A theory of coalition formation in committees

Journal of Mathematical

A THEORY

Economics 7 (1980) 115-134. 0 North-Holland

OF COALITION

FORMATION

Publishing Company

IN COMMITTEES

Bezalel PELEG The Hebrew University, Jerusulem, Israe?‘

Received February 1979, accepted May 1979

Let G= (N, W) be a strong weighted maiority game and let A be a set of alternatives. Denote by the set of linear orders on A. A social choice function F :I!?’-+A is a representurion of G if the simple game G*$‘! associated with F equais G. A coalition S is determining in G if it satkfies the following condition. Let F be a representation of G and let RN~K”. Then, if a simple majority of the members of S consider an alternative x to be their best choice, then S can ‘enforce’ x to be a Nash equilibrium payoff in the resulting non-cooperative voting game g(F,RN). In th’IS paper we generalize the above notion of a determining coalition to committees (i.e., proper and monotonic simple games), and give a complete characterization of the set of determining coalitions of a committee. Furthermore, we discuss our notion of a determining coalition in the light of some real-life data on formation of coalitions in town councils in Israel. L

1. Introduction This work is devoted to an attempt of a characterization of the coalitions that form in committees. Specifically, for each committee G we distinguish a set of ‘determining’ coalitions in G, D(G), and we claim that only coalitions in D(G) are likely to form in G. Our theory is valid under the following assumptions. Let G be a committee (i.e., a proper simple game). We assume that G is expected to resolve a certain family 4p of choice problems. The family cp may not be completely specified at the time when a coalition forms in G. All that is required is that each member of q will be resolved by a ‘regular’ voting procedure (see Definitions 3.9 and 5.1). Actually, the members of cp may be identified with regular social choice functions. Let FE cp. We assume further that when the time arrives to resolve F, then this is done by the members of G by playing a non-cooperatioe voting game g(F, R’ ) (see Definitiotl 2.5 and Remark 7.14), where R” is the profile of preferences of the members of G over the set of alternatives of the choice prob!em associated with F. Now, a coalition S in G is c;il!ed determining if it satisfies the following condition. Let FE q and let g(F,RN) be, as above. a ‘realization’ of F (see Definition 5.2). Then, if according to RN a ‘simple majority’ of the members of S consider an alternative x to be their best choice, then S can ‘,enforce’ x to be a Nash equilibrium payoff in the game g(F, R” ) (see

Definitions 7.1 ar ;d 7.8). We claim ‘hat, under the above assumptions, only determining coalitions form in G (see Hypothesis 7.13). We now review briefly the contents of the paper. Section 2 contains the necessary definitions concerning social choice functions. The various simple games associated wit;, a social choice function are introduced in section 3, and some of their properties are discussed. In section 4 we consider the converse problem: with each proper simple game we associate ‘nice’ social choice functions. Section 5 consists of a heuristic discussion of determining coalitions. A desirability relation for coalitions is introduced and discussed briefly in section 6. It enables us to give a rigorous definition of a determining coalition in a proper simple game, in section 7. Section 7 contains also a mathematical investigation of determining coalitions which leads to a complete characterization of the set of determining coalitions of a proper simple game (see Theorem 7.6 and Remark 7.9). We conclude section 7 with our hypothesis concerning the formation of determining coalitions (see Hypothesis 7.13). in section 8 we list the coalitions that formed in 26 towns in Israel after the municipal elections of 1965, 1969 and 1973 (see Table 8.1). The real-life data seem to support our hypothesis (see Table 8.2). Also, the limitations of our theory are discussed in order to provide an explanation of the formation of non-determining coalitions in several real-life cases (see Remark 8.4). The last section, section 9, contains several remarks on possible applications and further investigations of our theory. For a recent comprehensive essay based on a different approach to the formation of coalitions in cooperative games, the reader is referred to Shenoy (1977). I am grateful to M. Maschler for several helpful discussions, and to R.J. Aumann for several valuable remarks.

2. Consistena social choice functions Let A be a set of alternatives which contains at least two members. A linear order on A is a complete, reflexive, transitive and antisymmetric binary relation on A. We denote by L the set of linear orders on A. Let N be a finite set with n members. N is called a society, members of N are called voters, cr players, and non-empty subsets of N are called coalitions. For a colaition S we deiiote by I! the set of all functions from S to L. Definition 2.1.

A sociul choicefiwrion

(SCF) is a function from p to A.

We shall be interested in the following properties of SCF’s. Definition 2.;?.

A SCF F is monotonic if it satisfies:

B. Feleg, Coalition formation

(M)

117

If F(R’“) =x, Ry E LN, and for all a, k A - {xi and all i E N, LIR%~NR; h and xR’u=xR~a, then F(R’;J)=x.

Dejinition 2.3.

(p)

irt commi:tees

A SCF F is Parvtiun if it satisfies:

If RNELN, XJEA,

X#J

and xR’y for all WV, then F(RN)fy

Definitiolz 2.4. Let F be a SCF. A permutation z of N is a symzetry of F if for all RN= (R’ ,..., RN) in LN F(R*)= F(R”(‘) ,..., Rx(“)). The group of a21 symmetries of F will be denoted by sym (F). F is anonymous if sym (F) = S,, where S,, is the symmetric group of the set of the players.

The str&tegic aspects following definitions.

of a voting

situation

are made

precise by the

bejbGtion 2.5. Let F be a SCF and let RI’% LN. The gume mwciatcd trnd R” is the n-person game in normal form g(F, R" ) where

with F

(a) L is the set of s?rategies for each player k IV, (b) F is the outcome function, (c) R’ is the preference relation of player i E N on the outcome space A. Dejinit iorl 2.4. Let F be a SCF and let R” E L? Q” E L” is an equi/ihriuw point (e.p.) of g(F, R”) if, for each ie N,

F(QN)RiF(QM-Li’,7”)

for all

7% L.

