Minimal cooperation in bargaining

Economics Letters 34 (1990) 311-316 North-Holland

311

Minimal cooperation in bargaining Youngsub Chun * Vanderbilt

University, Nashville,

Seoul National

TN 37235, USA

University, Seoul, Korea

Received 30 March 1990 Accepted 27 April 1990

Mid-point domination requires that the solution outcome dominate or be dominated by a reference point. By investigating two different formulations of the condition, we present characterizations of the Nash and the Kalai-Rosenthal solutions for n-person bargaining problems.

1. Introduction

With a minimal amount of cooperation, agents can be expected to agree on at least the average of their extreme positions: the worst and the best positions. Consequently, it is reasonable to require that, if the average is feasible, then the final compromise should dominate the average, and of the average is not feasible, then the final compromise should be dominated by the average. This property of mid-point domination has been introduced for bargaining problems by Sobel (1981) ’ and further studied by Thomson (1981), Moulin (1983), and Peters (1986). ’ Although the motivation for the requirement of mid-point domination is very clear, questions arise on the definition of extreme positions. In the bargaining problem, the worst position is undoubtfully the disagreement point. What is the best position? Here we study two possibilities. The first possibility, which has been discussed in the literature mentioned above, is to choose the maximal feasible utility level of each agent guaranteeing other agents the utility levels of the disagreement point. The second possibility is to choose the maximal feasible utility level of each agent. By adopting the abstract formulation of the bargaining problem, we will show that, combined with other properties, the first mid-point domination property characterizes the Nash (1950) solution, whereas the second the Kalai-Rosenthal (1978) solution. 3

* I am grateful to Professor William Thomson for his valuable comments. However, I have full responsibility for any shortcomings. ’ Under the name of symmetric monotonicity. ’ Thomson (1988) provides an excellent review of the literature that arouse out of Nash’s paper. 3 This result indicates that, although the definition of the Kalai-Rosenthal solution is somewhat related to that of the Kalai-Smorodinsky (1975) solution, its properties are of independent interest. 0165-1765/90/$03.50

0 1990 - Elsevier Science Publishers B.V. (North-Holland)

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in bargaining

2. Preliminaries

An n-person bargaining problem, and d is a point in S, such that: (1) (2) (3) (4)

or simply a problem,

is a pair (S, d), where S is a subset of W”

S is convex and closed, a,(S) = max{ X, ( x = (x,, . . . , xn) E S} exists for all i, S is comprehensive, i.e., for all x E S and for all y E Iw”, if y 5 x, 4 then y E S. there exists x E S with x.

S is the feasible set. Each point x of S is a feasible alternative. The coordinates of x are the utility levels, measured in some von Neumann-Morgenstern scales, attained by the n agents through the choice of some joint action, d is the disagreement point. The intended interpretation of (S, d) is as follows: The agents can achieve any point of S if they unanimously agree on it. Otherwise, they end up at d. Let C be the class of all problems and r be the class of feasible sets satisfying (l), (2) and (3). A solution is a function F: 2 + Iw” such that for all (S, d) E 2, F( S, d) E S. F( S, d ), the value taken by the solution F when applied to the problems (S, d), is called the solution outcome of

(S, d). The following notation and terminology will be used frequently. Given (S, d) E 2, let 1R(S, d) = {x E S ( x 2 d } be the set of individually rational points of (S, d). WPO( S) = {x E S 1for all x’ E Iw”, x’ > x implies x’ 4 S} is the set of weakly Pareto optimal points of S. Similarly, PO(S) = (xESlforal1 x’EIW”, x’ 2 x implies x’ e S} is the set of Pareto optimal points of S. S is smooth at x E WPO(S) if there exists a unique hyperplane supporting S at x. (S, d) is a smooth problem if, for all x E WPO(S), S is smooth at x. Given x1,. . . , xk E R”, camp{ x1,. . . , xk } is the comprehensive hull of these points (the smallest comprehensive set containing them). Given S c r, int(S) is the interior of S, and r. int( PO(S)) is the relative interior of PO(S) with respect to WPO(S). Next we define the Nash (1950) and the Kalai-Rosenthal (1978) solutions.

