an arbitration game and the egalitarian bargaining solution€¦ · game theory has provided two...

Soc Choice Welfare (1995) 12:29-41

© Springer-Verlag 1995

An arbitration game and the egalitarian bargaining solution*

Walter Bossert 1, Guofu Tan 2

Department of Economics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3Gt 2 Department of Economics, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z1

Received: 7 May 1993/Accepted: 27 June 1994

Abstract. We analyze a simple arbitration procedure which is a multi-stage variant of Nash's demand game. In the absence of discounting, all Nash equilibria of the game yield the egalitarian solution in the first stage. The crucial feature of our arbitration procedure is that, in the case of incompatible demands, the game is allowed to continue and the player who demands the higher gain over the disagreement point is penalized by restricting her or his feasible demands in the following stage. Suitable modifications of the arbitration game yield the lexicographic extension of the egalitarian solution, resp. the proportional solutions. Journal of Economic Literature Classification Numbers: C72, C78.

1. Introduction

Game theory has provided two approaches to bargaining problems, namely, axiomatic and strategic models of bargaining. In the axiomatic approach origin- ating in Nash (1950), solution concepts are derived from properties that are considered desirable. By avoiding details of bargaining processes, this approach provides great generality. 1 In the strategic approach, bargaining problems are formulated explicitly as extensive form games (see Rubinstein, 1982). The outcomes of bargaining situations are the subgame-perfect equilibria (or other types of noncooperative equilibria) of the extensive games. Recently, the connections

* Financial support through grants from the Social Sciences and Humanities Research Council of Canada and the University of Waterloo is gratefully acknowledged. Earlier versions of the paper were presented at McMaster University and Brock University. The authors thank Nejat Anbarci, Michele Piccione, Venkatraman Sadanand, an editor, and the referees for their useful comments. 1 See Thomson (1994) for a comprehensive survey.

30 W. Bossert, G. Tan

between these two approaches have also attracted much attention in the literature. 2 Several bargaining solutions have been shown to be consistent with the equilibrium outcomes of specific extensive form games.

The relationship between axiomatic bargaining solutions and equilibria of strategic models has first been studied by Nash (1953) in his pioneering paper. He considers a strategic model of bargaining that supports his axiomatic solution, namely, the Nash (1950) bargaining solution. His game consists of a single stage in which the two players simultaneously announce "demands" in terms of utilities. If these demands are compatible given the set of feasible utility vectors, then each player receives the amount he or she demanded; otherwise the disagreement event occurs. This game has many Nash equilibria. In order to refine the set of equilibria, Nash further considers a perturbed demand game and shows that if the disagreement outcome is excluded, the only equilibrium that is robust with respect to certain perturbations in the structure of the game yields the Nash bargaining solution. The refinement used by Nash bears some similarity to Selten's (1975) "trembling hand" perfection.

Nash's demand game has been extended and modified in other contributions. For example, Binmore (1987) has proposed a bargaining game of alternating offers and shown that the Nash bargaining solution is the limit of the unique subgame-perfect equilibrium of his game as the probability of the negotiation process breaking down approaches zero. Carlsson (1991) has studied a variation of the perturbed demand game by assuming that the players make errors in choosing their actions in the bargaining process. He shows that the equilibrium outcome converges to the Nash solution when errors go to zero. Osborne and Rubinstein (1990, Ch. 4) have provided more detailed discussions on these extensions. The purpose of all these studies is to examine the strategic foundations of cooperative bargaining solutions. In most cases, justifications of the Nash solution are obtained.

Another interesting cooperative bargaining solution is the egalitarian solution developed by Kalai (1977) and Myerson (1977). Without using Nash's axiom of independence of scale of utility they have provided alternative axioms under which interpersonal comparisons of utility must be possible. These axioms lead to the proportional solution and, in a symmetric bargaining situation, the egalitarian solution. One of the main axioms in this approach is a monotonicity axiom with respect to expansions of the feasible set. An alternative is a condition that involves a step-by-step negotiation process. This axiom imposes an invariance condition under decomposition of the bargaining process into several stages. As Kalai and Myerson pointed out, step-by-step negotiation has at least two advantages. First, it makes it easier to implement a solution since the negotiation can be broken up into several stages. Second, the players do not have incentives to change the order of the negotiations. This process is likely to be observed in actual negotiations.

