u c t u r e o f t h e game and w e l l - d e f i n e d c o n s i s t e n t , and s t a b l e p r e f e r e n c e s .

2.2.4

KILITY

I n t h e p r e c e d i n g s e c t i o n i t h a s a l r e a d y been i n d i c a t e d t h a t t h e c o n c e p t o f u t i l i t y w a s i n t r o d u c e d t o measure p r e f e r e n c e s f o r d i f f e r e n t outcomes o f a game. G a m e t h e o r y assumes t h a t s u c h measurements c a n be made and t h a t t h e p a y o f f s t o t h e a c t o r s a r e made i n u n i t s o f u t i l i t y . I t does n o t assume, however, t h a t i n a l l games t h e u n i t s a r e i n t e r p e r s o n a l l y comparable o r t h a t u n i t s o f u t i l i t y can b e t r a n s f e r r e d from one a c t o r t o a n o t h e r s ( s e e s e c t i o n 3 . 4 ) . An i m p o r t a n t b r a n c h o f game t h e o r y , u t i l i t y t h e o r y , i s c o n c e r n e d w i t h

m e a s u r i n g u t i l i t y and w i t h t h e s c a l e p r o p e r t i e s of t h e s e measurements.

U t i l i t i e s a r e o n e - d i m e n s i o n a l i f t h e a c t o r s a r e a b l e t o make c o n s i s t e n t c h o i c e s between a l l p a i r s o f p o s s i b l e outcomes o f t h e game. Then an o r d i n a l u t i l i t y s c a l e can be d e r i v e d . I n games w i t h c l e a r - c u t outcomes a c t o r s a r e f o r c e d t o make p r e f e r e n c e c h o i c e s anyway, so i t seems n o t u n n a t u r a l t o assume a o n e - d i m e n s i o n a l s c a l e o f p r e f e r e n c e s . Measurement a t a n o r d i n a l l e v e l i s s u f f i c i e n t f o r a meaningful d e f i n i t i o n of g a m e - t h e o r e t i c c o n c e p t s s u c h a s dominance, e q u i l i b r i u m , and s a d d l e p o i n t . These c o n c e p t s w i l l be introduced i n s e c t i o n 2 . 4 .

I f a c t o r s want t o randomize t h e i r s t r a t e g i e s ( e . g . , u s e one s t r a t e g y w i t h p r o b a b i l i t y . 7 5 and a n o t h e r w i t h .25), i t is no l o n g e r enough

to infer

u t i l i t y measurements from c h o i c e s between p u r e outcomes b e c a u s e t h e u s e o f randomized s t r a t e g i e s l e a d s t o random o u t c o m e s . Von Neumann and M o r e e n s t e r n d e v e l o p e d a l o t t e r y method t o d e t e r m i n e u t i l i t i e s o f outcomes, i n which a c t o r s are a s k e d t o c h o o s e between random outcomes. Under c e r t a i n c o n d i t i o n s

W.J.van der Linden and A. Verbeek

42

this (1) leads to utility measurement on a (one-dimensional) interval scale, while ( 2 ) the behavior of actors can be described as aiming at maximizing their expected utility. This was originally proven by von Neumann and Moreenstern (1944, chap. 3; see also Luce & Raiffa, 1957, chap. 2). We feel that this is a remarkably strong and important theorem in the foundations of preference measurement. A different lottery method has been proposed byNovick and Lindley (1979). Although in general utility must not be equated with money, situations arise where the monetary aspect is so important that other aspects, such as the pleasure of social interaction or skilful play, can safely be ignored. But even then the actors' utilities will in general not be linear with money andmay resemble those in Display 5 (see the next page). For large sumsof money, it is often assumed that utility in increasing linearly inthe logarithm of money, or even more slowly. The fact that large losses are often experienced as disproportionally disastrous makes many insurances profitable both to the insurance company and the insurance taker. Consider a large loss L < < 0 which has some small probability p of occurrence. If utility u(.)

decreases faster than linearly between 0 and L, it holds that

pu(L) < u(pL) < 0 , i.e. that the expected utility is lower than the utility of the expected loss. Because an insurance company sells large numbers of policies, it can apply the law of large numbers to predict the total amount of claims to be paid. The company will therefore charge their clients this expected loss plus some overhead, to be paid to its employees and shareholders but it will not charge as much as their expected utility. Hence, for the clients the utility of the premium is larger (less negative) than their expected loss of taking no insurance. As a final comment, we repeat that the utilities of the outcomes are assumed to remain constant during negotiations and decision making. 2.2.5

OTHER ASPECTS OF PLAYING GAMES

In game theory, psychological aspects of game playing are usually not formalized into the model. Nevertheless, their effects can sometimes be clearly demonstrated. In many situations the outcomes are greatly affected, for example, by display of fear o r firmness, assertiveness, or a firm decision timely stated. The main problem is that these aspects are hard to measure

Chapter 2 1 Coalition formation: a game-theoretic approach

43

Utility

0

r 0

o

m

-

p

O

O

O

O

0

0

m -

0

O

0 V

"

L

8

P 8 0

& 8 o

0

(in

log $)

I0 0 0

o P 0

thun linear in $

Display 5 . The a u t h o r s ' impression of t h e i r u t i l i t y for money. For small amounts i t i s about l i n e a r , and t h e u t i l i t y o f a l o s s i s minus t h e u t i l i t y o f t h e same g a i n . For l a r g e r amounts t h e i n c r e a s e of u t i l i t y f o r g a i n s becomes slower than l i n e a r ; t h e decrease

o f u t i l i t y f o r l o s s e s becomes f a s t e r than l i n e a r .

W.J.van der Linden und A. Verbeek

44

and formalize. However, some Pttempts have been made to w e game-theoretic notions in models also formalizing these more substantive aspects, for example in models for cliques and balance in sociometric graphs, and minimal resource theory of coalition formation in politics. In sections 4.6, 5.4 and 5.6-5.8 it will be shown how rather general forms of threat

or blackmail can be incorporated in game theory. 2 .2.6

IMPERFECT KNOWLEDGE

An important assumption referred to earlier is that of each actor's perfect knowledge about the strategies, outcomes, values of the coalitions, and preferences of all other actors. In many games, however, the actors have no knowledge of the actual plans and intentions of the others and it seems hardly possible to assess the values of the coalitions. In various situations keeping one's strategy secret can be essential. The assumption o f perfect knowledge has often been critized as very strict and liable to unrealistic results. This is correct, especially when games are complex and the actors naive. On the other hand, the assumption can also be valued positively; explicating the assumptions of game theory forces us to realize how little we really know about decision making under conditions of imperfect knowledge, and how frequent imperfect understanding of the situation plays an essential role. Very little is known about the formalization of game-like situations with imperfect understanding of the possible outcomes or even of the rules o f the game. For some examples of misconception about the structure of games, see Snyder (1971). 2.2.7

GAMES AS ISOLATED EVENTS

Another implicit assumption in game theory is that

all consequences of

a

certain strategy o r a certain manner of negotiating have been incorporated in the actor's preferences for the various possible outcomes. An important example of such a consequence is harm to the actor's reputation, which generally will be kept as small as possible in view of future games with the same actors or with others who might learn about one's behavior in the present game. Once all consequences have been incorporated in the utilities, a game can be treated as an isolated event where decisions are merely based on the

Chapter 2 1 Coalition formation: a game-theoretic approach

45

outcomes o f t h e game. T h i s makes game t h e o r y much more t r a c t a b l e . However, i n p r a c t i c a l a p p l i c a t i o n s i t i s a l m o s t i m p o s s i b l e t o a s s e s s . t h e consequences f o r o n e ' s r e p u t a t i o n as w e l l as a l l o t h e r p o s s i b l e consequences. mainly due t o t h e f a c t t h a t i n r e a l - l i f e

T h i s is

s o c i a l i n t e r a c t i o n s w e s i m p l y do

n o t know what games w e w i l l b e i n v o l v e d i n and who o u r c o - a c t o r s o r oppon e n t s w i l l be n e x t month, n e x t y e a r , o r i n t e n y e a r s '

t i m e . Perhaps t h i s

makes p e o p l e behave a l i t t l e more " d e c e n t l y " i n many p r a c t i c a l s i t u a t i o n s t h a n would b e o p t i m a l i n i s o l a t e d games. On t h e o t h e r hand, i n t h e h i s t o r y of mankind l y i n g and c h e a t i n g a l s o have t h e i r p l a c e , a p p a r e n t l y b e c a u s e t h e a c t o r s deemed i t p r o f i t a b l e t o behave i n t h i s manner.

2.3

SOME CHARACTERISTICS OF GAMES

2.3.1

INTRODUCTION

In s e c t i o n 2 , an o u t l i n e of t h e b a s i c a s s u m p t i o n s of normal form a s w e l l a s c h a r a c t e r i s t i c f u n c t i o n game t h e o r y h a s been p r e s e n t e d . These a s s u m p t i o n s h o l d f o r a l l games. T h i s , however, i s n o t t h e case w i t h t h e p r o p e r t i e s of games d i s c u s s e d i n t h i s c h a p t e r . Each of t h e f o l l o w i n g s e c t i o n s d i s c u s s e s one or more o f t h e s e p r o p e r t i e s t o g e t h e r w i t h , whenever p o s s i b l e , t h e i r importance f o r t h e a p p l i c a t i o n of game t h e o r y t o c o a l i t i o n f o r m a t i o n . T o g e t h e r , t h e s e p r o p e r t i e s p r o v i d e t h e dimensions f o r a c l a s s i f i c a t i o n of games; some of i t s s u b c l a s s e s w i l l be of major i n t e r e s t i n t h e r e m a i n i n g pa rt of t h e chapter .

2.3.2

CONSTANT-SUM GAMES AND RELATED NOTIONS

A c h a r a c t e r i s t i c f u n c t i o n game w i t h a c t o r s I,

...,

n and c h a r a c t e r i s t i c

f u n c t i o n v i s c a l l e d c o n s t a n t sum i f t h e r e i s a c o n s t a n t k s u c h t h a t f o r each c o a l i t i o n S c

11,...,n }

and i t s complement Sc =

11,...,n}\S

v ( S ) + v(Sc) = k I f k = 0 t h e game i s c a l l e d z e r o sum. Each constant-sum

game i s , i n a

s e n s e , e q u i v a l e n t t o a zero-sumgame. T h i s i s f u r t h e r e x p l a i n e d i n Appendix 2 . I f v ( S ) non-constant

+

v(Sc) is v a r i a b l e i n S, t h e n t h e game i s c a l l e d

sum.

S i m i l a r l y , a normal form game is c a l l e d c o n s t a n t sum i f f o r e a c h p o s s i b l e

W.J. van der Linden and A. Verbeek

46

outcome t h e sum of t h e p a y o f f s t o a l l a c t o r s i s e q u a l t a c o n s t a n t k . Again, i f k = 0 , t h e game i s z e r o sum, and i f a game i s n o t constant-sum, it i s c a l l e d non-constant

sum. S t r a t e g i c e q u i v a l e n c e can a l s o be d e f i n e d

f o r normal form games, and a l l constant-sum games a r e s t r a t e g i c a l l y e q u i v a l e n t t o a zero-sum game. Note t h a t t h e s e d e f i n i t i o n s imply t h a t p a y o f f s of s e p a r a t e a c t o r s may be added u p , i . e . , t h a t each a c t o r ' s p r e f e r e n c e s a r e measured on an i n t e r v a l s c a l e and t h a t t h e s e s c a l e s a r e c o m p a r a b l e i n a w a y t h a t makes a d d i t i o n meaningf u l . T h i s assumption i s i m p l i c i t in t h e d e f i n i t i o n of c h a r a c t e r i s t i c f u n c t i o n games b u t i t must be made e x p l i c i t l y i n t h e class of normal form games ( s e e s e c t . 2.3.6). I n s e c t i o n 2 . 2 . 1 i n e s s e n t i a l games have been d e f i n e d ; i t was a l s o argued t h a t i n i n e s s e n t i a l games no c o a l i t i o n can o b t a i n more t h a n when i t s actors operate i n i s o l a t i o n . In t h e sequel, w e w i l l t her ef or e only c o n s i d e r e s s e n t i a l games, i . e . , assume t h a t v(SIU S2) > v(S1) + v(S2) f o r a t l e a s t one p a i r of d i s j o i n t c o a l i t i o n s S1 and S 2 . An example of an e s s e n t i a l zero-sum game i n c h a r a c t e r i s t i c form i s given i n Display 6 .

D i s p l a y 6 . Example of a zero-sum c h a r a c t e r i s t i c f u n c t i o n game. T h i s game i s e q u i v a l e n t ( s e e Appendix 3 ) t o t h e game d e s c r i b e d i n s e c t i o n 2 . 5 . 2 and Appendix 7 . 2.3.3

NUMBER OF ACTORS

C o a l i t i o n f o r m a t i o n and i t s f o r m a l i z a t i o n as a c h a r a c t e r i s t i c f u n c t i o n game r e q u i r e a t l e a s t t h r e e a c t o r s t o be o f any i n t e r e s t . Although of c o n s i d e r a b l e complexity, t h e t h e o r y o f t h r e e - a c t o r games i s a r e l a t i v e l y simple p a r t of game t h e o r y . A complete t r e a t m e n t i s given i n von Neumann and Morgenstern (1944, sect. 60.3). The s i m p l i c i t y o f t h e s t r u c t u r e o f p o s s i b l e c o a l i t i o n s i n games w i t h a s few as t h r e e a c t o r s i s i n s h a r p c o n t r a s t t o t h e s t r u c t u r e of games w i t h , s a y , over t e n a c t o r s . Not o n l y

Chapter 2 f Coalition formation: a game-theoretic approach is t h e r e a l a r g e d i f f e r e n c e i n t h e number of p o s s i b l e s o l u t i o n s , b u t when t h e number o f a c t o r s i s i n c r e a s e d e n t i r e l y new phenomena o c c u r a s w e l l (von Neumann & M o r g e n s t e r n , 1944, s e c t . 53). F o r a s u r v e y of t h e many d i f f e r e n t p o s s i b i l i t i e s f o r co-operation,

t h r e a t , and b l a c k m a i l i n

two-actor normal form games, t h e r e a d e r i s r e f e r r e d t o Rapoport and Guyer (1966) who show t h e v a r i o u s p o s s i b l e b i - m a t r i x

games w i t h o n l y two

s t r a t e g i e s f o r each a c t o r .

2.3.4

REPETITIONS OF THE GAME

A s a l r e a d y n o t e d i n s e c t i o n 2 . 7 . , game t h e o r y d e a l s w i t h games a s i s o l a t e d e v e n t s and assumes t h a t u t i l i t i e s i n c l u d e t h e e v a l u a t i o n of f u t u r e consequences of t h e b e h a v i o r o f t h e a c t o r s . I f , however, t h e a c t o r s know b e f o r e h a n d how many times a game i s g o i n g t o be p l a y e d , a d i f f e r e n t f o r m a l i z a t i o n o f t h e i n f l u e n c e of t h e p r e s e n t b e h a v i o r of one a c t o r on t h e f u t u r e behavior of t h e o t h e r s i s p o s s i b l e. W e can t h e n model t h e s i t u a t i o n

as a supergame i n which t h e a c t o r s c h o o s e a s u p e r s t r a t e g y which i n d i c a t e s how t h e y w i l l p l a y i n e a c h i n d i v i d u a l game. The i n f l u e n c e of p r e v i o u s on f u t u r e b e h a v i o r is now n o t f o r m a l i z e d v i a t h e u t i l i t i e s , b u t v i a t h e f a c t t h a t each a c t o r simultaneously optimizes h i s / h e r s t r a t e g i e s f o r t h e whole series of games, a l l o w i n g f o r t h e o t h e r a c t o r s t o do t h e same. Beware, however, of t h e f o l l o w i n g m a g n i f i c e n t "paradox",

discussed a t

g r e a t e r l e n g t h i n Luce and R a i f f a (1957, sect. 5 . 5 ) .

C o n s i d e r a t w o - a c t o r game where c o - o p e r a t i o n i s f r u i t f u l b u t o n e - s i d e d c h e a t i n g even more a d v a n t a g e o u s f o r t h e c h e a t e r . A t y p i c a l example i s t h e p r i s o n e r ' s dilemma.

I f t h e game i s p l a y e d 50 times, it i s t e m p t i n g

t o e x p e c t a t e n d e n c y f o r c o - o p e r a t i o n s i n c e c h e a t i n g c a n always be p u n i s h e d i n s u b s e q u e n t p l a y s . However, c h e a t i n g i n t h e 5 0 t h p l a y c a n n o t be p u n i s h e d , so t h a t t h i s p l a y i s , i n f a c t , s t r a t e g i c a l l y e q u i v a l e n t t o an i s o l a t e d game. B u t , t h e n , t h e same argument h o l d s f o r t h e 4 9 t h p l a y , and f o r t h e 4 8 t h ,

...,

etc.

N o t i c e t h a t t h e paradox i s a b s e n t , o r a t l e a s t weaker, i f t h e a c t o r s do n o t know t h e number of r e p e t i t i o n s b e f o r e h a n d ; however, w e f e e l t h a t t h e c r u x of t h e paradox i s n o t knowledge of t h e number of r e p e t i t i o n s b u t t h e q u e s t i o n of what k i n d of s t r a t e g y w i l l e v e r make t h e opponent change h i s unwanted b e h a v i o r . A p u n i s h i n g s t r a t e g y seems t o b e e f f e c t i v e

47

48

W.J. van der Linden and A. Verbeek

only if it lowers the opponent's payoff rather than the actor's One may feel that in real situations with 50 repetitions the final play is not isolated because of an established co-operation or antagonism. If this happens, the assumed constancy of the utilities is violated. In other words, this possibility should have been incorporated in the utilities of the supergame, which then becomes much more complex than a mere repetition of 5 0 identical plays. Repetition of games also entails the possibility of communication via patterns of strategic choices. For example, one may try to influence the other actors by showing certain persistent choices. This form of communication can occur instead of or in addition to preplay communication (see section 2.3.5). It is not immediately obvious that a repetition of a characteristic function game can be modeled as a characteristic function game. However, each characteristic function game can be modeled as a normal form game with comparable, transferable utilities and with the possibility of preplay communication (see Appendix 4 ) , and repetitions of normal form games constitute a normal form supergame. 2.3.5 COMMUNICATION AND CO-OPERATION

In characteristic function games, communication is an essential ingredient of the game because the actors have to form coalitions by the process of addressing the others, making and receiving offers, and entering into agreements. In normal form games, on the other hand, communication may be restricted or regulated in many ways. In two-actor, constant-sum games, communication is of no use and can only have the effect of "psychological warfare". In other normal form games, the rules may permit binding agreements or may allow for no more than a "statement of intent". If the game is repeated, non-verbal communication via persisting patterns of strategic choices is always possible (cf. section 2 . 3 . 4 ) . Luce and Raiffa (1957) define a co-operative game as "a game in which the actors have complete freedom of preplay communication to

make joint binding agreements" (p. 8 9 ) . Real-life communications are Often aimed at changing the other persons' preference schemes or at

Chapter 2 f Coalition formation: a game-theoretic approach

49

obscuring o r overemphasizing some aspects of the situation. It should be noted that this is not paralleled

in the co-operative models of game

theory. AS noted earlier, game theory assumes constancy of preferences and perfect knowledge of the game with all actors. The reader is urged to spend some thoughts on the huge difficulties involved in modeling, for example, misconceptions about the structure of the game, changing utilities, and convincing but not binding agreements. Let us review a few aspects of communication more closely. First, consider those highly competitive situations where only one coalition will receive a prize while the actors left out receive nothing. In those situations it is often advantageous to be talkative and aggressive and to try to lead the discussion. An example is a market with several stalls in a row selling the same commodity--vegetables, for example. Here coalitions are (two-actor) seller-buyer coalitions, and the stallholders are professionals at the game, fully aware of the importance of highly-voiced invitations to negotiate. Another aspect of communication is that in some situations a well-timed and convincing statement of intended behavior can strongly reduce the choices of the others. ("Whatever you do, I'll

. . . ' I ,

o r , "At the beginning I want to make it

perfectly clear that I'll never

. . . ") .

This behavior is especially

advantageous in games with several locally optimal solutions, which each require co-operation of all actors but are preferred differently by different actors. It can be partially incorporated in the game by including a binding statement of intent in the set of possible strategies. (See the example of the battle of the sexes discussed in Appendix 6.) However, factors as becoming the first to make such a statement and the shift from a binding to a convincing statement seem much harder to formalize.

In the above situations, the actors who state their intentions not convincingly enough or at the wrong time are in a disadvantageous position. In certain situations, they may even wish they had never participated in the communication. Consider the game in Display 7, for example, which can be found in Luce and Raiffa (1957, p . 111).

In this two-actor game actor B would probably refuse to come to the

W.J.van der Linden and A. Verbeek

50

strategies for actor B

B1

82

strategies for actor A A1

(1,2)

(3,1)

A2

(0,-200)

(2,-300)

D i s p l a y 7 . (Luce and R a i f f a (1957) s e c t i o n 5 . 1 1 , page 1 1 1 ) . Example of a bi-matrix game. Because A 1 dominates A 2 , and B 1 dominates B2, (Al, B1) i s an e q u i l i b r i u m ( f o r dominance and e q u i l i b r i u m see 4 . 3 and 4 . 4 ) .

If,

however, p r e p l a y communication i s p a r t of t h e game, then a c t o r A can demand t h a t B chooses B2, by t h r e a t e n i n g t o choose A2 o t h e r w i s e . So B i s u n l i k e l y t o come t o t h e n e g o t i a t i o n s , u n l e s s f o r c e d t o do s o . conference t a b l e because t h i s may o n l y weaken h i s / h e r p o s i t i o n . Co-operation

i s a concept beyond communication. One i l l u s t r a t i o n i s t h e

p o s s i b i l i t y of j o i n t l y randomizing s t r a t e g i e s . For i n s t a n c e , i f two

actors p r e f e r a d i f f e r e n t l o c a l l y o p t i m a l payoff v e c t o r , t h e y can d e c i d e t o a l t e r n a t e o r t o choose each v e c t o r with a c e r t a i n p r o b a b i l i t y .

In t h e

l a t t e r c a s e t h e y u s e randomized b u t c o - o r d i n a t e d s t r a t e g i e s . Again t h e b a t t l e of t h e s e x e s is a s t a n d a r d example. Another a s p e c t of c o - o r d i n a t i o n

i s t h e p o s s i b i l i t y of an exchange o f u t i l i t i e s ( s i d e payment o r " b r i b i n g " ) , discussed i n t h e next s ect io n . 2.3.6

COMPARABILITY AND TRANSFERABILITY OF UTILITY

I n a c h a r a c t e r i s t i c f u n c t i o n game, t h e members of t h e c o a l i t i o n d i v i d e

t h e v a l u e of t h e c o a l i t i o n among themselves. So, t h e p a y o f f s are t r a n s f e r a b l e by d e f i n i t i o n . S i n c e p a y o f f s a r e measured on a u t i l i t y s c a l e , u t i l i t i e s c a n a l s o be c o n s i d e r e d t r a n s f e r a b l e . However, t h i s d o e s n o t imply t h a t each a c t o r h a s t h e same u n i t of u t i l i t y .

I n o t h e r words,

t r a n s f e r a b i l i t y does not n e c e s s a r i l y imply c o m p a r a b i l i t y . The d e f i n i t i o n of t h e c h a r a c t e r i s t i c f u n c t i o n game a l s o assumes t h a t each a c t o r ' s u t i l i t i e s are s t r i c t l y i n c r e a s i n g i n t h e p a y o f f s .

Chapter 2 / Coalition formation: a game-theoretic approach I n normal form games where communication i s a l l o w e d , an a c t o r may t r y t o i n f l u e n c e t h e o t h e r s by p r o p o s i n g s i d e payments. An i m p o r t a n t d i s t i n c t i o n i s w h e t h e r o r n o t t h e s i d e payments are a r e d i s t r i b u t i o n o f t r a n s f e r a b l e payoff.

I f t h e payoff i s n o t t r a n s f e r a b l e ( e . g . , t h r e e

y e a r s o f imprisonment or a p r o p o s a l of m a r r i a g e ) , s i d e payments may b e i n money o r i n any o t h e r t r a n s f e r a b l e medium. I f t h e s i d e payments a r e made i n a n o t h e r medium t h a n t h e p a y o f f s t h e n t h e accountancy becomes q u i t e c o m p l i c a t e d ; e a c h a c t o r h a s t o keep t r a c k of t h e u t i l i t i e s o f h i s / h e r d i r e c t p a y o f f s , o f s i d e payments r e c e i v e d , and of s i d e payments made t o the others.

2.3.7

RESTRICTIONS ON POSSIBLE COALITIONS

T h i s s e c t i o n o n l y d e a l s w i t h c h a r a c t e r i s t i c f u n c t i o n games. H i t h e r t o ,

i t h a s been assumed t h a t i n t h e s e games any c o a l i t i o n can b e formed and t h a t a l l c o a l i t i o n t r a n s i t i o n s a r e a l l o w e d d u r i n g n e g o t i a t i o n s . However, i n c e r t a i n s i t u a t i o n s n o t a l l c o a l i t i o n s and t r a n s i t i o n s a r e p o s s i b l e . R e s t r i c t i n g t h e c l a s s of p o s s i b l e c o a l i t i o n s d o e s n o t t a k e u s o u t s i d e t h e formalism of c h a r a c t e r i s t i c f u n c t i o n games; w e have o n l y t o d e f i n e t h e v a l u e of t h e c o a l i t i o n s e x c l u d e d a s u n p l e a s a n t l y low a s , s a y , 0 or -100,000.

R e s t r i c t i n g t h e p o s s i b l e t r a n s i t i o n s from one c o a l i t i o n t o

a n o t h e r d o e s i n t r o d u c e a new a s p e c t b e c a u s e t h u s f a r w e have n o t f o r m a l i z e d any p a r t of t h e p r o c e s s of n e g o t i a t i o n . The d i s c u s s i o n i s postponed t o s e c t i o n 5 . 9 . 2.3.8

SIMPLE GAMES, SYMMETRIC GAMES, QUOTA GAMES, AND APEX GAMES

I n some games t h e c h a r a c t e r i s t i c f u n c t i o n h a s a v e r y s i m p l e , r e g u l a r s t r u c t u r e . I n t h i s s e c t i o n , f o u r examples of s u c h c l a s s e s a r e g i v e n . Many of t h e e x p e r i m e n t s t o be d i s c u s s e d i n s e c t i o n 6 have u s e d experimen-

t a l games from one of t h e s e classes. A c h a r a c t e r i s t i c f u n c t i o n game i s c a l l e d s i m p l e i f t h e c h a r a c t e r i s t i c

f u n c t i o n c a n t a k e o n l y two v a l u e s . U s u a l l y , t h e s e v a l u e s are d e n o t e d by 1 and 0 and i n t e r p r e t e d a s "winning" and " l o s i n g " ,

r e s p e c t i v e l y . There

i s an e x t e n s i v e l i t e r a t u r e on s i m p l e games, g o i n g back a s f a r a s von Neumann and Morgenstern (1944, pp. 420-503).

A major f i e l d of a p p l i c a t i o n

i s p o l i t i c a l c o a l i t i o n f o r m a t i o n and v o t i n g b e h a v i o r . O b v i o u s l y , a s i m p l e game i s c h a r a c t e r i z e d by t h e c o l l e c t i o n o f minimal winning c o a l i t i o n s ,

51

W.J. van der Linden and A. Verbeek

52

t h a t i s , of s u b s e t s S having t h e v a l u e v(S) = 1 f o r which e v e r y p r o p e r s u b s e t S'c S h a s t h e v a l u e v ( S ' ) = 0. An important s u b c l a s s of simple games are t h e weighted m a j o r i t y games. Here each a c t o r i h a s a weight w

i

and a s u b s e t o f a c t o r s , S , i s winning

i f and o n l y i f i t h a s more t h a n h a l f o f t h e t o t a l sum of w e i g h t s :

c 1. E Sw i

> 1/2 z.wi.

