n-person nash bnargaining with variabel threats

The Japanese Economic Review Vol. 47, No. 3, September 1996

N-PERSON NASH BARGAINING WITH VARIABLE THREATS*

By MAMORU KANEKOt and WEN MAO$ TUniversity of Tsukuba, and $Villanova University

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash’s variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large.

1. Introduction

An important constituent of bargaining is the specification of the payoff vector when players fail to achieve an agreement. This payoff vector is called a disagreement point. With the major exception of Nash (1953), a disagreement point is assumed to be exogenously given. I ) In some economic examples, exogenous disagreement points such as an endowment point in an exchange economy are naturally determined. In general game situations and in economic situations with externalities, however, we may not find a natural definition of an exogenously given disagreement point. This fact requires us to consider a bargaining model with an endogenous determination of a disagreement point. Nash (1953) gave such a model with two players. The purpose of this paper is to investigate the behaviour of this model with n players. We compare, from both axiomatic and noncooperative game theoretic viewpoints, the n-player extension of Nash’s model with an alternative model where disagreement points are also endogenously determined.

Nash’s (1953) model - the first model in this paper - is as follows. An underlying environment is described by a noncooperative strategic game G. The players are allowed to cooperate for obtaining higher payoffs and bargain over possible cooperative payoffs, subject to the possibility of failing to achieve an agreement. If the players fail, they return to the original noncooperative game. For this possibility, each player has to choose a strategy in the case of disagreement, prior to bargaining.

* The authors thank J. J. Kline, A. Okada, J. Wako and two referees of this journal for helpful comments on earlier drafts.

I ) The role of a disagreement point has been extensively studied in axiomatic bargaining theory with the assumption of an exogenously given disagreement point, see Chun and Thomson (1990) and Peters and Van Damme (1991).

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The Japanese Economic Review

The model has two stages: each player chooses and announces a strategy as a threat strategy to the other players - threat stage; and then the players bargain for cooperative payoffs - demand stage. In the threat stage, a player threatens his opponents with a strategy to be played in the case of disagreement so as to gain an advantage in the demand stage. Here it is presumed that each player commits to play his threat strategy in the case of disagreement.

A commitment is, however, not always possible. When it is not available, a player may not use his threat strategy if disagreement actually occurs. The second model in this paper describes such a situation. In the second model the players bargain for cooperative payoffs without a pre-chosen disagreement outcome; if they do not agree on a cooperative outcome, they go to and play the original game G noncooperatively. This means that the players bargain in the first stage; and if they do not achieve an agreement, they go to the second stage of the noncooperative game G. The main difference between these two models is that commitments to play threat strategies are available in the first but not in the second. The primary purpose of the introduction of the second model is to characterize the behaviour of the first model by making comparisons between them.

We formulate these two models as extensive games. Solutions are strategy combinations in these extensive games that satisfy the subgame perfect equilibrium property (Selten (1975)) and Nash’s axioms except Pareto optimality - the Subgame Perfectness Axiom together with the other axioms implies Pareto optimality. In either model, we fully characterize the solution by these axioms. Since our axiomatizations can be regarded as ones in noncooperative game theory, they give a unified view to the axiomatic and noncooperative approaches. Sections 2 and 3 are devoted to the descriptions and characterizations of the two models.

In the two-person case, Nash (1953) proved that the first model gives a unique equilibrium payoff outcome, and moreover, that it becomes a strictly competitive game when we restrict our attention to the threat stage. In Section 4, we argue that these properties are not preserved for games with more than two players.

In Section 5 , we show that for a game with sidepayments and a large number of players, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large.

2. A bargaining model with commitments to threat strategies

We denote a finite n-person strategic game by G = (N, { X ; } , { h ; } ) , where N = { 1, . . ., n} (n 2 2) is the player set and for each player i E N, X; is a finite set of pure strategies and hi: niE,X, + R is a payoff function. We denote the set of all mixed strategies of player i in game G by Mi(G), and the product set njcNMi(G) by M(G). The set of all jointly feasible payoff vectors is denoted by F(G), i.e., F(G) is the convex hull of the set { h(x): x E n,,X;}. We call F(G) the feasible region generated by game G. This means that the players are allowed to coordinate their jointly mixed strategies but no transfer of goods is allowed. The finiteness assumption of the pure strategy space Xi is made for simplicity. The class of all finite games with the fixed player set N is denoted by 6.