In this paper, we follow the analysis by Kalai (1977) and Myerson (1977). Instead of assuming their axioms, we directly assume interpersonal comparability of utility and analyze a sequential arbitration game. We consider an arbitrator who is responsible for helping two players to resolve their dispute. The arbitrator may

2 See, for example, Moulin (1984), Binmore et al. (1986), Binmore (1987), Osborne and Rubinstein (1990), and Peters et al. (1991) for more detailed discussions. An excellent interpretation of these two theoretical models of bargaining is provided by Rubinstein (1992).

An arbitration game 31

want to offer a fair recommendation. One simple procedure he or she could propose is to let th e players submit their demands. If the demands are compatible then each player receives what he or she demanded. If the demands are not compatible, however, the arbitrator has reason to believe that the player (e.g., player 1) who asked to enjoy a higher gain over the disagreement point than the other player (player 2) asked for too much. Therefore, the arbitrator would propose to penalize player 1 either by imposing an outcome which gives player 1 the disagreement utility and player 2 what he or she demanded or by using this outcome as a new disagreement point for another round of negotiation where the players revise their demands. We consider the latter one, but both procedures yield the same equilibrium outcome as we will see in Sect. 4. This arbitration game shares features similar to the step-by-step negotiation described above. Player 2 can guarantee himself or herself a relatively higher utility in the second round by committing to a "moderate" strategy in the first stage. In our arbitration game, however, the players might not recognize the axioms proposed by Kalai and Myerson, but agree that the arbitration procedure is simple and fair. They might also anticipate that the game leads to the egalitarian solution which provides them with higher payoffs than the disagreement utility. Therefore, they might agree to play the game. In Sect. 4, we show that the Nash equilibrium outcome in this arbitration game is consistent with the egalitarian solution. This result provides additional support for the egalitarian solution to general bargaining problems.

The arbitration game we consider is a multi-stage variant of Nash's demand game. The first stage is the same as in Nash's demand game, but the game continues in the case of incompatible demands. The player who demands the higher gain over the disagreement point is penalized in the second stage. In particular, the disagreement point is moved by guaranteeing the player with the lower demand her or his demanded utility, and the resulting new ideal point provides - as before - the upper bound for the players' demands. This restriction seems to be reasonable and plausible in order to penalize the player who asked for "too much" in the case of incompatible demands. The game continues until either the game stops after a finite number of stages or there is "perpetual incompatibility," in which case the disagreement outcome is implemented. We show that in the absence of discounting all Nash equilibria of the arbitration game lead to the egalitarian bargaining solution in the first stage. We do not need any refinement of the equilibria of the strategic game.

It should be noted that the game as described above is formulated in terms of the utilities of the players. In this respect, we have a framework analogous to the one used by many earlier authors such as, for example, Nash (1953), van Damme (1986, 1987), Peters et al. (1991), Anbarci (1993), among others (see also Thomson 1994, Ch. 8 for a discussion). In particular, van Damme (1987) defines a one-stage simultaneous-demand game that provides a justification of the Kalai- Smorodinsky (1975) solution. This game can also be used to support the egalitarian solution if the ideal point is not normalized. The end of van Damme's game is forced after the initial demands are formulated. In contrast, our arbitration procedure is a multi-stage game that takes into account the possibility of incompatible demands. Howard (1992) analyzes a game where the game designer need not know the players' utilities, but the players have to know each other's utility functions.

The main conclusion to be drawn from our paper is not the mere fact that the egalitarian bargaining solution has some strategic justification, but rather, that it is consistent with the noncooperative outcome of a simple arbitration procedure. This result is not trivial. As we will see in the discussion in Sect. 4, if discounting is


introduced, then there exist multiple equilibria of the arbitration game which may not yield the egalitarian solution, If the criterion for the penality rule is not formulated in terms of the gains over the redefined disagreement points, the egalitarian solution may not be an equilibrium outcome of the arbitration game. Of course, there may exist other simple arbitration procedures which lead to the egalitarian solution. Also, our game does not provide an implementation of the egalitarian solution, since the arbitrator has to know the players' utilities. This feature is shared by many earlier contributions (see the above mentioned exam- pies), including Nash's (1953) original demand game. The issue of implementing the egalitarian solution seems to be problematic, due to the need for interpersonal comparisons of utility.