(Here Q’-jir is the restriction of QN to N- (ii.1 Definit ioiz 2.7. Let F be a SCF and let R” E J!?. Q” EL’ is an VXW/ e.p. of g(F, R”) if Q* is an e.p. of g(F, RN), ud F(QN)= F(R”). DefiMo/l 2.8. A SCF F is fextrctiy) has an (exact) e.p.

consistent

if, for each R’E b;“, g( F, R”)

Dqfinit ion 2.9. Let F be a SCF and let R” E LN. Q’ E L? is a strorzg e.p, of g(F, RN) if for every coalition S and for every P% k, there exists a player

in S such that F(QN)RiF(QNbS,Ps). N - S).

(Here Q”-’

is the restriction

of Q” to

A SCF F is e?ctrc~~~u~d strongly cortsisterzt if there exists a function E:LN-+LN such that for each RNeLNj E(R”) is a strong e.p. of g(F, RN), and th e composition F o E = F. Definition

2.10.

118

B. Peleg, Coalition formution in committees

For motivation and a discussion of Definitions 2.8 and 2.10 the reader is referred to Peleg (1978a). 3. The simple games associated with a SCF In this section we associate three simple games with every SCF. The first simple game associated with a SCF F is that specified in Definition 3.2 in Peleg (197sb). The other two are derived from the notions of 01and ‘) -effectiveness for cooperative games without side payments [see Aumann we investigate the relationship between the three (1961)]. Furthermore, simple games associated with a SCF F and the consistency properties of F (see Definitions 2.8 and 2.10). For the sake of completeness we recall several definitions which are used in the sequel. For a comprehensive study of simple games the reader is referred to Shapley t 1962). A simple gum is an ordered pair G = (N, W), where N is a society and W, the set of winning cocrlitions, satisfies [SE Wand TX S]*TE

W (monotonicity).

Let G = (N, W) be a simple game. G is propt~ if

G is

nui: if

W=9.

Let l”:L”-+A be a SCF (see Definition 2.1). We associate with F the first simple game G*(F) in the following way. .Dt$ttit iorz 3.1.

A coalition 5 is wiming [with respect to (w.r.t.) F], if

[R%!?,xEA

and XR’J for all iES and all I~EA]=+‘(R~)=.Y.

Df$t~it im 3.2.

The jbst simple gurne crssocicrted with F is the game G*(F) - (N, W), where W is the set of winning coalitions w.r.t. F.

RcJtTztrrk3.3. Since A contains at least two alternatives G*(F) is proper. Also, if F is Paretian (see Definition 2.3) then V E W: Thus, G*(F) is non-null if F is Paretian. LkJinit iota 3.4.

Let G = (N, W) be a simple: game. G is \\‘L’llkif

The members of I/ are called z?etOpluyers.

119

The following simple result is true. Theor~?r 3.5. Let F: LN-+A be a SCF. l” G*(F) (see Definitim 3.2), bus no seto phq’ers then F is exact/y consistent (see Definition 2.8). P moj: Let R”E f?’ and let F(RN)=x. Let P% LN satisfy XP’J for all J E A and all kN. Since G*(F) has no veto players, N - ii), E W for all i E N. Hence, by monotonicity, NEW Thus, F(PN)=x (see Definition 3.1). Furthermore, for each ie N, F( p” - if T’ ) = _x for all

T’EL.

Thus, P” is an exact e.p. for g( F, RN) (see Definition 2.7). The reader is referred to Theorem 1 in Dutta and Pattanaik proof of a mere general result.

(1978) for a

Let F be a SCF. In order to define the second simple game associated with F we recall the definition of a-effectiveness. Definition 3.6.

Let F: LN-+A be a SCF. Let x54 and let S be a coalition. S is a-@‘kfir~~ for x if there exists QS EL? such that for all P” -’ E L”-’ F(Q”, PN-‘)= x. S is cc-n+nnirzgif S is z-effective for every y E A. with F is Defittitimr 3.7. Let F be a SCF. The second :hpfu gtrtne mswiated the gama G,(F)= (N, W,), where W, is the set of x-winning coalitions w.r.t. F.

R~nurk 3.8. Let F be a SCF. Then G,(F)= (N, W,) is proper. Furthermore. if G*(F)= (N, W) (see Definition 3.2), then WC M$ Definition 3.9. Let F be a SCF and let G*(F)=(N,W) F is reglilur if W = W,.

and G,(F)=(N,W,).

kV\: show now that Borda’s rule may yield an irregular SCF. We start with a definition which will be used in the sequel. D $nit ion 3.10.

A simple game is ,s~*rnnwtricif its symmetry the symmetric group of its set of players.

group equals

Rent& 3.11. It follows from Definition 3.10 that a symmetric simple game can be described by a pair (II, k) of natural numbers, where 11is the number of players and k is the size of a minimal winning coalition.

B. P&g, Coalition ,formatiuu in committees

120

Let A= ~s~,x~,x~, 1 be a list of three different alternatives and if x is 21;. Let w:LxA -+(OJ,Zj be defined by: w(R,x)=&i let N = { 1,2,. . ., the ith alternative in the order R. For R* E I! and x E A we denote E.wmpk

3.12.

We now define a SCF F by the foll;>wing rule. Let RNe I!?‘.F@)=x, iff (1) i < t~W(RN, xi) < w(RN, x,), and (“-1 i > t*i~(P, xi) ,5 w(P, x,). Clearly, F is monotonic, Paretian and anonymous (see Definitions 2.2, 2.3 and 2.4 respectively). Furthermore, as the reader can easily verify, G*(F) = (21,15) while G,(F)= (21,13) (see Remark 3.11). Examples of regular SCF’s will be given m the next section. Now, before investigating sufficient conditions for regularity of SCF’s, we find it convenient to introduce the third simple game associated with a SCF. Dejbiition 3.13. Let F: p-+ A be a SCF. Let x E A and let ’ be a coalition. S is jheffectiw for x if for each PN% tips’ there exist 0% I? such that F(QS, PNmS)=x. S is [hiwzing if S is b-effective for every JE A.