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Mamoru Kaneko and Wen Mao: N-Person Bargaining with Variable Threats

A bargaining problem is given as a pair (G, 7') of an n-person game G E G and a compact convex subset T of R" with F(G) C i? The set T is called the bargaining region. We denote the set of all bargaining problems (G, 7') by El. The bargaining region T coincides with the feasible region F(G) when the cooperative payoff vectors in T are obtained purely from the coordination of jointly mixed strategies without any transfer of goods. When some transfer of a commodity is allowed as well as the coordination of their mixed strategies, the bargaining region T is larger than the feasible region F(G). One example is a bargaining problem with sidepayments: the bargaining region T is defined by

T = u E R" : x u i S max x h ; ( p ) and ui 3 min hi@) for all i E N . ( I )

Here the players maximize the total utility measured by money (a composite good) and may transfer some amount of money among themselves. This is often called the transferable utility assumption (see Kaneko and Wooders (1 994) for the underlying assumptions for transferable utility). The boundary condition ui 3 minpEM(G) hi(p) is a kind of arbitrary choice. Bargaining problems with sidepayments and large numbers of players will be considered in Section 5.

We associate an extensive game T(G, 7') with each bargaining problem (G, 7') in B, which we call the associated bargaining game (with commitments). The associated bargaining game T(G, 7') has two stages - Stage 1 is the threat stage and Stage 2 is the demand stage. The game tree of T(G, 7') is described in Figure 1.

{ i pEM(G) j PEM(G) 1

Stage I : Each player i E N chooses a mixed strategy pi from Mi(C) independently, and announces it to the other players. Stage 2: Each player i E N independently chooses a utility demand u; from R. If the vector u = (u1, . . ., u,) belongs to the bargaining region T, the final outcome is u = ( U I , . . ., u,,), and otherwise, it is h(p) = (hl(p) , . . ., h,(p)). The final outcome gives a payoff to each player.

FIGURE I . T(G. r ) (Simultaneous Move Form).

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Stage 1

Stage 2

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Note that threat strategies P I , . . ., p,, are announced as mixed strategies to be played in the case of disagreement. Hence the disagreement payoffs are given as the expected payoffs h(p) = (h l (p) , . . ., h,(p)), from the ex ante point of view.

The game T(G, 7') has two types of subgames. For each n-tuple p = ( P I , . . ., p,,) of strategies, the demand stage forms a proper subgame of T(G, T), which we denote by T(G, r; p ) , and the other is T(G, 7') itself.

In T(G, T), a strategy of player i is a pair of a point q; E Mi(G) and a function ly; from M(G) to R. That is, player i chooses q; in Stage 1 and announces it to the other players. In Stage 2, he chooses a demand ui =+;(q) depending upon the announcement q = (41, . . ., q,,). The function lyi gives an action to the subgame T(G, T p ) for each p = ( P I , . . ., p,,) E M(G). An n-tuple of strategies (q , ly) = ((41,. . ., q,,), ( l y ~ , . . ., ly,,)) in T(G, 7') is called a strategy combination. The payoff function H;(q, I)) of player i E N in T(G, 7') is defined by

Hi(% ly) = lyi(4) if 9%) E T

= hi(q) otherwise. (2)

The induced payofl function %(u) of player i on the subgame T( G, p ) is defined by * ( u ) = u; if u E T and %(u) = h,(p) otherwise.

A solution function Y is a function which assigns to each bargaining problem (G, 7') in B a strategy combination (q, ly) in T(G, T). Now we consider the following axioms.

Axiom 1 (Feasibility). For any (G, 7') in B with Y(G, 7') = (q, ly), ly(p) E T fbr all

Axiom 2 (Subgame Perfect Equilibrium). For any (C, 7') in B, Y(C, 7') = (q, ly) is a subgame perfect equilibrium in T(G, T).

P E W G ) .

Here a strategy combination (q, ly) is called a subgame perfect equilibrium iff for every subgame of T(G, T), the restriction of (q, ly) is a Nash equilibrium in that subgame. In the present context, this means that (1) (q, ly) is a Nash equilibrium in T(G, 7') and (2) q ( p ) is a Nash equilibrium in the subgame T(G, T; p ) for each P E W G ) .

Axiom 3 (Invariance under Affine Transformations). Let (G, 7') E B with W(G, 7') = (q, ly). Then q(aC + b, a T + b)) coincides with (q, aly + b) for any u > 0 and b in R".

Here aG + b denotes the game G' = ( N , { T } , { A , ! } ) obtained from G by affine transformations of payoff functions, i.e., h:(x) = a;h;(x) + b; for all x E n j E ~ X ; and all i E N, aly + b is (al ly , ( . ) + 61, . . ., u,,ly,,(.) + b,,), and aT+ b = {(alvl + 61, . . ., a,v, + b,,): u E T } .

Axiom 4 (Anonymity). Let ( G , 7') E B with Y(G, 7') = (q, V) and n any permutation of N = { 1, . . ., n } . Then Y(xC, x7') = (xq, xly).

