multilateral bargaining vijay krishna; roberto …...aspects of the theory, respectively. the...

Multilateral Bargaining

Vijay Krishna; Roberto Serrano

The Review of Economic Studies, Vol. 63, No. 1. (Jan., 1996), pp. 61-80.

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Review of Economic Studies (1996) 63, 61-80 01996 The Review of Economic Studies Limited

Multilateral Bargaining VIJAY KRISHNA

The Pennsylvania State University

and

ROBERTO SERRANO Brown University

First version received May 1993;$nu1 version accepted July 1995 (Eds.)

We study a multilateral bargaining procedure that extends Rubinstein's alternating offer game to the case of n players. The procedure captures the notion of consistency in the sense familiar in cooperative game theory and we use it to establish links to the axiomatic theory of bargaining.

1. INTRODUCTION

Game theoretic analyses of bilateral bargaining adopt one of two approaches-the axio-matic or the strategic. These reflect, for the most part, the normative and the positive aspects of the theory, respectively. The axiomatic approach originates in the elegant characterization of the Nash solution (1950). The strategic approach is best exemplified by Rubinstein's (1982) equally elegant analysis of a non-cooperative, alternating offer, bargaining game in extensive form. In Rubinstein's game players' time preferences determine a unique subgame-perfect equilibrium. Surprisingly, the two approaches can be neatly related. When players are patient, the equilibrium agreement of Rubinstein's game approximates the agreement dictated by Nash's axiomatic theory (Binmore, Rubinstein and Wolinsky (1986)). In this sense, the alternating offer game implements the Nash bargaining solution.

Nash's axiomatic theory of bilateral bargaining extends unchanged to inultilateral situations. Thus, given an n-person bargaining problem, a solution satisfying Nash's four axioms (scale invariance, efficiency, symmetry and independence of irrelevant alternatives) must maximize the product of players' utilities over the set of feasible payoffs.

Attempts to extend the strategic theory to the case of more than two players have been less successful. A possible extension of Rubinstein's game to the case of three or more players, which seems quite natural at first blush, is the following. Player 1 proposes an agreement. If all other players accept, the game ends. If any player rejects 1's proposal, play moves to the second period in which player 2 proposes an agreement, and so on. With these rules, any player can veto a proposal and only unanimous agreements can be executed. The game, of course, reduces to Rubinstein's game when there are exactly two players. Unfortunately, as pointed out by Shaked (reported by Sutton (1 986) and Osborne and Rubinstein (1990)), this game has many perfect equilibria. Indeed, if players are sufficiently patient, any feasible agreement can be achieved in a perfect equilibrium. Chang- ing the order of moves, the simultaneity of responses, etc., does not alter this conclusion.

This paper attempts to reconcile the axiomatic and strategic approaches to the multilateral bargaining problem. We suggest an n-person bargaining procedure different from

61

62 REVIEW O F ECONOMIC STUDIES

the "unanimity game" described above. Our procedure appears as natural as the unanimity game and, for n = 2, it also reduces to Rubinstein's. However, it has an advantage over the unanimity game in that it has a unique perfect equilibrium, for any number of players. Moreover, for all n, the unique equilibrium is characterized exactly as in the case of two players and the equilibrium agreement approximates the n-player Nash bargaining solution when players are patient. The uniqueness of the equilibrium is accomplished by allowing players to "exit" with "partial agreements".

We take as our starting point Lensberg's (1988) modern axiomatization of the Nash solution for multilateral bargaining problems. Lensberg's characterization relies on the idea, attributed to Harsanyi (1959), that a bargaining solution must handle problems with varying numbers of players in a "consistent" manner. The notion of consistency, described below, will play an important role in the interplay between the axiomatic and strategic theories for multilateral bargaining problems.

In related work Jun (1987) and Chae and Yang (1988, 1994) have proposed similar extensions to Rubinstein's game each of which also has a unique perfect equilibrium. Their procedures ask players to engage in a series of biluieral negotiations. Any player that reaches a satisfactory agreement may "exit". While the possibility of exit will also play an important role in our analysis, our mechanism is distinct from the Jun/Chae and Yang mechanisms in that offers are made to all the players simultaneously and thus the bargaining is wzultilaterul.

2. CONSISTENT SOLUTIONS

The following informal story motivates Lensberg's consistency axiom.' Suppose that the solution is interpreted as incorporating the decisions of an arbitrating court and that it prescribes a utility vector (u, , u2, u3) for some three player problem. Suppose that player 3 is satisfied with the amount u3 awarded by the court and "exits" the game with u3. Players 1 and 2, however, wish to appeal the decision by resubmitting the now "reduced" two-person problem to the court. Consistency requires that the court not change its decision on appeal. Of course, what it means for player 3 to "exit" the game and how the "reduced" two-player game is defined is crucial. A formal definition of consistency that specifies these notions precisely is given below.

The following notation will be adopted. Given a vector z= (z l , z2 , . . . , z,,), and A c N = (1, 2 , . . . , n), zA will denote the vector ( z ~ ) ~ ~ ~ and z(A) will denote the sum xjsAzi. Also, (pi , z - ~ ) will denote the vector z with its i-th component replaced by yi.

A bargaining problern among the n players in N is defined by a set B c IW: which is a compact, convex and comprehensive set of feasible utilities. It is assumed that the disagreement point has been normalized to 0 and that B contains a point u>>O. Let C ndenote the class of all n person bargaining problems and let C denote the union over all n of En.

A bargaining solution is a function Fdefined over C such that for every BEE, F(B) E B. F is consistent if, for all N, B and S, T cN, such that Sn T= a,and Sv T= N,

where B J , = r ( B ) = { ~ S J ( ~ S , FT(B))eB), is the section of the original set B along the axes in S when the utilities of players in Ta re held fixed at levels prescribed by F(B), provided,

1 . Lensberg (1988) uses the term "stability". We use the term consistency in order to be, well, consistent with similar usage in cooperative game theory.

KRISHNA & SERRANO MULTILATERAL BARGAINING

u21

of course, that BI,,.,,,EC.. This is illustrated in Figure 1 for the case when N = {l ,2 , 31, S = { I , 2) and T= ( 3 ) .

The Nash solution may easily be seen to be consistent in this sense. Lensberg (1988) shows that the Nash solution is the only one that satisfies the axioms

of scale invariance, efficiency, anonymity (a slight strengthening of Nash's symmetry axiom) and consistency. The power of the result comes from the fact that by considering problems with varying numbers of players, it is able to replace the axiom of independence of irrelevant alternatives with the axiom of ~onsistency.~

For our purposes, the import of Lensberg's theorem is the following. First, for a bargaining procedure to implement the n-person Nash solution, the equilibrium agreement (if unique) must be consistent. Second, if the extensive form of the bargaining game has subgames with varying numbers of players which are "isomorphic" to the original game, then the requirement of subgame perfection ensures consistency. These two observations suggest that the consistency axiom may be a valuable guide to the design of an appropriate bargaining procedure.

Consider the bargaining problem of dividing a dollar among three players. Suppose player 1 proposes a division x = (x,,x2, x3). If both players 2 and 3 accept the proposal the game ends with that division. If both reject, in the second period player 2 proposes a division y and players 3 and 1 must respond. If player 3 accepts x and 2 rejects, 3 can "exit" the game with an amount x3 while players 1 and 2 are left to bargain over the division of 1 -x3 in period 2. This bargaining now proceeds as in the two player alternating offer game with player 2 proposing a division of 1 -x3.

The game outlined above can also be interpreted as one where player 1 offers to purchase the right(s) to represent the accepting player(s) in any future negotiations. Thus in the example given above, player 1 purchases the right to represent player 3 in future

2. In Lensberg's (1988) original formulation the number of potential agents is infinite. A variant of the result is available, however, in which number of potential agents is bounded. This requires an additional axiom, that the solutioil be continuous. See Thompson and Lensberg (1989) for details.