I t is somewhat s u r p r i s i n g t h a t n o t e v e r y s i m p l e game

h a s t h i s form; t h e s m a l l e s t counter-example h a s 6 a c t o r s (von Neumann & Morgenstern, 1944, sect. 53.2.3). A symmetric game i s a game i n which a l l a c t o r s have e x a c t l y t h e same

p o s s i b i l i t i e s i n t h e s e n s e t h a t i f two a c t o r s are i n t e r c h a n g e d , t h e n t h e c h a r a c t e r i s t i c f u n c t i o n remains unchanged. I t can e a s i l y be deduced t h a t t h i s h o l d s i f and only i f f o r each S v(S) only depends on t h e number of a c t o r s i n S . Hence, t h e o r y c a n o n l y s a y something of t h e s i z e of a p o s s i b l e c o a l i t i o n and n o t h i n g about i t s a c t u a l c o m p o s i t i o n . A s a l l a c t o r s a r e considered e q u a l from t h e formal p o i n t o f view, w e do not have enough information t o model t h e more i n t e r e s t i n g a s p e c t s of c o a l i t i o n f o r m a t i o n . Important a p p l i c a t i o n s of symmetric games i n c l u d e economic models o r marketing b e h a v i o r . Some games have a s t r u c t u r e i n which e a c h a c t o r h a s a w e i g h t , wi, and t h e r e e x i s t s a f u n c t i o n u ( S ) which can o n l y t a k e t h e v a l u e s 0 or 1. T h e c h a r a c t e r i s t i c f u n c t i o n can be d e s c r i b e d by v ( S ) = u ( S ) CiESwi,sometimes w i t h t h e e x c e p t i o n of such sets as t h e grand c o a l i t i o n . See C r o t t and Albers (1981, p. 288) f o r a p r e c i s e and s l i g h t l y more g e n e r a l d e f i n i t i o n . There seems t o be no g e n e r a l l y accepted d e f i n i t i o n , however. G a m e s with t h i s s t r u c t u r e a r e c a l l e d q u o t a games and t h e w e i g h t s wi are known a s t h e a c t o r ' s q u o t a s . The b e s t known c a s e i s t h e one w i t h u(S) = 1 f o r a l l two-element v ( { i , j } ) = wi

s u b s e t s of S. I n t h i s case, f o r any two a c t o r s i and j

+

w . . I t seems obvious t h a t a q o u t a r e p r e s e n t s an a c t o r ' s J s t r e n g t h i n a c o a l i t i o n . However, a q u o t a can be n e g a t i v e , w h i l e t h e

s t r e n g t h o f an a c t o r i n a winning c o a l i t i o n c e r t a i n l y must be p o s i t i v e . No a c t o r would pay f o r e n t e r i n g a winning c o a l i t i o n i f he/she does n o t l o o s e a n y t h i n g by s t a y i n g s i n g l e . Quotas a r e t h u s no s a f e measures of an a c t o r ' s s u b j e c t i v e s t r e n g t h . Another o b j e c t i o n is t h a t i t is h a r d t o imagine t h a t , f o r example, i n a 1 0 - a c t o r game t h e a c t o r s would indeed

53

Chapter 2 f Coalition formation: a game-theoretic approach s o l v e 1 0 e q u a t i o n s i n 1 0 unknowns t o l e a r n e a c h o t h e r ' s s t r e n g t h . However, a c t o r s might l e a r n t h e i r q u o t a s by t r i a l and e r r o r i n a series of s u c c e s s i v e p l a y s and t h i s might i n f l u e n c e t h e d i v i s i o n of p a y o f f s i n l a t e r rounds.

"Me and my a u n t " , i n t r o d u c e d i n s e c t i o n 2 . 1 . 3 , i s an example o f an apex game. The s t r u c t u r e of apex games was a l r e a d y i n v e s t i g a t e d i n von Neumann and Morgenstern (1944, pp. 473-503) Horowitz'

b u t h a s become w i d e l y known s i n c e

(1973) t r e a t m e n t i n which a t e r m i n o l o g y was e s t a b l i s h e d and

remarkable s o l u t i o n s g i v e n . An apex game i s a n - a c t o r

s i m p l e game,

n 2 3 , whose c o a l i t i o n s c o n s i s t of (1) a l l t h o s e c o a l i t i o n s which i n c l u d e a c e r t a i n a c t o r c a l l e d Apex and ( 2 ) t h e c o a l i t i o n formed by t h e o t h e r n-actors

c a l l e d Base a c t o r s (Horowitz, 1973). For an example of an apex

game c h a r a c t e r i s t i c f u n c t i o n , see D i s p l a y 1 where "My a u n t " i s t h e Apex a c t o r and " I " , P , Q , and R a r e t h e Base a c t o r s . The p o s i t i o n o f t h e Apex a c t o r r e s e m b l e s t h a t o f a m o n o p o l i s t . The o n l y t h i n g h e / s h e h a s t o d o i s t o p e r s u a d e one of t h e o t h e r s i n t o a c o a l i t i o n . I n d o i n g s o , h e / s h e can p l a y o f f t h e Base a c t o r s a g a i n s t e a c h o t h e r t o s t i p u l a t e f a v o r a b l e terms. Each Base a c t o r h a s t w o o p t i o n s which p l a c e s him/her i n a d e l i c a t e p o s i t i o n : e i t h e r u n i t e w i t h t h e o t h e r s a g a i n s t t h e Apex a c t o r o r compete w i t h them f o r t h e Apex a c t o r ' s f a v o r . The f o r m e r o p t i o n seems b e n e f i c i a l t o a l l Base a c t o r s b u t i n v o l v e s t h e r i s k t h a t one of t h e o t h e r s y i e l d s t o an a d v a n t a g e o u s o f f e r from t h e Apex a c t o r . I n t h e l a t t e r case, c o m p e t i t i o n w i l l b e h i g h and a l l w i l l have t o d e a l w i t h a s t r o n g Apex actor.

Note t h a t , r e s t r i c t i n g t h e p o s s i b l e c o a l i t i o n s t o t h e Base and 2 - a c t o r Apex c o a l i t i o n s , apex games c a n be r e p r e s e n t e d a s q u o t a games. I f t h e game i s 0-1 and h a s n a c t o r s , t h e n t h e q u o t a s are e q u a l t o l / ( n - l ) t h e Base a c t o r s and ( n - 2 ) / ( n - l )

for

f o r t h e Apex a c t o r .

Apex games have drawn much a t t e n t i o n b o t h i n t h e t h e o r e t i c a l and e x p e r i m e n t a l l i t e r a t u r e and are p r o b a b l y o n l y s u r p a s s e d i n p o p u l a r i t y by t h e p r i s o n e r ' s dilemma game (Appendix 5 ) .

54

W.J. van der Linden and A. Verbeek

2.3.5 CHANCE, NATURE, THE BANKER, THE UMPIRE, THE MONOPOLIST, THE VETO ACTOR, THE APEX ACTOR, AND OTHERS

In normal form games and c h a r a c t e r i s t i c f u n c t i o n games c e r t a i n a c t o r s have r e c e i v e d s p e c i a l names. Many p a r l o r games, such a s poker and backgammon, can only be modeled i n normal form i f w e a l l o w f o r an " a c t o r " who r e p r e s e n t s t h e random d e v i c e used i n t h e game v i a a known p r o b a b i l i t y d i s t r i b u t i o n . T h i s a c t o r could be c a l l e d Chance and is e v i d e n t l y i n d i f f e r e n t t o t h e outcome of t h e game. There is a n o t h e r a c t o r who i s i n d i f f e r e n t a s t o t h e outcome of t h e game and of whom w e do n o t know how h i s s t r a t e g i e s a r e s e l e c t e d . This a c t o r i s o f t e n c a l l e d Nature. In t h e d e c i s i o n - t h e o r e t i c approach t o s t a t i s t i c s , i t is assumed t h a t N a t u r e ' s s t r a t e g i e s a r e t h e s e l e c t i o n of parameter v a l u e s

f o r c e r t a i n p r o b a b i l i t y d e n s i t y f u n c t i o n s . The s t a t i s t i c i a n p l a y s a g a i n s t N a t u r e , and h i s t a s k i s t o select d e c i s i o n r u l e s ( e s t i m a t o r s o r test s t a t i s t i c s ) t h a t are o p t i m a l a g a i n s t N a t u r e ' s c h o i c e s . A less p o t e n t actor is one who h a s o n l y one p o s s i b l e s t r a t e g y . Such an

a c t o r is o f t e n added t o make a game c o n s t a n t sum, and t h i s a c t o r i s t h e r e f o r e c a l l e d t h e Banker. Thus i n p r i n c i p l e , any n - a c t o r non-constant-

sum game i s e q u i v a l e n t t o an ( n + l ) - a c t o r constant-sum game. However, f o r c h a r a c t e r i s t i c f u n c t i o n games t h e a d d i t i o n a l assumption i s needed t h a t t h e Banker does not t a k e p a r t i n t h e n e g o t i a t i o n s s i n c e o t h e r w i s e t h e s t r a t e g i c p o s s i b i l i t i e s o f t h e o t h e r a c t o r s would n o t remain t h e

same (see Appendix 4 ) . The Umpire d o e s n o t t a k e p a r t i n t h e game a s an a c t o r . H i s r o l e h a s o n l y been i n t r o d u c e d t o f o r m a l i z e t h e n e c e s s i t y o f h o l d i n g t h e a c t o r s t o t h e r u l e s of t h e game and t o b i n d i n g agreements. T h i s obedience may e x p l i c i t l y be f o r c e d by a person or an i n s t i t u t i o n , or more i m p l i c i t l y by, f o r example, s o c i a l p r e s s u r e . I n some s i m p l e games t h e r e is an a c t o r who must be i n c l u d e d i n any winning c o a l i t i o n , t h a t i s , no c o a l i t i o n can win u n l e s s i t i n c l u d e s t h i s a c t o r ; t h e o t h e r s l a c k s u f f i c i e n t power t o win by themselves. A group o f such a c t o r s can be c o n s i d e r e d a s i n g l e a c t o r , who f o r obvious r e a s o n s i s c a l l e d

Chapter 2 J Coalition formation: a game-theoretic approach t h e M o n o p o l i s t . H e i s a l s o c a l l e d t h e Veto a c t o r s i n c e h e h a s t h e power t o v e t o any coalition. The Apex a c t or, i n t r o d u c e d i n t h e p r e c e d i n g s e c t i o n , r e s e m b l e s t h e m o n o p o l i s t b u t is somewhat l e s s powerful b e c a u s e t h e r e i s one winning c o a l i t i o n i n which h e i s n o t i n c l u d e d : an Apex a c t o r c a n n o t v e t o t h e Base c o a l i t i o n .

In some games i t may make s e n s e t o suppose a power continuum a l o n g which a c t o r s c a n be o r d e r e d .

I n s i m p l e games, however, t h e r e l a t i v e number of

minimal winning c o a l i t i o n s an a c t o r i s i n may s e r v e as an o p e r a t i o n a l i z a tion.

I n q u o t a games, t h e a c t o r s ' q u o t a s c a n have t h i s f u n c t i o n . Funk,

Rapoport, and Kahan (1980) o f f e r a c o n c e p t o f p o s i t i o n a l power which i s simply t h e number of c o a l i t i o n s of which an a c t o r may be a member.

2.4

SOLUTION CONCEPTS FOR NORMAL FORM GAMES

2.4.1 RANDOMIZATION For t h e t w o - a c t o r normal form games from D i s p l a y 4 i n s e c t i o n 2.2.2, i t i s r a t h e r o b v i o u s t h a t c o n t i n u o u s l y c h o o s i n g t h e same s t r a t e g y i n a series of r e p e t i t i o n s i s d i s a d v a n t a g e o u s . I t i s somewhat less o b v i o u s t h a t when t h e game i s p l a y e d o n l y o n c e , n e i t h e r A n o r B have an o p t i m a l s t r a t e g y . T h i s c a n be shown a s f o l l o w s : I f A knew t h a t , f o r i n s t a n c e , showing "heads" was o p t i m a l for B and i f f o r t h a t r e a s o n B c o u l d d e f i n i t e l y choose "heads",

t h e n A c o u l d b e a t B by showing " t a i l s " .

I n f a c t , B's

power l i e s i n h e s i t a t i n g between t h e t w o s t r a t e g i e s and A ' s l a c k of knowledge of B's p l a n s . Von Neumann had t h e b r i l l i a n t i d e a t o e x t e n d A ' s s t r a t e g i e s by i n c l u d i n g randomized s t r a t e g i e s :

A randomly c h o o s e s "heads up" w i t h p r o b a b i l i t y PA and " t a i l s up" w i t h p r o b a b i l i t y 1-P

A

Similarly, B randomly c h o o s e s "heads up" w i t h p r o b a b i l i t y PB and " t a i l s up" w i t h p r o b a b i l i t y 1-P

B

The c h o i c e s a r e assumed t o be made i n d e p e n d e n t l y , so t h e p r o b a b i l i t i e s of t h e outcomes a r e a s g i v e n i n D i s p l a y 8 .

55

W.J.van der Linden and A. Verbeek

56

strategies for actor B marginal show heads

show t a i l s

probabilities for actor A

strategies for actor A show heads

A ' "B

show t a i l s

( 1-PA) PB

marginal probabilities

PB

l-Pg

1

for actor B Display 8 . P r o b a b i l i t i e s i n t h e randomized v e r s i o n of t h e game shown i n D i s p l a y 4 . A c t o r s A and B choose t h e i r s t r a t e g i e s independently. The expected payoff i s shown i n D i s p l a y 9. The expected outcome of t h e game, E , is:

= -2

+ 3pA + 5pB

-

8pApB.

Now A w i l l choose pA so as t o maximize E , and B w i l l choose pB s o a s t o minimize E . (The f o c u s i n g on expected outcomes i s a consequence of t h e way u t i l i t y i s measured, assuming t h a t people behave as i f t h e y maximize

t h e i r expected u t i l i t y ; see s e c t i o n 2 . 2 . 4 ) . Hence i t i s worthwile c o n s i d e r i n g E a f u n c t i o n of pA and pB. The f u n c t i o n can b e r e p r e s e n t e d g r a p h i c a l l y i n s e v e r a l ways. One p o s s i b i l i t y is given i n D i s p l a y 9 , where E = -2

+ 5pA + 3pB

-

8pApB is d e p i c t e d a s a f u n c t i o n of pA f o r s e v e r a l

v a l u e s of pB. I t i s e a s y t o see t h a t B can f o r c e E below z e r o , t o h i s / h e r advantage, by choosing pB s l i g h t l y o v e r 1/3. E d o e s n o t depend on p

B chooses pg = 5/8. Then E = -2 + 15/8 = -1/8.

if A In Display 9, t h i s corres-

Chapter 2 1 Coalition formation: a game-theoretic approach

51

+3

B pays A

+2 f

-1 A pays B -2

D i s p l a y 9. The e x p e c t e d payoff of t h e game shown i n D i s p l a y 8 a s a Note t h a t t h e f u n c t i o n of pA f o r v a r i o u s v a l u e s o f p B' e x p e c t e d payoff i s n e g a t i v e f o r a l l v a l u e s of pA i f pB i s s l i g h t l y o v e r 1/3. So a c t o r B can f o r c e t h e e x p e c t e d p a y o f f below z e r o , t o h i s a d v a n t a g e . I f a c t o r A p e r s i s t s i n a pA f

. 6 2 5 , B can f o r c e t h e e x p e c t e d payoff even f u r t h e r down,

e i t h e r by c h o o s i n g pB = 0 o r by c h o o s i n g p

B

= 1.

ponds t o a h o r i z o n t a l l i n e , 1/8 below t h e P A - a x i s . Whenever B makes t h i s c h o i c e , A c a n n o t p r e v e n t B from winning 1/8 on a v e r a g e , even when h e / s h e l e a r n s about B ' s p r o b a b i l i t i e s , f o r example by e x p e r i e n c e . F o r any o t h e r c h o i c e of pB, A c a n do b e t t e r i n t h e l o n g r u n a s he g r a d u a l l y l e a r n s about B ' s c h o i c e by e x p e r i e n c e . The c h o i c e pB = 5/8 i s c a l l e d t h e minimax c h o i c e

of B for r e a s o n s e x p l a i n e d i n 2 . 4 . 5 . S i m i l a r l y , E d o e s n o t depend on pB i f A c h o o s e s pA = 3/8, and t h e n E = -2

+

15/8 = -1/8 a g a i n . A p p a r e n t l y , B c a n f o r c e E down t o -1/8 by

c h o o s i n g pB = 5/8 b u t a t t h e same t i m e A c a n f o r c e E t o b e a t l e a s t -1/8 by c h o o s i n g pA = 3/8. The e x p e c t e d outcome of -1/8 i s p r o p e r l y c a l l e d t h e

W.J. van der Linden and A. Verbeek

58

v a l u e of t h e game. I t i s a r e a s o n a b l e measure of how much t h e game i s worth t o A and B. The g e n e r a l i z a t i o n s about games i n v o l v i n g more t h a n two s t r a t e g i e s and/or a c t o r s a r e s t r a i g h t f o r w a r d , although i t i s much h a r d e r t o produce c l e a r graphs i n such s i t u a t i o n s . The e x i s t e n c e of minimax s t r a t e g i e s and values is discussed i n s ecti o n 2 . 4 . 5 . 2.4.2

DOMINANCE OF STRATEGIES

I f an a c t o r compares two p o s s i b l e s t r a t e g i e s , s a y A 1 and A2, and d i s c o v e r s t h a t f o r any combination of s t r a t e g i e s of t h e o t h e r a c t o r s A 1 g i v e s an outcome a t l e a s t e q u a l t o A2, t h e n w e s a y t h a t A 1 dominates A2. I t seems r e a s o n a b l e t o suppose t h a t an a c t o r never chooses a dominated s t r a t e g y , b u t t h i s i s r e a s o n a b l e only under t h e a d d i t i o n a l assumption t h a t t h e a c t o r s do n o t co-operate.

I f t h e y do c o - o p e r a t e ,

then, surprisingly, a

combination of dominated s t r a t e g i e s c a n be more f a v o r a b l e t o each a c t o r t h a n a combination o f undominated s t r a t e g i e s . T h i s l e a d s , however, t o r a t h e r an u n s t a b l e s i t u a t i o n which f o r t h e two-actor case i s known as t h e p r i s o n e r ' s dilemma. An example of t h i s u n s t a b i l i t y i s g i v e n i n Display 10;

see a l s o Appendix 5 . I f t h e a c t o r s do n o t c o - o p e r a t e , one can argue t h a t t h e y need n o t c o n s i d e r dominated s t r a t e g i e s . Consider t h e example i n D i s p l a y 1 0 . Here A 3 is dominated by A2 and B4 by B 1 . I f A 3 and B4 are removed, B 3 ' i s dominated by B 1 ' and can a l s o be removed.0ne may wonder whether, i n g e n e r a l , t h e o r d e r i n which a c t o r s e l i m i n a t e dominated s t r a t e g i e s from t h e game d e t e r m i n e s t h e f i n a l subgame. I t i s e a s y t o prove t h a t i t does n o t .

The

r e s u l t i s always a unique subgame independent of t h e o r d e r used i n t h e elimination process. S e r i o u s problems w i t h dominance may a r i s e when t h e r e a r e i n f i n i t e l y many unrandomized s t r a t e g i e s ( s e e Appendix 8 ) . However, i n t h e m a j o r i t y of s o c i a l s c i e n c e a p p l i c a t i o n s t h e number of unrandomised s t r a t e g i e s i s finite.

Chapter 2 1 Coalition formation: a game-theoretic approach

59

Strategies for Betor B 81'

l o r actor

A

A1

t:)

+

82'

-2

+

-2

3

-2

-2

1

-2

3

2

elimination of

0

-2

-2

2

the donlnated

the dominated

strategies

strsteglE.8

A1'

-2

A2'

1

-2

3

elimination of

Bl"

82"

-2 1

-2

*l"

A2"

D i s p l a y 1 0 . E l i m i n a t i o n of dominated s t r a t e g i e s . I n t h e o r i g i n a l game A 3 and B4 a r e dominated. A f t e r removing t h e s e , B 3 ' i s dominated and removed. The r e s u l t i n g game c o n t a i n s no dominated s t r a t e g i e s any more.

I t i s t h e maximal submatrix-without-dominated-

s t r a t e g i e s of t h e o r i g i n a l game m a t r i x . 2.4.3

DOMINANCE OF PAYOFF VECTORS AND PARETO OPTIMALITY

I n t h i s s e c t i o n w e s h a l l d e a l w i t h normal form a s w e l l a s c h a r a c t e r i s t i c f u n c t i o n games because of t h e i r s i m i l a r i t y w i t h r e s p e c t t o dominance of payoff v e c t o r s . One p a y o f f v e c t o r , w i t h one component f o r e a c h a c t o r , i s s a i d t o dominate a n o t h e r i f e v e r y a c t o r i s b e t t e r o f f or a t least as w e l l o f f w i t h t h e f i r s t . I n more t e c h n i c a l terms, dominance i s t h e c o o r d i n a t e -

w i s e or p r o d u c t o r d e r on t h e s p a c e of payoff v e c t o r s . T h i s p r i n c i p l e i s well-known

i n v a r i o u s f i e l d s o f mathematics and i t s

a p p l i c a t i o n s a s P a r e t o o p t i m a l i t y , c o l l e c t i v e r a t i o n a l i t y , maximal ( b u t n o t n e c e s s a r i l y l a r g e s t ) p o i n t i n a p a r t i a l o r d e r , o r a d m i s s a b i l i t y . A payoff v e c t o r i s u n a c c e p t a b l e i f i t is dominated by a n o t h e r f e a s i b l e v e c t o r . For a normal form game " f e a s i b l e " means t h a t t h e r e is a c o m b i n a t i o n of s t r a t e g i e s l e a d i n g t o t h e payoff v e c t o r . F o r a c h a r a c t e r i s t i c f u n c t i o n game

i t means t h a t t h e sum o f a l l p a y o f f s i n t h e v e c t o r d o e s n o e t e x c e e d t h e value v ( { l , , . . , n } ) of t h e grand c o a l i t i o n .

In game t h e o r y a P a r e t o o p t i m a l

s o l u t i o n i s o f t e n s a i d t o be c o l l e c t i v e l y r a t i o n a l , a l t h o u g h s c e p t i c s may r e g a r d it as "an n - f o l d Combination o f w i s h f u l t h i n k i n g " pp. 218).

( J o n e s , 1980,

W.J.van der Linden and A. Verbeek

60 2.4.4

STABILITY

I n a normal form game a combination of s t r a t e g i e s , one f o r each a c t o r , i s c a l l e d s t a b l e or an e q u i l i b r i u m p o i n t i f t h e f o l l o w i n g h o l d s : (5)

No one a c t o r can i n c r e a s e h i s / h e r outcome by choosing a n o t h e r s t r a t e g y while t h e o t h e r a c t o r s s t a y p u t .

The f o l l o w i n g r e s u l t , s t a t i n g t h e e x i s t e n c e of s t a b l e p o i n t s i n normal form games, i s one of t h e o u t s t a n d i n g mathematical achievements of t h i s century : (6)

I f random s t a t e g i e s a r e allowed, t h e n any n - a c t o r normal form game h a s a t least one s t a b l e p o i n t .

(von Neumann, 1928, f o r n=2; Nash, 1951, f o r g e n e r a l n ) . A s t a b l e p o i n t can be i n t e r p r e t e d as a l o c a l maximum. No a c t o r can improve

h i s payoff by a s m a l l change, i . e . o n l y i n h i s own s t r a t e g y . I n two-actor constant-sum games s t a b l e p o i n t s have t h e f o l l o w i n g t w o p r o p e r t i e s , which g r e a t l y enhance t h e i r importance: (7)

I f t h e s t r a t e g i e s ( A i , B j ) and ( A i ' , B j ' ) l e a d t o s t a b l e p o i n t s , t h e n ( A i , B j ' ) and ( A i ' , B j ) a l s o l e a d t o s t a b l e p o i n t s .

(8)

A l l s t a b l e p o i n t s have t h e same payoff v e c t o r .

The l a t t e r p r o p e r t y can e a s i l y be d e r i v e d from t h e former. Also, i t is e a s y t o show t h a t an a c t o r ' s payoff f o r any s t a b l e p o i n t is t h i s a c t o r ' s "maximin" v a l u e of t h e game ( t o be d e f i n e d i n t h e n e x t s e c t i o n ) . The above p r o p e r t i e s make t h e s t a b l e p o i n t i n two-actor

constant-sum games

i n t o a widely a c c e p t e d s o l u t i o n concept which always e x i s t s , ( 6 ) , and i s , i n a s e n s e , u n i q u e , ( 8 ) . For 2 3 - a c t o r games and non-constant-sum

games,

s t a b l e p o i n t s are s t i l l noteworthy but f a l l s h o r t of an a c c e p t a b l e , g e n e r a l s o l u t i o n concept. I f a c t o r s c a n communicate, s e v e r a l a c t o r s may d e c i d e t o change s t r a t e g i e s s i m u l t a n e o u s l y , which s e r i o u s l y i m p a i r s t h e i n t e r p r e t a t i o n of s t a b l e p o i n t s a s s o l u t i o n c o n c e p t s . Moreover, D i s p l a y 11 shows t h a t s t a b l e s o l u t i o n s c a n have r a t h e r u n f a v o r a b l e p r o p e r t i e s . T h i s bi-matrix game h a s two s t a b l e p o i n t s : (A2,B2) and (A3,B3). I t is e a s y t o

see t h a t t h e y a r e s t a b l e among t h e unrandomized s t r a t e g i e s , b u t theorem ( 6 ) h o l d s f o r a s e t t i n g where randomization is allowed and i n t h a t s e t t i n g t h e t w o p o i n t s are a l s o s t a b l e , N o t e , however,

t h a t p r o p e r t i e s ( 7 ) and ( 8 )

are v i o l a t e d f o r t h e s e two p o i n t s . The s i t u a t i o n i s even worse, though: (A1,Bl) dominates both s t a b l e p o i n t s i n s p i t e of t h e f a c t t h a t A 1 1s

Chapter 2 1 Coalition formation: a game-theoretic approach

61

strategies for actor B

83

B1

B2

A1

( 2 , 2)

(-4, 3 )

(-4,-4)

A2

(3,-4)

(1, 0 )

(-4,-4)

st r a t egies for actor A

(-4,-4)

A3

D i s p l a y 11. A b i - m a t r i x

(-4,-4)

( 0 , 1)

game w i t h two s t a b l e p o i n t s ,

(A2,B2) and

(A3,B3). N o t e t h a t A w i l l p r e f e r t h e f o r m e r , and B t h e l a t t e r ! Moreover i n many ways (A1,Bl) is t h e b e s t s o l u t i o n . But i t i s q u i t e u n s t a b l e , for A2 d o m i n a t e s Al,

and 82 d o m i n a t e s B 1 .

(See a l s o Appendices 5 and

6.)

dominated by A2 and B 1 by B2. So (A1,Bl) is h i g h l y u n s t a b l e , b u t i n many ways i t is t h e b e s t s o l u t i o n . The s i t u a t i o n , which i s t y p i c a l of t h e p r i s o n e r ' s dilemma game, w i l l b e d i s c u s s e d a t g r e a t e r l e n g t h i n Appendix 5 .