We first define two-person bargaining problems in the next section. A multi- stage arbitration game is introduced in Sect. 3. Our main result shows that if we restrict attention to strictly comprehensive bargaining problems, the unique Nash equilibrium outcome of our arbitration game is the egalitarian solution outcome. This result is stated and proven in Sect. 4. Sect. 5 contains a discussion of some extensions. In particular, if we drop the strict comprehensiveness assumption and consider general comprehensive problems, we show that a straightforward modifi- cation of our arbitration game leads to the Leximin solution (a lexicographic extension of the egalitarian solution). Similarly, the proportional (or weighted egalitarian) solutions can be obtained. We also illustrate the problems that occur concerning an extension of our results to games involving more than two players. Limitations of our model and its relationships to the strategic implementation of cooperative bargaining solutions are discussed in Sect. 6.

2. Bargaining problems

A two-person bargaining problem is a pair (S, d) where S c ~x 2 is the feasible set of utility vectors and d e IR 2 is the disagreement point. ~

The problem (S, d) is comprehensive (strictly comprehensive) if and only if, for all x, y ~ ]R 2,

x ~ S a n d x > y > d =~ y e S ( a n d 3 z ~ S s u c h t h a t z > > y ) .

is the set of bargaining problems (S, d) such that

(i) S is compact and convex; (ii) (S, d) is comprehensive;

(iii) 3x ~ S with x ~> d.

The set of strictly comprehensive bargaining problems is denoted by ~, that is, ~ is the subset of ~ that is obtained if comprehensiveness is strengthened to strict comprehensiveness in condition (ii) above.

The ideal point of a bargaining problem (S, d) ~ ~, a(S, d), is defined by

a i ( S , d ) = m a x { x i l x ~ S , x > d } V i = 1 , 2 .

A bargaining solution is a function F : N ~ IR 2 such that F(S ,d)eS for all (S, d) e ~ .

3 Notation: IR (IR + ÷) denotes the set of all (all positive) real numbers, N is the set of positive integers, and our notation for vector inequalities is > , > , ~>.


The egalitarian solution E (see Kalai 1977) is defined by letting, for all (S, d) ~ ~, E(S, d) be the unique point x e S such that

xl - dl = x2 - d2 and ~]y ~ S such that y >> x.

A generalization of the egalitarian solution is the class of proportional (or weighted egalitarian) solution E ~ (with a ~ IR+ +), also due to Kalai (1977). For (S, d) ~ ~ , E~(S, d) is defined as the unique point x e S such that

xl -- dl = a ( x 2 - d2) and ~]y ~ S such that y ~> x.

Clearly, a -- 1 leads to the egalitarian solution. For bargaining problems (S, d) that are comprehensive but not necessarily

strictly comprehensive, the egalitarian solution can yield outcomes that are not strongly Pareto optimal. To avoid this problem, a lexicographic extension of E, the Leximin solution L (see, for example, Imai 1983) can be used. To define the Leximin solution, let x' ~ IR 2 be the rank-ordered permutation ofx ~ IR 2 such that x~ < x~. The lexicographic maximin ordering ~ t o n ]R 2 is defined by

x> ' zy "*~ [ x ~ > y ~ o r x ~ - - y ~ andx~_>y~]

for all x, y ~IR 2. For all (S, d) E ~, L(S, d) is defined as the unique point x s S such that

( x - d ) ~ _ t ( y - d ) for all y e S.

If (S, d) e ~, L(S, d) coincides with E(S, d). For notational convenience, we assume d = (0, 0). This can be done without

loss of generality if the utilities are translation invariant, which is an assumption that is (explicitly or implicitly) used in virtually all axiomatic approaches to the bargaining problem - see Thomson (1994) for further details. To simplify notation, we will write S ~ ~ instead of (S, (0, 0))~ ~ to denote a bargaining problem. Furthermore, the egalitarian solution can be written as E(S) = (e(S), e(S)) in this case.

3. A multi-stage arbitration game

In this section, we restrict attention to bargaining problems in ~ , that is, strictly comprehensive problems, and analyze a simple arbitration game that yields the egalitarian solution as the unique Nash equilibrium outcome. The arbitration game we consider is a multi-stage simultaneous-demand game, which follows Nash's (1953) approach. Various modifications and extensions of Nash's original demand game have been discussed in the literature. 4 In most cases, the Nash (1950) bargaining solution is justified using these games. The purpose of the present paper is to show that the egalitarian solution is consistent with the outcome of a simple arbitration game.

