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Rationality of bargaining solutions
Journal of Mathematical Economics 33 Ž2000. 389–399 www.elsevier.comrlocaterjmateco
Rationality of bargaining solutions M. Carmen Sanchez ´
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Departamento de Fundamentos del Analisis Economico, UniÕersidad de Murcia, Ronda de LeÕante 10, ´ ´ 30008 Murcia, Spain Received 17 March 1998; received in revised form 5 July 1999; accepted 8 July 1999
Abstract The aim of this paper is to obtain the rationality of two-person bargaining solutions by imposing usual assumptions in cooperative bargaining theory. In particular, it is proved that the conjunction of Independence of Irrelevant Alternatives and either Pareto Continuity or Strong Monotonicity implies this rationality. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: C78 Keywords: Rational choice; Bargaining solutions
1. Introduction A bargaining solution is a special case of a single-valued choice function that selects an outcome for each choice situation Žbargaining problem., which is attained by means of cooperation by the agents involved. One of the interpretations of these solutions is to consider that the outcomes are given by the recommendations of an impartial arbitrator, whose preferences, in some way, represent the preferences of the agents as a group. Thus, the agreement reached in each bargaining problem may be thought of as the most preferred alternative
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Tel.: q34-968-363-756; E-mail: [email protected]
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 3 3 - 6
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within the set of feasible outcomes according to the arbitrator’s preferences. Moreover, with this interpretation, the axioms used to characterize these solutions can be interpreted as rationality requirements for the arbitrator. It makes sense, therefore, to analyze the rationality of bargaining solutions from the point of view of general choice functions since it would reinforce this interpretation. This requires finding a binary preference relation the maximization of which determines the outcome selected in each bargaining problem. Among the works that must be mentioned along these lines, are those of Lensberg Ž1987., Peters and Wakker Ž1991. and Bossert’s Ž1994. who, in different contexts, provide a set of sufficient conditions Žall of them standard in cooperative bargaining models. which ensures the rationality of bargaining solutions. Peters and Wakker Ž1991. assume that the domain of the solutions is given by 2 the family of convex and compact subsets of Rq , and look for conditions under which a choice function maximizes a real-valued function. To do this, they prove that the rationality of two-person bargaining solutions may be derived from Pareto Optimality, Independence of IrreleÕant AlternatiÕes and Pareto Continuity. Bossert Ž1994. has provided an alternative proof of this result by requiring bargaining problems to be also comprehensive and by imposing Continuity instead of Pareto Continuity. He obtains the existence of a rationalizing ordering that can be endowed with some regularity properties. As Bossert Ž1994. mentions, the result of his work provides sufficient, but not necessary, conditions for the rationality of a bargaining solution. Thus, some rational solutions that are not covered by his result are the egalitarian solution ŽKalai, 1977., monotone path solutions, or their lexicographic extensions. Formal definitions and axiomatic characterization results of these solutions can be found in the book of Thomson Žforthcoming.. The purpose of the present paper is to show that the combination of Independence of IrreleÕant AlternatiÕes and Pareto Continuity also ensures the rationality of two-person bargaining solutions in the context of Bossert’s work. This result generalizes the rationality part of Bossert Ž1994. and enlarges the family of two-person bargaining solutions in which rationality can be derived from standard bargaining axioms, usually invoked to axiomatize bargaining solutions. Moreover, an alternative rationality result that provides some structure for the rationalization is given by replacing Pareto Continuity with a monotonicity assumption Ž Strong Monotonicity ..
2. Preliminaries 2 A two-person bargaining problem is described by a pair Ž S, d . where S : Rq 2 is the set of feasible utility vectors that individuals can attain and d g Rq represents the disagreement point, that is, the vector of utilities that individuals
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would obtain if they did not reach an agreement. Given x s Ž x 1, x 2 ., y s Ž y 1 , y 2 . 2 , the vector notation used is the following: x ) y if and only if x i ) y i for g Rq all i s 1,2 and x G y if and only if x i G y i for all i s 1,2. As usual in bargaining theory, henceforth it is assumed that d s Ž0,0. and the domain considered is given by 2 < D s S : Rq S is convex, compact, comprehensive and
there exists s g S such that s ) 0 4 . 2 To simplify the notation, for every subset A : Rq , the comprehensiÕe and ² : conÕex hull of A is denoted by A , that is, 2 < ² A: s l B : Rq A : B, B convex and comprehensive4 .
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Formally, a two-person bargaining solution on D is a function that selects an 2 outcome for each bargaining problem in D, that is, F: D Rq such that for all Ž . Ž . S g D, F S g S. Throughout the work, F D denotes the image of F, that is, 2 < F Ž D . s x g Rq 'S g D such that F Ž S . s x 4 .
Since a bargaining solution is a particular case of a single-valued choice function, it is possible to analyze its rationality and to look for sufficient conditions to ensure the existence of a binary relation the maximization of which determines the selected outcome in each bargaining problem. As usual in the related literature, some standard properties are required of the rationalizing binary 2 . relation Ž RU .. In particular, reflexiÕity Ž xRU x for all x g Rq , transitiÕity Ž xRU y U U 2 . and yR z, implies xR z for all x, y, z g Rq and completeness Ž xRU y or yRU x 2 for all x, y g Rq with x / y . are required. Definition 1 F is rational if there exists a reflexive, complete and transitive binary relation RU such that ;S g D ,
F Ž S . 4 s s g S < sRU y ; y g S 4 .
F is said to be rational on F Ž D . if ;S g D ,
F Ž S . 4 s s g S < sRU y ; y g S l F Ž D . 4 .
The (direct) reÕealed preference relation, which is derived from F, is formally 2 defined as follows: for all x, y g Rq , x / y, xPy
m 'S g D such that x , y g S and F Ž S . s x.
This relation is irreflexive, and not generally complete. The non-existence of cycles of length 2 for this relation is known as the Weak Axiom of ReÕealed
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Preference (WARP), while the Strong Axiom of ReÕealed Preference (SARP) states its acyclicity. Axiom 1 Strong Axiom of ReÕealed Preference (SARP): 2 ; x 1 , x 2 , . . . , x t g Rq , if x 1 Px 2 , x 2 Px 3 , . . . , x ty1 Px t , then not w x t Px 1 x .