2.4.5

MAXIMIN STRATEGIES AND VALUES

A somewhat p e s s i m i s t i c way t o assess t h e v a l u e of a game t o an actor i s

t h e f o l l o w i n g o n e : To e a c h of t h e a c t o r ' s ( p o s s i b l y randomized) s t r a t e g i e s

i t s l o w e s t p o s s i b l e outcome i s a s s i g n e d . The a c t o r i s s e c u r e t o o b t a i n a t l e a s t t h i s minimum. I t i s o b v i o u s t o choose t h e s t r a t e g y w i t h t h e l a r g e s t minimum. Such a s t r a t e g y i s c a l l e d maximin. The minimum outcome of t h e maximin s t r a t e g y is c a l l e d t h e a c t o r ' s s e c u r e v a l u e o r maximin v a l u e o f t h e game. I f an a c t o r h a s more t h a n one maximin s t r a t e g y , t h e a s s o c i a t e d s e c u r e v a l u e s a r e e q u a l , so t h e maximin v a l u e is p r o p e r l y d e f i n e d . F o r example, i n t h e m a t r i x game from D i s p l a y 4 t h e maximin v a l u e f o r A i s - 2 , whereas i t i s -1 f o r B ( B ' s outcomes are t h e n e g a t i v e s of t h e e n t r i e s i n t h e m a t r i x ) . I f r a n d o m i z a t i o n is a l l o w e d , t h e maximin s t r a t e g i e s are a s c a l c u l a t e d i n s e c t i o n 4 . 1 : PA = 5/8 and PB = 3/8 w i t h maximin v a l u e s of -1/8 f o r A and +1/8 f o r B.

O b v i o u s l y , a f i n i t e s e t of unrandomized s t r a t e g i e s always h a s a maximin s t r a t e g y . F o r r a n d o m i z a t i o n s of a f i n i t e s e t o f s t r a t e g i e s t h i s i s n o t so o b v i o u s , h u t i s i s t r u e . F o r i n f i n i t e s e t s of o p t i o n s t h e r e need n o t b e a

62

W.J. van der Linden and A. Verbeek

maximin s o l u t i o n , n o t even i n a o n e - a c t o r game: I f you may choose a n a t u r a l number and your outcome is t h a t amount of d o l l a r s , t h e r e simply i s no maximum ( c f . Appendix 8 ) .

For two-actor constant-sum games a s shown i n D i s p l a y 9 , i t i s e a s y t o check t h a t t h e maximin s t r a t e g i e s of t h e two a c t o r s t o g e t h e r form an e q u i l i b r i u m p o i n t , and, c o n v e r s e l y , t h a t any e q u i l i b r i u m p o i n t c o r r e s p o n d s t o a combination o f maximin s t r a t e g i e s . Now ( 6 ) ,

( 7 ) , and ( 8 ) combine i n t o t h e s o - c a l l e d minimax (or maximin)

theorem: (9)

In a two-actor constant-sum game w i t h a f i n i t e number of unrandomized s t r a t e g i e s i n which randomization is a l l o w e d , e a c h a c t o r h a s a t l e a s t one maximin s t r a t e g y ; each p a i r of maximin s t r a t e g i e s , one f o r each a c t o r , corresponds t o a s t a b l e payoff v e c t o r ; c o n v e r s e l y , each s t a b l e p o i n t corresponds t o a p a i r of maximin s t r a t e g i e s ; and i f t h e r e are more t h a n one s t a b l e p o i n t s , t h e n a l l have i d e n t i c a l payoff v e c t o r s .

In t h e p r e v i o u s s e c t i o n , w e have a l r e a d y s e e n t h a t no g e n e r a l i z a t i o n s are a v a i l a b l e w i t h r e s p e c t t o non-constant-sum 2.4.6

or

2

3 - a c t o r constant-sum games.

THREAT

I n two-actor constant-sum games, a c t o r s a r e d i r e c t opponents: e v e r y penny one wins i s p a i d by t h e o t h e r . In o t h e r m a t r i x games c o n f l i c t s may be less d i r e c t . On t h e one hand, a c t o r s c a n o f t e n p r o f i t by some form o f coo p e r a t i o n , as i n t h e p r i s o n e r ' s dilemma game, but on t h e o t h e r hand, t h e y can sometimes o b s t r u c t c e r t a i n s t r a t e g i e s of t h e o t h e r s i d e by u s i n g t h r e a t s ( a c t i o n s t h a t "may h u r t you more t h a n t h e y h u r t m e " ,

J o n e s , 1980,

pp. 225). Note t h e apparent c l a s h w i t h our d e f i n i t i o n o f u t i l i t y : t h r e a t e n i n g a c t o r s seem t o d e f y t h e i r u t i l i t i e s i n o r d e r t o nag o t h e r a c t o r s . But t h i s i s o n l y seemingly so; u t i l i t i e s r e f e r t o t h e a c t o r ' s p r e f e r e n c e s f o r t h e v a r i o u s p o s s i b l e outcomes and t h r e a t e n i n g h a s no a d d i t i o n a l u t i l i t y i n i t s e l f . Malicious p l e a s u r e , f o r example, should have been i n c o r p o r a t e d i n t h e u t i l i t y f u n c t i o n ( j u s t as p l e a s u r e i n coo p e r a t i o n ) . By t h r e a t e n i n g , an a c t o r i s supposed o n l y t o f o r c e o t h e r s i n t o a mode of b e h a v i o r t h a t o f f e r s him/her b e t t e r p e r s p e c t i v e s . A d r a m a t i c example is g i v e n i n Display 12. Here, B1 dominates 9 2 , b u t i f A i s c e r t a i n t h a t B p l a y s 91 t h e n A can s a f e l y p l a y A2 and win 1 0 . But by t h r e a t e n i n g

Chapter 2 f Coalitionformation: a game-theoretic approach

63

strategies for actor B B2

B1

strategies for actor A A1 A2

( 0 , 0)

( 0 , 10) (10,

0)

(-1 000 000,-1)

D i s p l a y 1 2 . “ T h i s may h u r t you more t h a n i t h u r t s me“ ( t h i s = B2, you = A , m e = b ) . A b i - m a t r i x game w i t h a t h r e a t s t r a t e g y , B2. Nota t h a t B2 i s q u i t e v a l u a b l e t o B , e v e n though i t i s dominated by B1.

t o p l a y 8 2 , B c a n t r y t o f o r c e A i n t o p l a y i n g A 1 and B wins 1 0 .

The f i r s t example i n D i s p l a y 13 (McKinsey, 1 9 5 2 ) r e v e r s e s t h e s i t u a t i o n .

strategies for actor B B1

B2

Actor A h a s o n l y one s t r a t e g y : A 1

( 0 , -1000)

( 1 0 , 0)

D i s p l a y 13. ( c f . McKinsey, 1 9 5 2 , Example 1 7 . 9 , page 351)

I n t h i s b i - m a t r i x game a c t o r A h a s o n l y one s t r a t e g y , i . e . no c o n t r o l a t a l l . Y e t A seems t o be i n a b e t t e r p o s i t i o n t h a n B: Wouldn‘t you r a t h e r p l a y a s A?

A h a s o n l y one s t r a t e g y and no c o n t r o l a t a l l . Y e t , A seems t o be i n t h e s t r o n g e s t p o s i t i o n b e c a u s e A w i l l win 1 0 , u n l e s s B i s w i l l i n g t o a c c e p t a l o s s o f 1 , 0 0 0 a t no compensation a t a l l . P e r h a p s B c a n t r y l u r i n g A i n t o a s m a l l payment f o r p l a y i n g B2 b u t B ‘ s t h r e a t t o p l a y B 1 c a n n e v e r b e c o n v i n c i n g . T h i s example r e l i e s h e a v i l y on t h e i m p l i c i t a s s u m p t i o n t h a t A and B ‘ s u t i l i t i e s a r e g i v e n i n comparable u n i t s . I f one u n i t of u t i l i t y

f o r A was e q u i v a l e n t t o 1,000 u n i t s f o r B , t h e game would be q u i t e d i f f e r e n t and become a s shown i n t h e second example. Now B ’ s t h r e a t t o p l a y B 1 i s much more s e r i o u s . The dependence o n a c e r t a i n c o m p a r a b i l i t y of u t i l i t y

i s i n h e r e n t i n many t h r e a t s i t u a t i o n s . Hence, t h e p h r a s e “ i t may h u r t you more t h a n i t h u r t s m e ” .

W.J. van der Linden and A. Verbeek

64

I n r e a l l i f e b l a c k m a i l , s t r i k e s , b o y c o t s , wars, and imprisonments a r e obvious examples of t h i s t y p e o f t h r e a t . T h e i r implementations a r e examples of what was c a l l e d u n a c c e p t a b l e payoff v e c t o r s i n s e c t i o n 4 . 3 . A concept r e l a t e d t o t h r e a t , b l u f f i n g , i s e l u c i d a t e d i n von Neumann and

Morgenstern (1944, s e c t . 1 9 . 2 ) .

2.5

SOLUTION CONCEPTS FOR CHARACTERISTIC FUNCTION GAMES

In t h i s s e c t i o n s e v e r a l s o l u t i o n s f o r c h a r a c t e r i s t i c f u n c t i o n games a r e g i v e n . In p r a c t i c e , t h e s e games t y p i c a l l y t a k e t h e form of s e v e r a l rounds of n e g o t i a t i o n , r e s u l t i n g i n t o a f i n a l d i s t r i b u t i o n of p a y o f f s among t h e a c t o r s . Most game t h e o r e t i c s o l u t i o n s f o c u s on p r e d i c t i n g t h e s e d i s t r i b u t i o n s r a t h e r t h a n t h e c o a l i t i o n s t h a t w i l l form. T h i s seems r e a l i s t i c and c o n s i s t e n t w i t h t h e f a c t t h a t i n s o c i a l l i f e c o a l i t i o n s a r e no e n d s i n themselves b u t v e h i c l e s used by i n d i v i d u a l s maximizing t h e i r p a y o f f s .

2.5.1

PAYOFFS, IMPUTATIONS, AND DOMINANCE

A payoff v e c t o r i n a c h a r a c t e r i s t i c f u n c t i o n game i s an n-vector x satis-

fying

and

Condition 1 0 i s c a l l e d i n d i v i d u a l r a t i o n a l i t y because i t assumes t h a t no a c t o r a c c e p t s less t h a n he can o b t a i n on h i s own. C o n d i t i o n 11s i s c a l l e d t h e f e a s i b i l i t y c o n d i t i o n ; i t shows t h a t t h e payoff v e c t o r ( x l ,

...,x

) can

always be r e a l i z e d , i f n e c e s s a r y by forming t h e grand c o a l i t i o n . The concept of P a r e t o o p t i m a l payoff v e c t o r s i n normal form games h a s t h e f o l l o w i n g c o u n t e r p a r t i n c h a r a c t e r i s t i c f u n c t i o n games. A payoff v e c t o r x

i s c a l l e d an i m p u t a t i o n i f i t s a t i s f i e s (10) and (llb)

x1

+

... + x

= v(I1,

...,n1)

Chapter 2 Coalition formation: a game-theoretic approach

65

I n s e c t i o n 4 . 3 . i t w a s s e e n t h a t a payoff v e c t o r x d o m i n a t e s a v e c t o r y i f e a c h a c t o r i s b e t t e r o f f ( o r a t l e a s t as w e l l ) w i t h x t h a n w i t h y , t h a t

i s , i f xi

2

yi f o r e a c h i . O b v i o u s l y , no i m p u t a t i o n c a n dominate a n o t h e r ;

f r o n ( l l b ) i t f o l l o w s t h a t i m p u t a t i o n s a r e P a r e t o o p t i m a l (undominated).

2.5.2

DOMINANCE BY COALITIONS AND THE CORE

The above c o n c e p t of dominance d o e s n o t r e f l e c t t h e f a c t t h a t s u b s e t s of a c t o r s , and n o t n e c e s s a r i l y i n d i v i d u a l a c t o r s , c o l l e c t t h e i r v a l u e ( o n c e t h e y have r e a c h e d an agreement on t h e d i v i s i o n ) . The f o l l o w i n g more demanding c o n c e p t of dominance d e a l s w i t h t h i s a s p e c t .

A c o a l i t i o n S i s s a i d t o dominate payoff v e c t o r x i f

( f o r an e q u i v a l e n t see e . g . J o n e s , 1980, s e c t . 4 . 5 . ) .

The i n t e r p r e t a t i o n

i s s i m p l e : There i s no r e a s o n f o r t h e members of S t o a c c e p t x as a f i n a l payoff ( p r o v i d e d , of c o u r s e , t h a t t h e y a r e w i l l i n g t o form a c o a l i t i o n and r e a c h an agreement on t h e d i v i s i o n o f v ( S ) ) .

A s an example, c o n s i d e r t h e game where 1, 2 , and 3 may d i v i d e $3,000,000 on c o n d i t i o n t h a t a m a j o r i t y a g r e e s on t h e payoff v e c t o r ( s e e Appendix 9 ) .

A r e a s o n a b l e i m p u t a t i o n seems t o be $ 1 , 0 0 0 , 0 0 0 for e a c h a c t o r s i n c e , a f t e r a l l , t h e s i t u a t i o n i s f u l l y symmetric. But 3 shrewdly w i s p e r s i n 2 ' s e a r t h a t t o g e t h e r t h e y c a n a l s o e f f e c t u a t e t h e i m p u t a t i o n x = (O,l&M,l&M), f o r

{2,31 d o m i n a t e s t h e symmetric s o l u t i o n . Now 1 g e t s t e n s e and o f f e r s y = (1M,2M,O) t o 2 , which is p o s s i b l e b e c a u s e [1,21 d o m i n a t e s x . #en

2 is

about t o a c c e p t t h i s , 2 and 3 n o t i c e t h a t t o g e t h e r t h e y dominate y and t h e y r e i t e r a t e t h e n e g o t i a t i o n s . And s o o n , and so f o r t h .

So dominated i m p u t a t i o n s a r e q u i t e u n s t a b l e i n t h e sense d e s c r i b e d above. Hence, i t seems r e a s o n a b l e t o r e j e c t them a s p o s s i b l e s o l u t i o n s and t o r e q u i r e t h a t t h e e v e n t u a l payoff v e c t o r s h o u l d be i n t h e

core, i . e . ,

the

s e t of a l l i m p u t a t i o n s t h a t a r e n o t dominated by any c o a l i t i o n . The d e f i n i t i o n of t h e c o r e i s narrow, i n f a c t even t o o narrow f o r g e n e r a l a p p l i c a t i o n b e c a u s e many games s i m p l y have no undominated i m p u t a t i o n s (see

66

W.J. van der Linden and A. Verbeek

t h e above example and p r o p e r t y (16) below). In such games t h e f i n a l payoff v e c t o r is n e c e s s a r i l y dominated. Thus, actors sometimes simply have t o a c c e p t u n s t a b l e s o l u t i o n s . Examples of c o r e s a r e given i n Display 14.

coalition

characteristic function v

value i n game ( a )

{1,2,31

P

value i n game ( b )

value i n game ( c )

4

4

4

2

1

0

3

1

0

3

4

0

0

0

0

0

0

0

0

0

0

0

0

0

D i s p l a y 14. Three 3 - a c t o r c h a r a c t e r i s t i c f u n c t i o n games. In games ( a ) and (c ) o n l y t h e grand c o a l i t i o n l e a d s t o a P a r e t o optimal p a y o f f . I n game ( a ) t h e c o r e c o n s i s t s of o n l y one i m p u t a t i o n , (1,1,2). In game ( h ) o n l y t h e c o a l i t i o n s t r u c t u r e {l}, { 2 , 3 1 l e a d s t o i m p u t a t i o n s i n t h e core.

Here t h e c o r e is t h e set of a l l payoff v e c t o r s

(O,h,4-h) w i t h 1

S

S

3. I n game ( c ) any d i s t r i b u t i o n

of 4 o v e r t h e a c t o r s 1, 2 , and 3 i s i n t h e c o r e , i n c l u d i n g such unreasonable o n e s a s (4,0,0)and ( 0 , 2 , 2 ) , where some a c t o r s r e c e i v e no c r e d i t f o r t h e i r i n dispensable participation. T h i s s e c t i o n c o n c l u d e s w i t h a more formal d e s c r i p t i o n of some e l e m e n t a r y p r o p e r t i e s of dominance and o f t h e c o r e . L e t x be a n i m p u t a t i o n and S a non-empty s u b s e t of a c t o r s . Then, (13)

"S dominates x" i s e q u i v a l e n t t o

Chapter 2 / Coalition formation: a game-theoretic approach (14)

67

t h e r e i s an i m p u t a t i o n y s u c h t h a t

tics

Yi

5 V(S)

and

x. < yi f o r a l l iES C o n d i t i o n (13) i s commonly d e s c r i b e d as "y dominates x w i t h r e s p e c t t o S" ( s e e , e . g . Luce & R a i f f a , 1 9 5 7 , s e c t . 9 . 1 . ) .

N o t e t h a t , a s opposed t o o t h e r dominance-like

(15)

concepts,

t h e r e l a t i o n of dominance is i n g e n e r a l n o t t r a n s i t i v e i n t h e set of i m p u t a t i o n s .

I n t h e above example u = (0,1M,2M) d o m i n a t e s v = (2M,0,1M) w i t h r e s p e c t t o { 2 , 3 } ; v d o m i n a t e s w = (1M,2M,O) w i t h r e s p e c t t o {1,3} b u t w d o m i n a t e s

u w i t h r e s p e c t t o ( 1 , 2 } . Hence, n e g o t i a t i o n s may be " c i r c u l a r " . (16)

The c o r e i s e q u a l t o (imputation x: = (payoff

'ieS

x: CiEs

ci

xi

t V(S) f o r a l l S }

x i 2 v ( S ) f o r a l l S and xi = v ( l 1 , .

. . ,n})}

T h i s means t h a t t h e c o r e i s a compact, convex s u b s e t of Rn.

Two o t h e r

r e s u l t s on dominance are:

(17)

Every i m p u t a t i o n i n e v e r y constant-sum

game i s dominated by

some o t h e r i m p u t a t i o n , ( i . e . t h e core of a constant-sum game

i s empty). (18)

For e v e r y payoff v e c t o r x i n any game t h e r e is a p a y o f f v e c t o r y t h a t i s n o t dominated by x .

I n o t h e r words, one p a y o f f v e c t o r n e v e r d o m i n a t e s a l l o t h e r payoff v e c t o r s . (Remember t h a t o n l y e s s e n t i a l games are c o n s i d e r e d ; see s e c t . 2 . 2 . 1 ) .

W.J. van der Linden and A. Verbeek

68

It does not take many steps to prove these properties. (15) is proven by the example given, while (16) is also trivial. The other proofs can be looked up in textbooks such as Jones (1980). 2.5.3

VON NEUMANN-MORGENSTERN SOLUTIONS AND SUBSOLUTIONS

The solution to be discussed in this section is important mainly for three reasons. The first is a historical reason: it has received much attention in game theory ever since its geneses. A second reason is that the notions employed in its definition are standard and have proved satisfactory in many parts of mathematics. A final reason is that it is surprisingly unsatisfactory as a general solution concept for characteristic function games.

A set X of imputations is called stable if (19)

for all x,y

&

X it is not true that x dominates y

and (20)

for each imputation y g X there is an x E X such that x dominates Y.

Now the von Neumann-Morgenstern solution is defined as the collection of all stable sets of imputations pertaining to the game. One interpretation of this solution concept is the following. If the negotiation arrives at an imputation y outside a stable set X, there is a coalition that prefers some imputation x in X by (20) (but other coalitions may still prefer other payoff vectors inside or outside X). If the negotiation focuses on an imputation in X, no coalition will propose another imputation in X (although possibly some coalitions will strive for an imputation outside X). Suppose the negotiations consist of two rounds. In round one the actors agree on a certain stable set in which the solution must lie. In round two the ordinary negotiations start. Then it is always possible to move from any payoff vector into the stable set, and once the actors are there they probably stay there.

Chapter 2 / Coalitionformation: a game-theoretic approach

69

Some e l e m e n t a r y p r o p e r t i e s of s t a b l e s e t s and t h e von Neumann-Morgenstern s o l u t i o n are:

(21)

Every s t a b l e set c o n t a i n s t h e c o r e ;

(22)

The c o r e i t s e l f i s s t a b l e i f and o n l y i f i t s a t i s f i e s ( 1 9 ) ;

(23)

I f t h e c o r e i s s t a b l e , t h e n i t i s t h e o n l y s t a b l e set.

An example of a s t a b l e c o r e i s t h e 3 - a c t o r game w i t h v({1,2,3}) = 1 v ( { 1 , 2 ] ) = v ( { 2 , 3 } ) = v ( { 1 , 3 } ) = c1 v ( { i l ) = v t l 2 l ) = ~ ( { 3 } )= 0 , and c1 5 1 / 2 ( c f . J o n e s , 1980, p. 208; V o r o b ' e v , 1977, s e c t . 4 . 1 5 )

(24)

A c l a s s o f games w i t h more t h a n one s t a b l e set i s formed by t h e

s i m p l e games ( s e c t . 3 . 8 . ) .

I f S is any "minimal winning"

c o a l i t i o n ( v ( S ) = 1 and v ( S ' ) = 0 f o r e v e r y p r o p e r s u b s e t S ' of S ) , t h e n e a c h of t h e f o l l o w i n g sets i s s t a b l e :

xS

= { i m p u t a t i o n x : x. = = { p a y o f f v e c t o r x:

x. = (25)

o

for i g

s}.

C

o iES

for a l l i

S}

x. = 1, x . Z 0 and 1

A s t a b l e s e t c a n n e v e r c o n s i s t of a s i n g l e i m p u t a t i o n ( c f . 1 8 ) .

The von Neumann-Morgenstern s o l u t i o n e n t a i l s some s e r i o u s problems which c a n be summarized a s f o l l o w s :

(26)

The n o t i o n of a s t a b l e s e t s t i l l d o e s n o t i n d i c a t e a u n i q u e

(27)

Many games d o n o t have a u n i q u e s t a b l e s e t (see 2 4 ) .

i m p u t a t i o n as a s o l u t i o n b u t a

set of

imputations.

So i f one p r e d i c t s t h a t t h e f i n a l payoff v e c t o r w i l l be i n a s t a b l e s e t , t h e n one i n f a c t o n l y p r e d i c t s t h a t i t w i l l be i n t h e u n i o n o f a l l s t a b l e

sets. Unions of ( a t l e a s t two) s t a b l e sets a r e l a r g e r a g a i n , and t h e y are c e r t a i n l y n o t s t a b l e b e c a u s e s t a b l e sets a r e inclusion-maximal

with respect

t o ( 1 9 ) . For 3 - a c t o r constant-sum games, f o r i n s t a n c e , t h i s u n i o n is e q u a l t o t h e set of a l l imputations.

(28)

Some games have no s t a b l e set a t a l l .

70

W.J. van der Linden and A. Verbeek

Lucas (1967, quoted by J o n e s , 1980, p. 209) g i v e s an example of a 1 0 - a c t o r game without a s t a b l e s e t . Von Neumann and Morgenstern (1944) have shown t h a t e v e r y 4 - a c t o r constant-sum game h a s a t l e a s t on s t a b l e s e t . A s o l u t i o n concept c l o s e l y r e l a t e d t o t h e von Neumann-Morgenstern

s o l u t i o n i s R o t h ' s (1976) s u b s o l u t i o n . The concept of s u b s o l u t i o n s i s a l s o based on t h e i d e a of s t a b i l i t y , b u t i f d i f f e r s from t h e d e f i n i t i o n i n (19) (29)

-

(20) i n t h a t t h e l a t t e r is r e p l a c e d by:

f o r e a c h i m p u t a t i o n y g X t h a t dominates a n x' E X t h e r e i s an x

E

X such t h a t x dominates y.

Note t h a t i n (29) n o t a l l e l e m e n t s o u t s i d e X are dominated by an element i n X, but t h a t t h i s o n l y h o l d s f o r elements o u t s i d e X t h a t j e o p a r d i z e an element i n X . The f o l l o w i n g p r o p e r t i e s hold f o r s u b s o l u t i o n s : (30)

Every s o l u t i o n i s a s u b s o l u t i o n ;

(31)

Every s o l u t i o n c o n t a i n s t h e c o r e ;

(321

Every game h a s a s u b s o l u t i o n ;

(33)

S u b s o l u t i o n s are n o t n e c e s s a r i l y u n i q u e .

T h i s c h a p t e r i s concerned w i t h t h e p r e d i c t i o n of the f i n a l c o a l i t i o n s t r u c t u r e and i t s payoff v e c t o r . T h e r e f o r e , t h e c o n c l u s i o n i s t h a t s t a b l e

sets are mainly u s e f u l when t h e r e i s o n l y one s t a b l e s e t , i n p a r t i c u l a r when t h e c o r e i s a s t a b l e set ( c f . 23). 2.5.4 PARASITISM AND SELF-SUPPORTING SUBSETS

The p r e c e d i n g s e c t i o n s d e a l t w i t h sets o f r e a s o n a b l e i m p u t a t i o n s ; w e w i l l now t u r n t o t h e r e s u l t i n g c o a l i t i o n s t r u c t u r e .

F i r s t w e w i l l d i s c u s s a form of i n s t a b i l i t y which can even occur i n t h e c o r e . I n game ( a) i n Display 1 4 , a c t o r s 1 and 2 t o g e t h e r r e c e i v e 2 by t h e s i n g l e i m p u t a t i o n i n t h e c o r e . T h i s i s no more t h a n i f t h e y j o i n e d i n c o a l i t i o n {1,2}. I f t h e y d i d so, 3 would r e c e i v e n o t h i n g and t h e r e s u l t would n o t be P a r e t o o p t i m a l . C l e a r l y , 1 and 2 can t r y t o blackmail 3 i n t o

an imputation ( 1 + ~ ,1+6, 2-~-6),w i t h

E

and 6 > 0 , and, p r o b a b l y ,

E

= 6.

However, such an i m p u t a t i o n i s n o t i n t h e c o r e ; it i s dominated by {1,3)

Chapter 2 f Coalition formation: a game-theoretic approach

71

and a l s o by {2,31. T h i s p o s s i b i l i t y c a n by f o r m a l i z e d a s f o l l o w s .

A n o n t r i v i a l s u b s e t S i s s a i d t o be p a r a s i t i z e d a t i m p u t a t i o n x i f

and

The s t r i c t c o r e i s d e f i n e d a s t h e s e t of i m p u t a t i o n s x f o r which no s u b s e t S

i s p a r a s i t i z e d a t x.

C o n d i t i o n (34) means t h a t t h e actors o f Sc need S i n o r d e r t o r e c e i v e a s much a s t h e y d o . B u t (35) i m p l i e s t h a t t h e a c t o r s o f S a r e j u s t a s w e l l o f f i f t h e y d o n o t c o - o p e r a t e w i t h Sc. A s t r i c t i n e q u a l i t y i n (35) i s e q u i v a l e n t t o d o m i n a t i o n o f x by S and i m p l i e s ( 3 4 ) . Hence, dominance i m p l i e s p a r a s i t i s m , and

(36)

t h e s t r i c t c o r e is a s u b s e t of t h e c o r e .

I n game ( b ) i n D i s p l a y 14 t h e s t r i c t c o r e i s empty, b e c a u s e t h e s i n g l e p o i n t i n t h e c o r e a l l o w s p a r a s i t i s m . The s t r i c t c o r e s of games ( b ) and (C) a r e t h e s e t of i m p u t a t i o n s (O,k,4-k)

w i t h 1 < k < 3 and t h e set of

d i s t r i b u t i o n s o f 2 o v e r a l l a c t o r s so t h a t e a c h a c t o r r e c e i v e s a p o s i t i v e amount.

These examples are t y p i c a l i n t h a t i f t h e s t r i c t c o r e i s non-empty t h e n the core is the s t r i c t core plus its l i m i t points. Nevertheless, i t is p r e c i s e l y t h i s minor r e s t r i c t i o n t h a t l e a d s t o an i n t e r e s t i n g form of s t a b i l i t y i n c o a l i t i o n f o r m a t i o n , namely t h e p r e s e n c e o f " s e l f - s u p p o r t i n g c o a l i t i o n s " . I n d e a l i n g w i t h t h i s and r e l a t e d c o n c e p t s below, o n l y games w i t h a non-empty s t r i c t c o r e w i l l be c o n s i d e r e d . A s a consequence of (17), t h i s i m p l i e s games t h a t are non c o n s t a n t s u m .