Let F be a SCF. The third simple gmw ussociatcti with F is the game G,(F)= (PI. M$), where I?,( is the set of [&winning coalitions w.r.t. DclJinition 3.14, F. Remark 3.25. Let F be a SCF and Irt G,(F)= (N, W,) and G,(F)= (N, W,]). Then, clearly, I&k3 W,. It is well knowvl that b-effectiveness is a weaker concept than z-effectiveness [see Aumann (1961)-J. Hence, it seems to us unnecessary to produce an example of a SCF which satisfies W$# W,. Also, we do not know whether GJF) is always proper. D&nitiotz

3.16. Let F be a SCF = (IV,CVp).F is tight if W = W,].

and

let

G*(F)=

(N, W)

and

G@(F)

Clearly, a tight SCF is regular. The following theorem ties the work in Peleg ( 1978a, b) with tile present discussion.

PraoJ Let G*(F)=!N, W) and G,#)=(N, Wp). Let SE Wfi. We have to show that Se W Assume, on the contrary, that S efW. Then there exist R,’ E L?’ and us A su(zh that aR’x for all & S and all x E A, and F(RN)= b,

where b#ti (see Definition 3.1) Since F is exactly and strongly consistent the game g(F,R”) (see Definition 2.51, has a strong e.p, Q” (see Definition 2.9), such that F(QN)= h. Since SE Wfl, S is p-effective for CI. Hence there exists PS& such that F(PS,QNeS)= (1. Since LIR% for all ifz S, QN is not a strong e,p, of g(F, RN), which is the desired contradiction. The following theorem shows that a large class of SCFs are tight. First we need tke following definition. A simple game G = (N,

De~irlitiorr 3.18.

is strong

if

SBW*N-SEW Tht rirer?r 3.19. 3.2).

:<*

utrong

Let F: L* + A be u SCF. fj’ G*(F) = (N, W ) tight (see Dfzjinitioll 3.16).

(SLJC

Dilfihit

iw

tlIe?t F is

P~oojI Let G,(F)= (IV, Wfi) (see Definition 3.14). Let Sq! W We have to show that S $ Ws. Since C*(F) is strong T=N -S is winning. Let LIE. , and let Q% LT satisfy aQ’x for aI1 in Tand ali x E A. Since 7% W, for every f” E c F(fS, QT) = Q. Let h E A, b # (1. Then we have that S is not &effective for h. Thus, S$ wfl.

4. TWOexamples In this section we introduce a modified, (weaker), version of the notion of a representation of a committee as specified in Definition 4.1 m Peleg 61978b). Then, we proceed to construct examples of tight representations of committees which possess most of the properties introduced in section 2. Dcfinitio~~ 4.1. Let G= (N, W) be a simple game and let A be a set of alternatives. A SCF F:&+A is a vepresentntion of G if G”(F)=G (see Definition 3.2). F is a representation of order m of G, if A is a finite set with m elements. &FU.Z& 4.2. A ‘representation’ in the sense of Definition 4.1 of Peleg (1978b) is an exactl!~ ami strongly corzsister~rrepresentation (see Definition 2.10), according to the present definition. For motivation and a discussion of rct,resentations the reader is referred to section 4 of Peleg (1978b). We describe now our first example. Let G= (N, W) be a proper simple game which is tzon-null (i.e., N E W) and let A = {xl,. . .;x”, > be a list of m different alternatives, rnz 2. Let further @‘.E p and X,J’E A, .X# _Y..Y,no-

II. Peleg,

122

Coalition Jormcition

in commi;tees

y w.r.t. R N if (&N IxR’y} is winning (i.e., it is d member of W). The core of A, w.r.t. G and RN, is the set of uniJominated alternatives in A, and is denoted by C(A,N,W,RN)=C(RN). Let p=min{/Si&IW}, where, here and in the sequel, if B is a finite set then II?1denotes the number of members of B. For p =
Let now RN E LN. We denote

C,(RN)=C(A,N, W,,RN). Note that, since N E W, C,(RN) is the set of Pareto-maximal members of A, and, since A is finite, it is non-empty. Thus, for each RN ELM there exists a natural number k=k(RN), &k$~, such that C,(RN)=Q, for psl
C‘,W”;)#9. Remark

4.3.

Let RN EL”. If c E A dominates every x E ,4 - {c*)w.r.t. RN, then c is called a Condorcet alternatice (w.r.t. RN). Clearly, if R” E LN and c is a Condorcet alternative w.r.t. RN then C(R”) = (c). Remark 4.3 motivates the following definition. De$nitinn

4.4.

The generalized Condorcet .function (GCF) for the game G and the set A, is the SCF a(G,A)=a:LN -+A defined by the following rule: For each R” E LN, o(G, A, RN) =g(RN) is the first alternative (according to the list A) in C,(RN), where k= k(R”) is defined above. Remark 4.5. Clearly, if CE A is a Condorcet alternative w.r.t. RN E LN, then a(R” )=c. In this sense o is a ‘generalization’ of Condorcet’s rule. For

different generalizations of Condorcet’s rule the reader is referred to Young (1977) and to Young and Levenghck (1977). Before stating the properties of 0 we need the following definition. Let F : LN -+ A be a SCF. F is fait&u1 if sym (F)= sym (G*(F)), where sym(G*(F)) is the symmetry group of the game G*(F) (see Definitions 2.4 and 3.2). Definitioll 4.6.

Theorem 4.7. Let G = (N, W) be a non-null (i.e., N E W) proper simple game and let A= ~x,,...,x ,,,i he u list of m different alternatives, nzz 2. The GCF o = a( G, A ) (see Definitioft 4.4) is a moilotonic, Paretiajl, faithful and tight representat iorr 0j’ or&~ ~II of G (SLV Dejinitions 2.2, 2.3, 4.6 and 3.16 respect it’4133 ).