The Strong Axiom of ReÕealed Preference was introduced by Houthakker Ž1950. in the context of consumer choice models, where he proves that it is equivalent to rationality by means of a transitive binary relation Žthis result was independently obtained by Ville, 1946.. Richter Ž1966. provides more general results for rationality of choice functions Žnot necessarily single-valued and in general domains.. He also proves that, in the single-valued case, SARP is equivalent to rationality by means of a reflexive, complete and transitive binary relation ŽRichter, 1971; Corollary 1, p. 37.. Therefore, to obtain the rationality of two-person bargaining solutions, within the set-up of this paper, it will be enough to prove that SARP is satisfied. Finally, standard bargaining axioms, that will be used to obtain the rationality of these solutions, are formally stated. Axiom 2 Independence of IrreleÕant AlternatiÕes (IIA): ;S1 ,S2 g D ,
if S1 : S2 and F Ž S2 . g S1 , then F Ž S1 . s F Ž S2 . .
This axiom, originally introduced by Nash Ž1950., is equivalent to WARP when the domain is closed under intersection Žsee Hansson, 1968.. It is also important to recall that IIA is necessary for the rationality of a Žsingle-valued. choice function, although it is not sufficient for the rationality of two-person bargaining solutions Žsee Peters and Wakker, 1991.. Thus, none of the Kalai-Smorodinsky solution ŽKalai and Smorodinsky, 1975., equal–loss solution ŽChun, 1988. or rational equal–loss solution ŽHerrero and Marco, 1993. are rational. With respect to continuity assumptions, the Hausdorff topology over D is 2 and S1 ,S2 g D, d denotes the Euclidean distance in considered. Given x g Rq 2 Rq , dŽ x,S1 . s inf dŽ x, s .< s g S1 4 , and r Ž S1 ,S2 . s sup dŽ s,S2 .< s g S14 . Then, the Hausdorff topology is defined by the following metric, ;S1 ,S2 g D ,
d Ž S1 ,S2 . s max r Ž S1 ,S2 . , r Ž S2 ,S1 . 4 .
The Continuity assumption usually requires that small changes in the set of alternatives presented for choice imply small changes in the selected outcome. The one that will be used in this paper is that of Pareto Continuity, introduced by Peters Ž1986. and also used by Peters and Wakker Ž1991.. This assumption is
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weaker than Continuity and takes into account not only changes in the set of alternatives presented for choice, but also in its Pareto Optimal boundary, which is defined as follows: for every S g D, the Pareto Optimal boundary of S is given by PO Ž S . s s g S < y G s, y / s y f S4 .
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Axiom 3 Pareto Continuity (PC): For all sequences Sn4n g N , with Sn g D for all n g N, if Sn4 S and PO Ž Sn .4 PO Ž S ., both in the Hausdorff topology, then F Ž Sn .4 F Ž S ..
3. Rationality results The first result of the paper shows that by imposing only Independence of IrreleÕant AlternatiÕes and Pareto Continuity, the rationality of two-person bargaining solutions is obtained. This result generalizes the rationality part of the result in Bossert Ž1994.. Theorem 1 If F: D satisfies SARP. 1
™R
2 q
is a bargaining solution satisfying IIA and PC, then it
Proof. ŽSee Appendix A..
B
As a consequence of this result, it can be ensured that, in particular, Nash, egalitarian, monotone path solutions, as well as their lexicographic extensions are rational. Remark 1 Although in a different context Žin which bargaining problems are not required to be comprehensive., Peters and Wakker Ž1991. prove that the combination of IIA, Pareto Optimality and Continuity, ensures that the choice function F 2 R such maximizes a real-valued function. That is, there exists a function f :Rq Ž Ž .. Ž . Ž . that for every S g D, f F S ) f s for every s g S, s / F S . It is not difficult to prove that the same result can be obtained in our context by dropping Pareto Optimality and by imposing only Continuity and IIA ŽSanchez, 1996.. Therefore, ´ it is not only Nash’s solution that works as if maximizing a group utility function, but also the egalitarian and monotone path solutions.
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Finally, an alternative rationality result is presented by imposing Strong Monotonicity instead of Pareto Continuity. As in the result of Bossert Ž1994., it is obtained that the rationalization, when restricted to F Ž D ., satisfies some additional properties. Formally, 1 This result was initially proved by imposing also Weak Pareto Optimality Žsee Sanchez, 1996.. ´ The author thanks an anonymous referee of the series of working papers AD ŽIVIE and Universitat d’Alacant. who suggested a way of proving the result by dropping this assumption.
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Axiom 4 Strong Monotonicity (SM): ;S1 ,S2 g D , if S1 : S2 , then F Ž S1 . F F Ž S2 . . If RU is a rationalization of F on F Ž D . and P U is the associated strict preference relation Žthat is, xP U y if not w yRU x x., then it is said that the rationalization on F Ž D . satisfies: Monotonicity: If for all x, y g F Ž D ., x ) y implies xP U y. Strict Quasi-concaÕity: If for all x, y g F Ž D ., if xP U y and for some a g Ž0,1., w a x q Ž1 y a . y x g F Ž D ., then w a x q Ž1 y a . y x P U y. Upper Semicontinuity: If for all x g F Ž D ., y g F Ž D .< yRU x 4 is closed in F Ž D ..
