An equitable solution for multicriteria bargaining games

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Abstract

In this paper we study bargaining models where the agents consider several criteria to evaluate the results of the negotiation process. We propose a new solution concept for multicriteria bargaining games based on the distance to a utopian minimum level vector. This solution is a particular case of the class of the generalized leximin solutions and can be characterized as the solution of a finite sequence of minimax programming problems.

Introduction

A bargaining problem describes a decision situation in which agents must reach a unanimous decision. There is a set of feasible alternatives, any one of which can be the outcome of the bargaining problem if all the agents agree on it. In the event that no unanimous agreement is reached, some pre-specified disagreement outcome will be the result. These problems have usually been analyzed from a unicriterion perspective. However, in real-world negotiations, it is often the case that more than one issue is at stake. In this sense, multicriteria models provide a more realistic representation of these problems. Nevertheless, formal models of multicriteria bargaining are scarce in the literature. Krus and Bronisz, 1993, Krus, 2002 propose and characterize a generalized version of the Kalai–Smorodinsky solution (Kalai and Smorodinsky, 1975) for multicriteria bargaining problems. Korhonen et al. (1986) analyze multicriteria bargaining situations and develop interactive approaches for solving these kinds of problems. Related work can be found in Ehtamo and Hämäläine (2001). Recently, Hinojosa et al. (2005) have proposed a solution concept that is specific for this class of games, the generalized maximin solutions and a lexicographical refinement of this concept, the generalized leximin solutions. This notion of solution provides a set of outcomes for the multicriteria bargaining game which is consistent with the generalized Rawls’ principle of justice (Rawls, 1971). However, in general the leximin solution does not consist of a unique outcome, and it is necessary to apply further criteria in order to arrive at a single solution.

In this paper we propose a solution concept for the multicriteria bargaining game which induces a specific subset of the generalized maximin outcomes, the equitable solutions. It is based on minimizing a distance function which involves a compromise between the levels reached by different criteria. Compromise solutions have already been proposed in the literature for group decision problems. For single criterion bargaining problems, Yu (1973) first proposed a family of compromise solutions and demonstrated some general properties. For multicriteria bargaining games, in an early paper Bergstresser and Yu (1977) suggested an approach where the payoff for each criterion is considered as the payoff of a single player. This permits the multicriteria bargaining game to be interpreted as a scalar game with the number of players equal to the total number of criteria and the application of compromise solutions for conventional bargaining games. This approach is also considered in Hwang and Lin (1987). However, this setting does not take into account the fact that each player has to control several criteria when deciding which outcome is convenient to agree on and therefore the agreements obtained may not correspond to the preferences of the agents. In Hinojosa et al. (2004), a solution based on a compromise between the minimum levels of the criteria is proposed and also, the problem of selecting the disagreement point when each agent values different criteria is analyzed.

The analysis presented in this paper differs from the existing literature in that the concept of compromise solution, which we propose for the multicriteria case, maintains the multidimensional nature of each agents’ payoff. In particular we consider the distance corresponding to the Chebyshev metric in order to take into account the maximum regret of the agents with respect to the consensus achieved. We recursively apply the idea of minimizing the maximum regret of the group of players with respect to their criteria in order to define a solution concept which yields a unique consensus outcome: the equitable lexicographical solution. This solution concept is characterized as the solution of a sequence of minimax programming problems.

The rest of the paper is organized as follows. In Section 2 we describe the multicriteria bargaining game. In Section 3, we define and characterize the family of equitable solutions. In Section 4, we refine the proposed notion of equitable solutions and introduce the equitable lexicographical solution for which a lexicographic procedure is constructed. Section 5 is devoted to the conclusion. An example illustrates the concepts and results throughout the paper.

The following notation will be used with respect to given vectors x,yRn: x > y means that xj > yj, for j = 1,  , n; x  y means that xj  yj, for j = 1,  , n and x  y; and x  y means that xj  yj, for j = 1,  , n. If no confusion is likely, for a matrix XRm×n we will denote by Xj the jth row, and by Xi the ith column. We will also denote the dominance relations between matrices X,YRm×n, as follows: X  Y if xjiyji,i,j; X  Y if X  Y and X  Y; X > Y if xji>yji,i,j.

Section snippets

Multicriteria bargaining games

Let N = {1,2,  , n} be a set of agents, such that each agent i  N values the same set of criteria, {1,  , m}. An n-person multicriteria bargaining game is a pair (S, d), where S is a subset of an (m × n)-dimensional space, SRm×n. Each point in S represents a feasible outcome, that is, the payoffs with respect to all the criteria which the players can achieve by mutual agreement. In the event of disagreement, agents receive the status quo, dRm×n. The problem consists of supporting the agents to reach

Equitable solutions

In this section, we establish the concept of equitable solution for the multicriteria bargaining problem a propose a procedure to obtain them. The idea behind this solution concept is that the agents jointly agree on those outcomes that minimize the distance in the Chebyshev norm between the payoffs which can be attained by the agents in each criteria and a utopian minimum payoff vector.

In order to define this solution concept for the class of multicriteria bargaining games, each feasible

The equitable lexicographical solution

Once the set S(z1) is determined, if it is a singleton, S(z1) = {X}, the corresponding deviation vector is the unique lexicographical minimum for S, and therefore, X is an equitable solution for the multicriteria bargaining game. If there is a whole set of outcomes whose minimum payoff vector coincides at z1, then it is possible to refine the proposed notion of equitable solution, in the sense that once a minimum payoff vector whose vector of deviations is lexicographical minimum is assured to

Conclusions

The main contribution in this paper consists of proposing a well defined solution concept for multicriteria bargaining games: the equitable lexicographical solution. This solution is equitable in the sense that yields to a cooperative consensus result where the group of players obtain payoffs with respect to their criteria which cannot be improved simultaneously in a lexicographical way.

The equitable lexicographical solution provides a notion of solution for a general decision making problem

Acknowledgments

The research of the authors is partially supported by the Spanish Ministry of Science and Technology projects BFM2002-11282-E, BEC2003-03111, and CENTRA project ECO14-2005. The authors are grateful to an anonymous referee for some helpful comments.

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