A non-cooperative bargaining game with risk averse players and an uncertain finite horizon

Economics Letters 20 (1986) 9-13 North-Holland

A NON-COOPERATIVE AND AN UNCERTAIN

Joseph

E. HARRINGTON,

The Johns Hopkins Received

Unirwsity,

BARGAINING GAME WITH FINITE HORIZON

RISK AVERSE PLAYERS

Jr. * Balnmore,

MD 21218, USA

12 June 1985

For an alternating offer bargaining game with an uncertain horizon. the equilibrium payoff to a player averse is he relative to his opponent. As the horizon becomes infinite in length. the equilibrium partition bargaining solution.

is greater, converges

the less risk to the Nash

1. Introduction

The main attraction to the non-cooperative game approach to the bargaining problem is that it forces one to be concerned with the actual bargaining procedures available to the parties. Rubinstein (1982) has examined the non-cooperative equilibria for an infinite horizon bargaining game in which the two players alternate making offers. His main interest is in determining the influence of time preferences and bargaining costs on the equilibrium partition. Rubinstein (1982) finds that the more patient is a player (that is, the lower is his discount rate), the greater is his share. An issue which has yet to be properly examined in the non-cooperative game setting is the effect of risk preferences on the equilibrium partition. Going back to Zeuthen (1930), risk preferences have long been thought to be a major determinant of the outcome of bargaining situations. The axiomatic approach has generally shown that a player’s share is less, the more risk averse he is relative to his opponent. In order to investigate this issue in the non-cooperative game framework, we introduce uncertainty into the alternating offer model by making the final period of bargaining a random variable whose distribution is common knowledge. Since the purpose of the non-cooperative game framework is to make the bargaining procedure explicit. we feel an examination of risk requires that there be explicit uncertainty in the model. This approach is to be contrasted with that of Roth (1985) who investigates the effect of risk preferences in a non-stochastic bargaining game.

2. The model

Players 1 and 2 have to decide how to divide up $1. The bargaining procedure they will use is the alternating offer model described in Rubinstein (1982). In any period, a player is designated to make an offer X. This offer states that the other player gets X and the individual making the offer gets * I would like to thank Elaine Bennett for suggesting this research topic to me. The helpful comments of Charley Soo Hong and Steve Zucker are very much appreciated. Any remaining errors are, of course. my own.

0165-1765/86/$3.50

0 1986, Elsevier Science Publishers

B.V. (North-Holland)

Bates, Chew

1 - X. Thus all offers will be Pareto-efficient. The other player has the choice of either accepting or rejecting the offer. If he accepts the offer, the game ends and players 1 and 2 divide up the dollar as prescribed by the accepted offer. If he rejects the offer. the game then moves to the next period in which case the player who rejected the previous offer is now the one who makes the offer. The game continues until an offer is accepted or the game terminates, whichever occurs first. If the game terminates before an offer is accepted, the players then get their disagreement outcomes of u’, and bcz. We will assume w, = 0 and VV?= 0. The game will last for a maximum of T periods where T is even. Player 1 is assigned the first offer while player 2 will make the offer in period T. Generally, player 1 (2) makes the offer in all odd (even) periods. While the game can last for as long as T periods, players are uncertain as to the actual terminal date, i. We will assume the terminal date follows a uniform distribution over { 1, 2. ., T }. V’~E (1, 2, . . . . T}.

prob(i=t)=(l/T) This distribution generated by (1):

is common

prob(i=tIt^>t)

=(l/(T-I+l))

fortE

=o

for tE {1,2,

Each player is assumed

U,(y) =I‘-,

knowledge

(1)

to the players.

U,( ,v) =y”

where

0 <

function Y, ,

following

posterior

distribution

is

. . . . T},

{F, i+l.

to have a utility

The

(2)

. . . . i-l}. which exhibits

r, 5 1.

constant

relative

risk aversion. (3)

In order to concentrate on the influence of risk preferences, players are assumed to maximize undiscounted expected utility. We will let S,’ denote the strategy of player i in period t. Since player 1 makes the offer in all odd periods, his strategy set takes the following form:

s;: [OJ]‘~

+ [O,l]

: [O,l]‘+

{accept,

fortg reject}

{1,3.

. . . . T-l}.

for TV (2, 4, . . . . T}.

(4)

In odd periods, his strategy is a mapping from the set of past offers to the set of offers he can make in period t. When t is even, his strategy maps from the set of past offers including players 2’s current offer into the possible choices of ‘accept’ or ‘reject’. Similarly, player 2 faces the following strategy set: Si: [O,l]‘+

{accept,

: [O,l]‘_’

+ [O,l]

3. Subgame

reject}

for tE (1, 3, . . . . T-l}, (5)

perfect

fortE

{2,4.

. . . . T}.

equilibrium

The solution concept to be used is Selten’s subgame perfect equilibrium. Within each period there are two stages. In stage 1, one player makes an offer while in stage 2 the other player either accepts or rejects it. We will let X,’ denote the outcome to player i for an offer made in period t and 2,’ denote the subgame perfect equilibrium value of X,‘.

J. E. Harrington,

Theorem I. S; = [(T=acceptiff Si=acceptiff = [(T-

Jr. / Non-cooperatice

fortE

(1.3.

. . . . T-l},

X;>[(T-t)/(T-t+l)]““(i’;*‘)

fortg

{2,4.

. . . . T}.