64 REVIEW OF ECONOMIC STUDIES

negotiations at a price of x3.This amount is paid to player 3 immediately by player 1, who then bargains with player 2 over the whole pie. We will refer to this game, interpreted in either way, as the "exit" game. We explain and explore these two alternative views of the game in more detail below.

The exit game is different from Shaked's unanimity game in that a player satisfied with his or her share can "take the money and run". To appreciate the difference, suppose that n= 3, and that player i's payoff from receiving an amount zi in period t is 6'-'zi. We first argue that if 1/2 <6 < 1, (0, 1,O) is an equilibrium agreement of the unanimity game. This can be supported by the following strategies. In any period t, the player whose turn it is to offer, offers e(t)e {el, e2, e3}, (ei is the i-th unit vector) according to the following rules. In period 1, e(l)=e2 and both players 2 and 3 accept. In any period t, if the offer of e(t) is made and rejected, then e(t + 1) =e(t). In any period t, if an offer of x #e(t) is made, then e(t + 1) =e' where i is the index of a player who responded in period t and for whom x i s 1/2. Finally, if player j is a responding player in period t, and e(t + 1)=e ',then j accepts an offer x if and only if x ,26 . If e(t+ ])#el , j accepts any offer. It can be verified that these strategies form a subgame-perfect equilibrium of the unanimity game."

However, (0, 1,O) cannot be an equilibrium agreement of the game where a player can exit if satisfied with the amount he or she is offered. For any E>O, if player 1 offered (1 -6 -E, 0, 6 + E), player 3 would certainly accept and exit. Player 1's payoff would then be positive in the resulting subgame: an alternating offer game between 1 and 2 over a pie of size 1-6 - E.

3. A MULTILATERAL BARGAINING PROCEDURE

We consider situations where a set of players N= { l , 2, . . . , n) are bargaining over a "pie" of size q. An ugreerlzent or offer is a vector x= (xl,x2, . . . ,x,) in which xi is player i's share of the pie. The set of possible agreements is

X = {xelWn: for all i, x i 2 0 and C / t N ~ i = q } .

The extensive form of the bargaining procedure in which n players bargain over a pie of size q will be denoted by GI(N; q ) , and is defined recursively. For any set T= {i, j) consisting of two players and r s q , Gi(T; r) will denote Rubinstein's alternating offer game described as follows.

In period 1, player i proposes an agreement xr= (xi, xi) which j must either accept or reject. If j accepts, the game ends and players receive the amounts prescribed by xr. If j rejects, then in period 2, j proposes an agreement J ~ Twhich i must accept or reject. If i accepts, the game ends. If i rejects, then i must make an offer in period 3 and so on. Notice that if i's offer in Gi(T; r) is rejected then G,(T; r) is played.

For any SGN, i e S and r 5q, define Gi(S; r) as follows. In period 1, player i makes an offer xs. All players j # i respond by accepting or rejecting xs. The responses are made sirnultaneou.sl~~.If j accepts the offer xs, he or she receives the amount 3 i~nrnerliately. Let A cS \ i be the set of players who accept i's offer in period 1. If A =S \ i then all players, including i, receive their share immediately. If #AcS\ i , then in period 2, Gj(S\A; r-xs(A)) is played where j is the smallest index in S \ A that is greater than i (if i is the

3. It can be argued that if 6 > 1/2, any division of the pie can be supported as a subgame-perfect equilibrium of the unanimity game. There is, however, a unique stationary equilibrium agreement. In the case of three players with linear utilities, this results in a division proportional to (1, 6, 62).

65 KRISHNA & SERRANO MULTILATERAL BARGAINING

largest index in S\A, j is the smallest index in S\A). If A =@, then in period 2, Gj(S; r) is played ( j is determined in the same manner as above).

Figure 2 depicts the game tree associated with G I ( N ; q ) for the case when N = (1, 2, 3) . Some features of the game are worth drawing attention to. First, in any Gi(S; r ) , the

person making the offer, player i, receives a payoff in the first period if and only if all other players accept his or her offer. Second, while a player ~ E Areceives the amount xj immediately this can happen in two different ways which are formally equivalent. Player j could receive the amount from an existing "pie", as, for instance, in an estate division problem. Alternatively, it is possible that no "pie" exists unless all players agree, as might happen in the case of a firm bargaining with more than one union. The model can then be interpreted as one where the proposing player purchases the rights to represent players in A for a total of x ( A ) and pays this amount by borrowing at no cost outside the game. The idea of interpreting "exit" in this way also occurs in Jun (1987) and in Dow (1989). We discuss this issue in more detail in section 8 below.

It remains to specify how players evaluate outcomes of G I ( N ; q ) . Player i's payoff from receiving an amount xi in period t is 6'-lui(xi) , where u i :R++R+ is a strictly increasing, continuous and concave function satisfying ui(0)=0 and 0 <6 < 14. The payoff to a player from perpetual disagreement in some game Gi(S;I.) is assumed to be 0 .

4. EQUILIBRIUM IN THE TWO-PLAYER GAME

Our analysis of the n-person game, is founded upon Rubinstein's (1982) analysis of the two-person alternating offer game. We begin by restating Rubinstein's main result in a form that will facilitate comparison with what follows.

4. It would be enough, for our purposes, to assume that players' preferences over outcomes satisfy the assumptions Al to A6 used by Osborne and Rubinstein (1990). Payoff functions of the type S:.-'u,(x,) form a convenient and identifiable class satisfying these assumptions, and we have adopted this specification merely for ease of exposition. The additional assumption that all players' discount factors are the same results in no further loss of generality.


Theorem 0. Suppose players 1 and 2 are bargaining over apie of size q in the alternating ofer game GI({], 2) ;q). Let (x:, x:, a*) be the (unique) solution to:

The unique perfect equilibrium agreement in GI({1,2); q) is (x: +a* , x:) and is achieved in period 1.

First, notice that the amount a * may be interpreted as the premium obtained by each player when playing the role of proposer over that when playing the role of responder. It is the same for both players.

Second, if the pie over which the two players are bargaining increases in size, x? ,x; and a * all strictly increase. To see this, suppose that (x?, x:, a*) is the solution to (1) as above and let (z:, z:, y*) be the solution when the size of the pie is r > q . Suppose, for instance, that z: s x : . Then from (1) and the analogous equations when the pie is of size r, we must have that y* 5a * and hence also that z; sx ; . But adding these up implies that r s q , which is a contradiction. Similarly, if y * s a * , then z:sx? and z; sx:, which is impossible. The property that each agent's share is increasing in q has been called resource nzonotonicity (Roemer (1986), Chun and Thompson (1988)).

Third, it can be argued that the unique perfect equilibrium strategies, a*,i= l , 2 , are as follows: in Gi({l, 2 ) ; q), i makes the ofer (x) +a*, xi*) and player j # i accepts if and only if ofered an amount greater than or equal to xi*. Notice that the strategies are stationary.

Thus far, in the interests of notational simplicity, we have suppressed the dependence of the equilibrium on the discount factor 6. Let GI({], 2 ) ; q)' denote the alternating offer bargaining game when players use a discount factor of 6 to evaluate future payoffs and let (x?(6), x2*(6), a*(6) ) be the corresponding solution to (1). By Theorem 0, for each 6, there is a unique equilibrium agreement of GI({], 2) ;q)" which is (xf (6) +a*(6) , xz(6)).

Define B= {(ul(xl), u2(x2)) :X I +x2 5 9 ) . The Nash bargaining solution selects the agreement that maximizes uI (x l ) x u2(x2).

For two player games, the relationship between the strategic and the axiomatic theories is given by the following result (Binmore, Rubinstein and Wolinsky (1986)).

Theorem 0'. The lirnit, as 841, of the unique equilibriuni agreenzent (xT(6)+ a*(S), x:(S)) in GI({], 2 ) ; q)', is the agreement given by the Nash solution to the bargaining problern 3.