B. Peleg, Coalition formation

in committees

I23

Proof:

Let G,Ja)= (N, Wp) (see Definition 3.14). Clearly, wfi 2 W: In order to show that W= kl$ we first introduce the following notation. For R it denote by t(R) the best alternative according to R. Suppose now that S$ I+! Let T=N-S and let P%LT satisfy t(P’)=x, for all iE 7: Since S$ r/t: T is blocking. (A coalition in a simple game is blocking if its complement is losing.) Hence, for each Q” E k!?, xi E C@‘, PT) and, the; ;fore, x1 = c( QS, P’). Thus, S is not Q-effective for x,. Hence, S$ U;,. Let now G*(o)+& lV*) (see Definition 3.2). Clearly, WC IV* c k%$.Hence W = W* = M$, and CTis a tight representation of G (see Definition 4.1). The proof that (7 is monotonic, Paretian and faithful is straightforward and will be ‘omitted. Let G = (N, W) be a alternatives, #?I2 2. If G exactly consistent. The then o(G,A) may not definition.

proper simple game and let A be a finite set of m has no veto players then, by Theorem 3.5, cr(G,,A ) is following example shows that if G has veto players be exactly consistent. First we need the following

Definitiorz 4.8. Let G = (N, IV) be a simple game. G is a weighted mLljor’it> gwnr if there exist a quota q > 0 and weights w’ 2 0,. . ., \t”‘z 0 such that

The (II+ 1 )-tuple [q; WI,. . ., w’] is called a representation of G (not to be confused with Definition 4.1), and we write G= [y; w’, . . ., w”]. Let G=[3; 1,1,2]. 3 is a veto player of G. Let ,4 = ~.Y,,.Y~,.Y~~ and let cr=o(G, A) (see Definition 4.4). If R’ =R2 = (x2, x+x,) and R3 = (x3, x1, x2), then C(RN) = (x2, x3) and therefore a(RN) =-x2. If Q3= (x1,x3,x2) then, since 3 is a veto player, for every P(1*21~til$ G(P”*~‘,Q~)=x~. Since 3 prefers X, to _y2,g(a, RN) (see Definition 2.5) has no exact e.p. (see Definition 2.7). Thus, CTis not exactly consistent (see Definition Indeed, let Q’ = Q2 = Q” 2.8). Note, however, that d is consistent. = (x~,x~,x~). Then for each RN E c, QN is, trivially, an e.p. of g(c, RN). We now turn to our second example. Let, again, G = QV,w) be a proper simple game and let A = (x,, . . ., x,, : I< a list of m different alternatives, rnz2. Let further RNd?. Denote

Example

4.9.

Si(RN)=tjE

Nlt(R’)=sij-,

i=l,...,m,

where, for RE L, t(R) is the best alternative according to R (see the proof of Theorem 4.7).

124

Definition

R. Prleg, Cotrlitior; jbwution

4.1;).

in committees

1 he top Jilnst ion ( f F) fu5 the game G and the set A, is the

SCF s(G,A)=r:L” *A defined by the following rule: Let RN of?. If there exists an alternative .x;EA such that ,!$(@)E W then Z.((N’)=Xi. Otherwise, we let i=minblSj(RN)fQ)) and define X(RN)=Xi. Theorem 4.11. Let G= (N, W) he a proper simple game a& let A = (X1,...,X,“) ) be N list of m different alternatives, m 2 2. The TF r=z(G,A) (see Defirtitioll 4.20) is u monotonic, Paretian, @t&l and tight representation of G qf order m.

The proof, which is similar to the proof of Theorem 4.7, is omitted. E.utrmple 4.12.

Let G= (3,2) (see Remark 3.1 I), and let A =(x,, .Y,,sJ,s~). profile R’=(x~,.~~,~~,x~), R2=(x2,x,,.x,,.x,) and R3 = (x3, xj, x,, x2). Then s(R”) =.q while x4 is a Condorcet alternative for RN. Thus, T does not satisfy Condorcet’s rule.

Consider

Remark

the

4.23.

It is clear from the above discussion that for every proper simple game one can construct ‘nice’ tight representations of every arder mz2.

5. Determining coalitions -

A heuristic discussion

Dqfinitiorz 5.1. Let G = (N, W) be a committee (i.e., a proper simple game). A choice problem. or an issue, which is relerxmt to G, is a regular SCF F:L” -+A which is a representation of G (see Definitions 3.9 and 4.1).

Issues which are relevant to a committee G = (N, W) arise in the following way. Suppose that G has to choose one alternative out of a set A of alternaiives. Then, a regular representation F:I!“-+ A of G is, intuitively, a ‘nice’ voting procedure which reflects faithfully the ‘power’ of the various coalitions in G. DQ,JiCtion 5.2. Let G = (W, W) be a committee and let F:LN\-+A be an issue which is relevant to G. A realizatiorz of F is a game g(F, RN) (see Definition 2.5), where R” E I?.

Thus, a realization g(F, RN) of an issue F is a game in normal form which has to be played in order to resolve the corresponding choice problem. Clearly, RN stands for the profile of true preferences at the time when the issue is resolved. There may be a time lag between the presentation of the issue to the members of G, and the formation of RN. During that time the members of G study the issue and form their opinions. The process of

B. P&g,

Coulition .fwtnut ion irt commit tees

125

formation of preferences may be, of course, affected by factors which are, formally, not directly connected to the choice problem under consideration. We now address ourselves to the following problem. Let G be a commit tee which has to resolve an issue, or a sequence of issues. Assume that binding agreements (i.e., agreements whose violation entails high monetary penalties), are legally unpermissible or technically impossible. Under this assumption we ask, which coalitions are likely to form in G’? As emphasized by Rapoport (1970, p. 286) this problem is very important to the behavioral scientist. We now outline our solution to the above problem. In order to simplify the presentation let us assume first that G is a town council which has just been elected to serve for T years. Formilly, G is a strong weighted majority game (see Definitions 3.18 and 4.8), whose players Ltre purtius. Now, a coalition has to form in G in order to rule the town during the next T years. The formation of a coalition is completed when the parties involved sign an agreement. However, violation of such an agreement and, in particuiar, defections from the coalition, usually do not entail monetary penalties. Also. since the number and the nature of the issues that will confront G are 11ot completely specified at the time when a coalition is erected, possible differences ofopinion within the coalition slzolrld he' token into uccourtt. Hence, the coalition that is formed should consider the possible realizations of the issues that confront G as non-cooperutive games. Denote by S the coalition that is formed and let g(F#) be a realization of an issue that is relevant for G. We assume that g(F$“) is resolved by the folloawing two-stage process. First S decides, by simple mujority, which alternative x is ‘best’ for it. Then, in the second stage, S ‘enforces’ .I’ to be a Nash equilibrium payoff in g(F, RN). If S can decide every realization of an issue which is relevant for G by the above two-stage process, then S is called determirrir~g (see Definition 7.1). We claim that, under the above assumptions, only determining coalitions will form in G. In the next section we define in exact manner, for an arbitrary simple game, when does a subset T of a coalition S consist of a ‘simple maj0rit.y‘ within S. Then, the above two-stage decision procedure and the notion of a determining coalition can be generalized to arbitrary committees. The formal definitions are given in section 7. 6. A desirability relation for coalitions Let G = (N, W) be a simple game. Definition 6.1. Let S and T be coalitions. S is at least as desiruble LIS ‘r (w.r.t. G), written S k r if for every B c N such that B n (Su T)=g,

BuTEW-BUSEM!