The game the arbitrator recommends can be described as follows. The two players state their demands in terms of utilities simultaneously. If these are compatible, each player gets the utility she or he demanded. Otherwise, the game

4See, for example, Binmore (1987), Osborne and Rubinstein (1990), Carlsson (1991), Anbarci (1992), Thomson (1994).


proceeds to the next stage, where the players can demand utilities again. But this time the player who asked for a higher gain from d = (0, O) in the previous stage will be penalized by restricting his or her new demand. The game proceeds in a similar way as before. More precisely, the rules of the arbitration game are as follows:

Stage 1: Each player submits a demand in terms of utility x] e [d~, a~(S, d!)], i = 1, 2, where d 1 = d = (0, 0). If x I = (xl, xX~) e S, the game is over and player i receives x~. If x 1 ¢S, the game proceeds to stage 2.

Staoe t (t > 2): The players submit demands xI e Ida, as(S, d')], where d~, i = 1, 2, is determined as follows:

J'x~ -1 if x~ - l - d ~ - l < x ~ - 1 - d ~ -1, d~ = [ dtx- x otherwise;

J'x~ -1 if x~ -1 - d ~ -1 < x ] -a - d ] -1,

d~ = ida_ 1 otherwise.

Analogously to stage 1, if x' ~ S, the game ends and player i receives x~. If x '¢ S, the game proceeds to stage t + 1 and continues until either the game stops after a finite number of stages or there is "perpetual incompatibility," in which case both players receive the disagreement utility of zero. In Fig. 1, we illustrate an example of a pair of incompatible first-stage demands and the resulting restriction on player 2's demand in the second stage.

Note that the criterion which is used to decide which player (if any) should be penalized is independent of the shape of the feasible set. This is an interesting feature of our game and provides a plausible penalty rule that is easy to apply.

The procedure described above is similar in spirit to Nash's (1953) demand game which justifies the Nash bargaining solution as some sort of perfect equilibrium which resembles Selten's (1975) notion of perfection. The most important feature of both games is that the players observe the demanded utilities, the feasible utility set, and the disagreement point. Time does not play any role in either of the

~2

,4

a~

/ x~

X 1 ~- .~

Z 1

Fig. 1.


two games. It will turn out that our game ends after the first stage in equilibrium. The difference is how we treat incompatible demands by penalizing the player with the higher demanded gain, and allow the game to continue. Thus, our strategic model is a multi-stage, simultaneous-move game with no discounting.

It is worth mentioning that van Damme's (1986) "game of solutions" (see also Thomson 1994, Ch. 8) also is a multi-stage game defined in terms of the players' utilities. Instead of submitting demands, the players submit a whole solution within a given class, and the feasible set S is reduced in each stage of the game. Under certain conditions, this game can be used to provide a strategic foundation of the Nash solution. Chun (1989) has modified van Damme's approach to provide a noncooperative justification for the egalitarian surplus sharing.

To complete the formal specification of our demand game, let, for t e IN such that t > 2, X' = 1-[1 ~,'<t [d~, al(S, d")] x [d~, a2(S, d")]. Furthermore, let X 1 be the "null history". Then a strategy of player i is a sequence of functions 00i = {00[}, t ~ N, such that 00~: X ' ~ [dI, ai(S, dr)] for i = 1, 2, t • N. That is, in each stage, the players pick an action from their respective set of available actions, where the current action is a function of the history of the game. Let ZI and 22 be the strategy sets of the players.

Given a~ e Z~, denote by x~(00~) the demand selected by player i in stage t. Let xt(6) = (X~(0"I), X~(62) ) for a e 2;1 x 272. If x'(a)¢S for all t e N , the strategy pair 00 leads to perpetual incompatibility of the demands, in which case the disagreement event occurs and both players receive zero utility. Thus, the payoff functions ui: Z1 x 27 2 ~-~ IR, i = 1, 2, of the game are given by

l'xt*(00) if 3t • N such that xt(a) • S, u(a) = (u1(00), u2 (a)) = [.(0, 0) otherwise

for all 00 e Zl x Z2, where t* = min{t [ t e N, xt(00) • S}. For S • 9~, let Fs denote the above described demand game. A strategy pair a* • ZI x Z2 is a Nash equilibrium of Fs if it satisfies

u1(00") > u1(001, a~) V001 • S1 and U2(00" ) >_ U2(0"* , 002) V0"2 • Z2'