I t i s a l s o assumed t h a t t h e

s o l u t i o n w i l l b e i n t h e s t r i c t c o r e . Moreover, t h e c o n v e n t i o n v ( @ ) = 0 is needed.

W.J. van der Linden and A. Verbeek

12

A subset of actors, S, is called self-supporting if one (and hence each)

of the following equivalent properties holds (37)

there is an imputation x in the strict core such that

(38)

= v(S); 'iES xi for each imputation x in the strict core

CiES xi = v(S); (39)

v({~,

..., nI)

= v(~) + v(sC).

The interpretation is that a self supporting coalition gets only its value but no more, and that negotiations can therefore be restricted to self-supporting sets. The easy proofs of the equivalence of (37)

-

(39)

as well as other properties of self-supporting sets are given in Verbeek (1981). Let a minimal self-supporting coalition be a non-empty self-supporting set of which each non-trivial subset is not self supporting. It is easy to prove that self-supporting coalitions are precisely unions of disjoint minimal self-supporting coalitions and that each actor belongs to at least one minimal self-supporting set. Minimal self-supporting sets can therefore be regarded as basic negotiation units. They need not be disjoint themselves, however. If they are, the game can be decomposed into independent subgames that are "truly co-operative" in the sense that they do not have any self-supporting proper subset, i.e. for any class of disjoint coalitions S1,

..., SK

it holds that v({l,

...,n})

> v(S1)

+

... +

v(SK).

If the minimal self-supporting subsets are not disjoint, they can still be interpreted as stable coalitions showing up in any solution within the strict core. These and other points are further expanded in Verbeek (1981). 2.5.5

SHAPLEY AND HARSANYI

Shapley (1953) offers a solution based on the idea of arbitration. It consists of a unique payoff vector, x = p ( v ) , which satisfies the three axioms listed below and is defined for each characteristic function v. The axioms are:

Chapter 2 1 Coalition formation: a game-theoretic approach (40)

p i s symmetric: i f i n t e r c h a n g i n g t w o a c t o r s d o e s n o t change t h e

(41)

x = p ( v ) i s an i m p u t a t i o n ;

73

game, t h e n p a s s i g n s t h e same p a y o f f s t o t h e s e two a c t o r s ;

(42)

i f v and w are c h a r a c t e r i s t i c f u n c t i o n s d e f i n e d f o r t h e same set of a c t o r s , t h e n p i s a d d i t i v e w i t h r e s p e c t t o v and w: P(V+W) = P(V) + P(W).

Shapley showed t h a t t h e r e i s one and o n l y one p w i t h t h e s e p r o p e r t i e s , which h a s t h e f o l l o w i n g e x p l i c i t f o r m u l a

(43)

x. = 1

c

(n-s) !(s-1) ! n!

1

v(~)-v(~',iil)

,

where t h e summation i s t a k e n o v e r a l l s u b s e t s of a c t o r s c o n t a i n i n g i , and s i s t h e number of a c t o r s i n S . The v a l u e x . i n (43) i s c a l l e d t h e Shapley v a l u e o f a c t o r i . I t l e a n s h e a v i l y on t h e t h i r d axiom, a f a v o r i t e one among m a t h e m a t i c i a n s b u t a n y t h i n g b u t o b v i o u s from a p s y c h o l o g i c a l p o i n t o f view ( f o r d e t a i l s , see Luce & R a i f f a , 1957, s e c t . 1 1 . 4 ) . Shapley values have an i n t e r e s t i n g i n t e r p r e t a t i o n a s a measure of a n a c t o r ' s p i v o t a l power. The b r a c k e t e d f a c t o r i n ( 4 3 ) i s t h e a d d i t i o n a l payoff t h a t a c c r u e s t o t h e c o a l i t i o n S \ { i ] i f i j o i n s . Assuming t h a t a l l c o a l i t i o n s t r u c t u r e s are e q u a l l y l i k e l y , i t c a n be shown t h a t t h e o t h e r factor is the probability t h a t i w i l l join S\{i].

The summation i n ( 4 3 )

t h e r e f o r e g u a r a n t e e s t h a t x . i s t h e average g a i n involved i n i ' s j o i n i n g 1

a c o a l i t i o n . T h i s f a c t makes t h e S h a p l e y v a l u e i n t o an a t t r a c t i v e o p e r a t i o n a l i z a t i o n of an a c t o r ' s power i n t h e game as d e f i n e d by t h e c h a r a c t e r i s t i c f u n c t i o n . I t h a s been u s e d , f o r example, t o measure t h e d i s t r i b u t i o n of power i n t h e U.N. S e c u r i t y C o u n c i l ( s e e e . g . J o n e s , 1 9 8 0 , pp. 220-221).

For an n-person

apex game ( c f . 3 . 8 . ) one e a s i l y v e r i f i e s t h a t

t h e Shapley v a l u e f o r t h e Apex a c t o r i s 1-2/11 u n i t s , and f o r e a c h of t h e 2 o t h e r a c t o r s 2 / ( n -n) u n i t s . For t h e game Me and My Aunt ( c f . t h i s y i e l d s a v a l u e of $60 f o r Aunt and $10 f o r e a c h o f I , P , Q and R .

Harsanyi (1959) proposed a r a t h e r i n t r i c a t e d e f i n i t i o n o f s t a b i l i t y f o r normal form games based on an a r b i t r a t i o n scheme. I n t h e appendix t o h i s paper t h e scheme i s a l s o a p p l i e d t o c h a r a c t e r i s t i c f u n c t i o n games y i e l d i n g t h e same f o r m u l a a s S h a p l e y ' s w i t h one i m p o r t a n t m o d i f i c a t i o n : t h e

W.J. van der Linden and A. Verbeek

74

c h a r a c t e r i s t i c f u n c t i o n v i s r e p l a c e d by a "modified c h a r a c t e r i s t i c f u n c t i o n " which a l s o t a k e s account of t h e p o s s i b i l i t i e s of t h r e a t connected with each c o a l i t i o n . 2.5.6

B A R G A I N I N G CONCEPTS

N e g o t i a t i o n s are dynamic p r o c e s s e s o f t e n going through s t a g e s i n which t h e acceptance of a c e r t a i n payoff v e c t o r i s a t i s s u e . I f t h e payoff v e c t o r i s n o t a c c e p t e d , i t can be t a k e n a s t h e s t a t u s a n t e f o r f u r t h e r n e g o t i a t i o n s . In such p r o c e s s e s , t h e a c t o r s a r e r e g u l a r l y a s s e s s i n g t h e i r b a r g a i n i n g s t r e n g t h s and behave according t o t h e outcomes of t h e s e a s s e s s m e n t s . I n o r d e r t o a s s e s s h i s s t r e n g t h an a c t o r may choose from s e v e r a l p o s s i b i l i t i e s : he may contemplate a d e t a i l e d p r o p o s a l f o r an a l t e r n a t i v e c o a l i t i o n w i t h s p e c i f i e d p a y o f f s t o t h e i r members, a s s e s s h i s s t r e n g t h r e l a t i v e l y t o t h e c o a l i t i o n he belongs t o a t t h e moment ( p o s s i b l y v i a p a i r w i s e comparisons with each o f t h e o t h e r members of t h e c o a l i t i o n ) , o r adopt an e m p i r i c a l approach, t u n i n g h i s a s s e s s m e n t s , f o r i n s t a n c e , t o t h e number o f t i m e s o t h e r s t r y t o p u l l him i n t o t h e i r c o a l i t i o n s . However, whatever a s p e c t of t h e s i t u a t i o n t h e a c t o r may f o c u s on, i t i s c l e a r t h a t h i s b a r g a i n i n g s t r e n g t h u l t i m a t e l y depends on t h e p r o f i t a b l e n e s s and c r e d i b i l i t y of a l t e r n a t i v e c o a l i t i o n s open t o him. The concept of p r o f i t a b l e n e s s of an a l t e r n a t i v e c o a l i t i o n with r e s p e c t t o t h e payoff v e c t o r under c o n s i d e r a t i o n w i l l be f u r t h e r e l u c i d a t e d below.

Of t h e s e v e r a l r e a s o n s why a c e r t a i n payoff v e c t o r x i s n o t a c c e p t e d , t h e o u t s t a n d i n g reason is t h e p r o p o s a l t o form a c o a l i t i o n

S t h a t dominates

x, i . e .

( c f . s e c t . 5 . 2 . ) . In t h i s s e c t i o n we w i l l f o c u s on t h i s a s p e c t . I t i s important t o n o t e t h a t t h e e x i s t e n c e o f a dominating c o a l i t i o n is o n l y a n e c e s s a r y c o n d i t i o n f o r l e a v i n g a payoff v e c t o r c ; it d o e s n o t g u a r a n t e e t h a t x w i l l n o t be a c c e p t e d . Recall, f o r i n s t a n c e , t h a t constant-sum a s

w e l l as many o t h e r games have an empty c o r e , t h a t i s , have no undominated payoff v e c t o r s ( c f . 1 7 ) . Also i n t h i s s e c t i o n w e o n l y c o n s i d e r c h a r a c t e r i s -

t i c functions v satisfying

Chapter 2 1 Coalition formution:a game-theoretic approach v(Ti1) = 0 ,

f o r each a c t o r i ,

and

,

~ ( s )1 0

f o r each c o a l i t i o n S .

If a c h a r a c t e r i s t i c f u n c t i o n w d o e s n o t s a t i s f y t h e s e r e q u i r e m e n t s , t h e v a l u e s w({i}) c a n b e r e g a r d e d a s a p a r t i c i p a t i o n f e e t o t h e game, and

is s t r a t e g i c a l l y e q u i v a l e n t t o w ( s e e Appendix 2 ) Turning t o a more f o r m a l approach t o b a r g a i n i n g p r o c e s s e s , w e s t a r t w i t h a few d e f i n i t i o n s .

I n s e c t i o n 5 . 1 . t h e c o n c e p t of an i n d i v i d u a l l y r a t i o n a l

and f e a s i b l e payoff v e c t o r was d e f i n e d and i t was i n d i c a t e d t h a t s u c h a v e c t o r i s c a l l e d an i m p u t a t i o n i f and o n l y i f i t is P a r e t o o p t i m a l . The c o n c e p t of payoff v e c t o r i s now expanded i n t o payoff c o n f i g u r a t i o n . A

is c a l l e d a payoff c o n f i g u r a t i o n i f i t s a t i s f i e s

p a i r (x,B)

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x i s a payoff v e c t o r ;

B

is a set o f d i s j o i n t c o a l i t i o n s w i t h u n i o n { 1 , 2 , .

say

B

= {B1,B2,

f o r each j

. . . ,B 1; B

.. , n j ,

is c a l l e d t h e c o a l i t i o n s t r u c t u r e ;

CiEB, J

A payoff c o n f i g u r a t i o n (x,B) is c a l l e d c o a l i t i o n a l l y r a t i o n a l i f f o r e v e r y s u b s e t S o f a c o a l i t i o n B . c 8 , j = 1, J

...,

m, i t h o l d s t h a t

A payoff c o n f i g u r a t i o n i s t h u s c o a l i t i o n a l l y r a t i o n a l e i t h e r i f i t is undominated o r i f any d o m i n a t i n g c o a l i t i o n c o n t a i n s a t l e a s t two a c t o r s belonging t o d i f f e r e n t c o a l i t i o n s i n

B.

Suppose actor io b e l o n g s t o c o a l i t i o n B from payoff c o n f i g u r a t i o n (x,B). j An o b j e c t i o n of io a g a i n s t (x,B) i s a c o a l i t i o n S and a payoff v e c t o r f o r t h e members of S, (yi)iEs,

satisfying

W.J.van der Linden and A. Verbeek

16

(45a)

i

(45b)

v(S) =

0

(45c)

y.

(45d)

yi

i

E

2

Si

ciEs

x

i > x

yi

for all i i

0

E

S;

'

If il also belongs to 8 . but not to S, then il may raise a counterobjection, J

that is, a coalition T and a payoff vector (zi)iET satisfying (46a)

i B T, i E T ; 0 1

(46b)

v(T) = CiET zi;

(46~)

z. 2 xi

(46d)

zi 2 yi

for all i

E

T;

for all i

E

T n S.

Note that objections and counterobjections are realistic in that the members of S are better off under the objection than under x, while the members of T are better off than under the counterobjection (if they belong to S ) and under x. Note however, that the actors required to consent either to S or to T may still have better alternatives. Neither has been taken into account that actors not belonging to B may try to j prevent the formation of S or T by making proposals still more profitable to one of the members of S or T. Finally, note that S and T may be disjoint. In that case ( T , e ) and (S,y) may coexist. Consider the game Me and My Aunt (cf. 1.3.), and in particular the payoff vector xo = (a,lOO-a,0,0,0)with coalition structure {Aunt,I} {P,Q,R}. Here "I" can raise the objection x1 = (0,100-a+6,(a-6)/3,(a-6)/3),

against

which "Aunt" can raise a counterobjection such as x2 = (a,O,lOO-a,O,O)if and only if 100-a 2 (a-6)/3.

Against xo an objection such as

x3 = (a+6,0,100-a-6,0,0) can be raised by "Aunt". Then "I" can raise a counterobjection such as x4 = (O,lOO-a,a,O,O) if and only if a 2 100-a-6. Then any objection against x o can be countered if and only if 50

S

a

5

75.

At the end of 5.6. we return to this example. Any measure of the support that a possible coalition S gives its member i with respect to a payoff vector x is called a profit. It will be denoted by p(S,x,i),

or by p(S,x)

if it has the same value for all i

following section two examples will be given, namely the

E

S . In the

Chapter 2 1 Coalition formation: a game-theoretic approach excess (s,x)

(47)

V(S)

- zits

77

xi,

and t h e surplus ( S , x , i ) 1 v(S)/s - x.

(48)

1'

where s i s t h e number o f a c t o r s i n S. The e x c e s s is t h e t o t a l g a i n o f a l l members of S r e l a t i v e t o x . The s u r p l u s s u p p o s e s t h a t v ( S ) i s s h a r e d e q u a l l y among a l l members of x and t h u s measures p r o f i t a t t h e l e v e l of t h e i n d i v i d u a l a c t o r . Of c o u r s e , o t h e r ways of s h a r i n g e i t h e r v ( S ) or t h e excess a r e p o s s i b l e a s w e l l , f o r in s tan ce, p ro por t i onal l y t o t he payoffs i n x o r a c c o r d i n g t o an o f f e r of t h e a c t o r who proposed t o form S .

The f i n a l d e f i n i t i o n needed i s t h a t of t h e c o n c e s s i o n of i a t x. L e t A . be t h e set of c o a l i t i o n s e a c h of which c o n t a i n s i . The c o n c e s s i o n o f i a t

x r e l a t i v e t o .b, is d e f i n i e d as c o n c e s s i o n ( A . , x , i ) = max

(49)

The c o n c e s s i o n i s t h e maximum amount o f loss a c t o r i i n c u r s by a c c e p t i n g x and n o t e n t e r i n g i n t o a n o t h e r p o s s i b l e c o a l i t i o n .

2.5.7

BARGAINING SOLUTIONS

A set P o f payoff c o n f i g u r a t i o n s i s a s o l u t i o n t o a b a r g a i n i n g p r o c e s s i f i t h o l d s t h a t (1) s o o n e r or l a t e r i n t h e p r o c e s s a payoff c o n f i g u r a t i o n i n P w i l l be d i s c u s s e d , and ( 2 ) i f t h e n e g o t i a t i o n s r e a c h a p a y o f f c o n f i g u r a -

t i o n i n P , t h i s w i l l b e a c c e p t e d by t h e a c t o r s . I n t h i s s e c t i o n some of t h e most p r o m i s i n g s o l u t i o n s t o b a r g a i n i n g p r o c e s s e s w i l l be d i s c u s s e d .

BARGAINING SETS

C o n s i d e r t h e set of a l l payoff c o n f i g u r a t i o n s i n which f o r any o b j e c t i o n of any a c t o r t h e r e i s a c o u n t e r o b j e c t i o n from a n o t h e r a c t o r from t h e same c o a l i t i o n . T h i s set i s c a l l e d t h e b a r g a i n i n g s e t , d e n o t e d by M(i)

(the

s u p e r s c r i p t reminds u s t h a t a l l payoff v e c t o r s a r e i n d i v i d u a l l y r a t i o n a l ;

see sect. 5 . 1 . ) . The b a r g a i n i n g set is nonempty; P e l e g (1967) h a s shown t h a t f o r any c o a l i t i o n s t r u c t u r e 8 t h e r e e x i s t s a payoff v e c t o r x s u c h t h a t (x,B)

E

MCi). So i n a s e n s e M(i)

does n o t p r e d i c t any p a r t i c u l a r

W.J. van der Linden and A. Verbeek

78

coalition structure. If the bargaining set is intersected with the set of coalitionally rational payoff configurations (see 4.4.), the coalitionally rational bargaining set is obtained, denoted by M. Obviously, M is a subset of the bargaining set; it is trivially nonempty, because it always contains the very unattractive "degenerate" payoff configuration with payoff zero to every actor and "coalitions" only consisting of single actors. The two solutions were proposed by Aumann and Maschler (1964). A variant of the latter solution is obtained if it is not only required that the finally accepted payoff configuration is coalitionally rational, as Aumann and Maschler did, but also that the objections and counterobjections satisfy this property. Maschler (1963) also proposed a modified bargaining set, M(im), defined identically to M(i),

which is

with the exception that it is not based on the

characteristic function but on a function representing the "power" of every possible coalition. This approach has, however, not drawn much attention either from theoreticians or from empirical investigators. Horowitz (1973) has raised the important point that condition (46c) is not strong enough to make the counterobjections credible. Above it was assumed that all actors i f io, il in T show noncompetitive behavior and are willing to help il, ignoring possible coalitions with io in which they obtain more than in ills counterobjection. This led him to a generalization of M(i)

based on the idea of multiobjections, that is,

objections in which i threats simultaneously through a collection of 1 distinct coalitions. The set of all individually rational payoff configurations for which a counter-multiobjection exists for any multiobjection of any actor is called the competitive bargaining set, H(i). For formal definitions and proofs of the following basic properties, the reader should refer to Horowitz (1973): (50)

M = M(~), e.g. for apex games;

(51)

H(i)

(52)

HCi) C Mci), with identity e.g. for n = 3 and for (n-1)-quota

may be empty;

games ; (53)

for apex games the payoff vectors in

are extreme points

Chapter 2 1Coalition formation: a game-theoretic approach of t h e convex h u l l o f t h e p a y o f f - v e c t o r s

i n MCi).

The l a s t p r o p e r t y h a s o f t e n been used i n e x p e r i m e n t a l game t h e o r y t o t e s t Hci) a g a i n s t t h e k e r n e l , which i s d e f i n e d i n t h e n e x t p a r a g r a p h s .

KERNELS K e r n e l s a r e s o l u t i o n s based on p a i r w a i s e comparisons of t h e c o n c e s s i o n s each a c t o r h a s t o make w i t h r e s p e c t t o a g i v e n payoff c o n f i g u r a t i o n ( x , B ) . These comparisons a r e made o n l y f o r p a i r s of a c t o r s b e l o n g i n g t o t h e same coalition B .

J

E

5. When a c t o r i t h r e a t e n s a c t o r j , i ’ s s e t of a l t e r n a t i v e

coalitions is defined as

The c o n c e s s i o n of i r e l a t i v e t o

c., 1J

s13 . . is

d e n o t e d by

c o n c e s s i o n ( S . . , x , i ) = max sEs,. P ( S , X ) , 1J

1J

where p i s a p r o f i t measure t o be c h o s e n . Note t h a t c . . and c . . may b e 1J

J1

p o s i t i v e , z e r o , or n e g a t i v e , i n d e p e n d e n t l y o f each o t h e r . W e say that a c t o r i outweighs a c t o r j i f

(54)

x. > 0,

J

and

I f ( 5 5 ) h o l d s b u t x . = 0 , we d o n o t s a y t h a t i o u t w e i g h s j b e c a u s e j

J

cannot be f o r c e d t o a c c e p t a n e g a t i v e p a y o f f , i . e . l o w e r t h a n v ( { i } ) . The g e n e r a l i z e d k e r n e l i s t h e s e t of a l l payoff c o n f i g u r a t i o n s i n which no one a c t o r o u t w e i g h s a n o t h e r b e l o n g i n g t o t h e same c o a l i t i o n . For any c h o i c e of p r o f i t measure p , a d i f f e r e n t k e r n e l i s o b t a i n e d . Davis and Maschler (1965) i n t r o d u c e d t h e f i r s t s o l u t i o n a l o n g t h e above l i n e s c h o o s i n g t h e e x c e s s d e f i n e d i n ( 4 7 ) a s p r o f i t measure. T h e i r

79

W J . van der Linden and A. Verbeek

80

s o l u t i o n is b r i e f l y known a s t h e k e r n e l , denoted by K. Some b a s i c p r o p e r t i e s of K a r e :

(56)

K c M(i)

(57)

For e v e r y c o a l i t i o n s t r u c t u r e 8 t h e r e i s a payoff v e c t o r x s u c h t h a t (x,B)

(58)

E

K (Davis and M a s c h l e r , 1965).

f o r apex games t h e payoff v e c t o r s i n K a r e extreme p o i n t s of ti) t h e convex h u l l of t h e payoff v e c t o r s i n M (Horowitz, 1973).

I t should b e n o t e d t h a t Horowitz'

(1973) c r i t i c i s m of MCi) a l s o h o l d s f o r

g e n e r a l i z e d k e r n e l s : i t i s assumed t h a t t h e o t h e r a c t o r s a r e n o t c o m p e t i t i v e b u t w i l l i n g t o a c c e p t a s much as i p r o p o s e s t o g i v e them.

I n t h e case of

Davis and M a s c h l e r ' s K t h e s u g g e s t i o n even i s t h a t i c l a i m s t h e whole e x c e s s f o r h i m s e l f , l e a v i n g t h e o t h e r a c t o r s w i t h what t h e y would r e c e i v e u n d e r x. T h i s p a r t i c u l a r o b j e c t i o n , which o n l y r e l a t e s t o K , c a n e a s i l y be remedied by a n o t h e r c h o i c e of p r o f i t measure. For example, C r o t t and A l b e r s (1980) assume t h a t t h e a l t e r n a t i v e c o a l i t i o n i n

s1J. , w i l l

equally

s h a r e v ( S ) and choose t h e s u r p l u s i n ( 4 8 ) as p r o f i t measure. They c a l l t h e i r s o l u t i o n t h e equal d i v i s i o n k e r n e l . I f t h e payoff c o n f i g u r a t i o n ( x , B ) b e l o n g s t o K C A , and i

E

S

E

B , and x . > 0 f o r a l l J

is a s i m p l e e x p l i c i t formula f o r x :

j E S, then there

where s = number of a c t o r s i n S , and

A . , = max IJ

iES, j g s

e(S)

- max

e(S)

i & S , JES

( c f . C r o t t & A l b e r s , 1980, page 293 (4)) I n t h e above p r e s e n t a t i o n i t i s o n l y a t r i v i a l s t e p t o i n t r o d u c e a new k e r n e l . But t o do j u s t i c e t o C r o t t and A l b e r s we mention t h a t h i s t o r i c a l l y it happened t h e o t h e r way round. T h e i r e q u a l d i v i s i o n k e r n e l h a s l e d us

t o adopt t h e g e n e r a l i z e d approach p r e s e n t e d h e r e .

EXAMPLES Below one f i n d s t h e v a r i o u s

solutions

f o r t h e 5 p e r s o n apex game M e and My

Chapter 2 Coalition formation: a game-theoretic approach

81

Aunt, i n t r o d u c e d i n 2 . 1 . 3 . I n t h i s game t h e b a r g a i n i n g s e t M(i) c o n s i s t s of t h e payoff c o n f i g u r a t i o n s (x,B) with

x = (a,lOO-a,0,0,0)

w i t h 50 2 a < 7 5 and

8 = f { A u n t , I } , { P , Q , R } } , and v a r i a n t s o f x and 8 o b t a i n e d by I , P , Q , and R .

permuting

x = (0,25,25,25,25) with

Each p a y o f f v e c t o r x i n M(i) hence M = M(i)

8

= { { I , P , Q , R } , {Aunt

t r i v i a l l y i s c o a l i t i o n a l l y r a t i o n a l , and

( c f . ( 5 0 ) ) . F o r . H ( i ) see Horowitz (1979, p p .

275-276).

To d e m o n s t r a t e t h e i d e a s u n d e r l y i n g k e r n e l s , a g a i n c o n s i d e r t h e p a y o f f v e c t o r x o = (a,lOO-a,O,O,O). The e x c e s s o f { I , P , Q , R } i s a and t h e e x c e s s o f c o a l i t i o n s l i k e { A u n t , P } is 100

-

a . So xo i s i n t h e k e r n e l i f and

o n l y i f a = 50. N e x t , c o n s i d e r x5 = ( O , a , b , c , d ) . The e x c e s s e s a r e : b + c + d , a + c + d , a+b+d. a + b + c . T h e s e a r e e q u a l , and x5

E

K i f and o n l y i f

a = b = c = d = 2 5 . F i n a l l y c o n s i d e r t h e g r a n d c o a l i t i o n , and t h e p a y o f f v e c t o r x7 = ( a , b , c , d , e ) . The e x c e s s e s a r e : 100-a-b, 100-a-e,

and a . Then x,

E

100-a-c,

100-a-d,

K i f and o n l y i f a = 300/7 and b = c = d = e = 100/7.

Along s i m i l a r l i n e s one e a s i l y d e r i v e s t h e p a y o f f v e c t o r s i n C r o t t & A l b e r s kernel K

'

CA'

( 6 2 & , 3 7 3 , 0 , 0 , 0 ) and o t h e r s w i t h I , P , Q and R i n t e r c h a n g e d ; ( 0 , 2 5 , 2 5 , 2 5 , 2 5 ); and (40,15,15,15,15).

2.5.8

DYNAMIC B A R G A I N I N G MODELS

Komorita (1979) t a k e s a n i t e r a t i v e a p p r o a c h t o c o a l i t i o n f o r m a t i o n i n which e a c h s t e p i s c o n s i d e r e d a n a c t u a l s t a g e o f t h e b a r g a i n i n g p r o c e s s .

A t each s t a g e t h e a c t o r h a s a b a r g a i n i n g s t r e n g t h w i t h r e s p e c t t o e a c h of t h e Z n - l

p o s s i b l e c o a l i t i o n s he may j o i n . Here K o m o r i t a ' s a p p r o a c h ,

which he l a r g e l y p r e s e n t s i n t e r m s o f e x a m p l e s , i s f o r m a l i z e d and e x t e n d e d .

A f u l l e r t r e a t m e n t i s g i v e n i n Verbeek ( 1 9 8 2 ) . F i r s t some n o t a t i o n and d e f i n i t i o n s a r e needed. A t s t e p r ( r = O , l , Z ,

w e have f o r e a c h a c t o r i and c o a l i t i o n S c o n t a i n i n g i a v a l u e E r

1s

...)

which i

W.J. van der Linden and A. Verbeek

82

would r e c e i v e i f S formed a t t h a t s t e p . T h i s v a l u e i s c a l l e d t h e p o t e n t i a l p a y o f f . We r e q u i r e t h a t t h e s e v a l u e s s a t i s f y i n d i v i d u a l and S-wise rationality:

(60)

EIS 2 v({i}) = 0

(for all i

(61)

ZiEs

( f o r a l l S).

EiS

= v(S)

E S

and r ) ;

The v e c t o r (ErS)iES, w i t h both i and S as v a r i a b l e i n d i c e s , w i l l be c a l l e d a p o t e n t i a l payoff v e c t o r . The b a s i c i d e a of t h e approach, which assumes t h a t a l l a c t o r s know ( E r ) .