126

B. Peleg, Coulition formation

in committees

If Sfi-7; but ?-tS d oes not hold, then we write S E----T (Clearly, asymmetric.) We also agree that S ~8 for every coalition S.

E- is

Remark 6.2. The relation t w?s introduced in Lapldot (1968). It generalizes the relation of desirability for players [Definition 9.1 in Maschler and Peleg ( 1966)]. Remark 6.3. Lapidot has shown that F-- may be cyclic. Hence, in particular, C, may not be transitive. However, if G = Cy; WI,. . ., w”] is a weighted majority game (see Definition 4.8), then, as the reader can easily verify, e is complete and E-- is acyclic. Remark 6.4. Intuitively, if S &--Tand S n T=@ then S consists ol a ‘simple majority’ within S u ‘E Indeed, e.g., if G= [q; WI,. . ., MY”] is a weighted majority game (see Definition 4.8), and S & K then Ci,=swi > Xi, T byi. We shall make use of the following simple properties of the relation Lemma 6.5. Let G = (N, W) be u simple gume urtd let S ad IfSLIT (w.r.t. G) cd ifT1T,, ihert Sr;;-T,.

T be coalitions.

Proof: Let B c N satisfy B n (S u tipI) = (4 and B u T1E W. Then =Tu (B--7+ W. Hence SW (B-T)E W, and therefore SUBE W. Lemma 6.6. Let G = (N, W) be u simple game cd !f S Ez T (wxt. G) und if’s, =)S, tlzerl S, & 11 Proqf Let BcN satisfy Bn(S,uT)=Q) and therefore B u S, E M! Thus S, e r

k.

Bu T

let S urd T be coulitions.

and BuTEW.

Then BuSEW,

Corollary 6.7. Let G = (N, W) be a simple game und let S urzd T bs coalitions. lf S &--7’(\:‘.r.t. G) und if T =)T, (S, 3 S) then S E- TI (S, E- T). Remark 6.8. Let G= (NOW) be a simple game. If S E---T (w.r.t. G), then T-S contains no veto players (see Definition 3.4). 7. Characterization of determining coalitions Let A be a set of alternatives and let N be a society. If R E L and if R has a first element x, then we denote t(R) =x. Let now K” E p, XE A and let S be a coalition. We denote T(S,x, R’)= (i:E SIt(R’)=x;.

(1)

B. Peleg, Codition

.formation in c’ommittec3

127

Note that since A may be infinite, T(S, X, RN) may be empty for every x E A. We now come to the definition of the central concept of this work. Dt$nit ion 7.1. Let F:LN-+A be a SCF and let G*(F)=(N,W)=G (see Defmition 3.2). A coahition S is determining (w.r.t. F) if it satisfies the following condition. If RNE LN, XE A and T= T(S,x, RN) [see (l)], and if T E-S- T (w.r.t. G; see Definition 6.l), then there exists Q% L? such that for all P”-%!!?, (a) F(QS,PNwS)=x, and (b) (QS,PNsS) is an e.p. of the game g(F, RN) (see Definition 2.5). The set of all determining coalitionc w.r.t. F is dent ae by D(F). I’ * id, a coalition S is determining w.r.t. a SCF F, if whenever a ‘simple majority’ of the members of S consider an alternative x to be their best choice (see Remark 6.4), then S can enforce _Yto be a Nash equilibrium payoff. First we state a sufficient condition for a coalition to be determining.

Lemma 7.2. Let F: LN--) A be a SCF Defirlitiort 3.2). [f SE W aild $S satisfies L&S

and

thw S is detemirring

S-[ii

!+ii)]*S--:il

and

let

G = G* (F ) = (N, W ) (sw

EbVi

12)

rz,.r.t. F.

ProoJ Let R’EL?, XEA and T= T(S, x, R”) [see (1 )]. Suppose that T &--S - T. By Corollary 6.7, T &--{ i) for all i E S - T. Therefore, again by Corollary 6.7, S-{il &--iii for all YES-7: Let Q%p satisfy t(Q’)=x for i6. We

claim that for every PX -’ E EV-’ the profile (Q”, P’-‘) is an e.p. of the game g(F, R”) (see Definition 2.5), such that F(Q’, P4’-‘) =_Y. Indeed, let P”-‘E LN-‘. Since SE W F(Q”, P”-‘)= x (see Definition 3.1). Denote Qy =(QS,P”-“). Let MV and Pi&. If i$S then, since SE:W, F(Qy-“‘,P;)=s (see Definition 3.1). If in S- T then, as we have already shown, S- [ii &- iii. Hence, by (2), S- { ij E W Therefore, again F(Q, N-ii) , pf)= x. Finally, if i E x then t(R’)=x [see (I)]. Hence .xR’F(QlIV_“‘,P’,). Thus, (Q”, PM-S) is, indeed, an e.p. of g(F, RN). Next we prove that for a SCF F D(F)c W, (see Definitions 7.1 and 3.7). Lemma 7.3. L,et F: LN-+A be a SCF. If’ S is u drturminitlg coalition (see Definition 7.1 ), then S is Ix-winning (see Dejhitim 3.6 ). Pwof:

rzl.r.t. F

Let SE D(F). Assume, on the contrary, that S qi Wa. Then there exists XEA such :‘ + S is not a-effective for x. Let RN E I! satisfy t(R’)=?c for all k N. Then, 1‘(S, x, RN)= S and S E-@ (see Definition 6.1). Now, since S is not cc-effective for X, for every Qs &!? there exists PNws EL!? such that

128

B. Peleg, Coalition formation

in committees

F(QS, PNeS) # x (see Definition 3.6). Hence, S is not dcterminiilg, which is the desit cd contradiction. Let F be the SCF defined in Example 3.12. As the reader can easily verify, S E D(F) iff iSI 2 14. Thus, a determining coalition may not be winning. Exumple 7.4.