While the proof of Theorem 0' follows directly from examining the equations (1) in the statement of Theorem 0, the result is rather surprising. There is little in the structure of Rubinstein's game that portends such a connection between the strategic and axiomatic theories. We will argue later that once a multilateral extension of Rubinstein's game is considered, the consistency axiom leads to some insight on this issue.

5. EQUILIBRIUM IN THE n-PLAYER GAME

In this section, we establish n-player analogues of Theorems 0 and 0'.


Given a set of players N and a pie of size q, we will refer to the equations:

u,(xi)=6ui(xi+a) , i e N;

as the characteristic equations for (N; q). We have already seen in Theorem 0 that when N consists of just two players, the characteristic equations determine the unique equilibrium of Gi(N; q).

First, the characteristic equations have a unique solution (x*, a*). To see this, initially treat a as a parameter and let xu be the corresponding solution to the equations:

ui(xi) =6u,(xj+ a ) , EN.

Notice that for all a LO, there is a unique such xu, and that xu is a continuous function of a. Furthermore, xO=O and as a increases, each component of xu increases without bound. All of these properties are therefore true for the sum xu(N) +a . This implies that the characteristic equations have a unique solution.

Second, the solution (x*, a*) is strictly increasing in the parameter q. Thus, the solution to the characteristic equations satisfies resource nzonotonicity.

Third, if (x*, a*) is the solution to the characteristic equations for (N; q) and S c N , then (xz, a*) is the solution to the characteristic equations for (S; x*(S) +a*). Thus, the solution is consistent.

Finally, suppose that S c N and that (ys, P) is the solution to the characteristic equations for (S; r). Let i$S and let T = S v (i}. Suppose that (zT, y) is the solution to the characteristic equations for (T; r + w,). Define y, by ui(y,) =6ui( yi+ P ) . Then (i) w,= yi if and only if y =P, zj= y, and for all j eS , zj=j?; (ii) wi> yi if and only if y > P , wj> z,> y, and for all ~ E S , zj> yj; and (iii) wi<yi if and only if y < P , w,<zj<yi and for all j eS , z,<yi. Intuitively, the amount yi is such that if player i joins the players in S with an endowment of y,, the solution to the characteristic equations is unchanged. If a player i joins S with an endowment of IV; which exceeds y,, player i is "taxed" by the players in S and receives an amount z j< poi. On the other hand, if i's endowment w, is less than y,, player i is "subsidized" by the other players and receives an amount zi> w, . This is referred to as the redistributive property of the solution.

To summarize, the solution to the characteristic equations is unique, resource mono- tonic, consistent and recii~tributive.~

We can now state our main result.

Theorem 1. Suppose players irz N are bargaining over a pie of size q in the game GI (N; q). Let (x*, a*) be the solution to the cltaracteristic equations for (N; q) :

u;(x,) =6uj(xi+ a) , i e N;

The unique perfect equilibrium ugreernent in GI (N; q) is (x? +a*, xTl) and is achieved in period 1.

5. The bargaining solution implicit in the utility vector rr,(x*), i~ N, was studied by Lensberg and Thomson (1988) who call it the N6 ("Nash-like") solution. They show that the family of solutions Ng for SE[O, I ] may be characterized by the axioms of synmetry, scale invariance and consistency. Although Lensberg and Thomson's concerns are purely normative, we will argue below that these solutions also have interesting strategic properties.


The formal proof of Theorem 1, which follows in the next section, is somewhat involved and so we provide an outline of the argument.

We first show that the agreement (x: + a*, xEl), achieved in period 1 can be supported as a perfect equilibrium outcome. Lemma 1 is a preliminary step and shows that (x: + a*, x*,) maximizes player 1's payoff among all agreements that would be accepted by all responding players if all other responding players were to also accept. Lemma 2 then provides the equilibrium strategies. While the supporting strategies are somewhat complex, it should be noted that since GI(N; q) is not a game of perfect information, standard existence theorems (e.g. Harris (1985)) are not applicable and there is no pre- sumption that it even has a perfect equilibrium.

The proof that (x: + a*, x?,) is the unique equilibrium agreement in GI(N; q) is by induction on the number of players6 (We know from Theorem 0 that this is true for all two-player games.) First, we argue in Lemma 3 that player i's payoff in an equilibrium of G;(N; q) is at least ui(xi* + a*). Lemma 4 establishes that an equilibrium of the game cannot be such that an agreement is reached in two consecutive periods, with some players accepting and exiting in some period t and the remaining players reaching an agreement in the next. Thus every equilibrium agreement is the result of unanimous acceptance. Lemma 5 collects the arguments of Lemmas 1 and 4. Any delay will reduce player 1's payoff contradicting Lemma 3. Thus the only possible outcome is one with immediate agreement and Lemmas 1 and 2 show that it must be the solution to the characteristic equations.

6. PROOF O F THEOREM 1

The formal proof of Theorem 1 is incorporated in Lemmas 1 through 5 below. For any Swith more than two players, define A;(S; r) to be the set of offers by player

i in Gi(S; r) that would be accepted by all players j f i provided that all players in S \ i other than j were to accept also. Suppose that player i makes an offer of ws and that all players except j accept. If j rejects ws, then in period 2, Gj([i, j) ; w,+ wj) would be played. Playerj would accept ws if and only if ~bwere at least as large as the discounted value of j's payoff in G,({i, j ) ; w;+ wj). Formally, wseAi(S; r) if for a l l j f i , ~ ~ ~ z , w h e r e (zj, d,a ' ) satisfy:

It is easy to verify that if (x:, x:, . . . , x,*, a*) is the solution to the characteristic * equations for (N; q) then (x: + a*, x:, XJ , . . . , x:) eAI(N; q).

Lemma 1. Let (x*, a*) be the solution to the characteristic equations for (N; q) . Then (x* + a*, x") is the unique maxinzizer of i's payof in Ai(N; q).

Proof. Suppose there exists another offer in Ai(N; q), say w=(x) + a*+ci , (x,? + c,)~,~) such that ciL 0, c(N) = 0 and at least one cj#O.

6. Our proof concerns pure strategy equilibria only. A full consideration of mixed strategies would take us too far afield.


Consider any k such that ck<O and suppose that all other players accept w. If k rejects w, Gk({i,k ) ;r) would be played, where r =(s*+a*) + (x: +ci+ck). Let (zi, zk, y) characterize the equilibrium of this game.

(i) If ci+ ck20, then by the redistributive property, zk+ y >x: +a* and hence 6uk(zk+ Y) 2 6uk(xE;*+a*) >uk(xE;*+ck).

(ii) If ci+ck<0, then again by the redistributive property, zi <xi*. But this implies that

But then 6uk(zk +y) >6uk(x: +a* +ci+ck) >uk(xk*+ck).

Thus, k would reject the offer PV if all other players were to accept and hence w is not in Ai(N; 9). II

Lemma 2. The agreement (x?+a*, xFI), aclzieved inperiod 1, is apeifect equilibrium outconle of GI (N; q).

Proof. We construct the perfect equilibrium strategies as follows. Consider any Gi(S; r) and let ( y ,0 ) be the solution to the characteristic equations for (S, r).

Let x be an offer in Gi(S; r). First, for each j ~ S \ i define a number Y.' by: uj(xj) = 6uj(z:) = ij2uj(zj+ y j). We will now define the set of players A(x) c S \ i who will accept x. This will be done in two steps.

Step I. First, define the sets A', A' , . . . , A T recursively as fo~lows.~ Let A'= % and R0=S. For t 2 0 , given A', A ' , . . . , A', and RO, R' , . . . , R' define A"' { ~ E R ' := xj2y;) where (y', p') is the solution to the characteristic equations for (R', x(Rf)) and define R"' =s\(A' v . . . v A' v A'+'). Continue in this manner until there is a T such that for every jeRT\i, Since N is finite, such a Texists.