4. The main result

The game Fs provides a link between strategic models of bargaining and the egalitarian solution to strictly comprehensive problems in the sense that all Nash equilibria of Fs are characterized by both players demanding e(S) in the first stage of the game. Note that no refinement of the Nash equilibrium concept (such as subgame perfection) is needed for this result. Formally, our main theorem is

Theorem 1. For any S ~ ~ , a strategy pair 00 e S, l x z~, 2 is a Nash equilibrium of Fs if and only ifxl(001) = x~(002) = e(S).

Proof. "If": Let S e ~ . Suppose a e S I x ~2 is such that x~(001) = x21(002) = e(S), but a is not a Nash equilibrium of Fs. Then at least one of the players can be better off by deviating from a. Without loss of generality, suppose there exists ~1 e 2;1 such that ul(a l , 02) > u1(00) = x~(001). It must be the case that x~(61) > x~(001), which implies (xl (~1), x~(002))~S. Then the game continues for more than one stage. According to the definition of Fs, player l's demand in subsequent stages is bounded from above by e(S), since x~(61) > x~(002). Consequently, ul(al , 002) -< x~(001). This is a contradiction.


"Only if": We have to show that the strategy pairs such that each player demands e(S) in the first stage are the only Nash equilibria of Fs. Suppose there exists a Nash equilibrium a e 2;1 x I;2 such that xl(a):~ E(S). There are three possible cases:

Case I: x~(a) e S. Clearly, x I (a) cannot be Pareto dominated in S. Otherwise, one of the players will be better off by adding e > 0 (sufficiently small in order to ensure that the resulting pair of demands is still feasible) to his or her demand in stage 1. Hence, x l(a) is on the Pareto frontier of S.

If x l (a l ) < x21(a2), let tYl eZ1 be such that x~(61)= x~(al) + 8 with 0 < e < x~(a2) - x~(al), and x2(61)= xl(61). Given the upper bound imposed on player 2's demand in the second stage, the game ends after the second stage (no matter what a2 looks like), and it follows ul(61, tr2)> x l ( a l ) + 8= ul(a) + e > Ul(a), which leads to a contradiction. The case x l ( a l ) > x~(a2) is analogous. Hence, a is not a Nash equilibrium. See Fig. 2 for an illustration of the above argument.

Case 1I: xl(a)¢S and xl(a l ) ¢ x~(a2). Without loss of generality, suppose xl(a l ) < x~(a2). Hence, d 2 = (x l (61 ) , 0).

If Ux(a) < xl(al), let 61 e Zx be such that xl(6x) -- x2(61) = xl(al), which - together with x~(61) < x~(a2) - implies that the game ends after the second stage, where ux(6t, a2) _> xl(a l ) > ul(tr), contradicting the assumption that a is a Nash equilibrium.

If ul (a) > xl (al), the strict comprehensiveness of S implies u2(a) < a2(S, d2). In this case, let 82 e ~2 be such that X21(62): a2(S , d2), which implies u2(al,62) = a2(S, d 2) > u2(tr), and hence tr is not a Nash equilibrium.

If Ul(a) = xl(al) , there are two further subcases to consider:

Z2

4'

4(o~),

Fig. 2.


(i) Ifxl(tr l) < al(S), let gl ~ Z, be such that xl(#~) = x2(gl) = xl(al) + ~ with 0 < ~ < min{al(S) - xl(al) , x[(a2) - xl(al)}. Again, the game ends after stage 2, and it follows u1(~1, a2) > xl(ax) + ~ > ul(a), a contradiction.

(ii) If xl( th) = al (S), the strict comprehensiveness of S implies u2(a) = 0. Let g2 s 2~2 be such that x21(tY2) = x22(¢72) = ~ with 0 < e < xl(al) . The game ends after the second stage, and it follows u2(al, #2) > e > u2(a), which establishes that a is not a Nash equilibrium.

Case III: xl(a)(sS and x l ( a l ) = x~(a2). Clearly, in this case, we must have xl(trl) = x~(a2) > e(S). First, suppose ul(a) < e(S). Let t~l e $1 be such that xl (gl ) = x2(01) = e(S) + ~ with 0 < ~ < xl(a) - e(S). Clearly, xl(gl) < x~(a2), and hence, the game ends after stage 2 with ul(#~, a2) > xl(gl) > e(S) > u~(a), and hence, tr is not a Nash equilibrium.