We now characterize the set D(F) for a regular SCF F. Lemmd 7.5. Let F:L’-+ A be u regular SCF (see Definition 3.9) and let G*(F)= (hi, W). !f SE D(F) the/z SE W and S satkjks (2).

Lzt S E D(F). By Lemma 7.3, SE W,. Since F is regular, PV= WT. Hence SE W. We shall now prove (2). Assume. on the contrary, that there exists i E S such that S - \ i) &ii), and S - ii) 4. W We note first that S - \ il f- 1it implies that i is not a veto player (see Remark 6.8). Hence, N - { ii s W. Also, since F is regular, S - (i] 6 Wz. Therefore, there exists x E A such that S- (ii is not z-effective for x. Let RN E P satisfy t(Rk)=x for k # i, and JR’X for all JE A. Since N - [if E W, F(RN)=x, (see Definition 3.1). T(S,x,R”)=S-{ij [see (l)]. Let now QS& satisfy Furthermore F(Q’,P”-S)=x for all PN-S~LV-S. We shall show that there exists P*YSE~-S such that (QS, Py -‘) is not an e.p. of g(F, RN). Indeed, since S - f iJ. is not a-effective for x, there exist Pi EL and Py-” E I!?’ such that F(QS-“‘,P’,,Py-S)=y and y#x. By the choice of R’, #x. Hence, (Qs,PyeS) is not an e.p. of g(F,RN). Thus, S is not determining w.r.t. F, which is the desired contradiction. Prooj.

We now come to the main result of this work. Theorem 7.6. Let G = (N, W) be a proper simple game. If the SCF’s F1 and F, are regular representations of G (see Definitions 3.9 and 4.1) then D(F,) = D(F,) (see Definition 7.1). Proof

Lemmata 7.5 and 7.2.

Corollary 7.7. Let G = (IV, W) be a strong simple game (see Definition 3.18). If the SCF’s F, cad F, are representutions gf G (see De$nition 4.1), then D(F,)=D(F,). Proof.

The proof follows from Theorems 3.19 and 7.6.

Theorem 7.6 leads to the following definition.

B. Pcleg.

Codition

fbmation

itt rommittce.s

129

Defin it ioil

7.8 Let G = (N, W) be a proper simple game and let F be a regular representation of G. A coalition S is determirzi)zg in G if SO(F). The set of determi%rtg coalitions in G is denoted by D(G). Remark

7.9. Clearly, a coalition S is detl:rmillirig in a proper simple game G = (JV,W), iff SE W and satisfies (2).

Corollury NED(G).

7.10.

If

G = (N, W)

is cl non-null

proper

simple

game

thea

Proof. Since G is non-null N E W Also, if in N and N - iii E- [i), then N - (i) E W (see Definition 6.1). The determining

coalitions in symmetric games are easily identified.

Remark 7.11. Let G = (n, k) be a symmetric simple game (see Remark 3.11), such that n/2ckcn. If n=3 then S&(G) iff /S/22. If 0-3. then %9(G) iff Is I zkt1. Remark

U.2.

Let G= (IV,W) be a proper simple game. A coalition S is if S 4 D(G) but there exists TED(G) such that SI> ?I An example of an overdetermining coalition is the coalition that formed in the council of Jerusalem after the 1973 elections (see town no. 1 in table I). The council is [16; 14, 7, 4, 4, 1, 1-J and the coalition is S = i 1,2,3.4). Since ‘2 3 41 E { 1j and {2,3,4) is losing, S is not determining. However, i 1,2,3) 19’ is determining. owrdetermining

We now formulate our hypothesis about formation of coalitions in simple games. Hypothesis 7.13. Let G = (N, W) be II proper simple gume, u12d Ict (F,), t E IT: be a family of issues which ure relecant_for G (see D@nition S.1 ), swh that dl their redizatimzs (see Defidiort 5.2) are considered by the mtwbers of N us rsott-cooperative games. 11 S is N conlitiou which forms to resolve the isstws (F, ), f E ?: the12 SE D(G) (see Defildtiou 7.8). Remark 7.14. Let G= (N, W) be a committee and let g(F, RN) be a realization of an issue which is relevant for G. g(F, RI”) is to be considered by the players as a non-cooperative game if binding agreements are either (a) legally unpermissible (as is usually the case in political ga&es; set section 5), or (b) technically impossible (e.g., if the outcome of g(F, RN) is determined by a secret ballot; I am indebted to M. Maschler for this last example).

f?. Peleg, Coalition formation

130

in committees

In the next section we shall present and discuss some real-life data which seem to support our hypothesis. In section 9 we comment on the normative aspects of our hypothesis (see Remarks 9.2 and 9.3). 8. Coalitions formed in towIncouncils in Israel In the following table we list the coalitions that formed in 26 towns in Israel after the last three municipal elections. We use the following notation in table I. Let G = (I\/, U’) be a proper simple game. Then we denote by W”’ Table 1 Coalitions -hl0.

formed in 26 towns in Israel.