Since R0 =S, (yo, pO) = (y, p), the solution to the characteristic equations for (S, r). Observe that by the redistributive property of the solution to the characteristic equa-

tions we have that ~ = P O ~ P ~ Z P * ~ . . .>=PT. Next, consider the set RT and let keRT be the player with the lowest index higher

than i. Suppose that RT\{i, k} consists of P players relabelled so that

Step II. Now define

if ujl (xil 2 sujl(Y:) otherwise.

R ~ + ' = s \ ( A ' v . . . v A T v AT") and f o r p z 2 ,

if ~+~,(x,~,) and AT+"- '2 6 ~ , ~ ( y ~ + ~ - ' ) #(a

A""= otherwise.

where as above R T + " = ~ \ ( A ' v . . v A T v A T + ' v . . . v AT+") and (yT+", PT+") is the solution to the characteristic equations for (RT+", x(RT+")).

7. Of course, each A' depends on .x and strictly speaking, we should write A1(.u) instead of A'. Although, we are suppressing the argument for the sake of notational ease, this dependence should be understood.


+Finally, define the set ~ ( x ) = A ' u A2 u . . . u A T u A T + ' u u + u A T + and R(x) = S\A(x).

We can now write the strategies that support (x: + a*,XT,) in GI(N; q): for all S c N and r 5q, in the game Gi(S; r)

(i) player i n~akes the offer (yi+ p, ; (ii) following an ofer x by player i, j ~ S \ i accepts ifand only i f j ~ A ( x ) .

We now verify that the strategies given above form a perfect equilibrium. As usual, because of discounting, it is sufficient to verify that following any history, it is not profitable for any player to deviate once and conform to the suggested strategies thereafter.

Let (2, y) be the solution to the characteristic equations for (R(x), x(R(x))). When- ever R(x) #S, it follows that for all j ~ A ( x ) , y ' 2y and for all j~R(x)\{i, k}, y j < y.

First, we argue that every player j # i is at a best response. Consider the last player (1) that was included in the set A(x) (that is for all p > l ,

AT+"= a),1f ~ E A ~ + Iu . . . u AT+ ' as in Step I1 then 1 is at a best response because by definition of A', 1's payoff from accepting is at least as large as his payoff from rejecting. If I E A ' u A 2 u . . u A T as in Step I, then player 1 is at a best response because of the +

redistributive property because by rejecting he will join players in R(x) for whom y ' is lower.

By the redistributive property, the rest of players in j ~ A ( x ) for whom y ' 2 y ' are at a best response a fortiori.

Next, consider the players in R(x). Consider the "first" player f#k such that f#A(x). Clearly,f joins R(x) as a result of Step I1 "after" A ' , A2, . . . , A T have been determined. Hence, by looking at his acceptance rule, f is at a best response.

By the redistributive property, the other players in j~R(x) \{ i , k) are also at a best response since they have lower premia, that is, y ' 5y '.

Finally, player k is at a best response because he becomes the proposer the next period, and uk(xk) < uk(.d) = 6uk(y;+ pT).

To conclude, we show that player i is also at a best response. We argue by induction on the number of acceptors a(x) = I A(x)l of a proposal x. If

x is unanimously accepted, that is if A(x) = S\i, then X E Ai(S, r) since all the players j ~ S \ i are at a best response. By Lemma 1, the offer (yi+P, y-)) is the unique maximizer of player i's payoff in Ai(S, r). Thus player i cannot improve his payoff by making an offer x such that a(x) = s - 1 .

Suppose that all offers x with a(x) > a > 0 are not profitable deviations. We will argue than an offer x with a(x) = a cannot be profitable either.

We argue by contradiction. That is, suppose there is an offer x with a(x) = a which is a profitable deviation.

Case 1 . { i ,k) cR(x) and { i ,k) # R(x). Denote by f the player in R(x) associated with the biggest premium: yfz for all j~R(x)\{i, k3. If i makes the offer x, his payoff J

is 6ui(zi) and player f's payoff is 6u1(zf) where as above, (2, y) is the solution to the characteristic equations for (R(x), x('R(x))). Consider an alternative offer w, such that wf = zr- E, wi = xi+ xf -yf and w, = x, for all j# i, f. For a small enough E, every player jeA(x) will accept this offer for exactly the same reasons that j accepted the offer x. Furthermore f will also accept this offer and w will be rejected by the rest of players in R(x). (This again uses the redistributive property.) Hence A(w)= A(x) u { f ) and thus w is a profitable deviation with a(w) >a, which is a contradiction.

11

KRISHNA & SERRANO MULTILATERAL BARGAINING 71

Case 2. R(x) ={i, k). Then, choose w such that wk =zk+ y -E, wi=xi+ xk -yk and for all j# i, k, u;.= xi.By the same arguments as above, all the responders would accept w. Hence, w is a profitable deviation with a(w)=s- 1 , which is impossible.

Finally suppose that x is such that a(x) =0. Then by the equilibrium strategies, player i's payoff will be 6ui(yi) <ui(yi+p). Thus no such deviation is profitable. Notice that here we are applying the principle that only single deviations need to be ruled out.

To prove that (x:+a*, x?,) is the unique equilibrium agreement in G,(N; q), we proceed by induction on the number of players. Theorem 0 establishes this for all two- player games. Now suppose that all games G,(S; r) with S c N and 1.59 have a unique equilibrium agreenzent (y? +p*, Y:,~), where (y: ,p* ) is the (unique) solution to the characteristic equations for (S; r).

First, observe that perpetual disagreement cannot be an equilibrium outcome. If it were, then for small enough E>O, an offer of (E, q - E, 0,0, . . . , 0) by player 1 would certainly be accepted by player 2 and by the induction hypothesis player 1 would be guaranteed a positive amount in the resulting subgame.

Lemma 3 below establishes that player i's payoff in an equilibrium of G,(N; q) is at least ui(x) +a*) . Since the proof of Lemma 3 is rather involved we sketch the underlying idea briefly. The proof proceeds by asking player i to make the offer x'(K)= (x) +(1 -~ ) a * ,(x: + ( ~ / ( n- l))a*),,'). It is easy to argue that such an offer must either be unanimously accepted or unanimously rejected. If it is unanimously rejected it must be that players j # i expect the equilibrium in the ensuing subgame to be very advantageous. We then examine the resulting equilibrium and argue that this equilibrium is in turn supported by an equilibrium in which players expect even larger shares and so on. Eventu- ally the expectations of players become mutually incompatible and thus x'(K) could not have been rejected. Hence it must have been unanimously accepted. The formal proof of Lemma 3 follows this general line of reasoning but is rather complicated since at each step, the structure of the supporting equilibrium has to be inferred. In the interests of exposition, the proof of a technical step (Claim 3.4) has been relegated to an appendix.

Lemma 3. In any equilibrium of G,(N;q), player i's payof is at least ui(x) +a*).

Proof. For K >0 define x'(K) = (x) + ( 1 -~ ) a * , (xi* + ( ~ / ( n- ]))a*),,'). We will argue that for ail K >0 , the offer xi(^) will be unanimously accepted. Thus if player i's payoff in an equilibrium of Gi(N; q) is less than u,(x: +a*), i will be able to break the equilibrium by offering for some small K, x'(K) and having it unanimously accepted.

Suppose that all players follow o after the offer x'(K) is made and let the set of players that accept x'(K) be denoted by A . Define R = N\A. We first argue that for all K, x'(K) is either unanimously accepted or unanimously rejected.

Suppose there exists a K such that the corresponding A #(a and A # N\i. Let the game played next period be G,,(R; .x~(K)(R)). Notice that the size of the remaining pie in period 2, x'(K)(R) <x*(R) +a*. By the induction hypothesis, those players who reject the offer are not playing a best response to the other players' strategies.