Here nG = (A', {X,!}, { h : } ) is obtained from game G by giving player n(i) in G a new name i in the new game nG. Formally, X,! = X;r(;), h,! (xx( , ) ,

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Axiom 5 (Independence of Irrelevant Alternatives). Let (G, T), (G’, T ) E B with WG, T ) = (4, V ) , WG‘, T) = (q’, V’ ) , and p E M(G), p’ E M(G’). r f h’(p’) = h(p), T’ c T and ly(p) E T, then ly‘(p’) = ly(p).

Axiom 1 requires that the demand vector ly(p) be feasible for any p. Axiom 2 states that the strategy combination (4, ly) forms a subgame perfect equilibrium in T(G, r). One may think that Axiom 2 implies Axiom 1 since the subgame perfect requirement could eliminate the infeasible outcome. However, this is not exactly true: if every player i demands a very high utility level ui, then u = (UI, . . ., u,) is not feasible, but is possibly an equilibrium in subgame T(G,

The subgame perfect equilibrium requirement for proper subgames in Axiom 2 corresponds to Pareto optimality in Nash’s ( 1950, 1953) original axiomatization. In Figure 2, the point u = (UI, 242) is not an equilibrium point in the subgame, since player 1 (and 2) can improve his final payoff by demanding slightly higher u~ (and u2). In this example, any point in the Pareto frontier which locates in the northeast of the threat payoffs h(p) is an equilibrium point in the subgame. Axiom 2 does not, however, imply Pareto optimality for some games; the argument above is not applied to the game described in Figure 3. Nevertheless, Axiom 2 together with Axioms 1 and 5 implies that v ( p ) is Pareto optimal.

Axiom 3 requires the function 11, to be invariant under affine transformations of payoff functions. This axiom comes from the basic presumption that payoff functions are representations of preference relations satisfying the von Neumann-Morgenstern expected utility axioms (cf., Herstein and Milnor (1953)). Such representations are uniquely determined up to affine transformations. Bargaining problems (G, T ) and

p ) .

1 FIGURE 2.

/ FIGURE 3.

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(aC + b, aT+ b) are identical except for the payoff representations; consequently, the solutions must also be identified except for their payoff representations.

Axiom 4 requires that the solution function Y depend upon the game structure but not upon the names of players, i.e., the solution would be invariant if the names of players change.

Axiom 5 states that if the threat points are the same, if the bargaining region shrinks into a smaller set, and if the outcome of the original game remains feasible in the new bargaining problem, it is also the final outcome of the new bargaining problem. This is essentially the same as Nash's (1950, 1953) original axiom of the same name. This axiom together with Axiom 1 has an important implication: T = T and h'(p') = h(p) imply q ' (p ' ) = q ( p ) . Thus, the final outcomes are independent of the strategic structures of underlying games; once threat strategies are chosen, only the bargaining regions and threat strategies are relevant. This fact reflects the presumption that the players are committed to play those threat strategies.

To state the main theorem of this section, we define Nash's fixed threat bargaining problem. A fixed threat bargaining problem is a pair (S, d) of a convex compact subset S of R" and a point d in S. We denote the set of all fixed threat bargaining problems by IF. The Nash solution function is the function f N : IF -, R" defined by

f N ( S , d ) E S; and f N ( S , d ) maximizes n (xi - d;) over x E S with x 3 d, (3)

where N ( d ) = ( i E N x; > d; for some x E S with x 2 d } . The value f N ( S , d) is called the Nash outcome for (S , d). Since we need to allow degenerated bargaining regions S, the Nash product is taken over the relevant players N(d) . However, since there is an x E S with x 2 d such that x; > di for all i E N(d), we can apply the standard argument for the Nash outcome with the nondegeneracy assumption by restricting N to the relevant player set N(d).

icN(d)

Theorem 1. A solution function Y satisfies Axioms 1-5 if and only if for any bargaining problem ( G , 7') E B, Y(G, 7') = (q , ly) satisfies

(i) ~ ( p ) coincides with the Nash outcome f N ( r h(p)) for all p E M(G); (ii) q is a Nash equilibrium in q, i.e., q ; (q ) 3 Q;(q-;, pi) for all p; E M;(G) and all

(iii) ifY(G, 7') = (4 , q) and Y ( a G + b, aT+ b) = (q' , ly') for any a > 0 and 6, then

(iv) if Y(G, i") = (4 , ly) and Y(xG, xT) = (q' , ly') for any permutation x of N, then

i E N ;

4' = 9;

4' = xq.

This theorem states that if Y satisfies Axioms 1-5, then (i) Y gives the Nash outcome to each subgame T(G, C p ) of the second stage; (ii) the threat strategies q are chosen to be a Nash equilibrium in the first stage on condition that the Nash outcome is played in the second stage; (iii) Y is invariant under a positive linear transformation of utility vectors; and (iv) Y is invariant under a permutation of the names of players. The statements (iii) and (iv) claim invariance for threat strategies q, but the invariance of ly is implied actually, since ly is uniquely determined by (i). The solution described by (i) and ( i i ) is a generalization of what Nash (1953) described for the variable threat bargaining game with two players. We call q an equilibrium threat point.