1s 1 E S

( o r a t least t h e l a r g e s t element f o r e a c h i )

i s t h e f o l l o w i n g . If S i s a c t o r i ' s p r e f e r r e d c o a l i t i o n (because Er

1s

2

E r T f o r a l l T c o n t a i n i n g i ) but some j

E

S p r e f e r s another c o a l i t i o n ,

i a s k s t o o much. I n o r d e r t o make S p r e f e r a b l e t o j as w e l l , i should r+l . accept a lower v a l u e EiS i n f a v o r of an i n c r e a s e i n t h e v a l u e Er+'. JS

T h i s can be f o r m a l i z e d a s f o l l o w s . For each c o a l i t i o n S, each a c t o r i

E

S h a s a s e t of a l t e r n a t i v e s

Actor i can u s e c o a l i t i o n s from t h i s set as a t h r e a t when n e g o t i a t i n g with t h e o t h e r a c t o r s i n S about h i s payoff d i s t r i b u t i o n . The b e s t a l t e r n a t i v e t o S f o r a c t o r i i s d e f i n e d as h i s maximum p o t e n t i a l payoff i n t h i s s e t o f a l t e r n a t i v e s ; i t i s denoted by

(63)

A ; ~ = max TESiS E:T

The d i f f e r e n c e

i s , analogously t o ( 4 9 ) , c a l l e d t h e p o t e n t i a l concession of i a t S i n r .

F i n a l l y , w e need a r u l e t o r e - a s s e s s i ' s p o t e n t i a l payoff from S a t s t e p r + l , as a f u n c t i o n of t h e p o t e n t i a l p a y o f f s and b e s t a l t e r n a t i v e s of a l l members j

E S.

The r u l e is denoted by $;

Chapter 2 1 Coalition formation: a game-theoretic approach

In o r d e r t o make $ a r e a s o n a b l e r e - a s s e s s m e n t s a t i s f y (60)

-

83

r u l e it i s required t o

( 6 1 ) . Moreover, i f a t s t e p r some b u t n o t a l l a c t o r s i n

S make a p o s i t i v e c o n c e s s i o n , a t l e a s t t h e a c t o r w i t h t h e smallest

( n o n - p o s i t i v e ) c o n c e s s i o n s h o u l d g e t l e s s and t h e a c t o r w i t h t h e l a r g e s t ( p o s i t i v e ) c o n c e s s i o n s h o u l d g e t more.

In formulas:

Now w e c a n d e f i n e two n o t i o n s o f s t a b i l i t y f o r p o t e n t i a l p a y o f f v e c t o r s . 0 A p o t e n t i a l p a y o f f v e c t o r (E. ) . i s c a l l e d a f i x e d p o i n t of I$, i f f o r a l l 1s I E S

i and S

E1

1s

1 where EiS

0 = E.

1s

'

i s d e f i n e d by ( 6 4 )

is c a l l e d a

limit o f

-

( 6 5 ) . A p o t e n t i a l p a y o f f v e c t o r (E

is)iEs

$, i f t h e r e e x i s t s a s t a r t i n g p o i n t (E.1s1.1 E S s u c h

t h a t t h e e l e m e n t s E I S d e f i n e d by ( 6 4 ) - ( 6 5 ) s a t i s f y

(67)

E~~ = i i m

r-

~i~.

I t i s e a s i l y seen t h a t each fixed point is a l i m i t .

Further t h e converse

is e q u i v a l e n t t o t h e d e s i r a b l e c o n d i t i o n t h a t t h e f u n c t i o n $ i s c o n t i n u o u s . L i t t l e i s known y e t a b o u t t h e e x i s t e n c e of f i x e d p o i n t s , and t h e c o n v e r g e n c e o f ( E i s ) a s i n ( 6 7 ) . F o r more d e t a i l s , examples, and c o u n t e r e x a m p l e s see Komorita ( 1 9 7 9 ) . and Verbeek ( 1 9 8 2 ) .

Komorita (1979) makes t h e f o l l o w i n g c h o i c e s :

s b e i n g t h e number o f a c t o r s i n S . T h i s is t h e " e q u a l s h a r e s " p a y o f f . F o r t h e set of a l t e r n a t i v e s he p r o p o s e s

W .J. van der Linden and A. Verbeek

84

r a t h e r t h a n (62) (Komorita, 1979, pp. 372-373). H i s r e - a s s e s s m e n t r u l e i s :

Note t h a t v ( S ) =

z. JES

Er

Js

f o r a l l r (proof by i n d u c t i o n on r). K o m o r i t a ' s

i n t e r p r e t a t i o n i s a s f o l l o w s : a t s t e p r+l e a c h a c t o r p o t e n t i a l l y g e t s h i s b e s t a l t e r n a t i v e p l u s an e q u a l s h a r e of t h e e x c e s s of v ( S ) o v e r t h e

sum of a l l b e s t a l t e r n a t i v e s of t h e members of S . However, i f t h e game i s h i g h l y asymmetric, such as an apex game o r a q u o t a game w i t h unequal q u o t a , t h e i d e a of e q u a l s h a r e s seems u n r e a l i s t i c and a d i f f e r e n t s t a r t i n g p o i n t and re-assessment

r u l e might b e more a p p r o p r i a t e .

I t i s n o t n e c e s s a r y t o i n t e r p r e t t h e model from t h i s s e c t i o n a s a model

for a c t u a l s t e p s i n a b a r g a i n i n g p r o c e s s , as Komorita d o e s i n h i s p a p e r . Another a t t r a c t i v e i n t e r p r e t a t i o n i s t o r e g a r d t h e model, i . e . t h e c h o i c e s of

sis

and $, a s a model for d e f i n i n g s t a b l e p o i n t s . The i t e r a t i v e p r o c e s s ,

t h e n , i s merely a t r i c k t o f i n d t h e s e s t a b l e p o i n t s ( t h e f i x e d p o i n t s of

$1. The a d v a n t a g e of t h i s i n t e r p r e t a t i o n i s t h a t i f t h e model i s poor f o r t h e b a r g a i n i n g p r o c e s s i n g e n e r a l , but good when t h e p r o c e s s g e t s c l o s e to a f i x e d p o i n t , i t w i l l s t i l l g i v e a good a p p r o x i m a t i o n t o t h e s e f i x e d p o i n t s and y i e l d s o l u t i o n s f o r t h e a c t u a l b a r g a i n i n g p r o c e s s . G e n e r a l l y s p e a k i n g , a model f o r a c o n v e r g i n g i t e r a t i v e p r o c e s s should be good n e a r

i t s f i x e d p o i n t s . I f i t i s , it p r e v e n t s t h e i n e v i t a b l e approximation e r r o r s from a tendency t o accumulate a n d , t h e r e b y , t h e model from an i n a d e q u a t e a p p r o x i m a t i o n of t h e p r o c e s s .

2.5.9

MISCELLANEOUS SOLUTIONS

The p r e c e d i n g s e c t i o n s d e a l t w i t h t h e most i m p o r t a n t s o l u t i o n s f o r c h a r a c t e r i s t i c f u n c t i o n s known s o f a r . We w i l l now d i s c u s s some of t h e o t h e r s o l u t i o n s which have a t t r a c t e d o n l y m a r g i n a l a t t e n t i o n .

The f i r s t s o l u t i o n i n game t h e o r y , s u c h a s t h e von Neumann-Morgenstern s o l u t i o n and t h e c o r e , s t r e s s e d t h e d i s m i s s a l of c e r t a i n payoff v e c t o r s a s u n r e a s o n a b l e . T h i s l e d t o t h e r e s t r i c t i o n t o i m p u t a t i o n s , and l a t e r t o undominated i m p u t a t i o n s . T h i s d o e s n o t mean t h a t a l l i m p u t a t i o n s o r a l l

Chapter 2 / Coalitionformation: a game-theoreticapproach

85

p o i n t s i n t h e c o r e are l i k e l y or e q u a l l y l i k e l y . M i l n o r (1952) proposed t h r e e o t h e r r e s t r i c t i o n s which e a c h l e a d t o a d i f f e r e n t c l a s s of " r e a s o n a b l e outcomes" ( s e e a l s o Luce & R a i f f a , 1 9 5 7 , c h a p . 1 1 ) . One of t h e r e s t r i c t i o n s i s t h a t f o r e a c h i i t must h o l d t h a t

where t h e maximum i s t a k e n o v e r a l l s u b s e t s c o n t a i n i n g i . ( 7 1 ) c a n be m o t i v a t e d by o b s e r v i n g t h a t i f i b e l o n g s t o a c o a l i t i o n S which o n l y g e t s

i t s v a l u e v ( S ) , t h e o t h e r a c t o r s i n S would n o t be s a t i s f i e d w i t h l e s s t h a n v ( S \ { i } ) . A l l i m p u t a t i o n s i n t h e von Neumann-Morgenstern

solution,

t h o s e i n t h e c o r e and a l s o t h e Shapley v a l u e obey ( 7 1 ) . D e s p i t e t h i s f a c t , Luce and R a i f f a r a i s e s e r i o u s c r i t i c i s m of (71). For t h i s criticism a s

w e l l a s a d e s c r i p t i o n o f t h e o t h e r r e s t r i c t i o n s , t h e r e a d e r is r e f e r r e d t o c h a p t e r 11 o f t h e i r book.

I n a l l s o l u t i o n s s o f a r i t h a s been assumed t h a t any s u b s e t o f a c t o r s i n t h e game can form a c o a l i t i o n ( f o r most c h a r a c t e r i s t i c f u n c t i o n s some o f t h e s e c o a l i t i o n s w i l l l e a d t o poor p a y o f f s , t h o u g h ) . O f t e n , however, i t would be more r e a l i s t i c t o assume t h a t t h e s e t of p o s s i b l e c o a l i t i o n s and c h a n g e s from o n e c o a l i t i o n t o a n o t h e r a r e r e s t r i c t e d .

In p o l i t i c s , f o r

example, i t i s o f t e n p o s s i b l e t o r e p r e s e n t t h e p a r t i e s r e a s o n a b l y w e l l on a l i n e a r l y o r d e r e d s c a l e and i t seems r e a s o n a b l e t o r e q u i r e t h a t a c o a l i t i o n s h o u l d have no "gaps" on t h i s s c a l e . C o a l i t i o n c h a n g e s might be r e s t r i c t e d by t h e r u l e t h a t o n l y one (or a t most k , f o r some f i x e d k) a c t o r ( s ) can d e f e c t from a c o a l i t i o n a t one t i m e . More f o r m a l l y , a p a i r c o n s i s t i n g of an i m p u t a t i o n x and a c o a l i t i o n s t r u c t u r e $-stable

8 c a n be c a l l e d

i f , f o r e v e r y c o a l i t i o n S t h a t can be o b t a i n e d from a c o a l i t i o n

i n 8 by a n a d m i s s i b l e o n e - s t e p (or k - s t e p ) d e f e c t i o n , i t h o l d s t h a t

i . e . , c o a l i t i o n s r e a c h a b l e i n o n e (or k) s t e p ( s ) d o n o t d o m i n a t e x. For d e t a i l s , s e e Luce and R a i f f a ( 1 9 5 7 , c h a p . 1 0 ) .

An e l e g a n t and s i m p l e s o l u t i o n h a s been p r o p o s e d by Kemeny ( 1 9 5 9 ) . With

W.J.van der Liizden and A. Verbeek e a c h a c t o r he a s s o c i a t e s a f u n c t i o n measuring h i s / h e r b a r g a i n i n g power which is independent of h i s b a r g a i n i n g p a r t n e r s . From t h e s e f u n c t i o n s i t f o l l o w s how a winning c o a l i t i o n w i l l d i s t r i b u t e i t s v a l u e . Assuming t h i s t o be known b e f o r e h a n d , e a c h a c t o r i s a b l e t o r a n k a l l p o s s i b l e c o a l i t i o n s a c c o r d i n g t o h i s payoff and c a n be e x p e c t e d t o j o i n t h e c o a l i t i o n most f a v o r a b l e t o him. I n m o s t cases, t h i s l e a d s t o u n i q u e s o l u t i o n .

2.6

A SURVEY OF EXPERIMENTAL GAME THEORY

2.6.1

INTRODUCTION AND OVERVIEW

Although t h e f i r s t p u b l i c a t i o n s i n game t h e o r y can b e t r a c e d back t o t h e e a r l y f o r t i e s , i t was n o t u n t i l 1954 t h a t t h e f i r s t p u b l i c a t i o n on e x p e r i m e n t a l game t h e o r y a p p e a r e d . T h i s was t h e p a p e r by K a l i s h , M i l n o r , Nash, and Nering (1954) which became w i d e l y known among game t h e o r i s t s by

i t s i n c l u s i o n i n Luce and R a i f f a ' s G a m e s and d e c i s i o n s (1957, s e c t . 1 2 . 3 ) . N e v e r t h e l e s s , t h e s i x t i e s o n l y show a few e x p e r i m e n t s s c a t t e r e d i n t h e l i t e r a t u r e , and w e have t o w a i t u n t i l t h e s e v e n t i e s t o watch t h e b i r t h of e x p e r i m e n t a l game t h e o r y a s a new branch of b e h a v i o r a l s c i e n t i f i c res e a r c h . But seldom h a s a new o f f s p r i n g grown up f a s t e r ! S i n c e t h e mids e v e n t i e s , dozens of e x p e r i m e n t s on c o a l i t i o n f o r m a t i o n i n games have been r e p o r t e d i n j o u r n a l a r t i c l e s . Moreover, n o t a l l r e s e a r c h r e a c h e s t h i s form o f p u b l i c a t i o n , and i t seems s a f e t o assume t h a t a t t h e moment s e v e r a l hundreds of e x p e r i m e n t s have been conducted t o test g a m e - t h e o r e t i c s o l u t i o n s e m p i r i c a l l y . That i t t o o k so long t o make game t h e o r y e x p e r i m e n t a l

i s p r i m a r i l y due t o t h e ( f o r t u n a t e ) f a c t t h a t i t s f i r s t c o n t r i b u t o r s were predominantly m a t h e m a t i c a l l y i n c l i n e d . The p r e s e n t g e n e r a t i o n , however, shows a b a l a n c e d m i x t u r e of c o n c e r n w i t h m a t h e m a t i c a l modeling and e x p e r i m e n t a l s k i l l s . I t i s i n t e r e s t i n g t o compare t h i s w i t h t h e t r a d i t i o n of s o c i a l p s y c h o l o g i c a l r e s e a r c h i n t o c o a l i t i o n forming i n t h e t r i a d ;

i n i t s f i r s t p a p e r s (Caplow, 1 9 5 6 , 1959; C h e r t k o f f , 1 9 6 7 ; Gamson, 1 9 6 1 a , 1961b; Vinacke & A r k o f f , 1957) t h i s t r a d i t i o n a l r e a d y shows an a d m i r a b l e concern w i t h e m p i r i c a l t e s t s of t h e o r e t i c a l p r e d i c t i o n s b u t , a p a r t from an o c c a s i o n a l p a p e r ( e . g . , Shenoy, 1 9 7 8 ) , l a c k s t h e m a t h e m a t i c a l s o p h i s t i c a t i o n game t h e o r y h a s .

A t f i r s t s i g h t , i t may seem s i m p l e t o conduct an e x p e r i m e n t a l test of g a m e - t h e o r e t i c s o l u t i o n s . Nothing more seems t o b e needed t h a n t o p l a c e

Cliapter 2 J Coalition formation: a game-theoretic approach

87

s u b j e c t s i n a game-like s i t u a t i o n , t o o b s e r v e t h e c o a l i t i o n s formed and t h e way t h e p a y o f f s a r e d i s t r i b u t e d , and t o compare t h e s e outcomes w i t h p r e d i c t i o n s from game t h e o r y . A s w i l l be s e e n p r e s e n t l y , t h i s i s a g r o s s misconception.

In f a c t , e x p e r i m e n t a l game t h e o r y i s a c o m p l i c a t e d f i e l d

w i t h many u n s o l v e d i s s u e s . I n t h i s s e c t i o n we w i l l f i r s t d i s c u s s some of t h e s e i s s u e s which have t o d o w i t h t h e q u e s t i o n s o f how e m p i r i c a l game-theoretic

p r e d i c t i o n s a r e and w h e t h e r t h e y c a n b e t e s t e d e x p e r i m e n t a l l y .

Then, w e w i l l r e v i e w t h e e x p e r i m e n t a l d e s i g n s t h a t have been u s e d h i t h e r t o , d i s c u s s i n g t h e t y p i c a l t a s k s t h a t have been g i v e n t o s u b j e c t s , t h e t y p e o f games i n v e s t i g a t e d , t h e most p o p u l a r e x p e r i m e n t a l c o n d i t i o n s , t h e u s u a l s u b j e c t s , and t h e l i k e . Next, t h e most i m p o r t a n t e x p e r i m e n t a l f i n d i n g s with respect t o t h e game-theoretic s o l u t i o n s discussed i n s e c t i o n 5 w i l l be g i v e n . Some g e n e r a l comments and recommendations c o n c l u d e t h i s s e c t i o n .

Although i t is t e m p t i n g t o d o s o , w e w i l l n o t p r e s e n t e m p i r i c a l r e s u l t s from s o c i a l psychology o r p o l i t i c a l r e s e a r c h i n t o c o a l i t i o n b e h a v i o r . S o c i a l p s y c h o l o g y i n p a r t i c u l a r o f f e r s many i n g e n i o u s e x p e r i m e n t s . The i n t e r e s t e d r e a d e r s h o u l d c o n s u l t t h e above r e f e r e n c e s a n d , e . g . , Burhans ( 1 9 7 3 ) , C a l d w e l l ( 1 9 7 1 ) , C h e r t k o f f and Esser ( 1 9 7 7 ) , Nacci and T e d e s c h i ( 1 9 7 6 ) , Nydegger and Owen ( 1 9 7 7 ) , Roth ( 1 9 7 9 ) , S t a n f i e l d , J e n k s , and McCartney ( 1 9 7 5 ) , Walker ( 1 9 7 3 ) , and Wolf and Shubik ( 1 9 7 7 ) . An e x c e l l e n t r e v i e w i s g i v e n i n t h e c h a p t e r by W i l k e , w h i l e d e Swaan's c h a p t e r also d e a l s w i t h a s p e c t s o f e m p i r i c a l p o l i t i c a l r e s e a r c h . An i n t e g r a t e d r e v i e w of game-theoretic,

s o c i a l p s y c h o l o g i c a l , and p o l i t i c a l a p p r o a c h e s t o

c o a l i t i o n f o r m a t i o n i s g i v e n i n Murnighan ( 1 9 7 8 a ) . I n t h e f o l l o w i n g , t h e s e p a p e r s w i l l b e t o u c h e d upon o n l y i n as f a r as t h e y o f f e r a new p e r s p e c t i v e

on g a m e - t h e o r e t i c r e s u l t s or d e m o n s t r a t e a r e l e v a n t e x p e r i m e n t a l t e c h n i q u e . I t is a l s o t e m p t i n g t o o f f e r t h e r e a d e r a n e x h a u s t i v e e n u m e r a t i o n of a l l e x p e r i m e n t a l r e s u l t s on game t h e o r y . S i n c e s o much m a t e r i a l i s a v a i l a b l e t h i s w i l l n o t b e done e i t h e r ; i n s t e a d , w e w i l l t r y t o g i v e an i m p r e s s i o n of t h e g r e a t v a r i e t y o f e x p e r i m e n t a l a p p r o a c h e s , p r o b l e m s , and f i n d i n g s , and a t t h e same t i m e emphasize what i s t y p i c a l and what t r e n d s c a n be o b s e r v e d . Examples o f h i g h l y o r i g i n a l models and e m p i r i c a l t e s t s n o t d e a l t w i t h i n t h i s c h a p t e r c a n b e found i n a series o f p a p e r s by C a s s i d y and Mangold

(1965), F r i e n d , L a i n g , and M o r r i s o n ( 1 9 7 7 ) , and L a i n g and Morrison (1973, 1974).

88 2.6.2

W.J. van der Linden and A. Verbee k SOME PRELIMINARY QUESTIONS

I n f a c t , game t h e o r y i s n o t h i n g more t h a n r i g o r o u s mathematics added t o assumptions about t h e behavior o f actors i n s i t u a t i o n s o f ( p a r t i a l ) conf l i c t . From t h i s a b s t r a c t system s t a t e m e n t s c a n be d e r i v e d which hold t r u e i n a s f a r as i t s assumptions a r e m e t . The q u e s t i o n h a s o f t e n been r a i s e d whether t h e assumptions are not t o o f a r removed from r e a l i t y t o y i e l d s t a t e m e n t s t h a t e v e r can be used a s p r e d i c t i o n s of a c t u a l b e h a v i o r . In o t h e r words, i s game t h e o r y n o t t o o much of an i d e a l i z a t i o n t o be r e a l i s t i c ? The most important assumptions have a l r e a d y been mentioned. One assumption adopted i n a l l game-theoretic models i s t h a t f o r each actor t h e p a y o f f s c a n be r e p r e s e n t e d by a u t i l i t y f u n c t i o n . Another i s t h a t e a c h a c t o r i s assumed t o be c o g n i z a n t of a l l r u l e s of t h e game and of t h e u t i l i t y f u n c t i o n s of a l l o t h e r a c t o r s . T o g e t h e r , t h e s e two assumptions a r e o f t e n combined i n t o t h e s t a t e m e n t t h a t game t h e o r y assumes " r a t i o n a l " a c t o r s . I t h a s a l r e a d y been argued t h a t t h e f i r s t assumption r e q u i r e s t h e a c t o r s t o demonstrate c o n s i s t e n t c h o i c e s between t h e p o s s i b l e outcomes of t h e game. Experience with p r e f e r e n c e measurement shows t h a t t h i s is n o t always t h e case. The second assumption i s o b v i o u s l y t o o s i m p l e f o r s i t u a t i o n s i n which many a c t o r s are involved i n a m u l t i l a t e r a l c o n f l i c t . Other assumptions a r e , f o r i n s t a n c e , t h e " i s o l a t e d n e s s " o f t h e game, t h e constancy of t h e u t i l i t e s , and ( f o r c h a r a c t e r i s t i c f u n c t i o n games) t h e t r a n s f e r a b i l i t y of u t i l i t y . F o r a t h e o r y t o be d e s c r i p t i v e of p r e f e r e n c e c o n f l i c t s i t must be concerned w i t h a l l s o c i a l , economic, and p s y c h o l o g i c a l a s p e c t s and a l l o w f o r a l l f a c t o r s and mechanisms a t work i n such c o n f l i c t s ( f o r an enumeration of many of t h e s e f a c t o r s , see K a l i s h , M i l n o r , Nash, & Nering, 1954). Game t h e o r y , on t h e o t h e r hand, seems t o be concerned

s o l e l y w i t h i d e a l , c l o s e d systems i n which r e a l - l i f e

" d i s t u r b a n c e s " are

n o t allowed. How can such a t h e o r y e v e r be d e s c r i p t i v e ? Does i t make any

sense t o c o n s i d e r s t a t e m e n t s d e r i v e d from such a t h e o r y as p r e d i c t i o n s , and t o test them e x p e r i m e n t a l l y ? Some a u t h o r s have r e a c t e d t o t h e s e q u e s t i o n s by c l a i m i n g t h a t game t h e o r y

i s not d e s c r i p t i v e b u t normative. For example, Luce and R a i f f a (1957, s e c t s . 6.4. and 1 1 . 5 . ) have pleaded t h a t s o l u t i o n s t o c o - o p e r a t i v e games i n c h a r a c t e r i s t i c f u n c t i o n form should be seen as a r b i t r a t i o n r u l e s . T h i s amounts t o i n t e r p r e t i n g game t h e o r y from t h e p o i n t o f view o f an a r b i t e r who h a s been asked t o a r b i t r a t e t h e game and t o i n d i c a t e a f a i r s o l u t i o n ' t o

Chapter 2 1 Coalition formation: a game-theoretic approach

89

t h e problem o f d i s t r i b u t i n g t h e p a y o f f s . " F a i r " means h e r e t h a t t h e s o l u t i o n ( u n i q u e l y ) f o l l o w s from w e l l - d e f i n e d d e s i d e r a t a w i t h r e s p e c t t o t h e a r b i t r a t i o n r u l e t h a t a r e s h a r e d by t h e a c t o r s . These d e s i d e r a t a t h e n s e r v e as c o n s i s t e n c y r e q u i r e m e n t s g u i d i n g t h e s o l u t i o n o f t h e game. A s a n a r b i t r a t i o n r u l e i s o n l y f a i r i n so f a r a s t h e a c t o r s a c t u a l l y e n d o r s e t h e s e r e q u i r e m e n t s , i t seems b e t t e r t o c a l l t h i s p o i n t o f view c o n d i t i o n a l l y n o r m a t i v e . An example o f a c o n s i s t e n t set o f r e q u i r e m e n t s l e a d i n g t o a u n i q u e s o l u t i o n i s t h e s e t o f S h a p l e y axioms from s e c t i o n 5 . 5 . I t s h o u l d be n o t e d t h a t a f a i r s o l u t i o n i n t h e c o n d i t i o n a l l y n o r m a t i v e s e n s e d o e s n o t a u t o m a t i c a l l y l e a d t o an e t h i c a l l y j u s t p a y o f f ( f o r example, t o an e q u a l s h a r e ) . Axioms s u c h a s t h e S h a p l e y set o n l y produce a s o l u t i o n i n c o m b i n a t i o n w i t h a g i v e n game, t h a t i s , f o r a c t o r s of a c e r t a i n s t r u c t u r a l s t r e n g t h and w i t h g i v e n u t i l i t y f u n c t i o n s . I n g e n e r a l , t h e S h a p l e y s o l u t i o n g i v e s more

t o t h e s t r o n g e r and less t o t h e weaker a c t o r s . A t f i r s t s i g h t , t e s t i n g n o r m a t i v e s o l u t i o n s e m p i r i c a l l y makes no more s e n s e t h a n , f o r example, t e s t i n g an Act o f B i r d P r o t e c t i o n or t h e r u l e s o f f o o t b a l l ! B e f o r e t r y i n g t o s o l v e t h i s p r o b l e m , some o t h e r problems w i t h r e s p e c t t o g a m e - t h e o r e t i c p r e d i c t i o n s a r e mentioned.

The m a j o r g a m e - t h e o r e t i c models d e v e l o p e d t h u s f a r a r e a l l d e t e r m i n i s t i c i n n a t u r e . T h i s means t h a t t h e y p r e d i c t w i t h p r o b a b i l i t y one t h a t a p a y o f f v e c t o r o r a c o a l i t i o n s t r u c t u r e from a c e r t a i n s e t s h a l l o c c u r ; a l l o t h e r payoff v e c t o r s o r c o a l i t i o n s t r u c t u r e s a r e p r e d i c t e d t o have a p r o b a b i l i t y equal t o zero.