3. Definitions The Nash solution, N. in IR(S, d).

For each (S, d) E 2, N( S, d) is the maximizer

For each (S, d) E 2, KR(S, The Kalai-Rosenthal solution, KR. the segment connecting d and a(S). We are interested

in solutions

Weak Pareto OptimaIity In the following,

(W. P.0).

convergence

satisfying

the following

Continuity (CONT). For all sequences d”-+d, then F(S”, d’)-+ F(S, d).

of feasible {(S”, d”)}

4 Vector inequalities: Given x,y E R”, x > y, x 2 y, x > y.

d) is the maximal

ny= t( xi - d,)

point

of S on

properties.

For all (S, d) E 2, F(S,

of a sequence

of the product

d) E WPO(S).

sets is evaluated c 2

and

in the Hausdorff

for all (S, d) E 2,

topology.

if S” + S and

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Boundary (BOUND). For all (S, d) E 2 and for all e E R,,, that 11F(S, d) - d’ 11cc and F(S, d’) = F(S, d).

there exists

d’ E int(S)

such

W.P.0 requires that there be no feasible alternative at which all agents are better off than at the solution outcome. CONT requires that a small change in the feasible set and the disagreement point cause only a small change in the solution outcome. BOUND is a technical condition, requiring that in any neighborhood of a given solution outcome, F(S, d), there always exists a point d’ in the interior of S such that F(S, d’) is the same as F(S, d). ’ All of these axioms are very mild requirements that are satisfied by all well-known solutions. The next axiom has been widely discussed in the recent literature on the bargaining problem with uncertain disagreement points. 6 Weak Disagreement Point Linearity (W. D. LIN). For all (S’, d’), ( S2, d2) E 1 andfor all a E [O,l], ifS’=S2=S, F(S, d’)=F(S, d2)= x, and S is smooth at x, then F(S, ad’ + (1 - cx)d2) =x. W.D.LIN requires that if two problems, (S, d’) and (S, d 2), have the same solution outcome, then a problem obtained by randomizing between these two problems should result in the same solution outcome. It can be motivated on the basis of timing of bargaining. The smoothness assumption, which may appear technical, has a very natural economic interpretation: It simply says that utility transfers are possible at the same rate in all directions. 7

4. Main results We now introduce our main axioms, motivated in the introduction. Mid-Point

Domination

I ( M. P. D. I ). ’

Mid-Point

Domination

2 (M. P. D. 2).

For all (S, d) E 2, F(S, For all (S, d) E 2, F(S,

*

d) 2 (d + a(S, d) 2 (d + a(S))/n

d))/n or F(S,

d) s (d

+ a(S))/n. Next we present our main results. Theorem solution.

1.

A solution satisfied

W.P.0,

CONT,

W.D. LIN and M.P.D.l

if and only if it is the Nash

Proof. It is obvious that the Nash solution satisfies all four axioms. The proof of the converse statement is divided into three steps. * Alternative axiomatic characterizations of the solutions without imposing this condition will be given in the remark following the main results. 6 This axiom was introduced by Livne (1988) in his study of the Nash solution under the name of Expectations Property. Variants of this axiom were studied in detail by Chun (1987). Chun and Thomson (1990a,b,c), Peters (1986), and Peters and van Damme (1990). ’ See Aumann (1985), and Chun and Thomson (1990b). 8 M.P.D.1 has been introduced by Sobel (1981) and used by Moulin (1983) and Peters (1986) in their characterizations of the Nash solution, whereas M.P.D.2 is new to this paper. See Thomson (1988) for other possible formulations of the axiom. 9 To maintain the symmetry with M.P.D.2, the conclusion of the axiom could be stated in the following form: Either F(S, d) 2 (d + a(S, d))/n or F(S, d) 2 (d + a(S, d))/n. Note that, since S is assumed to beconvex, (d + a(S, d))/n E S. Therefore, together with W.P.0, the second part becomes vacuous. On the other hand, note that (d + a( S))/n may not belong to S.