If ul(a) > e(S), the strict comprehensiveness of S implies u2(a) < e(S), and the above argument can be repeated with players 1 and 2 interchanged. This completes the proof. []

The restriction we impose on the possible future demands of a player whose demanded gain exceeds that of her or his opponent in a given stage of the game is crucial for our result. Without such a restriction, the uniqueness concerning the Nash equilibrium outcomes of the game would disappear. In fact, even if we restrict attention to subgame-perfect equilibria, multiple equilibria still arise. Clearly, this is a strong assumption. We believe, however, that it is justified, since it represents a very plausible and natural way of penalizing players with excessive demands.

Notice that there are many ways to penalize the player who asks for "too much". For instance, in stage t (t > 2) of the game, suppose that d~ is defined as (1 - 2)xt, -1 + ,tal(S, d '-1) if x~ -1 - d~ -1 < x~ -1 - d~ -1 and d~ -1 otherwise, and that d~ is defined in a similar way, where 2 e [0, 1] is a constant. Then it can easily be shown that the conclusion of Theorem 1 still holds. The rules of our game described in Sect. 3 correspond to the case ), = 0. When 2 > 0, the game imposes a more severe punishment on the "greedy" player. In the extreme case where 2 = 1, the player who asks for more gets zero and the other player receives her or his ideal utility. Clearly, there are many games that lead to the egalitarian solution.

Multiple equilibria also rise if discounting is introduced. To illustrate that, consider the following example. Let S __c_ IR 2 be the convex hull of the points (0, 1), (1, 0), (0, 0). Suppose that we have a (common) discount factor ~ = 1/4, that is, the original payoffs of our game are modified by using 6 t ' - lxt*(a) instead of xt'(tr). In addition to E(S) = (1/2, 1/2), payoff vectors such as (1/4, 3/4) can be supported as an equilibrium with discounting. To see that a strategy pair a e 2~1 x $2 such that xl(trl) = 1/4, x~(t~2) = 3/4 is a Nash equilibrium in this case, observe first that player 2 cannot have any incentive to deviate. If player 1 deviates, the best payoffhe or she can possibly achieve is al (S) = 1. However, since this payoff can only be achieved in the second stage of the game or later (if at all, depending on a2), 6a1(S) = 1/4 is an upper bound for her or his payoff. Consequently, none of the players has an incentive to deviate.

The intuition behind this multiplicity in the presence of discounting can be explained as follows. In the above example, if there is no discounting of the players' payoffs in the game, player 1 could submit a higher demand than 1/4 but less than 3/4. The game would proceed to the second stage. Player 2 would then be penalized and player 1 would benefit. Since player 1 is "patient", it would be better for him or her to do so. With discounting, however, player 1 may not want to wait even if his or her payoffis less than player 2's. Thus, each player is trying to take advantage of


the other player's impatience and consequently multiple equilibria arise. This outcome is expected since, in our demand game, the player who demands a higher gain (in the case of incompatible demands) is penalized only in the subsequent stages. Therefore, when the players are more patient, the rules of the demand game are more effective in terms of penalizing "greedy" players.

Furthermore, it is essential to use the gains over the respective redefined disagreement points in each stage as the criterion for the penalty rule. If, instead, we would compare the gains over the orioinal disagreement point in each stage, our result would not go through. In fact, in this case, there would exist strategy pairs such that xl(a) = E(S) which are not Nash equilibria. For example, let S _~ IR 2 be the convex hull of(0, 1), (1, 0), (0, 0). Let a s S i x Zz be such that player 1 demands the utility 1/2 in the first stage, 3/4 in the second stage, and 1/4 in the third stage, if these demands are feasible according to the rules of the game, and player 2 demands 1/2 in each stage, if possible. Clearly, x l ( a ) = (1/2, 1/2)= E(S) and the game ends in the first stage if the players choose m However, e is not a Nash equilibrium. Indeed, without restrictions on the new disagreement points, player 2 can deviate by choosing 62 such that he or she demands 3/4 in the first stage, 1/2 in the second stage, and 3/4 in the third stage, if these demands are possible. Then player 2 receives the payoff 3/4 > 1/2. This observation is not too sur- prising, since the definition of the egalitarian solution is based on the relative gains of the players. It is only natural that a game that justifies this solution exhibits this feature as well.