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Theorem 2 If F: D R 2q is a bargaining solution satisfying IIA and SM, then it is rational on F( D) and the rationalization satisfies Monotonicity, Strict Quasi-concavity and Upper Semicontinuity. Proof. The first observation is that, under these assumptions, the revealed preference P, which is asymmetric by IIA, is also transitive and complete when it is restricted to F Ž D .. 2 Consider x, y, z g Rq such that xPy, yPz. By IIA, F Ž ² x , y 4 : . s x , F Ž ² y, z 4 : . s y, F Ž ² x 4 : . s x and F Ž ² y 4 : . s y, and by denoting w s F Ž² x, y, z 4:., it is clear that w f ² y, z 4: Žif not, by IIA, w s y and a contradiction with the asymmetry of P is obtained.. If w g ² x, y4: then, by IIA, w s x and transitivity is obtained. If w g ² x, z 4:, then by IIA, F Ž² x, z 4:. s w. But then, x F z implies ² x, y, z 4: s ² y, z 4: and so w s y, a contradiction; and, in other case, since by SM, w G x and w G z, the only possibility is w s x and transitivity is also obtained. To show that P is complete over F Ž D ., consider x, y g F Ž D . such that x / y and denote w s F Ž² x, y4:.. By IIA, F Ž² x 4:. s x and F Ž² y4:. s y, and then, by SM, it is obtained that x F w and y F w, so x G y or y G x, which respectively implies yPx or xPy. To ensure reflexivity, the binary relation is modified as follows, xRU y w x s y x or w xPy x , which is obviously a reflexive, complete and transitive binary relation rationalizing F on F Ž D .. Moreover, the associated strict preference relation P U coincides on F Ž D . with the revealed preference P. Finally, it is proved that this rationalization satisfies the above mentioned properties. Monotonicity: If x, y g F Ž D . such that x ) y, then, by IIA, F Ž² x 4:. s x and it is obtained that xPy. Strict Quasi-concaÕity: Consider x, y g F Ž D . such that xPy and a g Ž0,1. such that w a x q Ž1 y a . y x g F Ž D .. But then, it is not difficult to prove that, by
m
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IIA and SM, x ) y, so w a x q Ž1 y a . y x ) y and, by reasoning as above, that w a x q Ž1 y a . y x Py. Upper Semicontinuity: If x g F Ž D ., to show that y g F Ž D .< yRU x 4 is closed in F Ž D . it is enough to note that, by IIA and SM, y g F Ž D .< yRU x 4 s F Ž D . 2 < l y g Rq y G x 4 , which is obviously closed in F Ž D .. B Different rationality results can be obtained by imposing monotonicity assumptions weaker than SM and by combining them with Pareto optimality conditions ŽSanchez, 1996.. As a consequence of Theorem 2, in particular, it can be ensured ´ that the rationalizing orderings obtained for the egalitarian and monotone path solutions satisfy Monotonicity, Strict Quasi-concaÕity and Upper Semicontinuity when they are restricted to F Ž D .. In the presence of Weak Pareto Optimality, Thomson and Myerson Ž1980. show that SM and IIA can be replaced by the condition of Domination, which implies the existence of a vector-dominance relation between all elements of F Ž D .. By means of this axiom they characterize the strictly monotone path solutions. In Theorem 2, the condition of Weak Pareto Optimality has been dropped but, as shown in the proof, SM and IIA imply the same kind of vector-dominance relation. Therefore, any bargaining solutions satisfying these axioms must be a generalization of the monotone path solutions. Acknowledgements I thank J.E. Peris, B. Subiza, J.V. LLinares and M.C. Marco for their helpful comments and suggestions. Financial support from the Instituto Valenciano de ŽIVIE. is also acknowledged. A previous version of Investigaciones Economicas ´ this work was made while the author was visiting THEMA, Universite´ de Cergy Pontoise ŽFrance.. She wants to thank their hospitality. Appendix A In order to simplify the notation, henceforth, for all i, j g 1,2, . . . ,n4 , it is considered that iqj if i q j F n . i q j y n if i q j ) n 2 Moreover, given x s Ž x 1 , x 2 ., y s Ž y 1 , y 2 ., z s Ž z 1 , z 2 . g Rq , it is said that y is between x and z if and only if i[js
½
x 1 - y 1 - z 1 and x 2 ) y 2 ) z 2 or x 1 ) y 1 ) z 1 and x 2 - y 2 - z 2 . Proof of Theorem 1. Since in our context IIA and WARP are equivalent ŽHansson, 1968., there exists no cycle of length 2 and, from Bossert’s result Žsee Bossert, 1994; Theorem 1., there exists no cycle of length 3. So, consider n G 4
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and suppose that there exists no cycle of length k for all k - n and, by 2 contradiction, that there exists a cycle of length n. Let A s x 1 , x 2 , . . . , x n 4 : Rq be the set of distinct elements of this cycle. The first observation is that, by definition of P and by applying IIA, F Ž ² x i , x i[1 4 : . s x i and F Ž ² x i 4 : . s x i ,
; i g 1,2, . . . ,n4 .
Ž 1.
Moreover, it is clear that x j f ² x i 4 : ; i / j, ; i , j g 1,2, . . . ,n4 ,
Ž 2.
Žif not, a cycle of length k with k - n is obtained.. Next, it is proved that for all i g 1,2, . . . ,n4 one of the following situations must occur Žsee Fig. 1.: Ža. x i[1 is between x i and x i[2 , and x i[1 g ² x i , x i[2 4:_ PO Ž² x i , x i[2 4:.; Žb. x i is between x i[1 and x i[2 , and x i f ² x i[1 , x i[2 4:. To prove that either Ža. or Žb. are satisfied, first note that, by Ž1. and Ž2., one of these three elements has to be between the other two. Suppose that x i[1 is between x i and x i[2 . If x i[1 f ² x i , x i[2 4:_ PO Ž² x i , x i[2 4:., then ² x i , x i[1 , x i[2 4: s ² x i , x i[1 4: j ² x i[1 , x i[2 4: and F Ž² x i , x i[1 , x i[2 4:. g ² x i , x i[14: j ² x i[1 , x i[2 4:. But, by IIA, if F Ž² x i , x i[1 , x i[2 4:. g ² x i , x i[14:, then F Ž² x i , x i[1 , x i[2 4:. s x i , and if F Ž² x i , x i[1 , x i[2 4:. g ² x i[1 , x i[2 4:, then F Ž² x i , x i[1 , x i[2 4:. s x i[1 , and in both cases a contradiction is obtained Žin the first case a cycle of length n y 1 results and, in the second case, a cycle of length 2.. Now suppose x i is between x i[1 and x i[2 . But then if x i g ² x i[1 , x i[2 4: a cycle of length 2 is immediately obtained. It remains to be shown that x i[2 cannot be between x i and x i[1. By contradiction, suppose that x i[2 is between x i and x i[1. If x i[2 g ² x i , x i[14:, a cycle of length n y 1 is obtained. So, x i[2 f ² x i , x i[14:; thus, ² x i , x i[1 , x i[2 4:
Fig. 1.