We now argue that for all K, x'(K) cannot be unanimously rejected. Suppose there exists a K such that x'(K) is unanimously rejected. In that case, let Y'(K) be the agreement prescribed by o in the resulting subgame.

Suppose player i receives an amount y j ( ~ ) in period t. Suppose that player j gets his or her share y ; ( ~ ) in period tj (notice that for all j, t,=t- 1, t or t + 1 and ti> 1 ) . Since

REVIEW OF ECONOMIC STUDIES

player j # i rejected x'(K), y ; ( ~ )> x;(K) + (ti- l )a*>xj* +a*. So i's share, y : ( ~ ) < x j ( ~ ) - ( n -l )a*<x) -a* .

Claim 3.1. Suppose x ' ( K ) is unanimously rejected. There exists a K' 2 K such that xi(^') is also unaninzously rejected and y i ( ~ ' ) is reached without playing Gi(N; q) again.

Proof of Clainz 3.1. Suppose that y ' ( ~ )is reached after Gi(N; q) is played. For all j f i we must have that y ; ( ~ )> x ~ ( K ) <X: (K) l )a* < x ~ ( K + 1 ) .+na* and thus y j ( ~ ) -n(n-Since Gi(N; q ) is played, i can make the offer X ' ( K + 1) which must be unanimously rejected otherwise y ' ( ~ )could not be an equilibrium agreement. So we can consider a corresponding y ' ( ~+ 1 ) . 1f y ' ( ~+ 1 ) is also reached after Gi(N; q) is played again, X ' ( K +2 ) must also be unanimously rejected and so on. But this leads to a contradiction since eventually this would imply that there is an equilibrium agreement of the form y ' ( ~ " )in which i's share is negative. 11

Because of Claim 3.1 we can assume, without loss of generality, that K =K', that is if x ' (K)is unanimously rejected, y i ( ~ )is reached without playing Gi(N; q ) again.

Claim 3.2. y ' ( ~ )cannot satisfy any of the following:

( I ) y :(K)eAj(N; q) , for j # i ; (11) Y ' ( K ) is reached in periods t - 1 and t ;

(111) y ' ( ~ )is reached in periods t and t + 1 and if i deviates and rejects the offer in period t , the induction hypothesis may be applied.

Proof of Claim 3.2. To rule out (1) recall that by Lemma 1 , y : ( ~ )>xi* +a* is impossible and thus y ' ( ~ ) $A , ( N ; q).

By the induction hypothesis, since y j ( ~ )<xi*, for all players j that bargain with player i in period t, y j ( ~ ) <xi* +a*. Thus ( 1 1 ) is not possible.

To rule out (111) let the period t offer be denoted by x and notice that for all players j who accept x , x,= y ; ( ~ ) . We will argue that i can profitably reject x. Recall that in this case, i's share ~ : ( K ) < X ; ( K )t(n- l ) a* +aa*, where a<n-- 1 denotes the number of players who accept x. By the induction hypothesis, this is a profitable deviation for player i because after i's rejection, the share of each bargainer in period t+ 1 goes down by at least a*. 11

Claim 3.2 establishes that if the offer x ' ( K ) were made in Gi(N; q ) and unanimously rejected, the resulting equilibrium agreement y ' ( ~ ) ,has the following properties: (i) it is reached over two periods, say, t and t + 1 ; (ii) player i alone accepts the offer x in period t. We will refer to such equilibria as Type IV equilibria.

We now define a new equilibrium agreement z ' ( K ) as follows. Suppose that if i were to reject x the subsequent equilibrium agreement is x' and that

this is the result of an equilibrium that is not of Type IV. Then let z ' (K) =xt . If x' is of Type IV, let x" be the equilibrium that occurs if i rejects the first period offer that results in x'. If x" is not of Type IV let z ' ( K ) =xu .Otherwise consider the equilibrium that results if i were to reject the first period offer that results in x" and so on. In general z ' ( K ) denotes the agreement resulting from the first non-Type IV equilibrium following a succession of rejections by player i. Notice that z'(K)is the equilibrium agreement of a subgame reached after a sequence of deviations by i alone. Suppose that z i ( ~ ) ,is such that i gets his share


in some period s> t. We now study the various possibilities for the equilibrium agreement Z f(K ) .

Claim 3.3 There exists a K' 2 K such that z ' (K' ) is reached without playing G,(N; q) again.

Proof of Claim 3.3. Suppose that z'(K) is reached after G,(N; q ) is played again and thus s 2n + I . First, suppose s =n + 1 . Then, i's share z ~ ( K )<y j ( ~ ) + ( n+ 1 - t)a* <x; (K)- t (n - l )a*+aa*+(n+ 1 - t ) a * s x j ( ~ ) - a * . (Since t 2 2 and a=n-2.) This is because in each of the periods between t and n + I, the share offered to player i can go up at most by a*, otherwise i's deviations would be profitable and y ' ( ~ )could not be an equilibrium agreement. Thus i's payoff in G,(N; q) in the equilibrium agreement z ' ( K ) is less than u,(x;(K)-a* )=u , ( x j ( ~+ 1 ) ) . Now consider the case where s >n + 1. By the same argument as above, if i's payoff in the equilibrium agreement z'(K) were u; (x~(K+ I ) ) or greater, i would find it profitable to deviate from Y ' ( K ) in period t which is impossible. So z'(K) is an equilibrium agreement of Gi(N; q) in which i's payoff is less than u ~ ( x ~ ( K+ 1)).

So, the offer X,(K + I ) made by player i in Gi(N; q) must be unanimously rejected and we can re-start the beginning of the proof leading to the offer X ' ( K +2) that is unanimously rejected, and so on. Eventually, there will be an equilibrium such that player i's share is negative. Hence, for some K' the agreement z ' (K ' ) must be reached without playing G,(N; q) again. I/

Again, without loss of generality, we can assume that K'= K , that is z ' ( K ) is reached without playing G,(N; q) again.

Claim 3.4. z ' ( K ) cannot satisfy any of thefollowing:

( I ) z i ( ~ ) ~ A i ( N ;q),for j f i ; (11) z ' (K) is reached in periods s- 1 and s ;

(111) z ' (K ) is reached in periods s and s+ 1 and if i deviates and rejects the ofSer in periods, the induction hypothesis may be applied.

Proof of Claim 3.4. See Appendix. 11

Since by definition z ' ( K ) was not of Type IV, as a result of Claim 3.4 we have established that no such equilibrium can exist. This establishes that x ' ( K ) cannot be unanimously rejected. Thus it must be unanimously accepted. 11

We argue next that there are no equilibria in which an agreement is reached in two parts, with some players accepting in period t and the remainder reaching an agreement in period ( t+ I ) .

Lemma 4. There is no equilibriurlz of Gi(N; q) such that i's initial ofSer is accepted by some but not all players and the remaining players reach an agreement in the next period.

Proof. Without loss of generality let i= 1 . Suppose o is such an equilibrium. Let 1's initial offer be x and suppose it is accepted

by players in A cN\I, A $121. Let R =N\A and suppose that the game Gk(R;x ( R ) ) is


played in period 2. By the induction hypothesis, the unique equilibrium split in this game is defined by the solution to the characteristic equations for (R; x(R)), say, (y,, P).

By Lemma 3, 6ul(yl) zul(x: + a*). Define y = y, -xT -a* as the additional share received by player 1 over and above xT +a*. First, notice that the characteristic equations that define a* and P, together with the last inequality imply that a* <y < P . Now define for all ieA, ci=xf -xi, as the amount "conceded" by player i relative to x*. Second, notice that Lemma 3 also implies that the total amount conceded by the players in A, c(A)=-P+ Y .

Case 1 :A contains at least two plaj~ers. We will argue that if 1 were to make the offer z that gives every player an amount E

greater than his or share in the above equilibrium it would be unanimously accepted. For small enough E this would be a profitable deviation for 1 and hence o could not be an equilibrium.