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Nash (1950, 1953) provided two complementary approaches to the Nash bargaining solution: axiomatic and noncooperative approaches. They are often regarded as different, but some authors noticed that they are "quite similar in spirit" (Luce and Raiffa (1957, p. 143)). Our variable threat bargaining model described above is indeed a noncooperative game. An outcome for the associated bargaining game r(G, r ) is a strategy combination, and Axioms 1 through 5 are requirements for a strategy combination. Axiom 2 is a requirement in noncooperative game theory, while Axioms 1, 3, 4 and 5 are reformulations of the axiomatic approach to the Nash bargaining solution. Thus our approach shows that these two approaches are not only similar in spirit but also have an explicit relationship.

The axiomatic approach is often regarded as belonging to cooperative game theory. Since we claim that our axiomatization can be viewed as one from noncooperative game theory, we should give evaluations of these axioms from the viewpoint of noncooperative game theory.

We claim that the axioms other than 2 are neutral to the judgement of whether it belongs to noncooperative or cooperative game theory. Clearly, Axiom 1 is neutral in this respect. The underlying presumption for Axiom 3 - Invariance under Affine Transformation - is that the von Neumann-Morgenstern expected utiity theory is common knowledge among the players.*) Since it determines a utility representation up to a positive linear transformation, the bargaining theory should be consistent with this theory. Axiom 4 - Anonymity - means that the solution should be independent of the particular names, 1, . . ., n, of players. This axiom can be regarded as requiring our solution to be a symmetric subgame perfect equilibrium, which is essentially the same as symmetric Nash equilibrium as discussed in Nash (1951). Finally, Axiom 5 - Independence of Irrelevant Alternatives - may look to be a condition belonging to cooperative game theory. However, this is essentially a general requirement for decision making - it corresponds to the assumption of a complete preordering in the case of utility theory, as J. F. Nash pointed out in a private note (see Shapley and Shubik (1974) and Kaneko (1980)). Thus this does not especially have a cooperative nature.

Of course, the entire approach of our bargaining theory has some aspect of cooperation, but it is described in the rules of the bargaining game, instead of the solution concept.

Proof of Theorem 1. The If-part can be proved in a routine manner. For the Only-lf part, we have to show that our axioms imply Pareto optimality in each proper subgame, that is, V ( p ) is Pareto optimal. Suppose that Pareto Optimality is proved. Then Nash's (1950) original proof is applied to each proper subgame, and we obtain (i). Properties (ii), (iii) and (iv) follow from Axioms 2, 3 and 4, respectively.

p) . First, we choose a vector b = ( b , , . . ., b,) with b d ZI for all ZI E 7: We extend the bargaining region T to the set 7"" = { v E R": b G ZI G u for some u E T}. Now we have two problems T(G, r; p) and r(G, 7""; p). The set 7"" has the same Pareto optimal surface as that of 7: We denote

Now we prove that V ( p ) is Pareto optimal in T(G,

WG 7"") by (q * ,V* ) .

2) In fact, it is sufficient for this axiom that a utility theory which determines a utility function up to a positive linear transformation is common knowledge. The utility theory based on utility differences, such as Alt (1936). is an alternative to the expected utility theory.

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Now we show that v * ( p ) is Pareto optimal in p. Suppose that for some u E p,

vj = v,?(p*) if j # i = u; i f j = i

u 3 v * ( p ) and ui > v ? ( p * ) for some i. Then the vector D defined by

belongs to p because b S v * ( p ) S v S u. Hence, in the subgame T(G, T'; p), player i can improve his payoff v r ( p ) by demanding ui, a contradiction to Axiom 2.

Since the Pareto surface of p coincides with that of T, v * ( p ) is on the Pareto surface of T, which implies v * ( p ) E 7: Since T C p, 11/*(p) E T, Axiom 5 implies v ( p ) = v* (p ) . This implies that v ( p ) is Pareto optimal in 7:

3. A bargaining model without commitments to threat strategies

In the previous model, each player makes a commitment to play a threat strategy. Unless a commitment is available for a player, he might be expected to change his mind in the case of disagreement if his threat strategy hurts himself. Sometimes commitments are simply impossible. In such a case, players may make another strategy choice after disagreement occurs. In this section, we consider a bargaining model where no player can make a commitment to play a threat strategy. The primary purpose of the introduction of this model is to investigate the behaviour of the previous model by making comparisons with the new model. In Section 5 , such a comparison is made to show that these two models yield approximately the same results for large numbers of players.