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s ² x i , x i[ 2 4: j ² x i[1 , x i[ 2 4: and, by IIA, F Ž² x i , x i[1 , x i[ 2 4:. g ² x i , x i[2 4:_² x i[1 , x i[2 4: Žotherwise, a cycle of length 2 is immediately obtained.. But then, define, for all t g w0,1x, z Ž t . s tx i q Ž1 y t . x i[2 , AŽ t . s 2 ² x i[1 , x i[2 , z Ž t .4: and consider function g:w0,1x Rq such that g Ž t . s F Ž AŽ t ... It is clear that g is a continuous function Žsince it is a composition of continuous ones. and, by IIA, it can be ensured that for every t g w0,1x, either g Ž t . s F Ž AŽ t .. f AŽ0. or g Ž t . s x i[1 Žsee Fig. 2.. By denoting B s gy1 Ž x i[1 ., which is non-empty Ž0 g B ., closed Žby continuity of g . and different from w0,1x Ž1 f B ., the existence of a point t 0 g B that maximizes function g can be ensured. But then, for every t g w0,1x such that t ) t 0 , g Ž t . f AŽ0., a contradiction with continuity of g. Therefore, for all i g 1,2, . . . ,n4 , Ža. or Žb. are satisfied. In particular one of the following cases must occur: ŽI. x 2 is between x 1 and x 3 , and x 2 g ² x 1 , x 34:_ PO Ž² x 1 , x 34:.; ŽII. x 1 is between x 2 and x 3 , and x 1 f ² x 2 , x 3 4:. By considering the first case, it must be true now that either ŽI.1. x 2 is between x 3 and x 4 , and x 2 f ² x 3 , x 4 4:; or, ŽI.2. x 3 is between x 2 and x 4 , and x 3 g ² x 2 , x 4 4:_ PO Ž² x 2 , x 4 4:.. In case ŽI.1., it follows that either x 4 g ² x 1 , x 2 4:, which leads to a cycle of length n y 2; or x 1 is between x 3 and x 4 Žsee Fig. 3.. But if n s 4 this possibility is ruled out by applying to alternatives x 3 , x 4 and x 1 that either Ža. or Žb. must be satisfied; and if n ) 4, this argument can be applied repeatedly to alternatives x 5 , x 6 , . . . , x n4 and to conclude that either x ny 1 g ² x n , x 14: Žwhich leads to a cycle of length 2 since F Ž² x n , x 1 4:. s x n . or x 1 is between x ny1 and x n , again contradicting that either Ža. or Žb. must be
™
Fig. 2.
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Fig. 3.
satisfied when applied to alternatives x ny 1 , x n and x 1. Thus, for instance, if n s 5 and the configuration of alternatives x 1 , x 2 , x 3 and x 4 is that of Fig. 3, then the possibilities for x 5 by considering x 3 , x 4 and x 5 and by applying that either Ža. or Žb. must be satisfied, have been denoted in Fig. 3 by A and B. But if x 5 g A, it follows that x 1 is between x 4 and x 5 , a contradiction; and if x 5 g B, then x 4 g ² x 5 , x 14:, also a contradiction. In case ŽI.2. it follows that x 3 is between x 1 and x 4 and x 3 g ² x 1 , x 4 4: Žbecause x 2 is between x 1 and x 3 and ² x 2 , x 4 4: : ² x 1 , x 4 4:.. But if n s 4, a cycle of length 2 is immediately obtained, and if n ) 4 and there exists k g 5,6, . . . ,n4 such that x ky 2 is between x ky1 and x k and x ky2 f ² x ky1 , x k 4: the same reasoning as in case ŽI.1. can be applied to obtain a contradiction. If not, that is, if for all k g 5,6, . . . ,n4 , x ky 1 is between x ky2 and x k and x ky1 g ² x ky 2 , x k 4:_ PO Ž² x ky2 , x k 4:., then it follows that x ny1 is between x 1 and x n , and x ny 1 g ² x 1 , x n4:, so, since F Ž² x 1 , x n 4:. s x n , a cycle of length 2 is also obtained. Finally, consider case II. By reasoning as above there are two possibilities, either ŽII.1. x 2 is between x 3 and x 4 , and x 2 f ² x 3 , x 4 4:; or ŽII.2. x 3 is between x 2 and x 4 , and x 3 g ² x 2 , x 4 4:_ PO Ž² x 2 , x 4 4:.. But in these cases, the same contradictions as in cases ŽI.1. and ŽI.2., respectively, are obtained. B
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References Bossert, W., 1994. Rational choice and two-person bargaining solutions. Journal of Mathematical Economics 23, 549–563. Chun, Y., 1988. The equal–loss principle for bargaining problems. Economics Letters 26, 103–106. Hansson, B., 1968. Choice structures and preference relations. Synthesis 18, 443–458. Herrero, C., Marco, M.C., 1993. Rational equal–loss solution for bargaining problems. Mathematical Social Sciences 26, 273–286. Houthakker, H.S., 1950. Revealed preference and the utility function. Economica 17, 159–174. Kalai, E., 1977. Proportional solutions to bargaining situations: interpersonal utility comparison. Econometrica 43, 1623–1630. Kalai, E., Smorodinsky, M., 1975. Other solutions to Nash’s bargaining problem. Econometrica 43, 513–518. Lensberg, T., 1987. Stability and collective rationality. Econometrica 55, 935–961. Nash, J.F., 1950. The bargaining problem. Econometrica 18, 155–162. Peters, H., 1986. Simultaneity of issues and additivity in bargaining. Econometrica 54, 153–169. Peters, H., Wakker, P.P., 1991. Independence of irrelevant alternatives and revealed groups preferences. Econometrica 59, 1787–1801. Richter, M.K., 1966. Revealed preference theory. Econometrica 34 Ž3., 635–645. Richter, M.K., 1971. Rational choice. In: Chipman, J.S., Hurwicz, L., Richter, M.K., Sonnenschein, H. ŽEds.., Preferences, Utility and Demand. Harcourt, Brace, Jovanovic, New York, 29–58. Sanchez, M.C., 1996. Rationality of bargaining solutions, Working Papers A Discusion ´ ´ 96-09, IVIE and Universitat d’Alacant, Spain. Thomson, W., forthcoming. Bargaining Theory: The Axiomatic Approach, Cambridge Univ. Press, New York. Thomson, W., Myerson, R.B., 1980. Monotonicity and independence axioms. International Journal of Game Theory 9, 37–49. Ville, J., 1946. Sur les conditions d’existence d’une ophelimite ´ ´ totale et d’un indice du niveau des prix. Annales de l’Universite´ de Lyon, Section A 3, 32–39.