For all ieA, let Ri= R u (i). Define (w', qi) to be the solution to the characteristic equations for (R'; x(R')).

Proof of Clairtz 4.1. Consider player i and suppose that @a*. Notice that since xi<xf ,ui(xi)<6ui(xi+a*) 56ui(xf). We now show that player i could profitably reject x. Then, if i rejects x, the size of the pie in the next period will be x(Ri)=x(R)+xi= [x*(R)+a*] +[c(A) +xi]. If [c(A) +xi] <xT , then by the induction hypothesis, and the redistributive property of the resulting equilibrium, i's share, wj, will be at least [c(A) +xi] >xi+ a*. On the other hand, if [c(A) +xi] 2 x f ,wjzxf . In either case, i should reject x. Thus ci <a* and hence wj >xf . 11

For E>O, consider an offer z by player I such that (i) for ieA, zi=xi+ E; (ii) z k =

y k + P + & ; (iii) f o r j e R \ ( l , k J , zj=yj+&; and (iv) z l = y l - ( n - 1 ) ~ . (Recall that k is the player who makes the offer y in period 2 in the equilibrium.) We now argue that this offer will be unanimously accepted, and thus for small enough E, player 1 can profitably deviate.

Claim 4.2. z is not unanimously rejected.

Proof of Claim 4.2. If z were unanimously rejected, in the resulting subgame, G2(N;q), each player ieA must get a share greater than xf (because of Claim 4.1); player k must get a share greater than $+2a* ;and each player j~ R\( l , k) must get a share greater than x; +a*. By Lemmas 1 and 2 the unique unanimously accepted offer in a subgame perfect equilibrium is the solution to the characteristic equations. Hence the game cannot end with a unanimous agreement.

Thus the resulting agreement must be reached over two periods, say as follows, player 2 makes an offer of x' in period 2, this is accepted by the players in A' and the remaining players in R' reach an agreement (y',,, P') in period 3. By Lemma 3, applied to G2(N; q), it is not possible that player 1 rejects x' because then the concession of the set A', c'(A1) =

x*(A1) -x'(A1) would be positive and thus at least one player icA' must be receiving a share less than x f . Therefore, player 1 must be accepting a share less than x: - (2r+a- 2)a* sx: -na* in period 2. By Claim 4.1, it must be the case that A ' = (11.

Suppose player 1 rejected the offer x'. Lemma 3 applied to Gi(N; q) (2 5 i s n ) and Claim 4.1 show that the supporting equilibrium in the subgame must be of the same kind

KRISHNA & SERRANO MULTILATERAL BARGAINING 75

(player 1 must be the only accepting player). Recall that the share accepted by 1 in period 2 was less than xT -na* and consider a sequence of (at most) n -1 deviations made by player 1 in which he or she rejects the offers that he or she alone was accepting. This constitutes a profitable deviation because GI(N; q) will be played in n - 1 periods and, by Lemma 3, player 1's payoff is at least 6"-lul(x: +a*) > 6ul(x: -na*). Hence, z cannot be unanimously rejected. 1 1

Claim 4.3. z will be unaninlously accepted.

Proof of Claiilz 4.3. Suppose that S is the set of players who reject z, together with player 1 . By Claim 4.2, S f N and thus the induction hypothesis may be applied to the game in period 2. Let (v, z) be the solution to the characteristic equations for ( S ;z(S)). Notice that we must have z jp otherwise u l >yl and z is a profitable deviation for 1.

(i) All players j e R must accept z. This is because ss p implies that v js yj< zj. Next for all ieA, define yi by ui(yi)= 6ui(yi+p) and partition the set A as follows: A + = ( ieA: x izyi ) and A - = ( ieA: xi<yi).

(ii) All players l z ~ A + must accept z for the same reasons as in (i). (iii) Finally, all players in A- must accept z. Suppose not, then let ~ E A -be the player

with the lowest index who rejects z. Then we must have that P > qi because otherwise by the redistributive property x i> wj> yi , which is a contradiction.

Next, notice that qi> z. This is because first, for every jeR \ l , xi< yj and hence x l >yl . Second, all ~ E A +accept x. Third, by (i) every player jeR accepts zj>yj. Therefore, z(S) <x(S). Hence if player i accepted x he or she must accept z.

Thus all players will accept the offer z. 11

Hence we have argued that 1 can profitably deviate by offering z and o could not be an equilibrium.

Case 2: A is a singleton. Suppose A = (2). Since c2>P +y, player 2 would profitably deviate by rejecting player

1's offer. Similarly, if A = {i), this equilibrium could only be "self-supported". Since in each

of these equilibria, c i > p + y, a sequence of (at most) i- 1 deviations is profitable (by Lemma 3 applied to Gi(N; 9)). 1 1

Lemma 5. The unique equilibriunz agreement in GI(N; q) is (x: +a*, x ? ~ ) and is reached in period 1.

PiooJ Since i's payoff in Gi(N; q) is at least ui(x: +a*) and i's payoff in Ai(N; q) is at most ui(x) +a*), the only equilibrium outcome of Gi(N; q) with immediate agreement is for all other players to accept. But this implies that the only equilibrium outcome of G1(N; q) is for 1 to offer (x: +a*, x t l ) and for all other players to accept. Any unanimous agreement reached after delay would result in a payoff to 1 of less than ul(x: +a*), but by Lemma 3, that is impossible. 1 1

Lemmas 1 through 5 together complete the proof of Theorem 1. An interesting feature of the mechanism G is that the only asymmetry in the way

players are treated arises from the fact that one of the players is the proposer and reaps


the premium a*. More precisely, notice that for all distinct i, j and k, player k's share in Gi(N; q) is the same as his or her share in G,(N; q).8 Moreover, player k's share as a responding player is independent of the exact "protocol" that is followed in choosing the next proposer. For instance, player 3's share is the same whether the proposals are made by players in the order 1, 2, 3 or 1, 3, 2 or 2, 1, 3 or 2, 3, 1. This is in marked contrast to the shares in the unique stationary equilibrium of Shaked's unanimity game where the equilibrium agreement is sensitive to the exact protocol.

7. RELATIONSHIP TO THE NASH SOLUTION

We are now ready to establish a direct analogue of Theorem 0'. Define B = { ( ~ ~ ( x ~ ) ) ~ , , : x ( N ) g q ) to be the set of feasible utilities. Let F&(B) be the

utility vector prescribed by the unique equilibrium of GI(N; q)" (We are again making the dependence of the game and its equilibrium on the discount factor 6 explicit). Fa, interpreted as a solution operating on all such bargaining problems, is scale invariant and efficient. In the limit, as 6 4 1 , F6 is anonymous and consistent. In light of Lensberg's theorem, it is not surprising that the agreement prescribed by F8 is, in the limit, the same as the agreement prescribed by the Nash ~ o l u t i o n . ~ Thus the intriguing relationship between the strategic and axiomatic theories can be traced directly to the consistency property.

Our final result is an exact analogue of Theorem 0'.

Theorem 1'. The limit, as 6-1, of the unique equilibrium agreement (.xT(s)+a*(6), ~ " ( 6 ) ) in GI(N; q)' is the agreement given by the Nash solution to the bargaining problei~z B.

ProoJ For any 6 , from the characteristic equations for (N; q), for all i :

ui($ +a*) x nj+{ =uI(xT+a*) x nj+uj(x~).uj(x?) I

Furthermore, as 8 4 1 , the premium a*(6)+0. Thus, (x:(6)+a*(6), x*,(6)) con- verges to the maximizer of njGN 11uj(xj) over B.

8. ON BARGAINING AND EXIT

We have argued that the notion of consistency leads rather naturally to the consideration of bargaining mechanisms in which players may exit with their shares even though unanimous consent has not been obtained. As such, this seems to run contrary to the very idea that the players are facing a pure bargaining problem in which sub-coalitions have no power.