Xj>

fortE

(1, 3, . . . . T-l},

fortE

{2,4.

t)/(T-

t)/(T-

t + l)]““(

ii+‘)

[(T-t)/(T-t+l)]““(ii+‘) t + l)]“y

ri:+‘)

t)/(T-

t + 1,yy

,:+y

= 0

= [(T-

(6)

(7)

. . . . T},

and

,?“=l-[(T-t)/(T-t+l)]““(l-k:;‘+‘)

i;

11

game

A unique subgame perfect equilibrium exists which is defined recursioeiy bq

where TE (2, 4. . ..}

= [(T-

bargaining

fortE(l.3,

. . ..T-l}.

fortg

. . . . T-2},

{2,4,

(8)

for t = T, t)/(T-

t + l)]““(

forte

2;“)

{1,3,

=l-[(T-t)/(T-t+l)]““(l-k~t’)

fortE(2.4,

=1

for t = T.

. . . . T-l}, . . . . T-2},

(9)

Proof. S,’ calls for player 1 to accept iff X,’ 2 0 which is the unique best response as by rejecting offer he gets his disagreement payoff of zero. Given S’r, player 2’s best reply is set Xl’ = 0. Thus, and Sl form a subgame perfect equilibrium for the two-stage game in period T. By the method backward induction one need only show that Si and Si are subgame perfect given SI+’ and S;+’ subgame perfect. It is straightforward to show that this condition holds once one recognizes that equilibrium path emanating from period t + 1 has player i (i = 1 if t + 1 is odd and i = 2 if t + Q.E.D. even) making an offer of i,‘+’ (j # i) and player j accepting it.

According to these strategies, the equilibrium path has player 1 making and player 2 accepting it. Solving recursively for the equilibrium partition, i;

= ((T-

l)/T)“r’[l

-((T-

2),‘(T-

. . (1 - (2/3)““)(1

l))“r’(l

-((T-

-(I/2)““).

3)/(T-

. . ,].

an offer of 2;

the ST of are the 1 is

in period

1

2))“‘l

(10)

Taking the first derivative of 2, with respect to r, and r, (j # i), it is easily derived that .,?,I is increasing in r, and decreasing in y/ (i = 1. 2). Consistent with the results generated by the axiomatic approach, a player’s share is greater, the less risk averse he is and the more risk averse is his adversary. Intuitively, the less risk averse player i is, the more willing he is to reject player j’s offer so as to chance getting his turn to make an offer. Player j must then offer player i more in order to get him to accept. The more risk averse player j is, the more attractive it is for player i to reject player j’s offer as he can expect to extract a higher share when he gets to make an offer. Once again, player j must then give player i more if he is to have player i accept his offer.

.2 -i IAl .’ Ix, 0 o”‘t”l ! 2 3

4

5

6

7

8

9

I IO

I I ‘I I I 12 13 I4

3 I5

‘1 I6 I7

I I8

‘I19 201

Fig. 1

In the finite horizon model, a player also does better if he gets to make the initial back to eqs. (8) and (9). if player i makes the offer in period t + 1 then

ri,‘= [(T-t)/(T-t+l)]““(~~+‘)
(11)

Thus, player i earns a higher payoff in period t + 1 when he when player j (j # i) does. This asymmetric advantage equilibrium outcomes are shown for different time horizons. that player 1 (2) makes the initial offer when T is even (odd). averse than player 1: Y, = 0.75. r, = 0.50.

4. Convergence

makes the initial offer than in period f is made evident in fig. 1 where the Player 2 is assigned the final offer so For this example. player 2 is more risk

to the Nash bargaining solution

Theorem 2 below shows that as the horizon becomes infinite, the first offer disappears and the subgame perfect equilibrium bargaining solution. Theorem 2.

offer. Referring

lim , +r

X’ = (r,/(y,

the asymmetry imposed by having partition converges to the Nash

+ r:)).

In retrospect, it is not too surprising that the non-cooperative bargaining equilibrium converges to the Nash bargaining solution. The non-cooperative model can be interpreted as an explicit game representation of Zeuthen’s (1930) theory of bargaining. The probability of the game terminating is just what Zeuthen referred to as the probability of open conflict. The particular distribution we have assumed also implies that the probability of open conflict is an increasing function of the number of offers rejected, prob(conflict

1z offers rejected)

= (l/(

T+ 1- z

))

Thus, the more offers which are rejected, the more likely bargaining are forced to go to their disagreement outcomes.

(12) is to break down and the players

J. E. Hurrington, Jr. / Non-cooprruriw

hurgumrng gmne

13

For the example in fig. 1, the outcomes converge rather quickly to the Nash bargaining solution of X, = 0.6 and X, = 0.4. When the horizon is T= 20, the equilibrium partition is k,’ = 0.619 and Xi = 0.381 which is less than 5% from their steady state levels. Thus. the Nash bargaining solution is a good approximation for the equilibrium partition when the horizon is sufficiently long.

References Harsanyi. John C., 1956, Approaches to the bargaining problem before and after the theory of games: A critical discussion of Zeuthen’s, Hicks’. and Nash’s theories, Econometrica 24, 144-I 57. Harsanyi. John C.. 1977, Rational behavior and bargaining equilibrium in games and social situations (Cambridge University Press, Cambridge). Nash, John F.. 1950, The bargaining problem. Econometrica 18, 155-162. Roth, Alvin E., 1985. A note on risk aversion in a perfect equilibrium model of bargaining. Econometrica 53. 207-211. Rubinstein, Ariel, 1982. Perfect equilibrium in a bargaining model. Econometrica 50. 97-110. Selten, Reinhard, 1975, Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory 4, 25-55. Zeuthen,

Frederick.

1930. Problems

of monopoly

and economic

warfare

(Routledge.

London).