-

Town

Year

Council

1

Jerusalem

1965 1969 1973

1116; 16,6,4, 3, 1, l] [16; 14, 7, 4, 4, 1, l]

W-D-Wrn D-W” W-D-W”

2

Eitat

1965 1969 1973

[6; 7, 2, 1, 11 f6; 6, 2, 1, 1, 11 [6; 6,4, 13

D-W” D-W” D-W”

3

Ashdod

1963 1969 1973

[6; 3. 3, 2, 1, 1, l] [S; 6.5,2,2-j [S; 9., 3, 2, 1-j

W-D-W” W”-D D-W”

4

Beer Sheba

1963 1969 1973

[8; 5, 4, 2, 2, 1, 1-J f9; 9. 3, 3, 1, I] [9; 9. 4, 2, 1, l]

W-D-W” D-W” D-W”

5

1965 1969 1973

18; 5,3,3,2,2-j

Bnei Brak

[8; 5, 3, 3, 2, 21

W”-D D-W” D-W”

6

Bat Yam

1965 1969 1973

[8; 7, 4, 2, 1, 1) 18; 8,5,2] [ll; 10,9, 2)

D-W” D-W” DnW”

7

Givatayim

1965 1969 1973

[8; 8, 3, 1, 1, 1, 1J [8; 10, 3, 1, l] [8; 3,4 1, 1-J

D-W” D-W” D-W”

8

Herzliya

1965 1969 1973

[9; 7, 3, 2, 2, 1, 1, 11 19; 9, 5, 2, 11 19; 8,6, 2, 1-J

D-W” D-W” D-W”

9

Hadera

1965 1969 1973

[S; 6, 4, 2, 1, 1, q I% 736, 21 C8; 776, 21

W-D-W”’ DnW” DnW”

10

Holon

1965 1969 1973

[9; 8, 4, 2, 1, 1%11 [9; 10, 4, 2, l] [ll; 12.6, 2, 1-J

D-W” D-W” DnW”

11

Haifa

1965 1969 1973

[ll; 10, 5, .I, 2, 1, 1-J [13; 14, 6, 2, 2, l] 113; 15, 6, 2, 1, 1-J

D-W” D-W” D-W”

[8; 6,4,3,2]

Coalition

Type

B. Meg, Cocrfirion format iorr in commirltes

131

Table 1 (cont.) No.

Town

Year

Council

Coalition

Type

13

Tiberias

1965 1969 1973

[7; 5, 3, 2, 2, 11 [7; 5, 3, 2, 2, 11 [S: 5, 3, 3, 2, 1, 11

(2, 3,4} 11, 3, 4) { 1, 2, 3, 4, 5)

DnWrn D-W”

Kfar Sava

[7; 6,3,2, 1, 1] [7; 8, 3.21 [S; 10, 3,2]

{1,41

13

1965 1969 1973

W”-D D-W” D-W”

14

Netauya

1965 1969 1973

18; 5.4.4.21 18; 5, 4, 2. 2, 1, 1] :s; 5, 5, 4, 11

$3) (1, 2, 3, 5, 61

1965 1969 1973

18; 4, 3, 2, 2, 2, 1, 1] IS; 6, 3, 2, 1, 1, 1, I] :S; 7, 3, 2, 2, 1-J

{ 1, 2, 3, 4, 6: : 1. 2, 3, 4, 5)*

D-W”

(1, 3, 5;

D-W’”

15

Acre

{l, 2: N

{I, 3, 4)

D-W”

DnW”’ D--W” D-Wm

W-D-W”

16

Lod

1965 1969 1973

:7; 5, 2, 2, 2, 2] 18; 7, 3, 2, 1, 1, q 18; 7, 4, 1, 1, 1, l]

{l, 2, 3; (I. 2, 4; (1, 3, 4, 5;

D-W” D-W” D-W”

17

Nahariya

1965 1969 1973

[7; 6. 3, 2, 1, I] c7; 7, 3, 2, 11 CS; 7, 5, 2, l]

{l, 3, 4) N (1,4)

D-W” D-W”’ W”-D

18

Afula

1965 1969 1973

[6; 4, 2, 2, 2, l] C6; 5, 2, 2, 1, 11 16; 6, 3, 21

(1, 2, 5; :1. 2) N

W-D-W” W”-D D-W”

19

Petah Tikva

1965 1969 1973

[9; 6, 3, 2, 2, 1, 1, 1, 11 [9; 7.4, 2, 2, 1, 1] [lo; 7, 6, 3, 1, 1, 11

11. 3, 4, 5, 61 (1, 3, 5, 6) (1, 3, 4, 5;

D-W” D _ W”’

20

Safed

1965 1969 1973

[6; 3, 3, 2, 1, 1, 11 C6; 5, 3, 2, l] C6; 4, 2, 2, 2, 11

11, 3, 4) N {l, 2, 5;

W”-D D-W” ‘W_ D __W”

Ashkelon

1965 1969 1973

[S; 6,5,2,2] [S; 7, 5. 2, l] [S; 6, 4, 4, 11

(1, 3, 4;

21

{l, 3:

11, 2, 4;

D-W” W”-D D-W”

W-D-W”’

22

Kiryat Gat

1965 1969 1973

[6; 5, 3, 2, 11 C6; 6, 3, 1, f] L-7; 6, 5, 1, 11

{l, 31 (11 (1, 3)

W”-D 11n W”’ Wrn-D

23

Rishon Lezion

1965 1969 1973

[S; 5, 3, 2, 2, 2, l] IS; 8, 3, 2, 1, 11

[S; 11941

(1, 3, 4, 5, 6) {l, 3, 4) ill

D- W”’ 12-W” Dn W”

24

Rehovot

1965 1969 1973

&-8;6, 3, 2, 2, 1, l] IX 6, 5, 2, 1, 11 CR 10,3, 1, 11

f 1, 3, 5; 12, 3, 4; 111

W-DW” iY”-D Dn W”

Ramat Gan

1965 1969 1973

[ll; 9, 6, 2, 1, 1, 1. l] [12; 10, 6, 2, 2, 2, l] [12; 10, 5, 3, 2, 2, I]

(1, 3, 4, 5;

25

( 1. 3, 4, 5, 6;

D-W” D-W” D-W”’

1965 1969 1973

[:6; 12, 12, 3, 2, 1, l] [16; 13, 12, 3, 1, 1, 1] [16; 14, 11, 2, 2, 1, l]

26

Tel Aviv

(1, 3. 4, 5, 6) j 1, 3, 4, 5, 6; 11, 3, 4, 5, 6) ‘IL 3, 4, 51

D-W” D-W” D-W”

the set of minimul winning coalitions in G, and by D the set of determining coalitions in G (see Definition 7.8). The data of table 1 were compiled by M. Justman. The classification of the coalitions into types was done by the author. Table 2 Sunmary

--_-

of the results of table 1.