The bargaining situation is described by an extensive game A(G, 7') for each (G, in B, which we call also the associated bargaining game (without commitments). The game A(G, T ) has two stages: the first stage is the demand stage

Stage 2

FIGURE 4. A(G, 7).

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and the second is the noncooperative game stage. The game tree of A(G, 7‘) is described in Figure 4.

Stage I : Every player i E N independently chooses a utility demand ui from R and announces it to the other players. If u belongs to T the game is over with probability 1 and the final payoff vector is u = ( U I , . . ., un), but with probability 0, the game goes to Stage 2 (see Remark 1). If the vector u = (UI, . . ., u,) does not belong to T, the game goes to Stage 2. Stage2: The players simply play the noncooperative game G.

Remark 1. The bargaining game A( G, T ) has a subgame G in Stage 2 even in the case of agreement, though the probability of reaching the Stage 2 is zero. This is necessary, since the Nash outcome u is determined by the strategies @(u) chosen in the subgame occurring with probability 0. Here we should regard the case of agreement as the limit of a case where bargaining results in disagreement with a small probability even if u is feasible. This is essentially the same as the idea of trembling-hand perfection due to Selten ( 1975).3) Nevertheless, we do not consider this problem formally here.

In the bargaining game A(G, T), a strategy of player i is a pair of a point ui E R and a function c#i from R“ to Mi(G). That is, player i chooses and announces ui to the other players in Stage 1. If the game goes to Stage 2, player i chooses a strategy p i = @;(v) from Mi(G). A strategy combination is an n-tuple of strategies ( t i , @) = ((uI, . . ., Un), ($1 , . . ., @,,)). The payoff function Hi(u, @) of player i in A(G, 7“) is given as Hi(u, @) = ui if u E T and Hi(u, @) = hi(@(u)) otherwise. The induced payoff function T ( p ) of player i for the subgame determined by v = (DI, . . ., v,) E R” is given as Y ( p ) = hi(p).

A solution function @ is a function which assigns to each (G, 7‘) in B a strategy combination (u, @) in the associated bargaining game A(G, 7‘). We consider the following axioms.

Axiom 6 (Feasibility). For any (G, T ) E B with @(G, 2“) = (u , @), u belongs to 7: Axiom 7 (Subgame Perfect Equilibrium). For any (G, r) E B, @(G, T ) = (u , @) is a subgame perfect equilibrium in A( G, T). Axiom 8 (Invariance under Affine Transformations). Let (G, T ) E B with @(G, T ) = (u , @), let a, b be points in R” with a > 0, and @(aG + b, aT+ b) = (u ’ , @’). u @ ’ ( u ’ ) = @(u), then u’ = au + b. Axiom 9 (Anonymity). Let (G, 2“) E B with @(G, T ) = (u, @), n any permutation of N = { 1, . . ., n } , and @(nG) = (u’ , @’). V@’(u’) = n@(u), then u’ = nu. Axiom 10 (Independence of Irrelevant Alternatives). Let (G, T), (GI, r ) E B, @(G, 7) = (u , @), and @(G’, T‘) = (u ’ , @’). r f h’(@’(u’) ) = h(@(u)), 7” T and u E T’, then u’ = u.

Axioms 6-10 are apparently parallel to Axioms 1-5, except that 8 and 9 are formulated in weaker forms than 3 and 4. The main difference is that the second stage

3 ) Nash (1953) used a similar argument which he called the “smoothing procedure” to derive the Nash outcome. See Kaneko ( 198 I ) and Binmore ( 1987) for detailed arguments.

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of the previous model is the first stage of the present model. Since Axioms 1, 3-5 are about the outcome of the second stage in the previous model, Axioms 6, 8-10 apply to the behaviour of the outcome in the first stage of the present model.

We obtain the following theorem, but its proof is omitted.

Theorem 2. A solution function Q> satisfies Axioms 6 1 0 if and only if for any (G, r) in B with @(G, r) = (u , @),

(i) u coincides with the Nash outcome fN( d) where d = h(@(u)); (ii) @(v) is a Nash equilibrium in game G for any v E R";

(iii) for all i E N, u; 2 h;(@(u-i, v;)) for all v; E R with (u-;, vi) $! 7:

The first assertion states that the Nash outcome u relative to the disagreement point h(@(u)) results in Stage 1. The second states that @(v) is a Nash equilibrium in noncooperative game G for any V, including infeasible ones. Since 8 and 9 are formulated in weak forms, Nash equilibrium @(v) may vary with v if G has multiple equilibria. Hence we need the third condition, which states that no player can improve the bargaining outcome u by breaking cooperation and playing a disagreement (Nash equilibrium) point in the second stage. The following example illustrates the difference between the results of Sections 2 and 3.