Rationality of bargaining solutions M. Carmen Sanchez ´
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Departamento de Fundamentos del Analisis Economico, UniÕersidad de Murcia, Ronda de LeÕante 10, ´ ´ 30008 Murcia, Spain Received 17 March 1998; received in revised form 5 July 1999; accepted 8 July 1999
Abstract The aim of this paper is to obtain the rationality of two-person bargaining solutions by imposing usual assumptions in cooperative bargaining theory. In particular, it is proved that the conjunction of Independence of Irrelevant Alternatives and either Pareto Continuity or Strong Monotonicity implies this rationality. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: C78 Keywords: Rational choice; Bargaining solutions
1. Introduction A bargaining solution is a special case of a single-valued choice function that selects an outcome for each choice situation Žbargaining problem., which is attained by means of cooperation by the agents involved. One of the interpretations of these solutions is to consider that the outcomes are given by the recommendations of an impartial arbitrator, whose preferences, in some way, represent the preferences of the agents as a group. Thus, the agreement reached in each bargaining problem may be thought of as the most preferred alternative
)
Tel.: q34-968-363-756; E-mail: [email protected]
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 3 3 - 6
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within the set of feasible outcomes according to the arbitrator’s preferences. Moreover, with this interpretation, the axioms used to characterize these solutions can be interpreted as rationality requirements for the arbitrator. It makes sense, therefore, to analyze the rationality of bargaining solutions from the point of view of general choice functions since it would reinforce this interpretation. This requires finding a binary preference relation the maximization of which determines the outcome selected in each bargaining problem. Among the works that must be mentioned along these lines, are those of Lensberg Ž1987., Peters and Wakker Ž1991. and Bossert’s Ž1994. who, in different contexts, provide a set of sufficient conditions Žall of them standard in cooperative bargaining models. which ensures the rationality of bargaining solutions. Peters and Wakker Ž1991. assume that the domain of the solutions is given by 2 the family of convex and compact subsets of Rq , and look for conditions under which a choice function maximizes a real-valued function. To do this, they prove that the rationality of two-person bargaining solutions may be derived from Pareto Optimality, Independence of IrreleÕant AlternatiÕes and Pareto Continuity. Bossert Ž1994. has provided an alternative proof of this result by requiring bargaining problems to be also comprehensive and by imposing Continuity instead of Pareto Continuity. He obtains the existence of a rationalizing ordering that can be endowed with some regularity properties. As Bossert Ž1994. mentions, the result of his work provides sufficient, but not necessary, conditions for the rationality of a bargaining solution. Thus, some rational solutions that are not covered by his result are the egalitarian solution ŽKalai, 1977., monotone path solutions, or their lexicographic extensions. Formal definitions and axiomatic characterization results of these solutions can be found in the book of Thomson Žforthcoming.. The purpose of the present paper is to show that the combination of Independence of IrreleÕant AlternatiÕes and Pareto Continuity also ensures the rationality of two-person bargaining solutions in the context of Bossert’s work. This result generalizes the rationality part of Bossert Ž1994. and enlarges the family of two-person bargaining solutions in which rationality can be derived from standard bargaining axioms, usually invoked to axiomatize bargaining solutions. Moreover, an alternative rationality result that provides some structure for the rationalization is given by replacing Pareto Continuity with a monotonicity assumption Ž Strong Monotonicity ..
2. Preliminaries 2 A two-person bargaining problem is described by a pair Ž S, d . where S : Rq 2 is the set of feasible utility vectors that individuals can attain and d g Rq represents the disagreement point, that is, the vector of utilities that individuals
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would obtain if they did not reach an agreement. Given x s Ž x 1, x 2 ., y s Ž y 1 , y 2 . 2 , the vector notation used is the following: x ) y if and only if x i ) y i for g Rq all i s 1,2 and x G y if and only if x i G y i for all i s 1,2. As usual in bargaining theory, henceforth it is assumed that d s Ž0,0. and the domain considered is given by 2 < D s S : Rq S is convex, compact, comprehensive and
there exists s g S such that s ) 0 4 . 2 To simplify the notation, for every subset A : Rq , the comprehensiÕe and ² : conÕex hull of A is denoted by A , that is, 2 < ² A: s l B : Rq A : B, B convex and comprehensive4 .
™
Formally, a two-person bargaining solution on D is a function that selects an 2 outcome for each bargaining problem in D, that is, F: D Rq such that for all Ž . Ž . S g D, F S g S. Throughout the work, F D denotes the image of F, that is, 2 < F Ž D . s x g Rq 'S g D such that F Ž S . s x 4 .
Since a bargaining solution is a particular case of a single-valued choice function, it is possible to analyze its rationality and to look for sufficient conditions to ensure the existence of a binary relation the maximization of which determines the selected outcome in each bargaining problem. As usual in the related literature, some standard properties are required of the rationalizing binary 2 . relation Ž RU .. In particular, reflexiÕity Ž xRU x for all x g Rq , transitiÕity Ž xRU y U U 2 . and yR z, implies xR z for all x, y, z g Rq and completeness Ž xRU y or yRU x 2 for all x, y g Rq with x / y . are required. Definition 1 F is rational if there exists a reflexive, complete and transitive binary relation RU such that ;S g D ,
F Ž S . 4 s s g S < sRU y ; y g S 4 .
F is said to be rational on F Ž D . if ;S g D ,
F Ž S . 4 s s g S < sRU y ; y g S l F Ž D . 4 .
The (direct) reÕealed preference relation, which is derived from F, is formally 2 defined as follows: for all x, y g Rq , x / y, xPy
m 'S g D such that x , y g S and F Ž S . s x.
This relation is irreflexive, and not generally complete. The non-existence of cycles of length 2 for this relation is known as the Weak Axiom of ReÕealed
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Preference (WARP), while the Strong Axiom of ReÕealed Preference (SARP) states its acyclicity. Axiom 1 Strong Axiom of ReÕealed Preference (SARP): 2 ; x 1 , x 2 , . . . , x t g Rq , if x 1 Px 2 , x 2 Px 3 , . . . , x ty1 Px t , then not w x t Px 1 x .