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Step 1. Let F be a solution satisfying M.P.D.l. Also, let (S, d) E 2 be a problem such that there exist p E Iw:+ and c E R with, for all x E WPO(comp{ IR(S, d)}) n IR(S, d), x .p = c. Then F(S, d)=N(S, d). Proof Let (S, d) E 2 be a problem satisfying the hypothesis of step 1. Note that (d + a(S, d))/n E PO(S). Therefore, by M.P.D.l, we have F(S, d) = (d+ a(S, d))/n. Since N(S, d) = (d+ a)S, d ))/n, we conclude that F( S, d) = N( S, d ). Step 2. If a solution F satisfies W.P.0, problems (S, d)EZ, F(S, d)=N(S, d). Proof:

CONT,

W.D.LIN

and M.P.D.l,

then for all polygonal

The proof is divided into three cases.

Case (i). S is smooth at F(S, d) and F(S, d) E PO(S). Let d’E [d, F(S, d)[ be such that (S, d’) satisfies the hypothesis of step 1. Since S is smooth at F(S, d) and F(S, d) E PO(S), such a d’ always exists. By step 1, Ir( S, d’) = N(S, d’). Now suppose, by way of contradiction, that F( S, d) # N( S, d), which implies that F( S, d) # N( S, d ‘). Since S is smooth at F( S, d) and F( S,d) E PO(S), there exists d” E int( S) such that (S, d”) satisfies the hypothesis of step 1 and N( S, d”) = F( S, d). Moreover, from step 1 again, for all do [d”, F(S, d)], F( S, 2) = N( S, d”) = F(S, d). By W.D.LIN, for all X E [0, l] and for all do [d”, F(S, d)[, we have F(S, hd+ (1 X)d) = F( S, d). Finally, by choosing a sequence of disagreement points {d “} such that F( S, d “) = F( S, d) and d” --+d ‘, we obtain a contradiction to CONT. Case (ii). S is smooth at N(S, d). Let {d”} c [d, N( S, d)[ be a sequence of disagreement points such that d’ = d and d” + N(S, d). Since S is smooth at N(S, d), there exists V such that for all Y 2 V, (S, d”) satisfies the hypothesis of step 1. Then, for all v 2 V, (a) F( S, d”) = N(S, d”). On the other hand, from case (i), if F(S, d) f N( S, d), then (b) either F(S, d) P PO(S) or F( S, d) should be a Pareto-optimal point of S, whose supporting hypetplane is not unique. (a) and (b) cannot be satisfied together without contradicting CONT. Case (iii). If neither case (i) nor case (ii) occurs, we approximate the problem (S, d) by a sequence of polygonal problems, {(S”, d”)}, such that S’ is smooth at N( S”, d ‘), d” = d for all v and S” + S. By CONT and the fact that N satisfies CONT, we obtain the desired conclusion. Step 3.

If a solution F satisfies W.P.0,

CONT, W.D.LIN

and M.P.D.l,

then F = N.

Proof: Since an arbitrary problem (S, d) E 2 can be approximated in the Hausdorff topology by a sequence of polygonal problems, we conclude by CONT that F(S, d) = N( S, d) for all (S, d) E 2.

Remark 1. Characterizations of the Nash solution using W.D.LIN have been presented by Chun and Thomson (1990b), and Peters and van Damme (1990). The basic difference between those two results and the present one is that we use M.P.D.l, whereas the others the following three axioms: Symmetry, Scale Invariance, and Independence of Non-Individually Rational Alternatives. Also, Peters (1986) characterized the Nash solution for 2-person problems by using a stronger version of W.D.LIN. His result is similar to the first part of our Theorem 3. Theorem 2. A solution satisfies W. P.0, the Kalai- Rosenthal solution.