Example 1 (Battle of the Sexes): Let N = {1,2}, XI = {aI,a2}, X2 = { @ I , / % } and let payoffs be given as follows:

PI Pz

Assume that the bargaining region T is given as the feasible region F(G). Since this underlying noncooperative game has two pure strategy equilibria (( 1 , 0), (1,O)) = (al , PI), ((0, l), (0, 1)) = (a2, p2), and one mixed strategy equilibrium ((2/3, 1/3), (1/3,2/3)), the second model gives the corresponding three types of solutions;

(1) u = (2, I), @(u) = ((1, O ) , (190)); (2) u = (1,2), @(u) = ((0, 11, (0, 1)); (3) u = (1.5, 1.5), @(u) = ((2/3, 1/3), (1/3,2/3)).

The first model gives the unique solution to this game;

q = (( 1, 0), (0, 1)) - equilibrium threat; and v(q) = (1.5, 1.5) - the final outcome. The unique solution in the first model gives the same final outcome as the solution (3) in the second model, though the disagreement points are different. The solutions (1) and (2) are degenerated in the sense that the disagreement points and the bargaining outcomes coincide. This degeneracy occurs sometimes not only in the second model but also in the first model. The class of games discussed in Section 4 has this property. The necessity of the assertion (iii) of Theorem 2 can be observed in this example; if 9 in (3) has @(uI, v2) = ((0, I ) , (0, 1)) for some v2 > u2 - @(.) jumps from @(u) = ((213, 1/3), (1/3,2/3)) to another equilibrium ((0, l), (0, l)), then (u, @) is not a subgame perfect equilibrium.

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We formulated the present model as a two-stage model instead of an extension of the model of Section 2. However, we can add the threat strategy choice to the present model before the first stage. This addition of threat strategy choice is not substantive if no player can make a commitment. In the last stage of this new model, every player pursues his payoff instead of playing the threat strategy he announced; the threat is not credible. In this formulation, we obtain essentially the same result as Theorem 2 with appropriate modifications of strategies and of Axioms

4. The structure of the set of equilibrium threat points

The variable threat model of Section 2 is regarded as the game of threat choice, assuming the Nash outcome for each subgame r(G, p ) . In the two-person case, Nash (1953) observed that it is a strictly competitive game, which implies that the equilibrium payoff vector is unique, and that equilibrium threat strategies are maximum strategies and also satisfy interchangeability in Nash’s (1951) sense. The model of Section 3 does not share these properties, as was discussed in Example 1. In this section we argue that when the number of players is more than two, the variable threat model loses such properties, too, and show that the structure of equilibrium threat points in the case of n + I players may be as complex as that of Nash equilibria of a noncooperative game with n players.

The following proposition states that for a constant-sum game G with the bargaining region T = F(G), there is no difference between the bargaining models of Sections 2 and 3, and that the set of final bargaining outcomes in both models coincides with the set of Nash equilibrium payoffs in G.

Proposition 1. Let G be a constant-sum game, Then the set of Nash equilibria in G coincides with the set of equilibrium threat points in the associated bargaining game T(G, F(G)). Furthermore, the set of Jinal bargaining outcomes in T(C, F(G)) ( in A(G, F(G))) coincides with the set of Nash equilibrium payoff vectors in G.

ProoJ Since G is constant-sum, the Pareto surface of F(G) is F(G) itself. Since any threat point is on the Pareto surface, the Nash outcome for a given threat point is the threat point itself. Thus fN(F(G) , h(p)) = h(p) for any p = M(G). Hence a Nash equilibrium in G is also an equilibrium threat point in T(G, F(G)), and vice versa. The set of final bargaining outcomes in T(G, F(G)) coincides with the set of Nash equilibrium payoff vectors in G. (The same arguments apply to A(G, F(G)).) H

Proposition 1 implies that bargaining is redundant for a constant-sum game G; constant-sum games are regarded as special from the viewpoint of bargaining. Nevertheless, this together with the next proposition helps us understand the structure of the set of equilibrium threat points in the general case.

4) We can modify this game further so that the game includes the choice of a commitment to play a threat strategy or not. Mao (1993) considers such a model and shows that in the case of two players, each player chooses a commitment to play a threat strategy, i.e., the model of Section 3 results endogenously. It is, however, also shown that this result no longer holds for n 3 3.

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Proposition 2. For every n-person game G, there exists an ( n + 1)-person constant-sum game G* such that there is a bijection between the equilibrium sets of G and G*.

The reason for Proposition 2, due to von Neumann and Morgenstern (1944), is as follows: for any game G = (N, { X i } , { h i } ) , we add one fictitious player n + 1 having only one strategy to the player set N so that the payoff functions of the original players are unchanged and the new player’s payoff is the negative sum of the others’ payoffs. The extended game is an ( n + 1)-person zero-sum game. Since the structure of the extended game is identical to the original game except the nominal existence of a fictitious player, the set of Nash equilibria remains unchanged with this additional player.