The Strong Axiom of ReÕealed Preference was introduced by Houthakker Ž1950. in the context of consumer choice models, where he proves that it is equivalent to rationality by means of a transitive binary relation Žthis result was independently obtained by Ville, 1946.. Richter Ž1966. provides more general results for rationality of choice functions Žnot necessarily single-valued and in general domains.. He also proves that, in the single-valued case, SARP is equivalent to rationality by means of a reflexive, complete and transitive binary relation ŽRichter, 1971; Corollary 1, p. 37.. Therefore, to obtain the rationality of two-person bargaining solutions, within the set-up of this paper, it will be enough to prove that SARP is satisfied. Finally, standard bargaining axioms, that will be used to obtain the rationality of these solutions, are formally stated. Axiom 2 Independence of IrreleÕant AlternatiÕes (IIA): ;S1 ,S2 g D ,
if S1 : S2 and F Ž S2 . g S1 , then F Ž S1 . s F Ž S2 . .
This axiom, originally introduced by Nash Ž1950., is equivalent to WARP when the domain is closed under intersection Žsee Hansson, 1968.. It is also important to recall that IIA is necessary for the rationality of a Žsingle-valued. choice function, although it is not sufficient for the rationality of two-person bargaining solutions Žsee Peters and Wakker, 1991.. Thus, none of the Kalai-Smorodinsky solution ŽKalai and Smorodinsky, 1975., equal–loss solution ŽChun, 1988. or rational equal–loss solution ŽHerrero and Marco, 1993. are rational. With respect to continuity assumptions, the Hausdorff topology over D is 2 and S1 ,S2 g D, d denotes the Euclidean distance in considered. Given x g Rq 2 Rq , dŽ x,S1 . s inf dŽ x, s .< s g S1 4 , and r Ž S1 ,S2 . s sup dŽ s,S2 .< s g S14 . Then, the Hausdorff topology is defined by the following metric, ;S1 ,S2 g D ,
d Ž S1 ,S2 . s max r Ž S1 ,S2 . , r Ž S2 ,S1 . 4 .
The Continuity assumption usually requires that small changes in the set of alternatives presented for choice imply small changes in the selected outcome. The one that will be used in this paper is that of Pareto Continuity, introduced by Peters Ž1986. and also used by Peters and Wakker Ž1991.. This assumption is
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weaker than Continuity and takes into account not only changes in the set of alternatives presented for choice, but also in its Pareto Optimal boundary, which is defined as follows: for every S g D, the Pareto Optimal boundary of S is given by PO Ž S . s s g S < y G s, y / s y f S4 .
´
™ ™
™
Axiom 3 Pareto Continuity (PC): For all sequences Sn4n g N , with Sn g D for all n g N, if Sn4 S and PO Ž Sn .4 PO Ž S ., both in the Hausdorff topology, then F Ž Sn .4 F Ž S ..
3. Rationality results The first result of the paper shows that by imposing only Independence of IrreleÕant AlternatiÕes and Pareto Continuity, the rationality of two-person bargaining solutions is obtained. This result generalizes the rationality part of the result in Bossert Ž1994.. Theorem 1 If F: D satisfies SARP. 1
™R
2 q
is a bargaining solution satisfying IIA and PC, then it
Proof. ŽSee Appendix A..
B
As a consequence of this result, it can be ensured that, in particular, Nash, egalitarian, monotone path solutions, as well as their lexicographic extensions are rational. Remark 1 Although in a different context Žin which bargaining problems are not required to be comprehensive., Peters and Wakker Ž1991. prove that the combination of IIA, Pareto Optimality and Continuity, ensures that the choice function F 2 R such maximizes a real-valued function. That is, there exists a function f :Rq Ž Ž .. Ž . Ž . that for every S g D, f F S ) f s for every s g S, s / F S . It is not difficult to prove that the same result can be obtained in our context by dropping Pareto Optimality and by imposing only Continuity and IIA ŽSanchez, 1996.. Therefore, ´ it is not only Nash’s solution that works as if maximizing a group utility function, but also the egalitarian and monotone path solutions.
™
Finally, an alternative rationality result is presented by imposing Strong Monotonicity instead of Pareto Continuity. As in the result of Bossert Ž1994., it is obtained that the rationalization, when restricted to F Ž D ., satisfies some additional properties. Formally, 1 This result was initially proved by imposing also Weak Pareto Optimality Žsee Sanchez, 1996.. ´ The author thanks an anonymous referee of the series of working papers AD ŽIVIE and Universitat d’Alacant. who suggested a way of proving the result by dropping this assumption.
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Axiom 4 Strong Monotonicity (SM): ;S1 ,S2 g D , if S1 : S2 , then F Ž S1 . F F Ž S2 . . If RU is a rationalization of F on F Ž D . and P U is the associated strict preference relation Žthat is, xP U y if not w yRU x x., then it is said that the rationalization on F Ž D . satisfies: Monotonicity: If for all x, y g F Ž D ., x ) y implies xP U y. Strict Quasi-concaÕity: If for all x, y g F Ž D ., if xP U y and for some a g Ž0,1., w a x q Ž1 y a . y x g F Ž D ., then w a x q Ž1 y a . y x P U y. Upper Semicontinuity: If for all x g F Ž D ., y g F Ž D .< yRU x 4 is closed in F Ž D ..