Our view is that there is much to be gained by de-linking somewhat the power structure implied by the characteristic function of the underlying cooperative game and the mechanism that implements a particular allocation. In designing the mechanism one has an additional degree of freedom whereby the underlying power structure can be

8. For instance, in the case of three players with linear utilities, the shares prescribed by the equilibrium are ~roportional to (I, 6 , 6 1. Recall that the shares in the stationary equilibrium of Shaked's game are propor- . . - - -tionk to ( l , 6 ,6').

9. We cannot invoke Lensberg's (19881 result directlv becasue FA is onlv defined for ~roblems where a "pie" is being split. To be able to app&the'result, F8 wohd have to be defiied for all baigaining problems defined by closed, convex and comprehensive sets of feasible utilities. A referee has conjectured that the result may also hold on the restricted domain that we consider, but as far as we know, this is an open problem.


disturbed by balancing the power that players have in the mechanism with the incentives to exercise it. For instance, in the game GI((], 2,3) ;q), player 1 has the power to award shares to a particular player, say 2, without the need to obtain player 3's consent. However, the added power that is awarded to player 1 can be neatly balanced by the incentive he has to keep player 2's share as small as possible. Since player I cannot exit in period 1 unless all players agree, in the event of a rejection of 1's initial offer by 3, the cost of every addition to player 2's share is partially borne by player I-in the subgame that results this will reduce the pie over which 1 and 3 will bargain. We wish to emphasize that the argument outlined above relies not on the characteristics of the equilibrium alone but rather on the incentives of the players in the overall game. The incentives are preserved whether or not an equilibrium is being played.

Second, there are many circumstances where the bargaining is over a "pie" that already physically exists (for example, an estate) and thus it is possible to "exit" even though unanimous consent has not been reached. This feature is also present in the classical "How to Cut a Cake Fairly" procedure (Dubins and Spanier (1961)).

Third, the game GI(N; q) can also be reinterpreted so that it is applicable to situations where the "pie" does not physically exist until all parties reach an agreement (for example, a firm bargaining simultaneously with more than one union). This is accomplished as follows. Suppose that player 1's offer of x is interpreted as an offer to purchase the right to represent the remaining players in future negotiations, if any. Thus player 1 offers to purchase player i's "signature" or "power of attorney" for a price of xi to be paid immediately. If player i accepts and some others reject, then I pays i an amount xi. It is convenient to think of player 1 borrowing at no cost from outside the game in order to do this. He or she then represents i and all other accepting players in the bargaining from period 2 onwards. Of course, it is possible that 1 (and the players he represents collectively) is himself "bought out" at a subsequent stage. In this case the player who buys 1 out, now represents 1 and the players that 1 was representing, in further negotiations.

Formally, this interpretation relies on the following observation. Suppose that player 1 makes an offer of x in GI(N; q), which is accepted by the players in A and rejected by the players in R \ l . Let (yz, P*) be the solution to the characteristic equations for (R; 9 - 4 4 ) :

ui(yi)=6ui(yi+ P), i~ R ;

Y(R)+ P =q-x(A).

Let (zz, y*) be the solution to the equations:

UI(ZI-x(A)) =6ul(zl- x(A) +y);

ui(zi) =6ui(zi+y), i~ R \ 1 ;

Then z: =y: +x(A) ; for all i~ R \ I , z) =y) ;and y* =P*. The second system determines the equilibrium in the subgame after 1 has bought the right to represent the players in A by paying a total of x(A) out of his or her own resources. Notice that the equation for player 1 incorporates the fact that the amount borrowed, x(A), will be repaid.

We are making the assumption that it is possible to borrow at no cost from outside the game. While it may be desirable to include the costs of borrowing, it is unclear how to do this. One may be tempted to posit that the per-period cost of borrowing is r = (1 -6)/6. However, there is no reason why the cost of borrowing should be related to 6


at all. Recall that the fact that the 6's are the same for all players is just a convenient normalization and in any case, with non-linear utilities the true nature of time preference is not captured by 6 alone (see the related discussion in Osborne and Rubinstein (1990, p. 84)). Thus the costs of borrowing r, must be treated exogenously and our solution is an approximation for the case when r is small.

The formulation of the model in terms of the "rights to represent" is similar to that in Dow (1989) who studies a bargaining problem in which coalitions have non-zero power and postulates a mechanism that implements the Shapley value. Jun (1987) also interprets his model in similar fashion.

9. CONCLUSION

Our results reconcile the axiomatic and strategic theories of bargaining in multilateral situations. In doing so, the use of the consistency axiom has been a key idea. The general notion of consistency has been a very important tool in cooperative game theory. In particular, it has been used to obtain powerful characterizations of the nucleolus, the core and the Shapley value (Thomson (1990) provides a concise survey of this work). Our results suggest that it may, in addition, provide a guide to the design of mechanisms that implement these and other solutions as non-cooperative equilibria.''

APPENDIX

In this appendix we establish Claim 3.4 made in the proof o f Lemma 3. Lemma A l below, is used in the proof o f Claim 3.4. W e state and prove it only for the case o f three

players but both the statement and the proof hold in an obvious manner for any numbers o f players (including the case o f two players).

Lemma At . Slcppose tkat ( w ;q ) is the solution to the characteristic equations for ( N ; r ) ; ( IV ' ; q ') is the solution for ( N ;r ' ) ; and (w"; q") is the solution for ( N ; r"). Let b,, hJ and bk>O. Then at least orze ofthe follo~ving statetiletzts holds:

( i ) M., <h, implies that bv, <hi ; ( i i ) <0, implies tkat n,; <hk ;

(iii) ~ c f<hk itlzplies that IVY <b,.

Proof: Suppose that (i) does not hold, that is, w,<b, and ruizb,. I f ( i i ) does not hold either, we also have that n( <b, and bv; 2bk. Then 14<biJ IV, implies that r' < r . ; and bvi <bk$ I V ~implies that r" <r'. Thus r" < r.. Hence we have that rv: <wi<b,. 11

Claim 3.4. z ' (K) canrzot satisjj any o f t l ~ e f o l l o ~ ~ i n g :

(1) z; (K)EA,(N;q ) , for j # i ; (11) z ' (K) is reached in periods s - 1 atzd s ;

(111) z ' (K) is reached it1 periods s and s + 1 and if i deviates and rejects the oJSer in period s, the induction hypothesis may he applied.

Proof of Clainz 3.4. W e will refer to the equilibrium agreements in the statement as being o f Types I , I1 and 111 respectively.

Recall that z ' ( K ) is such that i gets his share in some period s > t. Then, i's share z : (K)< J ( ( K ) + (s- t)a* <x: ' (K)- <.Y:(K) t (n- I)a* +aa* and t(n- I)a* +aa* + ( s - t)a*. This is because i's share in period t, y f ( ~ ) -

in each o f the periods between t and s, the share offered to player i can go up at most by a*, otherwise i's deviations would be profitable and y ' ( ~ ) could not be an equilibrium agreement.

10. Mas-Colell (1988) and Hart and Mas-Colell (1994) provide game forms that implement the Shapley value along these lines.

KRISHNA & SERRANO MULTILATERAL BARGAINING

Step I . We first argue that if z ' (K)is of Type I, if rejected, it cannot be supported by another equilibrium of Type I , which is in turn supported by another equilibrium of Type I and so on indefinitely. The basic idea is the same as that in Claim 3.3 and we will argue that in a sequence of such equilibria we will reach a situation where the shares of the players are mutually incompatible. The argument is complicated by the fact that the identity of the proposing player may be different at each stage (Lemma Al is used here); and that the fall in a player's share at each step may be smaller than a * .