TYF

D-M;‘”

DnW”

W”-D

W-D-W”

Total

Total Percentage

49 62.82

9 11.54

IO 12.82

10 12.82

78 100

Remark 8.3. It is worthwhite to mention that among the ten coalitions in W -D - Wm only three were overdetermining (see Remark 7.12). These three coalitions are Jerusalem 1973, Ashdod 1963 and Acre 1969. They are marked by an asterisk. Thus, out of the 20 coalitions in W-D, 17 are too small to be determining (see the next remark). Remark 8.4. Our theory of coalition formation is purely combinatorial: we disregard the fact that parties represent ideologies. Indeed, when parties strive to achieve certain goals in accordance with thei; declarations during the electoral campaign, the following phenomena may occur: (a) The set of the issues that will actually (N, W). may be quite limited.

confront

the elected council

(b) If g(F,R”) is a realization of an issue which is relevant to (N, W) (see Definition 5.2), then some members of N may have already committed themselves to play certain strategies in g(F,RN) (i.e., to vote as they promised to do before the elections). Now (a) does not affect our theory, while (b) does. However, if realizations of issues are resolved by a secret ballot (see Remark 7.14), or are not completely known at the time of the electoral campaign (which is quite often the case), then (b) can also be ignored. Thus, we have to look for a different explanation for the 20 coalitions in 1%’ -D (sec., Remark 8.3). It seems to us that Axeh-od’s theory of coanecred coalitions [see .jxelrod (1970, p. t69)] p rovides a suitable explanation. As Axelrod assumes, and we agree with him, coalitions should be ‘connected’ ideologically. Now, it is quite obvious that, for a given committee, the set of determining coalitions may be different from‘the set of ‘connected’ coalitions. Hence, the desire to establish a determining coalition may be in conflict with the desire to have a ‘connected’ one. Thus, the process of erecting a coalition may end up with a ‘connected’ coalition which is not determining. Unfortunately, we do not have enough information to determine the

B. Peleg, Coalitiotl formutiorl

in commitcues

133

‘connected’ coalitions for the councils in table 1. We would like to add that there is always a determining coalition which is connected, namely, the grand coalition (see Corollary 7.10). However, usually the grand coalition does not form since it is too big (see Remark 8.6). Remark

8.:

de

would like to emphasize that we do root accept Axelrod’s assumption that coalitions should be minimal (w.r.t. the set of ‘connected’ winning sets). We would conjecture that minimal members of the set of ‘connected’ unn determining coalitions are most likely to form; but we do not have a theory to support such a conjecture. However, we would like to present an example which violates Axelrod’s hypothesis of minimality. The example we have is the present Israeli cabinet. The Israeli parliament (i.e.. the Knesset) after the 1977 elections *ras the following representation [61; 5, 2, 1, 32, 1, 1, 15, 12, 4, 46, l] in a (somewhat arbitrary) ‘left to right’ ordering. The coalition formed in two stages. First the minimal winning connected coalition {S,9,101 (which is not determining), formed, leaving 4 offices in the cabinet vacant in order to allow for a quick and ‘smooth adjoining of party 7. And, indeed, after several weeks, party 7 joined the cabinet to form a determining connected coalition. Renztrrk 8.6. The grand coalition formed only in seven cases out of those listed in table 1. (Those are the coalitions in table 1 which are denoted by N.) This fact seems to indicate that small determining coalitions are more likely to form than large ones. However, at present, we do not have a satisfactory theory of formation of mininal determining coalitions (see also Remark 7.12).

9. Concluding remarks Remark

9.2.

would be very instructive and desirable if Hypothesis could be tested against real-life data taken from additional countries different political systems. Unfortunately, such an investigation is beyond capacity of the author and it, therefore, remains as a challenge to interested reader. Remark

It

7.13 and the the

9.2. The theory of Determining coalitions might be used to speed up actual formation of coalitions in committees. Indeed, if G is a committee, then Remark 7.9 can be used to discover the ‘stable coalitions’, i.e., the members of D(G) (see Definition 7.3). Ther the players, during the process of forming a coalition (which may depend on factors such as ideologies, previous commitments, etc.), are able to restrict their considerations to members of D(G).

B. Peleg, Coalition. format ion in commit tees

134

Finally, we would like to emphasize the following, quite obvious, fact. Let G be a committee. Then a necessary condition for a coalition S to be ‘stable’ is that SE D(G). However, the set D(G) is usually too big to enable us to make a good prediction of the coalition that will actually form. Thus, a further investigation is required in order to narrow down the set of ‘stable’ coalitions (see Remark 8.4). However, the theory of determining coalitions can be used, under the assumption of Hypothesis 7.13, to reject several theories which predict formation of minimal winning coalitions [see Axelrod (1970, pp. 172-173)]. Remd

9.3.

References Aumann, R.J., 1961, The core of a cooperative game without side payments, Transactions of the American Mathematical Society 98. 539. 552. Axelrod, R., 1970, Conflict of interest (Markham, Chicago, IL). Dutta, P. and PK. Pattaizaik, 1978, On nicely consistent voting systems, Econometrica 46, 16.1 170. Lapidot, E., 1968, Weighted majority games and symmetry groups of games, MSc. thesis, in Hebrew (Technion, Haifa). Maschler, M. and 5. Peleg, 1966, A characterization, existence proof and dimension bounds for the kernel of a game, Pacific Journal of Mathematics lS, 289. 328. Pelcg, B., 1978a, Consistent voting systems, Econometrica 46, 153.-161, Peleg, B., 19 18b, Representations of simple games by social choice functions. l~~tcrnational Journal of Game. Theory 7. 81-94. Rapoport. A.. 1970, N-person game theory (University of Michigan Press, Ann Arbor, MI). Shapley, L. S., 1962, Simple games: An outline of the descriptive theory, Bchaviural Scicncc 7, 59 66. Shenoy, P., 1977. On game theory and coalition formation, TR no. 342 (School of Operations Research and Industrial Engineering, College of Engineering, Cornell UrliIcrsity, Ithaca, NY). Young, H.P., 1977, Extending Condorcet‘s rule, Journal of, Economic Theory 16. 333 353. Yormg, H. P. and A. Levenglick, 1977, A consistent extension of Condorcet’s t‘lection prmclplc, forthcoming.