™
Theorem 2 If F: D R 2q is a bargaining solution satisfying IIA and SM, then it is rational on F( D) and the rationalization satisfies Monotonicity, Strict Quasi-concavity and Upper Semicontinuity. Proof. The first observation is that, under these assumptions, the revealed preference P, which is asymmetric by IIA, is also transitive and complete when it is restricted to F Ž D .. 2 Consider x, y, z g Rq such that xPy, yPz. By IIA, F Ž ² x , y 4 : . s x , F Ž ² y, z 4 : . s y, F Ž ² x 4 : . s x and F Ž ² y 4 : . s y, and by denoting w s F Ž² x, y, z 4:., it is clear that w f ² y, z 4: Žif not, by IIA, w s y and a contradiction with the asymmetry of P is obtained.. If w g ² x, y4: then, by IIA, w s x and transitivity is obtained. If w g ² x, z 4:, then by IIA, F Ž² x, z 4:. s w. But then, x F z implies ² x, y, z 4: s ² y, z 4: and so w s y, a contradiction; and, in other case, since by SM, w G x and w G z, the only possibility is w s x and transitivity is also obtained. To show that P is complete over F Ž D ., consider x, y g F Ž D . such that x / y and denote w s F Ž² x, y4:.. By IIA, F Ž² x 4:. s x and F Ž² y4:. s y, and then, by SM, it is obtained that x F w and y F w, so x G y or y G x, which respectively implies yPx or xPy. To ensure reflexivity, the binary relation is modified as follows, xRU y w x s y x or w xPy x , which is obviously a reflexive, complete and transitive binary relation rationalizing F on F Ž D .. Moreover, the associated strict preference relation P U coincides on F Ž D . with the revealed preference P. Finally, it is proved that this rationalization satisfies the above mentioned properties. Monotonicity: If x, y g F Ž D . such that x ) y, then, by IIA, F Ž² x 4:. s x and it is obtained that xPy. Strict Quasi-concaÕity: Consider x, y g F Ž D . such that xPy and a g Ž0,1. such that w a x q Ž1 y a . y x g F Ž D .. But then, it is not difficult to prove that, by
m
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IIA and SM, x ) y, so w a x q Ž1 y a . y x ) y and, by reasoning as above, that w a x q Ž1 y a . y x Py. Upper Semicontinuity: If x g F Ž D ., to show that y g F Ž D .< yRU x 4 is closed in F Ž D . it is enough to note that, by IIA and SM, y g F Ž D .< yRU x 4 s F Ž D . 2 < l y g Rq y G x 4 , which is obviously closed in F Ž D .. B Different rationality results can be obtained by imposing monotonicity assumptions weaker than SM and by combining them with Pareto optimality conditions ŽSanchez, 1996.. As a consequence of Theorem 2, in particular, it can be ensured ´ that the rationalizing orderings obtained for the egalitarian and monotone path solutions satisfy Monotonicity, Strict Quasi-concaÕity and Upper Semicontinuity when they are restricted to F Ž D .. In the presence of Weak Pareto Optimality, Thomson and Myerson Ž1980. show that SM and IIA can be replaced by the condition of Domination, which implies the existence of a vector-dominance relation between all elements of F Ž D .. By means of this axiom they characterize the strictly monotone path solutions. In Theorem 2, the condition of Weak Pareto Optimality has been dropped but, as shown in the proof, SM and IIA imply the same kind of vector-dominance relation. Therefore, any bargaining solutions satisfying these axioms must be a generalization of the monotone path solutions. Acknowledgements I thank J.E. Peris, B. Subiza, J.V. LLinares and M.C. Marco for their helpful comments and suggestions. Financial support from the Instituto Valenciano de ŽIVIE. is also acknowledged. A previous version of Investigaciones Economicas ´ this work was made while the author was visiting THEMA, Universite´ de Cergy Pontoise ŽFrance.. She wants to thank their hospitality. Appendix A In order to simplify the notation, henceforth, for all i, j g 1,2, . . . ,n4 , it is considered that iqj if i q j F n . i q j y n if i q j ) n 2 Moreover, given x s Ž x 1 , x 2 ., y s Ž y 1 , y 2 ., z s Ž z 1 , z 2 . g Rq , it is said that y is between x and z if and only if i[js
½
x 1 - y 1 - z 1 and x 2 ) y 2 ) z 2 or x 1 ) y 1 ) z 1 and x 2 - y 2 - z 2 . Proof of Theorem 1. Since in our context IIA and WARP are equivalent ŽHansson, 1968., there exists no cycle of length 2 and, from Bossert’s result Žsee Bossert, 1994; Theorem 1., there exists no cycle of length 3. So, consider n G 4
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and suppose that there exists no cycle of length k for all k - n and, by 2 contradiction, that there exists a cycle of length n. Let A s x 1 , x 2 , . . . , x n 4 : Rq be the set of distinct elements of this cycle. The first observation is that, by definition of P and by applying IIA, F Ž ² x i , x i[1 4 : . s x i and F Ž ² x i 4 : . s x i ,
; i g 1,2, . . . ,n4 .
Ž 1.
Moreover, it is clear that x j f ² x i 4 : ; i / j, ; i , j g 1,2, . . . ,n4 ,
Ž 2.
Žif not, a cycle of length k with k - n is obtained.. Next, it is proved that for all i g 1,2, . . . ,n4 one of the following situations must occur Žsee Fig. 1.: Ža. x i[1 is between x i and x i[2 , and x i[1 g ² x i , x i[2 4:_ PO Ž² x i , x i[2 4:.; Žb. x i is between x i[1 and x i[2 , and x i f ² x i[1 , x i[2 4:. To prove that either Ža. or Žb. are satisfied, first note that, by Ž1. and Ž2., one of these three elements has to be between the other two. Suppose that x i[1 is between x i and x i[2 . If x i[1 f ² x i , x i[2 4:_ PO Ž² x i , x i[2 4:., then ² x i , x i[1 , x i[2 4: s ² x i , x i[1 4: j ² x i[1 , x i[2 4: and F Ž² x i , x i[1 , x i[2 4:. g ² x i , x i[14: j ² x i[1 , x i[2 4:. But, by IIA, if F Ž² x i , x i[1 , x i[2 4:. g ² x i , x i[14:, then F Ž² x i , x i[1 , x i[2 4:. s x i , and if F Ž² x i , x i[1 , x i[2 4:. g ² x i[1 , x i[2 4:, then F Ž² x i , x i[1 , x i[2 4:. s x i[1 , and in both cases a contradiction is obtained Žin the first case a cycle of length n y 1 results and, in the second case, a cycle of length 2.. Now suppose x i is between x i[1 and x i[2 . But then if x i g ² x i[1 , x i[2 4: a cycle of length 2 is immediately obtained. It remains to be shown that x i[2 cannot be between x i and x i[1. By contradiction, suppose that x i[2 is between x i and x i[1. If x i[2 g ² x i , x i[14:, a cycle of length n y 1 is obtained. So, x i[2 f ² x i , x i[14:; thus, ² x i , x i[1 , x i[2 4:
Fig. 1.