Propositions 1 and 2 imply that for any n-person game G, there is an ( n + 1)-person game G* such that the Nash equilibrium set of G is isomorphic to the set of equilibrium threat points of the associated bargaining game T(G*, F(G*)). Let us apply this implication to a bargaining game with three players. Let G be a two-person game with multiple equilibria. Then the three-person constant-sum game G* given by Proposition 2 has the same number of equilibrium threats as that of the Nash equilibria for G, and these equilibrium threats are already the final outcomes. This shows that the uniqueness of the equilibrium payoff vector for a variable threat bargaining game with two players is not preserved for games with two players. When we go to games with many players, the set of equilibrium threats may become as complex as the set of Nash equilibria of a normal form game.

Constant-sum games are “degenerated” from the viewpoint of Nash bargaining. We can, however, modify a game slightly so that they are not constant-sum but the above two propositions remain true. In this sense, the assumption of constant-sum is not very crucial for showing that the set of equilibrium threats for n + 1 players is possibly as complex as that of Nash equilibria of a noncooperative game with n players.

Here we give a concrete example of a nonconstant-sum game to illustrate the nonuniqueness of the equilibrium outcomes in the variable threat model with n L 3. For such an example and also for the purpose of Section 5, we state the following (known) lemma.

Lemma 1. Let a,b E R” with a > 0 and C b j / a , S 1. Consider a f i e d threat bargaining problem (S, d) where S = {x E R”: Cxj la j 1 and x 2 b } and d E S. The Nash outcome f N ( S , d) is given as

f y ( S , d ) = a; 1 - dj/aj / n + d; for all i E N ( F ) (4)

Example2. Let N = {1 ,2 ,3] , X I = { a l , a ~ ) , XZ = {PI,&}, X3 = { y l , y2} , and let payofls be given by the following matrices:

TABLE 1 PI P 2

TABLE 2 PI P 2

UI F] (12 0.0.0 o,o, 10 a2 3 , 5 , 2 10,0,0

Y2

u’ HEI YI

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Consider the bargaining region T = F(G), which is given as the convex hull of four vectors (0, 0, 0), (10,0,0), (0, 10,O) and (0, 0, 10). We can apply Lemma 1 to calculate the Nash solution for each triple of pure strategies. For example, strategy triple (al, Bz, y 1 ) determines the threat payoff vector (1, 1, 5). Thus, Lemma 1 implies that the Nash outcome for (F( G), (1, 1,5)) is (2,2, 6). In the case of (a, , PI, y2), since (4,0,6) is on the Pareto surface of F(G), (4,0,6) itself is the Nash outcome of (F(G), (4,0,6)). Similarly, we have the Nash outcomes relative to the other pure strategy triples:

TABLE 3 PI B2

TABLE 4 BI i32

YI Y2

Equilibrium threat points in the associated bargaining game T(G, F(G)) are obtained by calculating the Nash equilibria in the matrix game in Tables 3 and 4. This game has Nash equilibria ( a l , 8 2 , y l ) , (a2, B I , y l ) (in pure strategies) and the corresponding Nash outcomes are (2,2,6), (10/3, 10/3, 10/3). Thus this example shows that the variable threat model has at least two distinct final outcome^.^)

5. Equilibrium threats in large bargaining games with sidepayments

When the number of players is small, e.g., n = 2, 3, the models of Sections 2 and 3 give quite different disagreement points (and a fortiori, different final outcomes). That is, the equilibrium threat points in the first model are, in general, different from the Nash equilibria in the underlying noncooperative game. In choosing a threat strategy, there is a tradeoff between threatening the other players and guaranteeing one's own higher payoff. This can be observed in equation (4) in Lemma 1. We show that when the number of players is large, guaranteeing one's higher payoff becomes more important than threatening others. Formally, under a certain condition, when the number of players is large enough, any equilibrium threat point in the associated bargaining game T(G, r) for a bargaining problem (G, r) with sidepayments becomes approximately a Nash equilibrium in the underlying game G and vice versa.

We consider a sequence {(G", 7")) = {((A'", {q}, { hy}) , 7")) of bargaining problems with sidepayments defined by (1) with -+ +oo as Y -, +GO, and assume the following condition:

Max Max Max (hjY(s-;, s;) - h;(s-;, t;))/lN"I -, 0 as Y -, +oo. ( 5 ) I€"' .s-, .s,, f, j + ;

5) In this paper, we do not consider coalitional behaviour. If we introduce coalitional behaviour, we meet similar variable threat bargaining inside a coalition; if a coalition considers a deviation from a final outcome, the players in the coalition bargain for a new cooperative outcome based on a new threat point among themselves. An investigation of such coalition behaviour is possibly made in term of the "coalition-proof" Nash equilibrium by Bernheim. Peleg and Whinston (1987). In the above example, (a!.P2, y ~ ) is not coalition-proof, and the other equilibrium threat point is coalition-proof. I t is, however, easy to find another example which does not have a coalition-proof equilibrium.