CONT,

BOUND,

W. D. LIN and M. P. 0.2 if and only if it is

Y. Chun / Minimal cooperation

It is obvious that the Kalai-Rosenthal Proof. converse statement is divided into three steps.

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in bargaining

solution satisfies all five axioms. The proof of the

Step 1. Let F be a solution satisfying W.P.0, CONT, BOUND and W.D.LIN. Also, let (S, d) E 2 be a problem such that S is smooth at F( S, d ). Then for all x E [d, F( S, d )[, F( S, x) = F( S, d ). be a problem satisfying the hypothesis of step 1. Now suppose, by the way of Proof: Let (S, d) E L?Z be a contradiction, that there exists do [d, F(S, d)[ such that F(S, 2) # F(S, d). Let {e”} c R,, monotonically decreasing sequence such that e” + 0. By BOUND, for each c”, there exists d” E int( S) such that I] F(S, d) - d’ (1 cc” and that F(S, d’) = F(S, d). By W.D.LIN, for all X E [O,l], F( s, Ad + (1 - h)d’) = F(S, d). Now pick another sequence of disagreement points, (2”) c int(S), such that, for each u, d” E [d, d”] and d” + d. For all v, F( S, d”) = F(S, d). Since F( S, 2) # F(S, d), we obtain a contradiction to CONT. Step 2. If a solution F satisfies W.P.0, CONT, BOUND, smooth problems (S, d) E 2, F(S, d) = KR(S, d). Proof.

W.D.LIN

and M.P.D.2,

then for all

The proof is divided into three cases.

(d + a( S))/n E WPO(S). If (d + a( S))/n E r. int( PO( S)), then W.P.0 and M.P.D.2 Case (i). together imply that F(S, d) = (d+ a(S))/n. Since KR(S, d) = (d+ a(S))/n, we have F(S, d) = KR( S, d). Otherwise, we approximate the problem (S, d) by a sequence of smooth problems {(S”, d’)} c t: such that d” = d for all Y, (d + a(S”))/n E r. int( PO(S)) and S” + S. By applying CONT, we obtain the desired conclusion. Case (ii). (d+u(S))/nES. Since (d+u(S))/nPS, there exists d’Eint(S) such that de [d’, a(S)] and (d’+ u(S))/n E WPO(S). B y case(i), F(S, d’) = KR(S, d’). By step 1, F(S, d) = F(S, d’) = KR(S, d’). Since KR(S, d) = KR(S, d’), we obtain the desired conclusion. (d+ u(S))/n E int(S). Since Cuse (iii). that (d’ + u(S))/n E WPO( S). By case for all do [d, F(S, d)[, F(S, 2) = F(S, This is possible only if F(S, d) = KR(S, Step 3.

If a solution F satisfies W.P.0,

(d+ u(S))/n E int(S), there exists d’ E [d, F(S, d)[ such (i), F(S, d’) = KR(S, d ‘). On the other hand, by step 1, d), which implies that F(S, d’) = KR(S, d’) = F(S, d). d). CONT, BOUND,

W.D.LIN

and M.P.D.2,

then F= KR.