I n t h i s r e s p e c t t h e y d i f f e r from major social p s y c h o l o g i c a l

models which a r e s t o c h a s t i c i n t h a t t h e y p r e d i c t ( n o n - d e g e n e r a t e ) p r o b a b i l i t y d i s t r i b u t i o n s over t h e set of p o s s i b l e c o a l i t i o n s t r u c t u r e s ( s e e , e . g . , C h e r t k o f f , 1 9 6 7 ; Komorita, 1 9 7 4 ; Walker, 1 9 7 3 ) . Most o f t h e s e models a r e s t o c h a s t i c b e c a u s e t h e y assume i n f a c t t h a t c e r t a i n r e s p o n s e mechanisms a r e o p e r a t i n g , making a c t o r s p o s i t i v e t o some c o a l i t i o n s and i n d i f f e r e n t t o o t h e r s . Models c o u l d a l s o b e s t o c h a s t i c b e c a u s e some p a r a -

meters a r e random, o r b e c a u s e an e r r o r term h a s been added r e p r e s e n t i n g random d i s t u r b a n c e s . The d i f f e r e n c e between d e t e r m i n i s t i c and s t o c h a s t i c models i n v o l v e s i m p o r t a n t d i f f e r e n c e s when making i n f e r e n c e s from e m p i r i c a l d a t a . S t o c h a s t i c models c a n be t e s t e d s t a t i s t i c a l l y . The above s o c i a l p s y c h o l o g i c a l models a r e a l l o f t h e p a r a m e t r i c a l l y m u l t i n o m i a l t y p e f o r which g o o d n e s s - o f - f i t

t e s t s w i t h known p r o p e r t i e s a r e a v a i l a b l e . For

d e t e r m i n i s t i c models i t i s i m p o s s i b l e t o t e s t t h e g o o d n e s s o f f i t s t a t i s t i c a l l y . These models are s i m p l y f a l s i f i e d a s soon a s one outcome o c c u r s

W.J.van der Linden

and A. Verbeek

that does not follow from the theory. However, it is common experience that anything can happen in experiments on game theory. In fact all nontrivial experiments in the literature show some subjects producing

out-

comes not predicted by the theory. Some authors (e.g., Rapoport & Kahan, 1976; Riker, 1967) are satisfied with outcomes varying randomly around the

predictions, but it is unclear whether these variations are actually random and how to assess their influence. Strictly speaking, all major gametheoretic models are falsified and, logically, it is no use conducting further experiments to try verifying them as yet. (An exception must be made for some recent attempts at stochastic modeline; see Cassidv & Mangold, 1975; Friend, Laing, & Morrison, 1977; Laing 81 Morrison, 1973, 1974; Roth, 1977, sect. 6 . These, however, do not attempt to make the major solutions from this chapter stochastic.) The ideal of experimental research is a crucial experiment in which competing models are tested against each other. Game theory seems an excellent area for such experiments since it offers a great variety of different models from which predictions about solutions can be derived. However, a closer look at the situation shows that the ideal of two or more models each yielding a different, competing prediction is hard to realize. Some models do not yield any prediction at all for a wide class of games. An example is the core which is empty for many games, including all essential constant-sum games. For some games, some models do not yield a point prediction but imply a whole set of solutions, or, worse still, non-unique sets of solutions, An example of the latter is the von NeumannMorgenstern solution, while the former may occur, for example, for the bargaining set. Games for which one model predicts a point but another a whole set of solutions prohibit a simple comparative test. The situation can be more complicated, however, Predictions from different models can show an overlap or one of the predicted sets can even be included in the other. Examples are the kernel, which for co-operative games in characteristic function form is contained in the bargaining set, and the bargaining set itself, which for some games is a special case of the competitive bargaining set. Murnighan and Roth (1977) used a 3-actor characteristic function game with a veto actor (although to test the core and the Shapley value) of which the entire space of imputations is encompassed both by the von Neumann-Morgenstern solution and the class of subsolutions.

Chapter 2 Coalition formation: a game-tkeoretic approach O b v i o u s l y , t h e l a r g e r t h e s e t of p r e d i c t i o n s ,

91

the larger the a p r i o r i

p r o b a b i l i t y of an outcome i n t h i s s e t . I t seems t h e r e f o r e u n f a i r t o t e s t models w i t h p o i n t p r e d i c t i o n a g a i n s t models w i t h s e t p r e d i c t i o n s .

I f one

s e t of s o l u t i o n s i s a s u b s e t o f a n o t h e r , t h e n an e x p e r i m e n t a l outcome can never s u p p o r t t h e former and f a l s i f y t h e l a t t e r . The s i t u a t i o n i s n o t always c o m p l i c a t e d , however. I n H o r o w i t z ' s apex game, which h a s been e x t e n s i v e l y experimented with (e.g.

Horowitz & R a p o p o r t , 1974; Funk, R a p o p o r t ,

& Kahan, 1980; Kahan & R a p o p o r t , 1979; R a p o p o r t , Kahan, Funk, 81 Horowitz,

1 9 7 9 ) , t h e k e r n e l and t h e c o m p e t i t i v e b a r g a i n i n g s e t p r e d i c t two d i f f e r e n t p o i n t s o f t h e p a y o f f s p a c e ( a l t h o u g h b o t h are boundary p o i n t s of t h e b a r g a i n i n g s e t ) . In Michener and S a k u r a i ' s (1976) v e t o game, t h e k e r n e l awards a l l of t h e p a y o f f s t o t h e v e t o a c t o r , which makes t h e game e x c e l l e n t l y s u i t e d f o r t e s t i n g t h e k e r n e l a g a i n s t t h e Shapley v a l u e . The p o i n t i s , however, t h a t c r u c i a l e x p e r i m e n t s a r e n o t always p o s s i b l e and t h a t c o m p e t i t i v e tests a r e o f t e n hampered by t h e n a t u r e of t h e p r e d i c t i o n s f o l l o w i n g from t h e models.

The above problems pose s e r i o u s q u e s t i o n s w i t h r e s p e c t t o e x p e r i m e n t a l game t h e o r y . They a r e n o t u n i q u e , however, and a l s o a p p l y t o o t h e r p a r t s of psychology and t o o t h e r s c i e n c e s . I n f a c t , some o f them are well-known m e t h o d o l o g i c a l problems which a l l e m p i r i c a l s c i e n c e s have been s t r u g g l i n g w i t h a t one t i m e o r a n o t h e r . As t o t h e q u e s t i o n whether game t h e o r y i s n o r m a t i v e o r d e s c r i p t i v e , t h e s e two p o s s i b i l i t i e s do n o t seem m u t u a l l y e x c l u sive.

I f i n a s i t u a t i o n o f c o n f l i c t where a l l p a r t i e s a c c e p t t h e a s s u m p t i o n s

of a c e r t a i n model a s r e a s o n a b l e , i t seems n a t u r a l t o u s e i t s ( h o p e f u l l y

u n i q u e ) s o l u t i o n a s a means f o r s o l v i n g t h e c o n f l i c t . T h i s does n o t , however, e x c l u d e t h e p o s s i b i l i t y t h a t game t h e o r y c a n a l s o have d e s c r i p t i v e v a l u e . I t d e s c r i b e s and p r e d i c t s r e a l i t y o n l y i n as f a r a s p e o p l e a c t u a l l y behave a s h a s been supposed i n i t s a s s u m p t i o n s . That game t h e o r y c a n have n o r m a t i v e a s w e l l a s d e s c r i p t i v e a s p e c t s h a s a l s o been o b s e r v e d by Lieberman (1962) and Rapoport and Kahan ( 1 9 7 6 ) . However, i t s h o u l d b e n o t e d t h a t game t h e o r y i n i t s p r e s e n t form does n o t p r o v i d e models f i t t i n g most s o c i a l c o n f l i c t s . I t s models a r e h i g h l y i d e a l i z e d mappings d e a l i n g o n l y w i t h p o r t i o n s o f r e a l i t y . As a consequence, t h e y can o n l y be e x p e c t e d t o p r e d i c t r e s u l t s o b t a i n e d u n d e r i d e a l i z e d c o n d i t i o n s . Except f o r some v e r y s i m p l e s i t u a t i o n s o f s o c i a l i n t e r a c t i o n

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and c o n f l i c t , s u c h c o n d i t i o n s can p r o b a b l y o n l y be approximated i n c a r e f u l l y c o n t r i v e d e x p e r i m e n t s . The s i t u a t i o n i s n o t u n u s u a l i n a d e v e l o p i n g new branch of s c i e n c e . There i s h a r d l y any m e t h o d o l o g i c a l d i f f e r e n c e between t h e i n t e r e s t o f game t h e o r y i n t h e b e h a v i o r of " r a t i o n a l " a c t o r s , and c l a s s i c a l p h y s i c s s t u d y i n g t h e b e h a v i o r o f "mass p o i n t s " swinging on a p e r f e c t l y f l e x i b l e s t r i n g w i t h mass z e r o o r moving w i t h o u t f r i c t i o n a l o n g an i n c l i n e d p l a n e . I t i s o n l y a t a l a t e r s t a g e t h a t d e v i a t i o n s from t h e s e c o n d i t i o n s are s y s t e m a t i c a l l y i n t r o d u c e d and t h a t t h e i r e f f e c t a r e assessed. The models from game t h e o r y are i n d e e d d e t e r m i n i s t i c and do n o t p a s s t h e

t e s t when one s u b j e c t i n one experiment does n o t show t h e p r e d i c t e d behavior.

I n t h i s s e n s e , a l l models have a l r e a d y been f a l s i f i e d , b u t t h i s

d o e s not seem much of a p r o b l e m . I n o u r o p i n i o n , it i s t o o e a r l y t o a c t u a l l y

test models: t h e r e are too many o f them and i t seems p r u d e n t t o c o n s i d e r them a l l a p r i o r i f a l s e . These models s h o u l d r a t h e r b e s e e n as a p p r o x i m a t i o n s t o r e a l i t y , and our primary c o n s i d e r a t i o n i s t o a s s e s s which one comes c l o s e s t s o t h a t t h i s can s e r v e a s a s t a r t i n g p o i n t f o r f u r t h e r improvements.

T i t l e s of p a p e r s s u c h as "A t e s t of

. . . . . I '

( e . g . Horowitz, 1977; Kahan &

Rapoport, 1 9 7 4 ) , o r even "A c o m p e t i t i v e t e s t of

. . . . . ' I

( e . g . Michener,

S a k u r a i , Yuen, & Iiasen; Michener, Yuen 81 G i n s b e r g , 1 9 7 7 ) , seem a b i t p r e m a t u r e . What i s needed f i r s t i s a h e u r i s t i c p r o c e s s i n which t h e o r y

t e l l s u s what t o l o o k at and how t o improve o u r e x p e r i m e n t s , and e x p e r i m e n t s s u g g e s t u s which models are b e s t and how t o improve them.

We b e l i e v e t h a t a s soon a s a c l o s e approximation t o r e a l i t y h a s been found i t w i l l be u s e f u l t o make t h e model s t o c h a s t i c , n o t by r e s o r t i n g t o " p o p u l a t i o n s t a t i s t i c s " b u t by i n c o r p o r a t i n g r e s p o n s e models p r e d i c t i n g t h e p r o b a b i l i t y of a c t o r s showing c e r t a i n r e a c t i o n s t o c e r t a i n a s p e c t s of t h e game ( e . g . , payoff o f f e r s ) . T h i s may n o t o n l y make t h e model more r e a l i s t i c b u t a l s o t e s t a b l e i n a more s t a t i s t i c a l f a s h i o n .

We b e l i e v e t h e problem t h a t p r e d i c t i o n s from d i f f e r e n t models c a n n o t be s i m p l y compared f o r a l l games because of t h e i r r a n g e s and r e l a t i o n s h i p s t o b e o n l y a temporary m a t t e r ; i t w i l l d i s a p p e a r as soon a s the d i s c u s s i o n c o n v e r g e s t o one model. I t i s t o b e e x p e c t e d t h a t from t h a t moment o n l y r e f i n e m e n t s o f t h e model w i l l b e t e s t e d a g a i n s t e a c h o t h e r . T i l l t h e n w e

Chapter 2 f Coalition formution: a game-theoretic approach

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have t o be aware of t h e s e p e c u l i a r i t i e s when d e s i g n i n g e x p e r i m e n t s and interpreting their results.

2.6.3

EXPERIMENTAL DESIGNS

R i k e r (1971) h a s p u t forward t h a t g a m e - t h e o r e t i c

s o l u t i o n s a r e not based

on a l l r u l e s needed t o d e s c r i b e e m p i r i c a l games. I n a d d i t i o n t o t h e f o r m a l r u l e s o f t h e game, i n f o r m a l r u l e s must be d i s t i n g u i s h e d . These a r e t h e r u l e s t h a t d e s c r i b e t h e game "as it is b r o u g h t o u t of t h e e t e r n a l world o f mathematics i n t o t h e t e m p o r a l world of a c t u a l p l a y " ( p . 1 1 6 ) . According t o R i k e r , i n f o r m a l r u l e s , as f o r example t h e sequence i n which t h e a c t o r s have t o n e g o t i a t e o r t h e p h y s i c a l c i r c u m s t a n c e s , may b e a s s i g n i f i c a n t

for t h e outcomes of t h e game as t h e f o r m a l r u l e s . Hence, b e f o r e p r e s e n t i n g r e s u l t s from e x p e r i m e n t a l game t h e o r y , a s p e c t s of e x p e r i m e n t a l d e s i g n s , s u b j e c t s , and e x p e r i m e n t a l c o n d i t i o n s u s e d i n game t h e o r y a r e d e s c r i b e d so t h a t t h e r e a d e r can g e t an i d e a o f t h e i n f o r m a l r u l e s u n d e r which t h e r e s u l t s were o b t a i n e d .

EXPERIMENTAL TASKS The u s u a l f o r m a t i n s o c i a l p s y c h o l o g i c a l e x p e r i m e n t s o n c o a l i t i o n f o r m a t i o n i s t h e m o d i f i e d P a c h i s i game p r o c e d u r e i n t r o d u c e d by Vinacke and Arkoff ( 1 9 5 7 ) . I n t h i s f o r m a t , s u b j e c t s p l a y a game i n which e a c h moves h i s / h e r c o u n t e r a l o n g t h e s p a c e s of a P a c h i s i b o a r d . On e a c h t r i a l a d i e i s thrown and t h e number o f s p a c e s e a c h p l a y e r a d v a n c e s i s d e t e r m i n e d by

t h e number on t h e d i e and t h e m u l t i p l i c a t i v e weight t h a t h e / s h e h a s r e c e i v e d a t t h e b e g i n n i n g of t h e game. The f i r s t t o r e a c h t h e g o a l i s t h e w i n n e r . P l a y e r s may form a c o a l i t i o n i n which c a s e a s i n g l e c o u n t e r is p l a c e d i n a p o s i t i o n e q u a l t o t h e sum o f d i s t a n c e s t h e i n d i v i d u a l members have a l r e a d y a t t a i n e d . On subsequent t r i a l s , t h e weight of t h e c o a l i t i o n i s e q u a l t o t h e sum of t h e i n d i v i d u a l w e i g h t s . A s e r i o u s r e s t r i c t i o n o f t h i s format i s i t s l i m i t a t i o n t o games w i t h t h e same payoff f o r e a c h p o s s i b l e c o a l i t i o n . A l s o , t h i s f o r m a t s e p a r a t e s t h e b a r g a i n i n g p r o c e s s from t h e p r o c e s s of c o a l i t i o n f o r m a t i o n s i n c e t h e d i s t r i b u t i o n of t h e p a y o f f w i t h i n t h e c o a l i t i o n g e n e r a l l y t a k e s p l a c e o n l y a f t e r t h e c o a l i t i o n h a s formed (Kahan & R a p o p o r t , 1 9 7 4 ) . For t h e s e r e a s o n s , t h e P a c h i s i b o a r d format i s h a r d l y u s e d i n e x p e r i m e n t a l game t h e o r y .

A more f l e x i b l e f o r m a t , widely u s e d i n game t h e o r y , is t h e one i n which t h e

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game i s p r e s e n t e d d i r e c t l y i n c h a r a c t e r i s t i c f u n c t i o n form and t h e s u b j e c t s a r e asked t o d e c i d e on two t h i n g s :

( 1 ) which c o a l i t i o n t o form, and ( 2 )

how t o d i s t r i b u t e t h e v a l u e of t h e c o a l i t i o n s among t h e i r members. T h i s s e t - u p was a l r e a d y used by K a l i s h , M i l n o r , Nash, and N e r i n g

(1954);

a g r e a t v a r i e t y of r e f i n e m e n t s and m o d i f i c a t i o n s have f o l l o w e d s i n c e t h e n . I n some e x p e r i m e n t s , t h e assignment was simply i n t r o d u c e d as an e x e r c i s e i n b a r g a i n i n g i n which p e r s o n s have t o d i v i d e a c e r t a i n amount of money ( e . g . , Buckley & Westen, 1973, 1974; Lieberman, 1 9 6 2 , 1975; Michener & S a k u r a i , 1976) o r a number of p o i n t s ( e . g . , Michener, S a k u r a i , Yuen, & Kasen, 1979; Murnighan, 1 9 7 8 b ) . S e l t e n and S c h u s t e r (1968) i n t r o d u c e d

t h e i r e x p e r i m e n t a s an economic game i n which f i r m s have t o b a r g a i n f o r c e r t a i n mining r i g h t s o n t h e c o n t i n e n t a l s h e l f . An o r i g i n a l s e t - u p i s t h e "shoe game" i n t r o d u c e d by Murnighan and Roth (1977, 1 9 7 8 ) . I n t h i s game, s u b j e c t s p l a y a market c o n s i s t i n g of p a r t i e s who e a c h own one s h o e . One p a r t y i s t h e monopolist who owns a r i g h t shoe w h i l e t h e o t h e r p a r t i e s each have a l e f t shoe. S i n g l e s h o e s have no v a l u e s i n c e o n l y a p a i r of s h o e s c a n be s o l d . Only a c o a l i t i o n of t h e m o n o p o l i s t and one of t h e o t h e r p a r t i e s c a n e a r n money. T h i s f o r m a t i s , however, o n l y s u i t a b l e f o r games i n which one a c t o r i s a m o n o p o l i s t and a l l o t h e r s a r e ( i d e n t i c a l ) weak actors. The form of communication allowed i n t h e v a r i o u s e x p e r i m e n t s v a r i e s consederably. Occasionally, t h e i n t e r a c t i o n takes place f a s e t o face with s u b j e c t s s e a t e d around t h e " b a r g a i n i n g t a b l e " .

R i k e r (1967, 1971) u s e s

a p r o c e d u r e w i t h p a i r s of s u b j e c t s b a r g a i n i n g i n rounds o f c o n v e r s a t i o n . I n most e x p e r i m e n t s , however, t h e s u b j e c t s c a n n o t see e a c h o t h e r and a w r i t t e n form of communication i s u s e d . I n a series o f e x p e r i m e n t s , Murnighan ( 1 9 7 8 b ) , Murnighan and Roth ( 1 9 7 8 ) , and Murnighan and Szwajkowski (1979) u s e opaque p a r t i t i o n s t o s h i e l d t h e s u b j e c t s and t h e e x p e r i m e n t e r from each o t h e r ' s view. The i n t e r a c t i o n t a k e s p l a c e v i a " o f f e r s l i p s " on which s u b j e c t s make t h e i r o f f e r s and mark t h e i r a c c e p t a n c e o r r e j e c t i o n . The way i n which c o a l i t i o n s are r e q u i r e d t o make known t h e i r d e c i s i o n s also varies.

I n some e x p e r i m e n t s , t h e game i s o v e r as soon a s s u b j e c t s

i n d i c a t e t h a t t h e y have formed a c o a l i t i o n and a g r e e d on t h e s p l i t of t h e p a y o f f , f o r example, by s i g n i n g a c o n t r a c t (Michener 81 S a k u r a i , 1976; Michener, Yuen, & G i n s b e r g , 1 9 7 7 ) . Lieberman (1962, 1975) u s e s a p r o c e d u r e

Chapter 2 Coalition formation: a game-theoretic approach

95

i n which t h e s u b j e c t s s i m u l t a n e o u s l y t u r n o v e r c a r d s t o make known t h e i r c h o i c e s , and c o a l i t i o n s are d e f i n e d a s r e c i p r o c a l c h o i c e s . In S e l t e n and S c h u s t e r ' s (1968) e x p e r i m e n t , c o a l i t i o n s are r e q u i r e d t o r e g i s t e r and become d e f i n i t i v e o n l y a f t e r t e n m i n u t e s of r e g i s t r a t i o n t i m e .

A f u l l y s t a n d a r d i z e d p r o c e d u r e f o r e x p e r i m e n t s w i t h games i n c h a r a c t e r i s t i c

f u n c t i o n form i s C o a l i t i o n s , a set of computer programs w r i t t e n a t t h e T h u r s t o n e P s y c h o m e t r i c L a b o r a t o r y , U n i v e r s i t y o f North C a r o l i n a . T h i s p r o c e d u r e h a s a l r e a d y been u s e d i n c o u n t l e s s e x p e r i m e n t s ( e . g . , Funk, Rapoport, & Kahan, 1 9 8 0 ; Horowitz, 1 9 7 7 ; Horowita & R a p o p o r t , 1974; Kahan & R a p o p o r t , 1 9 7 4 , 1 9 7 7 , 1979; Medlin, 1976; Rapoport & Kahan, 1976;

R a p o p o r t , Kahan, Funk 81 Horowitz, 1 9 7 9 ) . I n C o a l i t i o n s , s u b j e c t s communicate by s e n d i n g s t a n d a r d messages coded i n keywords v i a t e l e t y p e -

w r i t e r s c o n n e c t e d t o t h e computer. The computer c h e c k s t h e l e g a l i t y of t h e messages and s e n d s them o n . Examples o f keywords a r e "Reject",

"

"Offer",

"Accept",

~ ( t h e ' s u b j e c t withdraws from t h e game t o r e c e i v e t h e

v a l u e of h i s / h e r 1 - a c t o r c o a l i t i o n ) , and " R a t i f y " ( t h i s makes t h e c o a l i t i o n f i n a l and t e r m i n a t e s t h e b a r g a i n i n g p r o c e s s ) .

In Coalitions t h e bargaining

p r o c e s s p a s s e s t h r o u g h t h r e e d i f f e r e n t s t a g e s : ( 1 ) t h e o f f e r s t a g e , (2) t h e a c c e p t a n c e s t a g e , and (3) t h e r a t i f i c a t i o n s t a g e . Messages g o i n g a g a i n s t t h i s o r d e r a r e d e c l a r e d i l l e g a l by t h e computer and n o t s e n t t o t h e i n t e n d e d r e c i p i e n t . O t h e r f e a t u r e s o f t h e C o a l i t i o n s p r o c e d u r e are a t h r e e hour t r a i n i n g s e s s i o n p r e c e d i n g t h e e x p e r i m e n t s , i n c l u d i n g s u p e r v i s e d p r a c t i c e games t o a s s u r e t h a t s u b j e c t s f u l l y u n d e r s t a n d t h e r u l e s of t h e procedure.

I n most of t h e e x p e r i m e n t s r e f e r r e d t o above, s u b j e c t s a l s o

t o o k t u r n s as n o n - p a r t i c i p a t i n g

"observers",

by t h e o t h e r s on a t e l e t y p e w r i t e r .

who watched t h e game p l a y e d

T h i s was t o allow s u b j e c t s t o warm up

( w i t h o u t knowing t h e o t h e r s u b j e c t s p l a y i n g t h e game). The C o a l i t i o n s p r o c e d u r e i s f u l l y computer c o n t r o l l e d , i n v o l v e s a s t a n d a r d i z e d i n t e r a c t i o n between t h e a c t o r s , k e e p s t h e a c t o r s anonymous t o e a c h o t h e r , and makes s u r e t h a t a l l a c t o r s a r e c o g n i z a n t b o t h of t h e f o r m a l and " i n f o r m a l "

(Riker,

1971) r u l e s of t h e game.

A c o m p l e t e l y d i f f e r e n t e x p e r i m e n t a l s e t - u p was chosen by Wolf and Shubik

(1977).

I n t h e i r e x p e r i m e n t , s u b j e c t s d i d n o t a c t u a l l y p l a y games b u t

were c o n f r o n t e d w i t h d e s c r i p t i o n s of t h r e e - a c t o r c o a l i t i o n problems and asked t o i n d i c a t e t h e l i k e l i h o o d of e a c h p o s s i b l e c o a l i t i o n .

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FORM OF THE GAME Not a l l of t h e experiments conducted t h u s f a r p r e s e n t e d t h e game t o t h e s u b j e c t s i n c h a r a c t e r i s t i c f u n c t i o n form. Exceptions a r e t h e e x p e r i m e n t s by Michener, S a k u r a i , Yuen, and Kasen (1979) and Michener, Yuen, and Ginsberg (1977). I n t h e s e experiments t h r e e - a c t o r games w e r e used i n which s u b j e c t s were provided with two-choice payoff m a t r i c e s i n d i c a t i n g t h e consequences of choosing h i s / h e r l e f t n e i g h b o r , r i g h t n e i g h b o r , o r b o t h . According t o t h e s e a u t h o r s , an advantage of t h e u s e of normal form games i s t h a t t h e s u b j e c t ' s problem i s posed i n terms nf c o n c r e t e o p t i o n s r a t h e r t h a n i n terms of c o a l i t i o n membership. EXPERIMENTAL GAMES I N USE I t i s no s u r p r i s e t h a t t h e t y p i c a l games used i n e x p e r i m e n t a l game t h e o r y

only have a small number of a c t o r s . Most games a r e t h r e e - of f o u r - a c t o r games. Larger games are n o t o n l y d i f f i c u l t t o h a n d l e i n e x p e r i m e n t s , but may a l s o e n t a i l t h e o r e t i c a l problems because i t i s not always c l e a r which s o l u t i o n s f o l l o w from t h e models o r how t o compute them. N o n e t h e l e s s , some games have been t r i e d o u t w i t h a l a r g e r number o r a c t o r s . The f o l l o w i n g are among t h e t y p i c a l games used i n game-theoretic e x p e r i m e n t s :

-

-

3-actor non-constant-sum

game (Michener, S a k u r a i , Yuen, & Kasen, 1979;

Michener, Yuen, & Ginsberg, 1977; M i l l e r , 1980); 3 - a c t o r non-constant-sum

q u o t a game (Kahan & Rapoport, 1974; Medlin, 1976;

Rapoport 81 Kahan, 1976; R i k e r , 1967, 1971). T h i s game was a l s o used i n Kahan and Rapoport (1977) w i t h v a l u e s l a r g e r t h a n z e r o f o r t h e o n e - a c t o r and grand c o a l i t i o n ;

-

3 - a c t o r non-constant-sum

m a j o r i t y game ( L i e b e m a n , 1962, 1 9 7 5 ) ;

3-actor s i m p l e game with one monopolist and two weak a c t o r s (Murnighan 81 Roth, 1977, 1978);

-

-

3 - a c t o r apex game (Funk, Rapoport, & Kahan, 1980; Kahan & Rapoport, 1 9 7 8 ) ; 4-actor non-constant-sum

game ( M i l l e r , 1 9 8 0 ) ;

-

4-actor s i m p l e m a j o r i t y game (Buckley & Westen, 1974). Murnighan and

-

4-actor weighted m a j o r i t y game (Michener, Fleisman, & Vaske, 1976;

Szwajkowski (1979) used a v e r s i o n of t h i s game i n v o l v i n g a v e t o a c t o r ; Michener, & S a k u r a i , 1976). Some of t h e s e experiments i n c l u d e d a v e t o actor;

-

4 - a c t o r apex game (Funk, Rapoport, & Kahan, 1980, Horowitz 81 Rapoport,

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1974; R a p o p o r t , Kahan, Funk, & Horowitz, 1 9 7 9 ) ;

- 4 - a c t o r 3-quota game (Horowitz, 1 9 7 9 ) ;

-

4 - a c t o r 2-3-quota

game ( R a p o p o r t , Kahan, Funk, & Horowitz, 1 9 7 9 ) ;

5 - a c t o r constant-sum-game

( K a l i s h , M i l n o r , Nash, & N e r i n g , 1 9 5 4 ) ;

5 - a c t o r m a j o r i t y game (Buckley & Westen, 1973; Murnighan, 1 9 7 8 b ) ; 5 - a c t o r apex game (Horowitz & R a p o p o r t , 1974; S e l t e n & S c h u s t e r , 1 9 6 8 ) ; 5 - a c t o r 4-quota game (Horowitz, 1 9 7 7 ) ;

- 7 - a c t o r constant-sum game ( K a l i s h , M i l n o r , Nash, & Nehring, 1 9 5 4 ) ; - 7 - a c t o r s i m p l e game w i t h one monopolist and s i x weak a c t o r s (Murnighan & Roth, 1978);

-

1 2 - a c t o r s i m p l e game w i t h one m o n o p o l i s t and e l e v e n weak a c t o r s (Murnighan, & Roth, 1 9 7 8 ) .

The r e a d e r i n t e r e s t e d i n s p e c i f i c r e s u l t s for t h e s e games s h o u l d c o n s u l t the references.