Suppose that for some j# i, z ~ ( K ) E A ' ( N ;q ) . Then z ; ( K ) <.Y:(K)-2a* (because t 5 2 , a s n - 2 and in this case, s 5 n ) and by Lemma I z ; (K)<.$ + a * . Thus an offer of the form x ' ( K ' )in G,(N; q ) must be unanimously rejected, leading after a series of unanimous rejections to an offer z J ( ~ ' ) e A k ( N ;q ) , for some k # j . This in turn will lead to an offer Z * ( K " ) E A ~ , ( N ;q ) and so on. Eventually we will reach a player that has already been considered and so, without loss of generality let h = i .

Temporarily suppress the dependence of variables on K . Let c;be defined by u,()vj)= 6u,(.u:)and let q' be defined by +t$+qi=.uf.For h # i define 1 4 by uI,(ir;,)= Grr,,(\v;+ q ' ) and consider (w:; q l ) as the solution to the characteristic equations for ( N ;q'), where q'= iv f (N)+ q'. Of course, q i < q and q i < a*.

Next observe that z:<Y:-2a* = rv:+ q' -2a* < IV: - q ' Since (wi ; q i ) is the solution to a set of characteristic equations, zj< w:- q' and z ieAi (N; q ) together imply that zi< )I$+q'. Consider a vector .u' of the same form as x' such that .Y,'= rvf+ q'. This must be unanimously rejected, leading to an equilibrium of Type IV with agreement ?' which in turn leads as before to an equilibrium zJeA,,(N;q)." AS before we can infer that z/<evi- qi and ;f < rvi+ q i . Again consider the vector xk of the same form as .u' such that .ut= rvi + q'. Again we will find a jk

and a z k e A , ( N ; q ) .As explained above, there is no loss of generality in assuming that k = i . As above, z i < i ~ , : - q' and ~ : < I L . ~ +qi. Thus we have inferred that among the three players i, j, k :

z leA,(N; q ) and z:< it!;- q';

z 'e Ak(N; q ) and z,!<w)- q i ;

z k ~A , ( N ; q ) and rt <M.; - qi.

From Lemma A1 we can infer that either zi <w: or z: < b1.f; or zf<rv:. Hence there exists a player h e { i ,j , k} such that there is an equilibrium of G/,(N; q ) where k's share is less

than it,;, . Recall that q' < a* and hence ,t';8 <.$. Now consider the offer .x"(K') such that .Y:(K') = and a z " ( ~ ' ) .ic;,. This will in turn lead to a y " ( ~ ' ) Suppose

that L"(K')is also of Type I . Otherwise define (v"; qh) by setting ul,(vi)= = and calculating the other players' shares Stt,,(ui+ qh) 6 u , , ( ~ , ; )

via the characteristic equations. By replicating the argument given above we will find a player, say I , such that there is an equilibrium of G l ( N ;q ) where 1's share is less than v: , 7)" < q'.

Since there are only a finite number of players there must be a player, say h again, and a sequence of corresponding premia such that q b ( l )> qh(2)> qh(3).. . ;and furthermore in the iteration from q h ( l )to qh(l+ I ) , h's share has fallen by more than q h ( l ) .This sequence must have a limit to zero because otherwise infinitely often it would be the case that 11's share in an equilibrium of Gl,(N; q ) falls by at least the limit.

Since the limit of these qh(l)'sis zero, so is the limit of the corresponding d ' s . Thus for all E>O, there exists an equilibrium of GI,(N; q ) in which 11's share is less than E . However, this is impossible because k could make the offer z such that : , ,=E, and z ,= .u:+[ ( .uf+a*-c) / (n- I)]. This must be unanimously accepted, because if it were unanimously rejected for small enough E , we would have an equilibrium in which k's share is negative. Hence the limit of the qh(l)'scannot be zero.

Thus we have established that we cannot have an infinite sequence of Type I equilibria alone.

Step 11. Next we consider a situation where Type I1 equilibria are in turn supported by Type I 1 equilibria and so on.

Suppose that z ' ( ~ )is the result of an agreement in periods s - 1 and s. Let 11 be the player who makes the offer in period s- I . As in step I suppress the dependence of the variables on K and write rvj+ qi=.ui. Then zj<wj+ q i - a * < ) u i . Hence by the induction hypothesis applied to the solution in period s , z;,<it$,. Thus we have found an equilibrium of G,,(N; q ) in which 11's share is less than iv;,.

Now as in step I define (v"; qh) by ul,(vi)= 6ul,(ui+ q 6 ) = 6ul,(tv;,). Now the argument can be iterated and thus we will, as in step I , obtain a sequence of premia leading to the same contradiction. This establishes that we cannot have a sequence of Type I1 equilibria alone.

Step 111. Next consider Type I11 equilibria supported by other Type 111 equilibria.

1I . We are implicitlty assuming that y ' and z' occur without G,(N; q ) being played again. By claims 2.1 and 2.3 there is no loss of generality in doing this.

You have printed the following article:

Multilateral BargainingVijay Krishna; Roberto SerranoThe Review of Economic Studies, Vol. 63, No. 1. (Jan., 1996), pp. 61-80.Stable URL:


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References

The Nash Bargaining Solution in Economic ModellingKen Binmore; Ariel Rubinstein; Asher WolinskyThe RAND Journal of Economics, Vol. 17, No. 2. (Summer, 1986), pp. 176-188.Stable URL:

http://links.jstor.org/sici?sici=0741-6261%28198622%2917%3A2%3C176%3ATNBSIE%3E2.0.CO%3B2-E

How to Cut A Cake FairlyL. E. Dubins; E. H. SpanierThe American Mathematical Monthly, Vol. 68, No. 1. (Jan., 1961), pp. 1-17.Stable URL:

http://links.jstor.org/sici?sici=0002-9890%28196101%2968%3A1%3C1%3AHTCACF%3E2.0.CO%3B2-N

Existence and Characterization of Perfect Equilibrium in Games of Perfect InformationChristopher HarrisEconometrica, Vol. 53, No. 3. (May, 1985), pp. 613-628.Stable URL:

http://links.jstor.org/sici?sici=0012-9682%28198505%2953%3A3%3C613%3AEACOPE%3E2.0.CO%3B2-2

The Bargaining ProblemJohn F. Nash, Jr.Econometrica, Vol. 18, No. 2. (Apr., 1950), pp. 155-162.Stable URL:

http://links.jstor.org/sici?sici=0012-9682%28195004%2918%3A2%3C155%3ATBP%3E2.0.CO%3B2-H

http://www.jstor.org

LINKED CITATIONS- Page 1 of 2 -

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http://links.jstor.org/sici?sici=0012-9682%28195004%2918%3A2%3C155%3ATBP%3E2.0.CO%3B2-H&origin=JSTOR-pdf

Equality of Resources Implies Equality of WelfareJohn E. RoemerThe Quarterly Journal of Economics, Vol. 101, No. 4. (Nov., 1986), pp. 751-784.Stable URL:

http://links.jstor.org/sici?sici=0033-5533%28198611%29101%3A4%3C751%3AEORIEO%3E2.0.CO%3B2-T

Perfect Equilibrium in a Bargaining ModelAriel RubinsteinEconometrica, Vol. 50, No. 1. (Jan., 1982), pp. 97-109.Stable URL:

http://links.jstor.org/sici?sici=0012-9682%28198201%2950%3A1%3C97%3APEIABM%3E2.0.CO%3B2-4

Non-Cooperative Bargaining Theory: An IntroductionJohn SuttonThe Review of Economic Studies, Vol. 53, No. 5. (Oct., 1986), pp. 709-724.Stable URL:

http://links.jstor.org/sici?sici=0034-6527%28198610%2953%3A5%3C709%3ANBTAI%3E2.0.CO%3B2-M

http://www.jstor.org

LINKED CITATIONS- Page 2 of 2 -

http://links.jstor.org/sici?sici=0033-5533%28198611%29101%3A4%3C751%3AEORIEO%3E2.0.CO%3B2-T&origin=JSTOR-pdf

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