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For this sequence of bargaining problems, the following theorem holds.

Theorem 3. For any E > 0, there is a positive integer no such that for any (G", 7") with IN"( 2 no,

(i) any Nash equilibrium p in G is an &-equilibrium threat point in r(G, 7"), i.e., ,for all i E N",

f y ( T " , h"@)) + E 3 f y ( T " , h"@-ir r;)) for all r; E M;(G");

(ii) any equilibrium threat point q in r(G", 7 y ) is an E-Nash equilibrium in game G, i.e., for all i E N",

h;(q) + E b hi(q-,, r;) for all ri E M;(G").

Theorem 3 asserts, under condition (5), that for a large bargaining problem (G", 7") with sidepayments, the equilibrium threat points in T(G, 7") and the Nash equilibria in the underlying noncooperative game G" are approximately equivalent. Thus the bargaining models of Sections 2 and 3 yield virtually the same outcomes - the advantage of a commitment to play a threat strategy almost disappears - when the number of players is large.

Condition ( 5 ) states that the total effect of a strategy change by a single player on the other players' payoffs becomes small relative to the number of players. This may be interpreted in several ways. For example, if one player affects the others equally, the per capita effect becomes negligible for a large game. Alternatively, a single player affects a constant number of players with the same magnitude even when n becomes large. Condition (5) excludes the possibility that the total effect remains comparable with the total number of players. This condition is used in the proof of Theorem 3 as follows: In formula (4), the first effect appears as the average of total effects on the other players' payoffs but the second effect appears directly. Hence if a sequence of bargaining problems satisfies (5), the effect of a threat strategy on the other players becomes negligible when the number of players is large.

The assumption of sidepayments is crucial in the proof of Theorem 3 in that the formula given by Lemma 1 is used as was stated above. Under the sidepayment assumption, the Nash outcome is linear in each player's threat strategy, a fortiori, threat payoff. Without this assumption, we cannot know the behaviour of the Nash outcome: even the existence of an equilibrium threat point remains open for the bargaining game T(G, r) when the number of players is greater than two.@ We conjecture that if the existence is proved, the limit theorem would be also proved in a similar generality.

Proof of Theorem 3. Since 7y is defined by (l), it can be described as 7" = ( u E RI"'1: C j E N ' . U j I A " =z 1 and ui 2 By for all i E N"}, where A" = max,,M(G') C ; h r ( p ) and BY = minpGM(GI,) hy(p) for i E N . By Lemma 1, the Nash outcome f N ( T " , d) is given by

6) Under the assumption of transferable utility, the existence can be proved by the standard fixed point argument for any n .

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Let E > 0 be given. Denote the numerator, maxiE”’ max,_, max,, ,, ‘&i(h;(S-i, s;) - h ; ( ~ - ~ , t i ) ) , of ( 5 ) by K”. By (9, we can take an integer no so that E > Kv/((iV“( - 1) for all Y 2 no. Consider an arbitrary game G” with Y 3 no. Let p be any Nash equilibrium in G”, and let q be an equilibrium threat point in r(G”, 2“). It suffices to show that for all i E N”, (i) f y ( T ” , h”(p)) + E 3 f y ( 2 “ , h“(p-;, r;)) for all r; E Mi(G”); and (ii) hr(q) + E 3 hY(q-;, r;) for all r; E M;(G”).

Consider inequality (i). Let r; be an arbitrary strategy in M;(G‘“). Since p is a Nash equilibrium, h;(p-;, r;) S h l (p ) for an arbitrary ri E M;(G“). Using (6) , the difference between fy(2“, h”(p-i, r;)) and fy(7“, h”(p)) is evaluated as follows:

fy(rV, h’b-i, Ti ) ) - ~ N ( T ” , h”b))

s K“/IN”I KV/( (NYI - 1) < E .

This implies inequality (i). Consider (ii). Let r; E Mi(G”). Since q is an equilibrium threat point, we have, by

(6),

o sf;(^", h”(q)) - f r ( ~ ” , h”(q-;, T i ) )

Thus we have

The left hand side is greater than or equal to -K“/IN“I, which implies

-K”/IN”I s (hY(q) - hY(q-;, ri))((N”I - I)/lN”l.

--E x ( IN”( - l ) /JN”J < (hY(q) - hY(q-;, r;))(lN”I - l)/IN’’l,

(7)

Since E > K“/(“/(INy( - l) , we have E x (IN“/ - 1) > K”. Hence we have, by (7),

i.e., hr(q) + E > hr(q-;, r;).

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Final version accepted August 12, 1995.

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