SUBJECTS I n a l m o s t a l l e x p e r i m e n t s s u b j e c t s were u n d e r g r a d u a t e males r e c r u i t e d i n t h e f a s h i o n u s u a l f o r e x p e r i m e n t s i n t h e b e h a v i o r a l s c i e n c e s . As a r u l e , measures were t a k e n so t h a t f r i e n d s o r c l o s e a c q u a i n t a n c e s d i d n o t p l a y t o g e t h e r i n t h e same g r o u p . I t i s c o n s p i c u o u s t h a t h a r d l y any f e m a l e s

were u s e d , owing t o t h e wide-spread b e l i e f t h a t o n l y m a l e s t e n d t o e x h i b i t c o m p e t i t i v e b e h a v i o r , and t h a t males are less i n c l i n e d t o accomodation and a l t r u i s t i c o f f e r s t h a n f e m a l e s (Vinacke, 1 9 7 1 ) . However, t h e a n t i c o m p e t i t i v e h y p o t h e s i s h a s been t e s t e d i n a few e x p e r i m e n t a l games w i t h a s e x f a c t o r which a l l f a i l e d t o show s u p p o r t f o r t h e h y p o t h e s i s (Michener, Yuen, & G i n s b e r g , 1977; Murnighan, 1978b; Murnighan & Szwajkowski, 1 9 7 9 ) . To o u r knowledge, o n l y t h r e e e x p e r i m e n t s i n t h e l i t e r a t u r e u s e d n o n - s t u d e n t s as s u b j e c t s . In

Kalish et a l .

' s h i s t o r i c experiment t w o housewives and a

t e a c h e r p a r t i c i p a t e d , w h i l e R i k e r (1967, 1971) u s e d businessmen a s s u b j e c t s , mainly t o i n c r e a s e t h e e x t e r n a l v a l i d i t y of h i s e x p e r i m e n t s .

The q u e s t i o n c a n b e r a i s e d t o what e x t e n t t h e u s e o f t h e s e s u b j e c t s approaches t h e g a m e - t h e o r e t i c

i d e a l of " r a t i o n a l " p l a y e r s . Rapoport

,

Kahan,

Funk, and Horowitz (1979) answered t h i s q u e s t i o n by d e c i d i n g t o a v o i d h i r i n g t h e t y p i c a l c o l l e g e u n d e r g r a d u a t e and t o u s e s o p h i s t i c a t e d , i n t e l l i g e n t b a r g a i n e r s f u l l y f a m i l i a r w i t h g a m e - t h e o r e t i c c o n c e p t s and s o l u t i o n s .

In

t h e i r e x p e r i m e n t t w e n t y s u b j e c t s p a r t i c i p a t e d who a l m o s t a l l w e r e g r a d u a t e

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s t u d e n t s i n q u a n t i t a t i v e psychology. A l l had s u c c e s s f u l l y completed two o r more semesters i n advanced s t a t i s t i c s . The experiment w a s conducted a f t e r a c o u r s e on n-person game t h e o r y i n which t e x t b o o k s as Luce and R a i f f a (1957) and Rapoport (1970) were used a l o n g w i t h a series of t h e o r e t i c a l and experimental p a p e r s . A s f a r as w e know, t h i s i s t h e o n l y s e r i o u s a t t e m p t a t d e s i g n i n g an experiment i n which t h e p l a y e r s are f u l l y aware of t h e formal r u l e s of t h e game and t h e i r i m p l i c a t i o n s . REWARDS AND PAYOFFS When i n t e r p r e t i n g r e s u l t s form e x p e r i m e n t a l games, it i s important to know how s u b j e c t s were rewarded f o r p a r t i c i p a t i n g i n t h e experiment and what p a y o f f s were used i n t h e game. The l i t e r a t u r e shows no uniform u s e of rewards and p a y o f f s ; i n some c a s e s , s t u d e n t s were rewarded f o r p a r t i c i p a t i n g by g i v i n g them c r e d i t toward a c o u r s e requirement ( e . g . , Buckley & Westen, 1973, 1974; Miller, 1980; Murnighan & Roth, 1977; Rapoport, Kahan, Funk, & Horowitz, 1 9 7 9 ) . Other e x p e r i m e n t s , however, o f f e r e d t h e i r s u b j e c t s

monetary rewards ( e . g . , Horowitz 81 Rapoport, 1974; Kahan & Rapoport, 1974; Medlin, 1976). The payoffs used i n experimental games g e n e r a l l y have t h r e e d i f f e r e n t forms. F i r s t , i n some experiments t h e p a y o f f s were made d i r e c t l y i n money ( e . g . , Lieberman, 1962; R i k e r , 1967, 1971; S e l t e n & S c h u s t e r , 1968). I n most c a s e s , t h e amounts of money t o be won were modest, although i n t h e experiments by e . g . S e l t e n and S c h u s t e r (1968) and Rapoport, Kahan, Funk, and Horowitz (1979) winning c o a l i t i o n s r e c e i v e d l a r g e r sums ($15, f o r example). Secondly, i n o t h e r experiments s u b j e c t s played f o r p o i n t s , t i c k e t s , o r f i c h e s t h a t could be exchanged f o r money a f t e r w a r d s ( e . g . , K a l i s h , M i l n o r , Nash & Nering, 1954; Murnighan, 1978b). I n t h e Michener, S a k u r a i , Yuen, and Kasen (1979) and Michener, Yuen, and Ginsberg (1977) experiments, t h i s o p p o r t u n i t y w a s a b s e n t , however, mainly t o p r e v e n t s u b j e c t s from c o l l u d i n g a g a i n s t t h e experimenter as t h e l a t t e r a u t h o r s observe ( p . 1 1 2 ) . T h i r d , i n some experiments i n which s t u d e n t s were rewarded by c r e d i t toward c o u r s e r e q u i r e m e n t s , t h e amount of c r e d i t w a s a f u n c t i o n of t h e i r performances i n t h e games, o r t h e s e i n f l u e n c e d t h e g r a d e t o be earned. An example i s t h e Murnighan and Sawajkowski (1979) experiment.

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EXPERIMENTAL VARIABLES A l l e x p e r i m e n t s had one o r more v a r i a b l e s t h a t were s y s t e m a t i c a l l y v a r i e d

t o c r e a t e d i f f e r e n t e x p e r i m e n t a l c o n d i t i o n s u n d e r which g a m e - t h e o r e t i c p r e d i c t i o n s were s t u d i e d . These v a r i a b l e s f a l l i n t o two c a t e g o r i e s : gamet h e o r e t i c and e x t r a - g a m e - t h e o r e t i c

v a r i a b l e s . To t h e f i r s t c a t e g o r y belong

such v a r i a b l e s a s t h e number o f a c t o r s i n t h e game ( H o r o w i t z , 1 9 7 7 ; Horowitz 6r R a p o p o r t , 1974; Murnighan & R o t h , 1 9 7 7 , 1 9 7 8 ) , and t h e v a l u e of s u c h c o a l i t i o n s a s t h e grand c o a l i t i o n (Medlin, 19761, t h e Apex and

Base c o a l i t i o n i n apex games (Horowitz & R a p o p o r t , 19741, o r t h e 1 - a c t o r c o a l i t i o n s (Kahan & R a p o p o r t , 1 9 7 7 ) . V a r i a b l e s p e r t a i n i n g t o c e r t a i n c h a r a c t e r i s t i c s of a c t o r s a l s o belong t o t h i s c a t e g o r y : examples a r e t h e a c t o r s ' q u o t a s i n q u o t a games (Kahan & R a p o p o r t , 1974; Medlin

,

1976),

t h e s t r e n g t h o f t h e Apex a c t o r (Kahan & R a p o p o r t , 1 9 7 9 ) , t h e impact of p o s i t i o n a l v e r s u s q u o t a power (Funk, R a p o p o r t , & Kahan, 1 9 8 0 ) , and t h e p r e s e n c e of v e t o a c t o r s (Michener, F l e i s h m a n , & Vaske, 1976; Michener & S a k u r a i , 1 9 7 6 ) . Michener, Yuen, and G i n s b e r g (1977) u s e d e x p e r i m e n t a l games

i n which t h e s i z e of t h e c o r e was a s y s t e m a t i c f a c t o r . The p u r p o s e o f s y s t e m a t i c a l l y m a n i p u l a t i n g t h e s e v a r i a b l e s i s , of c o u r s e , t o a s c e r t a i n t h e v a l i d i t y o f g a m e - t h e o r e t i c models f o r d i f f e r e n t c a t e g o r i e s of games.

Among t h e extra-game t h e o r e t i c v a r i a b l e s , communication v a r i a b l e s a r e most p o p u l a r . Examples a r e t h e o r d e r o f communication i n p a i r s i n R i k e r ' s (1971) e x p e r i m e n t , f i x e d v e r s u s f r e e o r d e r of communication (Kahan & R a p o p o r t , 1 9 7 4 ) , secret v e r s u s p u b l i c communication (Kahan & R a p o p o r t , 1974; Murnighan & R o t h , 1 9 7 7 , 1978; Rapoport & Kahan, 1 9 7 6 ) , and communicat i o n of weak a c t o r s b e f o r e v e r s u s a f t e r s t r o n g a c t o r s (Horowitz, 1 9 7 7 ) . I n t h e Murnighan and Szwajkowski (1979) and R i k e r (1971) e x p e r i m e n t s , b a r g a i n i n g t o o k p l a c e u n d e r d i f f e r e n t l e v e l s of t i m e p r e s s u r e . A s mentioned e a r l i e r , t h e e f f e c t o f s e x was s t u d i e d i n Michener, Yuen, and G i n s b e r g ( 1 9 7 7 ) , Murnighan ( 1 9 7 8 b ) , and Murnighan and Szwajkowski ( 1 9 7 9 ) . I t should b e o b s e r v e d t h a t s i n c e t h e s e v a r i a b l e s d o n o t o c c u r i n game t h e o r y , no p r e d i c t i o n s can be made a s t o t h e i r e f f e c t on c o a l i t i o n forming and t h e d i s t r i b u t i o n of p a y o f f s i n e x p e r i m e n t s . 2.6.4

EXPERIMENTAL FINDINGS

T h i s s e c t i o n p r e s e n t s f i n d i n g s from e x p e r i m e n t a l game t h e o r y , emphasizing g e n e r a l r e s u l t s and t r e n d s r a t h e r t h a n d e t a i l s . R e a d e r s i n t e r e s t e d i n

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more d e t a i l e d d e s c r i p t i o n s should r e f e r t o t h e s o u r c e s c i t e d

. The

section

is organized a c c o r d i n g t o t h e main game-theoretic s o l u t i o n s p r e s e n t e d earlier i n t h i s chapter. VON NEUMANN-MORGENSTERN SOLUTION

H i s t o r i c a l l y , t h i s is t h e f i r s t s o l u t i o n i n game t h e o r y and predominant i n t h e e a r l i e r experiments u n t i l it d i s a p p e a r e d from t h e e x p e r i m e n t a l gamet h e o r e t i c l i t e r a t u r e a f t e r t h e mid-seventies.

Relevant experiments are

r e p o r t e d i n Buckley and Westen (1973, 1 9 7 4 ) , K a l i s h , Milnor. Nash, and Nehring (1954), Lieberman (1962, 1 9 7 5 ) , S e l t e n and S c h u s t e r (1968), and Riker (1967, 1 9 7 1 ) , while r e s u l t s i n Horowitz and Rapoport (1974) throw some l i g h t on t h e behavior of t h e von Neurnann-Morgenstern s o l u t i o n f o r apex games. The r e s u l t s d o not unambiguously s u p p o r t t h e s o l u t i o n . The most n e g a t i v e r e s u l t s were o b t a i n e d i n S e l t e n and S c h u s t e r ' s experiment with a 5 - a c t o r apex game ("Me and My Aunt") i n which t h e von NeumannMorgenstern s o l u t i o n simply n e v e r o c c u r r e d . But on t h e o t h e r hand t h e r e a r e t h e experiments by Riker (1971), i n which t h e von Neumann-Morgenstern h y p o t h e s i s could n o t be r e j e c t e d i n 6 o u t of 7 c a s e s , and by Buckley and Westen (1973), who r e p o r t s l i g h t l y more t h a n 90% o f t h e i r 112 e x p e r i m e n t a l t r i a l s y i e l d i n g von Neumann-Morgenstern i m p u t a t i o n s . (Readers i n t e r e s t e d i n t h e games used i n t h e s e and o t h e r experiments a r e r e f e r r e d t o t h e s e c t i o n on e x p e r i m e n t a l games i n u s e . ) T y p i c a l r e s u l t s f o r t h e von NeumannMorgenstern s o l u t i o n have been o b t a i n e d by, f o r example, Riker (1967) and Lieberman (1962, 1975) who r e p o r t p e r c e n t a g e s of s u c c e s s r a n g i n g from 25 t o 75. In t h e Lieberman (1962) experiment, however, a s i m p l e model a s t h e e q u a l - s h a r e h y p o t h e s i s t u r n e d o u t t o be a b e t t e r p r e d i c t o r t h a n t h e von Neumann-Morgenstern s o l u t i o n . I n Horowitz and R a p o p o r t ' s (1974) e x p e r i m e n t , t h e von Neumann-Morgenstern s o l u t i o n was supported by t h e f a c t t h a t i t s p r e d i c t i o n s c o i n c i d e d with t h o s e of t h e c o m p e t i t i v e b a r g a i n i n g set f o r t h e apex games used i n t h e experiment. Murnighan and Roth (1977) p r e s e n t r e s u l t s showing outcomes more p o s i t i v e f o r R o t h ' s s u b s o l u t i o n t h a n f o r t h e von Neumann-Morgenstern s o l u t i o n . We do not know o f o t h e r experiments on t h e s u b s o l u t i o n , however, so t h a t no g e n e r a l c o n c l u s i o n s can be drawn.

Chapter 2 J Coalition formation: a game-theoretic approach

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CORE I n s p i t e of t h e f a c t t h a t t h e c o r e h a s been s u b j e c t of many t h e o r e t i c a l s t u d i e s , o n l y a s m a l l number of e x p e r i m e n t s have been conducted t o a s c e r t a i n i t s d e s c r i p t i v e v a l u e . T h i s may be due t o t h e f a c t t h a t t h e c o r e h a s t h e u n d e s i r a b l e p r o p e r t y of b e i n g empty f o r many games. E x p e r i m e n t a l r e s u l t s c o n c e r n i n g t h e c o r e c a n b e found i n Horowitz ( 1 9 7 7 ) , Michener, Yuen, and Ginsberg ( 1 9 7 7 ) , Murnighan and Roth (1977, 1 9 7 8 ) , and Murnighan and Szwajkowski ( 1 9 7 9 ) . Horowitz, i n h i s e x p e r i m e n t a l (n-1)-quota

games w i t h

one weak a c t o r , found c o n s i d e r a b l e s u p p o r t f o r t h e s t r o n g e r a c t o r s b u t

less f o r t h e o t h e r s . I n t h e Michener, Yuen, and G i n s b e r g e x p e r i m e n t o n l y 1 4 . 2 % of a l l outcomes f e l l w i t h i n t h e c o r e . These a u t h o r s s y s t e m a t i c a l l y v a r i e d c o r e s i z e i n t h e i r experiment b u t t h e number of outcomes f a l l i n g w i t h i n t h e core d i d n o t a p p e a r t o depend on t h i s f a c t o r . The p a p e r s by Murnighan and Roth and by Murnighan and Szwajkowski r e p o r t mixed s u p p o r t f o r t h e c o r e and show t h a t t h e a c t u a l

outcomes a r e i n f l u e n c e d by s u c h

f a c t o r s a s t h e number o f a c t o r s and communication r e s t r i c t i o n s .

SHAPLEY VALUE The Shapley v a l u e i s one of t h e p o p u l a r s o l u t i o n s i n e x p e r i m e n t a l game t h e o r y . Many e x p e r i m e n t a l p a p e r s r e p o r t a n a l y s e s comparing d i s b u r s e m e n t s t o i n d i v i d u a l a c t o r s w i t h t h e i r Shapley v a l u e s , i n c l u d i n g s u c h r e l e v a n t p a p e r s a s Horowitz ( 1 9 7 7 ) , Kahan and Rapoport ( 1 9 7 7 ) , K a l i s h , M i l n o r , Nash, and Nering ( 1 9 5 4 ) , Lieberman ( 1 9 6 2 ) , Medlin (19761, Michener, F l e i s h m a n , and Vaske ( 1 9 7 6 ) , Michener and S a k u r a i ( 1 9 7 6 ) , Murnighan ( 1 9 7 8 b ) , Murnighan and Roth (1977, 1 9 7 8 ) , Murnighan and Szwajkowski ( 1 9 7 9 ) , Rapoport and Kahan ( 1 9 7 6 ) , and S e l t e n and S c h u s t e r (1968). By i t s i n t e r p r e t a t i o n a s a measure of an a c t o r ' s p i v o t a l power, t h e S h a p l e y v a l u e i s t h e o n l y game-theoretic

r e s u l t t h a t h a s pervaded t h e s o c i a l p s y c h o l o g i c a l

l i t e r a t u r e and h a s been t e s t e d e x t e n s i v e l y a g a i n s t s o c i a l p s y c h o l o g i c a l predictions.

No g e n e r a l c o n c l u s i o n can b e drawn from t h e e x p e r i m e n t a l f i n d i n g s , and r e s u l t s even seem t o c o n t r a d i c t e a c h o t h e r now and t h e n . FOT example, b o t h Lieberman and Medlin a s w e l l a s Murnighan and Szwajkowski found t h a t t h e Shapley value o f f e r s l i t t l e h e l p i n p r e d i c t i n g average i n d i v i d u a l p a y o f f s a c r o s s series of t r i a l s . Rapoport and Kahan, t o o , were f o r c e d t o c o n c l u d e t h a t t h e S h a p l e y v a l u e i s much i n f e r i o r a s a p r e d i c t o r t o , f o r

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W.J. van der Linden and A. Verbeek

example, t h e i n d i v i d u a l l y r a t i o n a l b a r g a i n i n g s e t . The two Murnighan and Roth experiments, on t h e o t h e r hand, found t h a t t h e Shapley v a l u e p r e d i c t e d t h e i r average outcomes r a t h e r a c c u r a t e l y . The o t h e r p a p e r s show r e s u l t s between t h e s e two extremes. S e l t e n and S c h u s t e r r e p o r t r e s u l t s c l e a r l y i n d i c a t i n g t h a t weak a c t o r s r e c e i v e more and s t r o n g a c t o r l e s s t h a n t h e i r Shapley v a l u e , whereas K a l i s h e t a l . and Kahan and Rapoport found t h a t a c t u a l outcomes t e n d t o be less e g a l i t a r i a n t h a n t h e Shapley p r e d i c t i o n s . The experiments by Michener and S a k u r a i and Michener e t a l . s u g g e s t a moderator e f f e c t due t o t h e p r e s e n c e of a v e t o a c t o r . I n games w i t h a v e t o a c t o r t h e Shapley value p r e d i c t e d t h e payoff d i v i s i o n s b e t t e r t h a n t h e k e r n e l o r minimum r e s o u r c e t h e o r y b u t tended t o perform less w e l l i n t h e absence o f such an a c t o r . BARGAINING SETS The b a r g a i n i n g sets a r e as popular i n experimental game t h e o r y as t h e Shapley v a l u e . Recent l i t e r a t u r e i n p a r t i c u l a r shows a c o n s i d e r a b l e number of t e s t s of b a r g a i n i n g s e t s a g a i n s t o t h e r s o l u t i o n s . For t h e most p a r t , t h e s e experiments have been conducted u s i n g t h e C o a l i t i o n s format d e s c r i b e d i n an e a r l i e r s e c t i o n . Papers c o n t a i n i n g i m p o r t a n t r e s u l t s on t h e e m p i r i c a l v a l i d i t y o f b a r g a i n i n g sets a r e Funk, Rapoport, and Kahan (1979), Horowitz (1977), Horowitz and Rapoport (1974), Kahan and Rapoport (1974; 1977, 1 9 7 9 ) , Medlin (1976), Michener, S a k u r a i , Yuen, and Kasen (1979), Michener, Yuen, and Ginsberg (1977), S e l t e n and S c h u s t e r (1968), Rapoport and Kahan (1976), Rapoport, Kahan, Funk, and Horowitz (1979), and R i k e r (1967). With t h e e x c e p t i o n o f t h e Michener e t al., S e l t e n and S c h u s t e r , and Riker experiments, a l l of t h e above experiments used t h e C o a l i t i o n s format. Experimental r e s u l t s a r e a v a i l a b l e f o r t h e ( i n d i v i d u a l l y r a t i o n a l ) b a r g a i n i n g s e t , M(i), t h e c o a l i t i o n a l l y r a t i o n a l b a r g a i n i n g s e t , M , t h e competitive bargaining s e t ,

and a modified b a r g a i n i n g s e t based on

a Shapley v a l u e s t a n d a r d o f f a i r n e s s , M(im). I t i s r e c a l l e d t h a t M c M(i), and H(i) -

5 H(i)

so t h a t a p o s i t i v e r e s u l t f o r t h e former set always i m -

p l i e s s u p p o r t f o r t h e l a t t e r . The b a r g a i n i n g sets a r e uniformly s u c c e s f u l f o r a g r e a t v a r i e t y of experimental games and c o n d i t i o n s . In f a c t , i n a l l of t h e above experiments t h e b a r g a i n i n g sets proved t o be b e t t e r p r e d i c t o r s

o f t h e payoff c o n f i g u r a t i o n s agreed upon by t h e a c t o r s t h a n any o t h e r s o l u t i o n t h e y were compared w i t h . To g i v e a q u a n t i t a t i v e i m p r e s s i o n , S e l t e n and S c h u s t e r found t h a t i n e i g h t o u t of t h e i r twelve p l a y s t h e f i n a l

Cliapter 2 Coalition formation: a game-theoretic approach c o a l i t i o n s showed payoff d i v i s i o n s f a l l i n g i n M ( i ) .

103

I n Kahan and R a p o p o r t ’ s

(1979) e x p e r i m e n t , some 80 p e r c e n t o f a l l outcomes w e r e i n M(i) and some 60 p e r c e n t of t h e s e were e x a c t l y i d e n t i c a l t o H ( i ) , which is a p o i n t i n M(i) f o r t h e apex games used i n t h i s e x p e r i m e n t . O t h e r e x p e r i m e n t s show comparable r e s u l t s . As r e g a r d s t h e d i f f e r e n t b a r g a i n i n g s e t s , r e s u l t s s u g g e s t t h a t outcomes s u p p o r t Mci)\M,

r a t h e r t h a n M (Rapoport & Kahan,

1976) and t e n d t o f a l l c l o s e t o H(i) in Mci) f o r apex games ( e . g . , Horowitz & R a p o p o r t ; R a p o p o r t , Kahan, Funk, and H o r o w i t z ) . I n t h e M i c h e n e r , S a k u r a i , Yuen, and Kasen e x p e r i m e n t i t was a l s o found t h a t MCim) p r e d i c t s t h e outcomes i n 3 - a c t o r non-constant-sum games s i g n i f i c a n t l y b e t t e r t h a n MCi). B a r g a i n i n g sets a r e c e r t a i n l y t h e most p r o m i s i n g models i n game theory.

KERNEL E x p e r i m e n t a l r e s u l t s on t h e p r e d i c t i v e power o f t h e k e r n e l c a n be found

i n Funk, R a p o p o r t , and Kahan ( 1 9 8 0 ) , Horowitz ( 1 9 7 7 ) , Horowitz and Rapop r t ( 1 9 7 4 ) , Kahan and Rapoport ( 1 9 7 4 , 1 9 7 9 ) , Medlin ( 1 9 7 6 ) , Michener and S a k u r a i ( 1 9 7 6 ) , M i c h e n e r , Yuen, and G i n s b e r g ( 1 9 7 7 ) , R a p o p o r t , Kahan, Funk, and Horowitz ( 1 9 7 9 ) , and S e l t e n and S c h u s t e r ( 1 9 6 8 ) . A p a r t from t h e e x p e r i m e n t s i n t h e Michener e t a l . and S e l t e n and S c h u s t e r p a p e r s , a l l e x p e r i m e n t s were c o n d u c t e d u s i n g t h e C o a l i t i o n s p r o c e d u r e . I t i s r e c a l l e d that f o r the kernel, K, K

5 MCi)

h o l d s . On t h e whole, t h e k e r n e l d o e s

n o t perform s o w e l l compared t o o t h e r g a m e - t h e o r e t i c

solutions. In several

e x p e r i m e n t s (Funk, R a p o p o r t , & Kahan; H o r o w i t z ; Horowitz & R a p o p o r t ; Kahan & R a p o p o r t , 1 9 7 9 ; R a p o p o r t , Kahan, Funk, & Horowitz; S e l t e n & S c h u s t e r ) o n l y modest s u p p o r t h a s been found o r none a t a l l and t h e outcomes f a l l i n g i n M(i) a p p e a r e d t o b e i n f a v o r o f H(i) r a t h e r t h a n K . The wordt r e s u l t was o b t a i n e d by S e l t e n and S c h u s t e r i n whose e x p e r i m e n t t h e k e r n e l o c c u r r e d o n l y once i n t w e l v e p l a y s . M i c h e n e r , Yuen, and G i n s b e r g found more s u c c e s s e s f o r t h e m o d i f i e d b a r g a i n i n g s e t b a s e d on t h e S h a p l e y v a l u e t h a n f o r t h e k e r n e l . Michener and S a k u r a i had t o c o n c l u d e t h a t f o r t h e i r e x p e r i m e n t k e r n e l p r e d i c t i o n s were i n f e r i o r t o t h o s e by minimum r e s o u r c e and p i v o t a l power t h e o r y from s o c i a l p s y c h o l o g y , a l t h o u g h M e d l i n ’ s r e s u l t s were more s u p p o r t i v e o f t h e k e r n e l t h a n of t h e S h a p l e y v a l u e . I n Kahan and R a p o p o r t ’ s (1974) and M e d l i n ’ s e x p e r i m e n t s t h e k e r n e l and ( i n d i v i d u a l l y r a t i o n a l ) b a r g a i n i n g set p r e d i c t i o n s c o i n c i d e d f o r a l l and a l m o s t a l l games u s e d , r e s p e c t i v e l y . I n t h e l i g h t

W.J.van der Linden and A. Verbeek

104

of t h e above r e s u l t s f o r t h e b a r g a i n i n g s e t s t h e s u p p o r t f o r t h e s e p r e d i c t i o n s may be a c c o u n t e d f o r by t h e b a r g a i n i n g set r a t h e r t h a n t h e k e r n e l a s p e c t of s t a b i l i t y . I t a p p e a r s t h a t t h e k e r n e l s y s t e m a t i c a l l y o v e r p r e d i c t s t h e p a y o f f s t o s t r o n g a c t o r s and u n d e r p r e d i c t s t h e p a y o f f s t o weak a c t o r s . T h i s b i a s h a s been e s t a b l i s h e d by H o r o w i t z , Kahan and Rapoport ( 1 9 7 4 ) , and Michener, Yuen, and G i n s b e r g .

OTHER SOLUTIONS The s o l u t i o n s d e a l t w i t h above have been predominant i n e x p e r i m e n t a l game t h e o r y . O c c a s i o n a l l y , however, o t h e r s o l u t i o n s h a v e been e x p e r i m e n t e d w i t h . The above s o l u t i o n s have a l s o been c o n f r o n t e d w i t h p r e d i c t i o n s from s o c i a l p s y c h o l o g i c a l models. Examples o f t h e f o r m e r a r e M i l n o r ' s " r e a s o n a b l e outcomes" ( K a l i s h , M i l n o r , Nash, & N e r i n g , 1954) and S e l t e n ' s e q u a l s h a r e p r i n c i p l e , which i s n o t so much a g a m e - t h e o r e t i c model a s an i n d i c a t i o n t h a t t h e p r i n c i p l e of e q u a l d i v i s i o n i s a s t r o n g norm t h a t may be o p e r a t i n g i n many c h a r a c t e r i s t i c f u n c t i o n games (Funk, R a p o p o r t , & Kahan, 1980; M i c h e n e r , Yuen, & G i n s b e r g , 1977; R a p o p o r t , Kahan, Funk,

& Horowitz, 1 9 7 9 ) . The p a p e r s by C a s s i d y and Mangold ( 1 9 7 5 ) , F r i e n d ,

L a i n g , and Morrison ( 1 9 7 7 ) , and L a i n g and Morrison (1973, 1974) w e r e a l r e a d y r e f o r l e d t o . Examples o f t h e l a t t e r c a n be found i n Kahan and Rapoport ( 1 9 7 4 ) , Michener, F l e i s h m a n , and Vaske ( 1 9 7 6 ) , Michener and S a k u r a i (19761, and Rapoport and Kahan ( 1 9 7 6 ) .