Our game requires the observability of all relevant features of the bargaining problem, and hence, Theorem 1 is not an implementation of the egalitarian solution in the sense formulated by Thomson (1994, Ch. 8) - see Sect. 6 below for further discussion. However, this observation also applies to many other contributions, including Anbarci (1993) and further examples discussed in Osborne and Rubin- stein (1990) and Thomson (1994).

5. Extensions

The strict comprehensiveness of S plays an important role in the proof of Theorem 1. If we relax this assumption and consider bargaining problems in ~ , there exist S s ~ such that the egalitarian solution is not an equilibrium outcome of the corresponding game Is. However, Fs can be used to provide a justification of the Leximin solution L for all S ~ ~ . It is straightforward to see that if S ~ is such that a(S) >> E(S), L(S) coincides with E(S) and all arguments used in the proof of Theorem 1 go through without change. Hence, Theorem 1 can be generalized to the following result.

Theorem 2. For any S ~ ~ such that a(S) >> E(S), a strategy pair a ~ Z1 x Z2 is a Nash equilibrium of Fs if and only ifxl(trl) = x21(tr2) = e(S).

If S is such that al(S) = e(S) o r a 2 ( S ) --= e(S) - see Fig. 3 for an illustration of this situation - there exist further Nash equilibria. However, they all yield the Leximin solution as the unique outcome even though this outcome is not necessarily reached in the first stage of the game.

Hence, to complete our justification of the Leximin solution to problems in ~ , we state the following theorem.


Z2

~,(s) ,

~1 ~" ~'2

\

~ ~(s)

Fig. 3.

Theorem 3. (i) For any S E ~ such that al(S) = e(S), a strateoy pair ~ ~ Zx x Z2 is a Nash equilibrium of Fs if and only if

xl(~) = L(S) or xl(cr)¢S and x~(cq) = al(S) and x2(a) -- L(S).

(ii) For any S ~ ~ such that a2(S) -- e(S), a strategy pair ~r ~ ~1 x 252 is a Nash equilibrium of Fs if and only if

xl(~) = L(S) or xl(cr)¢S and xzl(cr2) -- a2(S) and x2(~r) = L(S).

Proof (i) Given that x~(al) -- al(S) <_ x~(a2), none of the players can improve upon L(S) by deviating from ~r. Hence, all strategy pairs described in part (i) are Nash equilibria. The proof that no other Nash equilibria exist is analogous to the proof of Theorem 1. Part (ii) of Theorem 3 can be proved in the same way. []

Our arbitration game can be generalized in a straightforward way to yield the proportional solutions E ~ as unique Nash equilibria. To do so, we can, for any S e ~ and cx ~ IR+ +, define a game F~ by replacing the definition of the d~ (i = 1, 2; t > 2) in Fs by

)'x~ -I ifx~ -1-d~ -l<u(x~ - l - d ~ -1) d~ = [ d~- a otherwise;

)'x~ -1 if cx(x~ - t - d ~ - 1 ) < x ~ - x - d ~ -1 d~ = ~d~_l otherwise.

We leave it to the reader to formulate results analogous to Theorems 1-3 that provide noncooperative foundations of the proportional solutions and their lexicographic extensions.


Similar extensions could be defined in order to modify our game into games that provide justifications of the area monotonic solution (see Anbarci and Bigelow, 1994) or the Nash (1950) solution. However, in these cases, a would have to depend on the shape of the feasible set S, which makes the resulting penalty rule considerably less intuitive and attractive than the original one used in Fs.

It is not obvious how our results could be extended to bargaining problems with more than two players; in general, strategic bargaining models are often considerably harder to deal with if more than two players are involved. The main reason for this is that the penalty rule used in Fs does not generalize very well to higher dimensions. To illustrate the difficulties arising in this case, suppose n = 3, and the players demand the utilities x~ = ai(S, dr), x~ = a2(S, dr), x~ = d~ in stage t s N of a simultaneous-demand game, and these demands are not feasible. If we want to use a procedure analogous to the one used in the two-players case, the new disagreement point is simply the old one, and there is no guarantee that the game will end after a finite number of stages. The problem is that a single player (here, player 3) can not necessarily force a feasible outcome by demanding her or his current disagreement utility repeatedly. Consequently, arguments such as the one used in Case I of the proof of Theorem 1 do not apply. This difficulty is another example for the well-known phenomenon that many properties of comprehensive sets in two dimensions do not have counterparts in higher dimensions.