Since an arbitrary problem (S, d) E 2 can be approximated in the Hausdorff topology by a Proof. sequence of smooth problems, we conclude by CONT that F(S, d) = KR( S, d) for all (S, d) E Z. 0 solution satisfies a stronger version of W.D.LIN, which does not Remark 2. The Kalai-Rosenthal require the smoothness of S at F(S, d). However, if n > 2, then the Nash solution does not satisfy the stronger version, as shown in Chun and Thomson (1990b). Remark 3. Two axioms in Theorem 2, BOUND and W.D.LIN, can be replaced by Disagreement Point Quasi-Concuuity (henceforth D.Q-CAV), introduced by Chun and Thomson (1990b) and further studied by Chun (1987). lo Again, as shown in Chun and Thomson (1990b), the Nash solution does not satisfy D.Q-CAV if n > 2. However, if n = 2, then W.D.LIN in the Theorem 1 can be replaced by D.Q-CAV. lo D.Q-CAV: F,(S, &)}

For all (S’, d’), (S’, for all i.

d’) E X and for all (I!E [0, 11,if S’ = S2 = S, then E;(S, ad’ +(l

- a)d’)

2 min( <(s,

d’),

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Remark 4. Another characterization of these solutions can be obtained by using the following family of solutions, introduced and characterized by Chun (1987) for 2-person problems. Definition. A solution is a linear solution, for all (S, d) E ,I? and for all cx E ] - co,l[, F( S, 1 - cx)d + aF( S, d)) = F(S, d) and F(S, d) E WPO(S). This family of solutions is fairly large, and in particular it includes the Nash and the Kalai-Rosenthal solutions. Let (S, d) E t: and a solution F be given. Following Thomson (1988), let F-‘(S, x) = {d E int(s) 1F(S, d) = x} l1 be the set of disagreement points of S, whose solution outcome is x. For 2-person problems, if F is a linear solution, then F-‘(S, F(S, d)) U { F(S, d)} is a cone with vertex F(S, d). Theorem 3. Let F be a linear solution. (i) It satisfies CONT and M. P. D. I if and only if it is the Nash solution. (ii> It satisfies CONT and M.P. 0.2 if and only if it is the Kalai-Rosenthal solution, Since the proofs of this theorem are similar to, but simpler than, those of the Theorems respectively, we omit them.

1 and 2

References Aumann, R.J., 1985, An axiomatization of the non-transferable utility value, Econometrica 53, 599-612. Chun, Y., 1987, Bargaining with uncertain disagreement points for 2-person bargaining problems, Discussion paper no. 99 (University of Rochester, Rochester, NY). Chun, Y. and W. Thomson, 1990a, Bargaining with uncertain disagreement points, forthcoming in Econometrica. Chun, Y. and W. Thomson, 1990b, Nash solution and uncertain disagreement points, forthcoming in Games and Economic Behavior. Chun, Y. and W. Thomson, 199Oc, Egalitarian solutions and uncertain disagreement points, Economics Letters 33, 29-33. Kalai, E. and R.W. Rosenthal, 1978, Arbitration of two-party disputes under ignorance, International Journal of Game Theory 7, 65-72. Kalai, E. and M. Smorodinsky, 1975, Other solutions to Nash’s bargaining problem, Econometrica 43, 513-518. Livne, Z., 1988, The bargaining problem with an uncertain conflict outcome, Mathematical Social Sciences 15, 287-302. Moulin, H., 1983, Le choix social utilitariste, Cole Polytechnique discussion paper, 1983. Nash, J.F., 1950, The bargaining problem, Econometrica 18, 155-162. Perles, M.A. and M. Maschler, 1981, The super-additive solution for the Nash bargaining game, International Journal of Game Theory 10,163-193. Peters, H., 1986, Characterizations of bargaining solutions by properties of their status quo sets, Mimeo. Peters, H. and E. van Damme, 1990, A characterization of the Nash bargaining solution not using HA, forthcoming in Mathematics of Operations Research. Sobel, J., 1981, Distortion of utilities and the bargaining problem, Econometrica 49, 597-619. Thomson, W., 1981, A class of solutions to bargaining problem, Journal of Economic Theory 25, 431-441. Thomson, W., 1988, Bargaining theory: The axiomatic approach, University of Rochester Lecture Notes (University of Rochester, Rochester, NY). ri Perles and Maschler

(1981) introduced

this set under the name of status quo set of S. Also, see Peters (1986).