2.6.5

CONCLUSION

The most i m p o r t a n t c o n c l u s i o n t o be drawn from some f i f t e e n y e a r s of i n t e n s i v e e x p e r i m e n t i n g w i t h c h a r a c t e r i s t i c f u n c t i o n games is t h a t t h e c l a s s o f b a r g a i n i n g sets have been most s u c c e s s f u l i n e m p i r i c a l c o n f r o n -

t a t i o n s . T h i s d o e s n o t mean, however, t h a t t h e b a r g a i n i n g sets are i n e v e r y way t h e b e s t ; t o o l i m i t e d a number o f games h a s been u s e d ( p r e dominantly t h r e e - ,

four-,or five-actor

apex o r q u o t a games) t o w a r r a n t

t h i s c o n c l u s i o n . Moreover, i t i s u s u a l i n e x p e r i m e n t a l game t h e o r y t o e v a l u a t e s o l u t i o n s by t h e i r power t o p r e d i c t a v e r a g e outcomes, and occasionally

i n d i v i d u a l outcomes have shown a c o n s i d e r a b l e v a r i a b i l i t y .

B u t , on t h e whole, t h e f a c t t h a t b a r g a i n i n g sets have behaved u n i f o r m l y b e t t e r t h a n any o t h e r s o l u t i o n i s an e n c o u r a g i n g r e s u l t . B a r g a i n i n g sets may be c o n s i d e r e d our b e s t c a n d i d a t e f o r r e a l i t y a t h a n d , and it seems s e n s i b l e t o make f u r t h e r e f f o r s t o f i n d a d d i t i o n a l e x p e r i m e n t a l e v i d e n c e

Chapter 2 1 Coulition formation: u game-theoretic approach

105

and theoretical refinements. A s argued earlier, game-theoretic models are highly Ldealized mappings

of reality, with predictions that can only be expected to hold under idealized conditions. F o r experimental games this involves rigorous experimental control. Although considerable progress has been made in this respect, important conditions have been left untouched. In our opinion, the introduction of the Coalitions procedure was a first step in the right direction. The extensive period of training prior to the experiment and the roles of non-participating observers guarantees that subjects know the formal and informal rules of the game. Moreover, the procedure is fully standardized. A second step was the introduction of "sophisticated players" in Kahan, Rapoport, Funk, and Horowitz (1979). This guarantees that the subjects not only know the rules of the game but also their implications and that they are able to work with them. The main conclusions the authors derive from a comparison of results for sophisticated and unsophisticated players are "...(a) that sophisticated players are less restricted in coalition choice; (b) that their payoffs are less variable, less egalitarian and less affected by variables not in the characteristic function game; and (c) that these differences between the two populations of subjects increase with the complexity of the characteristic function game" (p. 69). Especially the result that the payoffs are less variable accross replications is a significant finding and an indication of more rigorous experimental control. We believe that the above two steps are valuable attempts at realizing the cognitive aspect of the ideal of "rational" players. What has been ignored hitherto, however, is the utility aspect. In section 6.3. we saw that the usual payoffs in experiments are (various amounts of) money, credit toward course requirements, or just "points" or fiches. In using these, it is implied that the subjects have unidimensional utilities monotonically increasing in these payoffs and remaining constant during the game. Note that the utilities are supposed to reflect the actor's evaluations of all aspects and consequences of the game, and to combine these consistently into one dimension. This is a remarkably strong assumption on which game theory leans heavily. Nevertheless, we have not found any experiment that checked whether this condition was fulfilled.

106

W.J. van der Linden and A. Verbeek

Nor d i d w e f i n d any a t t e m p t a t s y s t e m a t i c a l l y i n t r o d u c i n g t h e r i g h t u t i l i t y f u n c t i o n s w i t h t h e s u b j e c t s . (The s i m p l e i n s t r u c t i o n t o s t r i v e t o a maximum amount of money o r p o i n t s h e l p s , b u t i s n o t enough i n our o p i n i o n ) . Hence, i t i s l i k e l y t h a t most e x p e r i m e n t a l r e s u l t s have been confounded by o t h e r f a c t o r s a t work t h a n t h e s u b j e c t s ' p u r s u i t o f a maximal amount of t h e e x p e r i m e n t a l p a y o f f . I n f a c t , Lieberman's

(1975)

f i n d i n g t h a t a d e s i r e t o belong t o a s t a b l e c o a l i t i o n and t r u s t are i m p o r t a n t f a c t o r s , and Kahan and R a p o p o r t ' s (1979) c o n c l u s i o n t h a t d i s t r i b u t i v e norms i n f l u e n c e t h e b e h a v i o r o f t h e a c t o r s are examples of v i o l a t i o n s of t h e u t i l i t y a s s u m p t i o n . More of t h e s e examples c a n r e a d i l y b e added.

From t h e above, w e conclude t h a t an i m p o r t a n t improvement of e x p e r i m e n t a l game t h e o r y c a n b e found i n a b e t t e r c o n t r o l o f t h e a c t o r ' s u t i l i t y f u n c t i o n s . T h i s w i l l i n v o l v e new and i n g e n i o u s p r o c e d u r e s of s e l e c t i n g and t r a i n i n g s u b j e c t s and p r o b a b l y a d d i t i o n a l m o d i f i c a t i o n s of t h e e x p e r i m e n t a l t a s k s . When t h i s h a s been a c h i e v e d , i t seems s e n s i b l e t o g a t h e r e m p i r i c a l e v i d e n c e f o r t h e b a r g a i n i n g s e t s from a wider r a n g e of games and s i t u a t i o n s T h i s is e x p e c t e d t o l e a d t o ( s t o c h a s t i c ) r e f i n e m e n t s of t h e b a r g a i n i n g

sets as w e l l as t o f u r t h e r s t a n d a r d i z a t i o n of t h e e x p e r i m e n t a l p r o c e d u r e s u n t i l t h e former h a s found s u f f i c i e n t e m p i r i c a l c o r r o b o r a t i o n . Only t h e n c a n game t h e o r y become more " r e a l i s t i c " , by s y s t e m a t i c a l l y i n t r o d u c i n g d e v i a t i o n s from t h e e x p e r i m e n t a l p r o c e d u r e and i n c o r p o r a t i n g t h e i r e f f e c t s i n t o t h e model. But i n o r d e r t o ever become r e a l i s t i c game t h e o r y f i r s t h a s t o show more c o n c e r n w i t h i t s i d e a l i z e d c h a r a c t e r .

Chapter 2 1 Coalition formation: a game-theoretic approach

107

APPENDICES

EQUIVALENT FORMALIZATIONS I N GAME THEORY

1. PARLOR GAMES, SEQUENCES OF MOVES, AND NORMAL FORM GAMES

For many p a r l o r games l i k e c h e s s , b r i d g e , and p o k e r , i t i s n o t i m m e d i a t e l y o b v i o u s t h a t t h e y c a n be modeled a s normal f o r m games. These games, and many o t h e r s , c o n s i s t o f s e q u e n c e s of moves w i t h r u l e s i n d i c a t i n g whose t u r n

i t i s , and which moves a r e a l l o w e d . A move may c o n s i s t of a r a n d o m i z a t i o n p r o c e d u r e l i k e t h e s h u f f l i n g o f c a r d s as f i r s t move i n most c a r d games or t h e r o l l i n g of d i c e a s , f o r example, i n p o k e r . The a c t o r s may have f u l l knowledge of t h e s t a t e of t h e game ( e . g . , c h e s s ) o r t h e r e may be u n c e r t a i n t i e s ( f o r example, a s t o t h e c a r d s d e a l t t o t h e o t h e r a c t o r s ) . A l l t h e s e s i t u a t i o n s c a n i n p r i n c i p l e be modeled a s normal form games. W e s a y " i n p r i n c i p l e " , b e c a u s e i t may be p r a c t i c a l l y i m p o s s i b l e b e c a u s e o f t h e enormous amount of s t r a t e g i e s open t o e a c h a c t o r . T h u s , a p r a c t i c a l a p p r o a c h i s n e c e s s a r i l y h e u r i s t i c -- d e f i n i t e l y p a r t o f t h e f u n o f p l a y i n g . The formal mechanism i n v o l v e d i n c o n s t r u c t i n g t h e normal form model i s v e r y i n t e r e s t i n g b u t t o o i n t r i c a t e and t o o f a r from c o a l i t i o n f o r m a t i o n t o w a r r a n t i n c l u s i o n i n t h i s book. Moreover, most books on game t h e o r y g i v e e x c e l l e n t i n t r o d u c t i o n s ( e . g . , Luce & R a i f f a , c h a p . 3).

2. LINEAR TRANSFORMATIONS OF PAYOFFS

Normal form games a r e based o n u t i l i t i e s measured o n an i n t e r v a l s c a l e ( s e e s e c t . 2 . 2 . 4 ) . U t i l i t i e s i n g e n e r a l d o n o t have a z e r o o r u n i t o f measurement w i t h a s p e c i a l i n t e r p r e t a t i o n , t h a t i s , t h e y are i n v a r i a n t u n d e r p o s i t i v e l i n e a r t r a n s f o r m a t i o n s (u' = a

+ bu w i t h b > 0 , w i t h a

d i f f e r e n t a and b f o r e a c h a c t o r . T h i s means t h a t t h e r e s u l t i n g game is s t r a t e g i c a l l y e q u i v a l e n t t o t h e u n t r a n s f o r m e d game. ( 1 ) If u t i l i t i e s a r e e x c h a n g e a b l e , a s i n c h a r a c t e r i s t i c f u n c t i o n games, t h e s t r a t e g i c equivalence is only preserved i f t h e s c a l e f a c t o r b is t h e

same f o r a l l a c t o r s . ( 2 ) U t i l i t i e s d i f f e r i n g by a n o r d e r o f magnitude may r e q u i r e d i f f e r e n t

measurement p r o c e d u r e s . U t i l i t i e s may b e l i n e a r i n money i n games w i t h i n n o c e n t p a y o f f s w i t h i n t h e r a n g e of l c t

-

$1, b u t q u i t e n o n l i n e a r when t h e

p a y o f f s become l a r g e enough t o c a u s e a n a c t o r ' s bankrupcy. D i s r e g a r d i n g s o c i a l t i e s o r p r e f e r e n c e s may be j u s t i f i e d w i t h s m a l l p a y o f f s , "because

108

W.J. van der Linden and A . Verbeek

so much i s a t s t a k e " . Formally, w i t h i n game t h e o r y t h e s c a l e s of u t i l i t i e s a r e i r r e l e v a n t , but i n p r a c t i c a l a p p l i c a t i o n s t h e g r a d u a t e d j u s t i f i c a t i o n of t h e a b s t r a c t i o n s u s u a l l y does depend on t h e s c a l e of u t i l i t i e s . (3) Changing t h e z e r o of t h e u t i l i t i e s , while l e a v i n g t h e s c a l e i n v a r i a n t ,

amounts t o paying a p a r t i c i p a t i o n f e e by o r a remuneration t o t h e a c t o r s . Again, l a r g e i n i t i a l p a y o f f s may have t h e same e f f e c t s a s t h e g r e a t change i n s c a l e d i s c u s s e d above. ( 4 ) For small i n i t i a l p a y o f f s and small changes of s c a l e , i t seems reason-

a b l e t o assume t h a t formal e q u i v a l e n c e of t h e s t r a t e g i c a s p e c t s of t h e game e x t e n d s t o i d e n t i c a l behavior of a c t o r s i n games which o n l y d i f f e r i n p a r t i c i p a t i o n f e e . Luce & R a i f f a (1957, pp. 263-264), however, a l r e a d y r e p o r t e d t h a t t h i s i s not always s o . Actors do n o t always e x p l i c i t e l y r e a l i z e t h e equivalence of c e r t a i n equivalent c h a r a c t e r i s t i c f unct i ons, nor do t h e y i n t u i t i v e l y develop t h e same c o a l i t i o n forming b e h a v i o r . ( 5 ) If d i f f e r e n t a c t o r s have u t i l i t i e s of a widely d i f f e r e n t magnitude, t h e n t h i s a f f e c t s t h e i r n e g o t i a t i o n s t r e n g t h . An a c t o r who cannot a f f o r d t o r i s k a l o s s i s i n a weak p o s i t i o n . The concept of t h r e a t t y p i c a l l y depend on t h e s c a l e of u t i l i t y .

("This may h u r t you more t h a n i t h u r t s m e " )

N e g o t i a t i o n s t r e n g t h and t h r e a t t y p i c a l l y depend on a more o r less a b s o l u t e i n t e r p r e t a t i o n o f t h e u t i l i t i e s i n v o l v e d , and a f o r t i o r i on i n t e r a c t o r c o m p a r a b i l i t y . See a l s o s e c t i o n s 4.6. and Appendix 9 . N e v e r t h e l e s s , p o s i t i v e l i n e a r t r a n s f o r m a t i o n s w i t h t h e s a m e scale f a c t o r f o r a l l a c t o r s a r e o f t e n i n t e r p r e t e d a s isomorphisms, t h a t i s , a s f u n c t i o n s o n l y changing i r r e l e v a n t l a b e l s of t h e p a y o f f s v e c t o r s . I n t h a t c a s e constant-sum games and zero-sum-games

are c o n s i d e r e d e q u i v a l e n t . Also,

constant-sum c h a r a c t e r i s t i c f u n c t i o n games can be s t a n d a r d i z e d t o t h e form where v(grand c o a l i t i o n ) = v(sing1e acto r)

0,

= -1,

f o r each a c t o r .

T h i s i s c a l l e d t h e reduced form or "-1,O-normalization".

For non-constant-

sum games t h e same s t a n d a r d i z a t i o n i s p o s s i b l e , b u t t h e f o l l o w i n g "0-1n o r m a l i z a t i o n " i s more popular: v(grand c o a l i t i o n ) = 1, v(sing1e act o r)

= 0,

f o r e a c h actor.

The 0 , l - n o r m a l i z a t i o n i s achieved by paying a remuneration t o a c t o r i equal t o ai = vcactor i ) ,

Chapter 2 1 Coalition formation: a gatne-theoretic approach

109

and then changing the scale of payoffs for all actors by a factor b = l/(v{l,

...,n))

- ZYzl v({j}).

From this normalization the reduced form can be obtained by changing the payoffs again, now by a factor n , and next charging an entrance fee of

1 (cf. Luce & Raiffa, 1957, pp. 188-189). These normalizations are useful in that they abstract from irrelevant aspects. 3. SUPERADDITIVITY IN CHARACTERISTIC FUNCTION GAMES In section 2.1. it was mentioned that the condition (l), for disjoint S LJS ) 2 V(sl) C V(sz), 1' 1 2 is not very restrictive. Pareto optimality excludes formation of S V(s

1

s2

if this inequality is not satisfied. Now any function u : (subsets}

-f

R

can be transformed into a characteristic function satisfying (1). Just put

+ . . . u(Sk)) 1 where the maximum is over all (exhaustive and exclusive) partitions v(S)

=

max ( u ( S

u . . . S (k=1,2,3, ...,j,where j is the number of actors in S ) . 1 k An interesting property is that v is the smallest characteristic function

S = S

satisfying both (1) and u 2 v. Also, if v(S) = U(S,)

+

...

U(Sk)

for partition S = S1 u . . . Sk, this means that S can collect v(S)

from

the umpire by addressing the umpire disguised as the k disjoint coalitions S1' 4.

. . . , sk . THE RELATION BETWEEN NORMAL FORM AND CHARACTERISTIC FUNCTION GAMES

A constant-sum normal form game with transferable utilities, unrestricted side-payments, and perfect communication can be considered a constant-sumcharacteristic function game (sect.2.3.2.)without icss of information. If a group of actors considers cooperation, they will strive to maximize the sum of their payoffs, because of the assumption of the transferability of utility. Because the game is zero-sum the actors left out are trying to minimize this sum and unite in one coalition too. The perfect communication makes it possible to effectuate this. The game is thus reduced to a twoactor, zero-sum game for which the "unique" maximin solution (sect. is the universally accepted solution concept. This pins down the value of the coalition under consideration to the unique maximin value of this two-actor game. Furthermore, the relative strength of the actors in the

110

W.J.van der Linden and A. Verbeek

coalition, as reflected in the final distribution of the payoffs, seems to depend on the payoff vector in the normal form game only through the values of alternative coalitions. W e have to say "seems" because we have been unable to find a more formal justification for this intuitively reasonable step. Similarly, for each non-constant sum game with transferable utilities and unrestricted sidepayments, w e can construct a characteristic function game by defining the value of a coalition as its maximin value. But in nonconstant sum games the maximin value is only an (often very pessimistic) lower bound to the strength of an actor (here: a coalition). Although the opponent is able to hold the coalition down at that level, it is, in general, not in his own interest, except, perhaps, as a threat strategy. Because there is no universally accepted value for each coalition, we cannot say either that the internal distribution of the coalition value will only depend on the values of alternative coalitions. In summary, we consider the characteristic function constructed a useful tool for analyzing original normal form games, but, in general, not an equivalent game. In section 3.9. it is pointed out that every n-actor non-constant-sum normal form game can be treated as a (n+l)-actor constant-sum normal form game by adding a "banker" who pays the total payoff to the other actors while having only one strategy (i.e., no alternatives). But this offers no way out of the problem outlined above, because it does make a difference whether the banker can take part in negotiations on coalition formation (as in the (n+l)-actor game) or not (as in the original n-actor game). Certainly, no coalition is willing to pay the banker for joining them, but conversely the banker may be willing to pay a coalition to accept him a8 a member, and this influences the relative strength of the other actors. For more details see von Neumann and Morgenstern (1944, sect. 56.3.3.). EXAMPLES AND PARADOXES 5.

THE PRISONER'S DILEMMA GAME

The bi-matrix game in Display 15a is called a prisoner's dilemma game if

Chapter 2 / Coalition formation: a game-theoretic approach (a) General 2x2 bi-matrix

(b) The prisoner's

game

111

(c) The battle of

dilemma

the sexes

strategies for actor B B1

B2

B1

B1

B2

strategies for actor A

Display 15. Two famous examples of the 78 qualitatively different types of 2x2 bi-matrix games: the prisoner's dilemma and the battle of the sexes. A mixture of these two types of conflict was given above in Display 11. For an enumeration of all types see Rapoport & Guyer (1966). A2 dominates A 1 (4, > p1 and s

>

TI),

B2 dominates B1 (r

>

and

s 4,) but (s1,s2) is much worse than (pl,p2). A numerical example is 2 given in Display 15b. There is an obvious generalization to n-actor games: Each actor has two strategies of which one dominates the other, but if every actor chooses the dominating strategy everyone is much worse off than if the other strategy had been chosen. A typical example is voluntary production restriction of a certain product with too many producers and a saturated market.

It should be emphasized that considerations of solidarity cannot be used to predict the socially desirable solution (Al, B2) in Display 15. These considerations may be present, we hope, but must be supposed to have been taken into account when the utilities were determined. Thus, even given strong feelings of solidarity, in situations that can be modeled as a prisoner's dilemma game bigger gains are at stake causing A2 and B2 to dominate A1 and B1, respectively. If the two actors are not in a position to make binding agreements on (Al, B1) there seems no way to escape the miserable (A2, B2). Luce and Raiffa (1957, pp. 94-97) conclude their lucid discussion of the prisoner's dilemma game as follows:

W.J.van der Linden and A. Verbeek

112

"The h o p e l e s s n e s s t h a t one f e e l s i n s u c h a game a s t h i s c a n n o t be overcome by a pLay on t h e words " r a t i o n a l " and " i r r a t i o n a l " ; i t is i n h e r e n t i n t h e s i t u a t i o n . "There should b e a law a g a i n s t s u c h games!" I n d e e d , some h o l d t h e view t h a t one e s s e n t i a l r o l e of government is to d e c l a r e t h a t t h e r u l e s of c e r t a i n s o c i a l "games" must b e changed whenever i t i s i n h e r e n t i n t h e game s i t u a t i o n t h a t t h e p l a y e r s i n p u r s u i n g t h e i r own e n d s , w i l l b e f o r c e d i n t o a socially undesirable position."

W e might add t h a t p r i s o n e r ' s dilemma games a r e examples of s i t u a t i o n s i n which l a i s s e z - f a i r e d o e s n o t a u t o m a t i c a l l y l e a d t o a s t a b l e optimum. S o c i o l o g i s t s have observed t h a t s u c h games may a l s o g e n e r a t e norms, r a t h e r than laws, t h a t p r o h i b i t t h e dominating s t r a t e g y .

6 . THE BATTLE OF THE SEXES

The b a t t l e of t h e s e x e s , d e p i c t e d i n D i s p l a y 1 5 c , i s an example o f a game i n which n e g o t i a t i o n is p o s s i b l e and t h e a c t o r s a c t u a l l y need e a c h o t h e r t o g e t an o p t i m a l r e s u l t . The b a s i c problem i s t h a t t h e r e a r e two s t a b l e optima and t h a t e a c h a c t o r p r e f e r s a n o t h e r optimum. A c l a s s i c a l i n t e r p r e t a t i o n of t h e game i s g i v e n by Luce and R a i f f a (1957, s e c t . 5.3.):

A man and w i f e d i s c u s s an o u t i n g . A 1 and B1 a r e t h e c h o i c e s t o go t o a p r i z e f i g h t , A2 and 82 t o t o a b a l l e t . They would l i k e t o go t o g e t h e r b u t a l s o adhere t o t h e i r s t e r e o t y p i c a l p r e f e r e n c e s . I n t u i t i v e l y reasonable s o l u t i o n s are c h o o s i n g each optimum a l t e r n a t e l y o r t o s s i n g a dime. I f s u c h i n t e r m e d i a t e s o l u t i o n s a r e n o t open, a n e g o t i a t i o n t r i c k may be t o s t a t e o n e ' s i n t e n t i o n immediately and a s f i r m l y as p o s s i b l e . I f , f o r example, A would do s o , t h e game would i n f a c t be reduced t o t h e f i r s t row l e a v i n g B l i t t l e c h o i c e b u t a c c e p t i n g B1.

( U n l e s s B g e t s s o mad a s t o change h i s / h e r

u t i l i t i e s , of c o u r s e , b u t t h i s would imply exchanging t h e p r e s e n t game f o r another; c f . s e c t . 2.4.). 7 . THREE I S A CROWD

Once upon a t i m e an e v i l k i n g d e c i d e d t h a t t h r e e o f h i s s u b j e c t s c o u l d have $ 1 M i f t h e y a g r e e d on i t s d i s t r i b u t i o n by m a j o r i t y r u l e

....

The i n t r i n s i c i n s t a b i l i t y of any i m p u t a t i o n i n t h i s game was a l r e a d y decried in section 2.5.2.

I t i s a p p a r e n t from t h e symmetry of t h e game t h a t

Chapter 2 / Coalition formation: a game-theoretic approach

113

game t h e o r y i s too b a r r e n h e r e t o p r e d i c t w h i c h c o a l i t i o n w i l l f o r m . The game i s a t h r e e - a c t o r c o n s t a n t - s u m game, e q u i v a l e n t t o any o t h e r t h r e e a c t o r c o n s t a n t - s u m game. I t s r e d u c e d form (Appendix 2 ) i s g i v e n i n Display 16.

(a) General conditions

v({1,2,.

..,nl)

v(S)

o

=

v({1,2,3})

= -v(Sc)

v ( { o n e a c t o r } ) = -1 v(empty s e t )

=

0

( c ) Four a c t o r s

(b) Three a c t o r s

= o

=

0

v({1,2,3,4})

v({two a c t o r s } ) =

1

v({three actors}) =

v({one a c t o r } )

= -1

vcempty s e t )

=

o

1

v({1,2})=-~({3,4})= p v(I1,33)=-v(t2,41)=

q

v ( { 1 , 4 1) =-v ({Z ,31) =

r

v({one a c t o r ] )

= -1

v(empty set)

= o

D i s p l a y 1 6 . Zero-sum c h a r a c t e r i s t i c f u n c t i o n games i n r e d u c e d f o r m . In (a) t h e general conditions a r e given; (b) gives t h e only 3-actor

game, and ( c ) shows t h e g e n e r a l 4 - a c t o r

game. Here p , q , and r must e a c h l i e b e t w e e n

-4

and

4

i n c l u s i v e l y . For r e a s o n s o f symmetry w e may s u p p o s e

-4 5

p

s q < r

5

4. For

a t h o r o u g h d i s c u s s i o n , see von

Neumann and M o r g e n s t e r n ( 1 9 4 4 , c h a p . 7 ) .

8. THE LARGEST NUMBER WINS: INFINITE GAMES One o f t h e s e e m i n g l y i n n o c e n t r e s t r i c t i o n s i n s e c t i o n 2 . 2 . was t h a t t h e number o f s t r a t e g i e s was f i n i t e . The f o l l o w i n g e x a m p l e shows how d e v a s t a t i n g t h e n e g l e c t o f t h i s a s s u m p t i o n c a n b e : Two a c t o r s may w r i t e a number o n a s l i p o f p a p e r ( f o r e x a m p l e , t h e y p r e t e n d t o b e c e r t a i n c o u n t r i e s and t h e number i s t h e i r p r o p o s e d m i l i t a r y b u d g e t ) . The l a r g e s t number w i n s . O b v i o u s l y , any s t r a t e g y ( w r i t i n g down n ) i s d o m i n a t e d ( b y w r i t i n g down n+l) !

T h e r e a r e i n t e r e s t i n g e x t e n s i o n s of game t h e o r y t o i n f i n i t e numbers o f s t r a t e g i e s ( s e e , e . g . , McKinsey, 1 9 5 2 , c h a p . 7 f f . ) . One f u r t h e r e x a m p l e

is g i v e n i n which i n f i n i t y d o e s n o t p l a y an i m p o r t a n t role: A l a r g e g r o u p o f a c t o r s i s n o t a l l o w e d t o communicate (or o n l y i n s m a l l g r o u p s ) . Each a c t o r p a y s a $1 e n t r a n c e f e e and writes a p o s i t i v e i n t e r g e r number o n a

114

W.J.van der Linden and A. Verbeek

p a p e r . I f two o r more a c t o r s w r i t e down t h e same number t h e i r s l i p s are removed. The s m a l l e s t of t h e r e m a i n i n g “unique” numbers wins t h e j a c k p o t .

9 . THE BEGGAR AND THE MILLIONAIRE

T h i s two-actor

c h a r a c t e r i s t i c f u n c t i o n game h a s t h e f o l l o w i n g t r i v i a l

s t r u c t u r e . If two p e r s o n s c o - o p e r a t e t h e y can a c q u i r e a c e r t a i n p r i z e , s a y $100, A 8 u s u a l c o - o p e r a t i o n

a l s o i m p l i e s t h a t t h e y a g r e e on how t o

s p l i t t h e p r i z e . Now suppose t h e p r i z e is of utmost importance t o one a c t o r , a beggar ( B ) , and of minor importance t o t h e o t h e r , a m i l l i o n a i r e (MI. In t e c h n i c a l terms t h i s means t h a t one u n i t o f money h a s a d i f f e r e n t u t i l i t y f o r B t h a n for M . T h i s f a c t p u t s B i n a weak p o s i t i o n and he i s l i k e l y t o end t h e n e g o t i a t i o n s w i t h much less t h a n $50 ( c f . Luce & R a i f f a , 1957, s e c t . 6 . 6 . ) .

For t h e R e f e r e n c e s t o Chapter 2 , see page 269.