6. Conclusion

The arbitration game discussed in this paper follows Nash's (1953) approach of providing noncooperative foundations of axiomatic bargaining solutions. As it is the case for Nash's original game and numerous other contributions in this area, our game is defined in terms of the players' utilities. Therefore, our results do not provide implementations of the solutions discussed. An implementation of an axiomatic bargaining solution (see Thomson 1994) would require that the game is defined in terms of a concrete problem that can be translated into an abstract bargaining problem. For example, Moulin (1984) has first offered an extensive- form game to implement the Kalai-Smorodinsky (1975) bargaining solution in subgame-perfect equilibrium. Peters et al. (1991) have provided an extensive game similar to Moulin's which also implements the Kalai-Smorodinsky solution. The Nash solution can be approximately implemented by using the Rubinstein (1982) alternating offers scheme, and an exact implementation can be found in Howard (1992); see also Rubinstein et al. (1992) for a discussion.

The assumption of utility information being available to the decision maker clearly is quite strong. Therefore, one way of extending the results of the present paper would be to work in environments with less demanding informational requirements. For instance, it would be useful to consider an environment in which the players know each other's utility, but the arbitrator does not know the players' utility functions. In general, the implementability of various bargaining solutions is an issue that deserves further investigation in the future.

References

Anbarci N (1992) Nash's demand game revisited. Mimeo, SUNY at Buffalo Anbarci N (1993) Noncooperative foundations of the area monotonic solution. Q J Econ

108:245-258


Anbarci N, Bigelow JF (1994) The area monotonic solution to the cooperative bargaining problem. Math Soc Sci (forthcoming)

Binmore K (1987) Nash bargaining theory II. In: Binmore K, Dasgupta P (eds), The Economics of Bargaining. Blackwell, Oxford pp 61-76

Binmore K, Rubinstein A, Wolinsky A (1986) The Nash bargaining solution in economic modelling. RAND J Econ 17:176-188

Carlsson H (1991) A bargaining model where parties make errors. Econometrica 59: 1487-1496

Chun Y (1989) A noncooperative justification for egalitarian surplus sharing. Math Soc Sci 17:245-261

Howard JV (1992) A social choice rule and its implementation in perfect equilibrium. J Econ Theory 56:142-159

Imai H (1983) Individual monotonicity and lexicographic maximin solutions. Eeonometrica 51:389-401

Kalai E (1977) Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45:1623-1630

Kalai E, Smorodinsky M (1975) Other solutions to Nash's bargaining problem. Econo- metrica 43:513-518

Moulin H (1984) Implementing the Kalai-Smorodinsky bargaining solution. J Econ Theory 33:32-45

Myerson R (1977) Two-person bargaining problems and comparable utility. Econometrica 45:1631-1637

Nash J (1950) The bargaining problem. Econometrica 18:155-162 Nash J (1953) Two-person cooperative games. Econometrica 21:128-140 Osborne M J, Rubinstein A (1990) Bargaining and Markets. Academic Press, New York Peters H, Tijs S, Zarzuelo J (1991) Consistency and implementation of the Kalai-

Smorodinsky bargaining solution. Reports in Operations Research and Systems Theory M 91-09, University of Limburg, Department of Mathematics

Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica 50:97-109 Rubinstein A (1992) On the interpretation of two theoretical models of bargaining. Tel-Aviv

University Working Paper No. 7-92. Rubinstein A, Safra Z, Thomson W (1992) On the interpretation of the Nash bargaining

solution and its extension to non-expected utility preferences. Econometrica 60:1171-1186 Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive

games. Int J Game Theory 4:25-55 Thomson W (1994) Bargaining Theory: The Axiomatic Approach. Academic Press, New

York (forthcoming) van Damme E (1986) The Nash bargaining solution is optimal. J Econ Theory 38:78-100 van Damme E (1987) Stability and Perfection of Nash Equilibria. Springer, Berlin

an arbitration game and the egalitarian bargaining solution€¦ · game theory has provided two...

Documents