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s ² x i , x i[ 2 4: j ² x i[1 , x i[ 2 4: and, by IIA, F Ž² x i , x i[1 , x i[ 2 4:. g ² x i , x i[2 4:_² x i[1 , x i[2 4: Žotherwise, a cycle of length 2 is immediately obtained.. But then, define, for all t g w0,1x, z Ž t . s tx i q Ž1 y t . x i[2 , AŽ t . s 2 ² x i[1 , x i[2 , z Ž t .4: and consider function g:w0,1x Rq such that g Ž t . s F Ž AŽ t ... It is clear that g is a continuous function Žsince it is a composition of continuous ones. and, by IIA, it can be ensured that for every t g w0,1x, either g Ž t . s F Ž AŽ t .. f AŽ0. or g Ž t . s x i[1 Žsee Fig. 2.. By denoting B s gy1 Ž x i[1 ., which is non-empty Ž0 g B ., closed Žby continuity of g . and different from w0,1x Ž1 f B ., the existence of a point t 0 g B that maximizes function g can be ensured. But then, for every t g w0,1x such that t ) t 0 , g Ž t . f AŽ0., a contradiction with continuity of g. Therefore, for all i g 1,2, . . . ,n4 , Ža. or Žb. are satisfied. In particular one of the following cases must occur: ŽI. x 2 is between x 1 and x 3 , and x 2 g ² x 1 , x 34:_ PO Ž² x 1 , x 34:.; ŽII. x 1 is between x 2 and x 3 , and x 1 f ² x 2 , x 3 4:. By considering the first case, it must be true now that either ŽI.1. x 2 is between x 3 and x 4 , and x 2 f ² x 3 , x 4 4:; or, ŽI.2. x 3 is between x 2 and x 4 , and x 3 g ² x 2 , x 4 4:_ PO Ž² x 2 , x 4 4:.. In case ŽI.1., it follows that either x 4 g ² x 1 , x 2 4:, which leads to a cycle of length n y 2; or x 1 is between x 3 and x 4 Žsee Fig. 3.. But if n s 4 this possibility is ruled out by applying to alternatives x 3 , x 4 and x 1 that either Ža. or Žb. must be satisfied; and if n ) 4, this argument can be applied repeatedly to alternatives x 5 , x 6 , . . . , x n4 and to conclude that either x ny 1 g ² x n , x 14: Žwhich leads to a cycle of length 2 since F Ž² x n , x 1 4:. s x n . or x 1 is between x ny1 and x n , again contradicting that either Ža. or Žb. must be
™
Fig. 2.
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Fig. 3.
satisfied when applied to alternatives x ny 1 , x n and x 1. Thus, for instance, if n s 5 and the configuration of alternatives x 1 , x 2 , x 3 and x 4 is that of Fig. 3, then the possibilities for x 5 by considering x 3 , x 4 and x 5 and by applying that either Ža. or Žb. must be satisfied, have been denoted in Fig. 3 by A and B. But if x 5 g A, it follows that x 1 is between x 4 and x 5 , a contradiction; and if x 5 g B, then x 4 g ² x 5 , x 14:, also a contradiction. In case ŽI.2. it follows that x 3 is between x 1 and x 4 and x 3 g ² x 1 , x 4 4: Žbecause x 2 is between x 1 and x 3 and ² x 2 , x 4 4: : ² x 1 , x 4 4:.. But if n s 4, a cycle of length 2 is immediately obtained, and if n ) 4 and there exists k g 5,6, . . . ,n4 such that x ky 2 is between x ky1 and x k and x ky2 f ² x ky1 , x k 4: the same reasoning as in case ŽI.1. can be applied to obtain a contradiction. If not, that is, if for all k g 5,6, . . . ,n4 , x ky 1 is between x ky2 and x k and x ky1 g ² x ky 2 , x k 4:_ PO Ž² x ky2 , x k 4:., then it follows that x ny1 is between x 1 and x n , and x ny 1 g ² x 1 , x n4:, so, since F Ž² x 1 , x n 4:. s x n , a cycle of length 2 is also obtained. Finally, consider case II. By reasoning as above there are two possibilities, either ŽII.1. x 2 is between x 3 and x 4 , and x 2 f ² x 3 , x 4 4:; or ŽII.2. x 3 is between x 2 and x 4 , and x 3 g ² x 2 , x 4 4:_ PO Ž² x 2 , x 4 4:.. But in these cases, the same contradictions as in cases ŽI.1. and ŽI.2., respectively, are obtained. B
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References Bossert, W., 1994. Rational choice and two-person bargaining solutions. Journal of Mathematical Economics 23, 549–563. Chun, Y., 1988. The equal–loss principle for bargaining problems. Economics Letters 26, 103–106. Hansson, B., 1968. Choice structures and preference relations. Synthesis 18, 443–458. Herrero, C., Marco, M.C., 1993. Rational equal–loss solution for bargaining problems. Mathematical Social Sciences 26, 273–286. Houthakker, H.S., 1950. Revealed preference and the utility function. Economica 17, 159–174. Kalai, E., 1977. Proportional solutions to bargaining situations: interpersonal utility comparison. Econometrica 43, 1623–1630. Kalai, E., Smorodinsky, M., 1975. Other solutions to Nash’s bargaining problem. Econometrica 43, 513–518. Lensberg, T., 1987. Stability and collective rationality. Econometrica 55, 935–961. Nash, J.F., 1950. The bargaining problem. Econometrica 18, 155–162. Peters, H., 1986. Simultaneity of issues and additivity in bargaining. Econometrica 54, 153–169. Peters, H., Wakker, P.P., 1991. Independence of irrelevant alternatives and revealed groups preferences. Econometrica 59, 1787–1801. Richter, M.K., 1966. Revealed preference theory. Econometrica 34 Ž3., 635–645. Richter, M.K., 1971. Rational choice. In: Chipman, J.S., Hurwicz, L., Richter, M.K., Sonnenschein, H. ŽEds.., Preferences, Utility and Demand. Harcourt, Brace, Jovanovic, New York, 29–58. Sanchez, M.C., 1996. Rationality of bargaining solutions, Working Papers A Discusion ´ ´ 96-09, IVIE and Universitat d’Alacant, Spain. Thomson, W., forthcoming. Bargaining Theory: The Axiomatic Approach, Cambridge Univ. Press, New York. Thomson, W., Myerson, R.B., 1980. Monotonicity and independence axioms. International Journal of Game Theory 9, 37–49. Ville, J., 1946. Sur les conditions d’existence d’une ophelimite ´ ´ totale et d’un indice du niveau des prix. Annales de l’Universite´ de Lyon, Section A 3, 32–39.