A bargaining situation refers to a scenario in which several individuals interacting through some institutional arrangement seek to reach an agreement on the way to allocate the proceeds of some cooperative venture. The main problem that affects individuals in a negotiation is the need to determine how to cooperate. In general, the establishment of an agreement is beneficial for all negotiators, and although this is known by them, it is not clear how they could reach such an agreement. The problem is that each individual wants the established arrangement to be the most favourable to him; this naturally happens at the expense of the welfare of the other negotiators. The Bargaining Theory analyzes how groups of individuals who have conflicting preferences should and could make joint decisions that affect them all.

Many economic interactions involve negotiations on a variety of issues. Wages and other prices are often the outcome of negotiations among the concerned parties. Mergers and acquisitions require negotiations over the price at which such transactions are to take place. Bargaining problems have been of great importance for economics, particularly those related to the optimal resource allocation. The systematic study of these problems dates back to (at least) Edgeworth’s (1881) work entitled Mathematical Psychics. In the second part of his book, Edgeworth analyzes a pure exchange economy with two consumers only able to control their own private resources (endowments). He wondered what would be the final allocation of goods that both individuals could achieve; if the exchange process was conducted through a bargaining scheme in which offers and counteroffers extend until no individual have any interest to change the proposed allocation. Edgeworth found that for any agreed final allocation, there cannot be other feasible allocation in which at least one consumer is better off and the other is not worse off. The idea is that if it is possible to improve the welfare of an individual without harming the other, then at least one consumer will find beneficial to change the allocation, without this change being rejected by the other individual. An allocation satisfying this optimality criterion is called Pareto efficient. The set of all Pareto efficient allocations forms a continuum denominated the Contract curve.

Clearly, if an allocation belongs to the contract curve, then no other feasible allocation can make both consumers better off. However, this does not mean that there are no other allocations that each consumer individually prefers and that he (or she) can guarantee by his (or her) individual endowments. The set of allocations of a pure exchange economy of two consumers, that cannot be improved neither jointly nor individually, was called by Edgeworth the set of final settlements. That is, an allocation is a final settlement if it belongs to the contract curve (i.e., it is Pareto efficient), and no individual prefers his private endowment to the specified allocation (i.e., it is individually rational). In the case of exchange economies with more than two consumers, the set of final settlements is formed by all allocations on the contract curve such that no consumer wants to renegotiate these allocations with some group of consumers, i.e. those efficient allocations that cannot be improved for any subset of consumers in the economy. This notion of stability was reinvented and generalized by Gilles (1959) in the context of the Cooperative Game Theory, and it is known as the Core. Edgeworth observed that as the number of consumers increases, the set of final settlements tended to shrink, leading him to speculate that the core of an exchange economy converges to set of Walrasian allocations when the number of agents

tends to infinity.

Although a formal proof of this conjecture was not provided by Edgeworth, his ideas contributed to its subsequent formalization. It was only until the early 1960s that sufficient conditions for the convergence of the core were provided (core equivalence theorems). On one hand, Debreu and Scarf (1963) considered a series of replicas of the original economy; on the other hand, Aumann (1964) considered an economy with a continuum of agents. Unfortunately, the equivalence results established by these authors show that the convergence of the core can only be guaranteed under conditions of regularity of the economy more restrictive than those considered by Edgeworth. However, these findings have become a theoretical justification for the concept of perfect competition in large markets.

Many economic models of resource allocation have been studied using the traditional analysis of equilibrium in the absence of strategic externalities, particularly those related to exchange economies. An example of this is Lindahl’s (1958) equilibrium analysis for the distribution problem in an exchange economy with public goods. Lindahl taxation requires, however, information about the marginal benefits to each individual. Such information is not available in general. Then, consumers can lower their tax cost by misreporting their true benefits derived from the public good. Incentives to tell the truth under Lindahl taxation resemble those of a free-riding problem. The classical literature also assumes that a “benevolent” government takes decisions so as to maximize the social welfare. See for instance Arrow and Kurz’s (1970) formulation of the problems of public expenditure in the context of the Growth Theory. Nevertheless, within a democratic system, a person can vote and try to influence the government’s decisions. Then, analyzing the government as subject to the influence of those who elected it introduces political aspects into the redistribution problem. Game Theory has proved to be a powerful tool for analyzing the strategic incentives in these and other economic problems. Although these two approaches, the traditional equilibrium analysis and the game theoretic analysis, are seemingly unrelated, Osborne and Rubinstein (1990) provide an extensive study of the connection between the theory of strategic bargaining and the fundamentals of Walrasian equilibrium.

John Nash (1950, 1953) developed two different (but related) approaches for the study of strategic bargaining. In his 1950 paper, Nash proposed a model of bilateral negotiations in which the result of the cooperation is determined solely by the preferences of the two negotiators. In his model, when individuals negotiate, the payoffs allocations that the two individuals ultimately get depend only on the utility they would expect if negotiations were to fail to reach an agreement, and on the set of utility allocations that are jointly feasible for the two negotiators in a cooperative agreement. The bargaining solution proposed by Nash to such problems of bilateral negotiation, is the unique feasible utility allocation that maximizes the sum of weighted utilities (utilitarian criterion) and then divide this maximal weighted-utility worth in such a way that each individual gets the same weighted-utility gain over the disagreement outcome (egalitarian criterion). Here the utility weights accommodate utility scales to make them interpersonally comparable. Nash solution was originally defined as the unique solution that satisfies a set of “reasonable” axioms justified in normative arguments (see Myerson, 1991 chap. 8). The idea is to consider several properties that would seem natural for the solution to have and then one determines if such properties actually determine a (unique) cooperative solution. This is known in the literature as the axiomatic (or cooperative) approach and it gave rise to the cooperative analysis of the strategic bargaining.

One might also consider a cooperative solution as the equilibrium outcome of a non-cooperative bargaining game in which all the details of the negotiations process are well specified. This observation led Nash in his article of 1953 to formulate a non-cooperative model of bilateral bargaining in which the unique “stable” equilibrium payoffs matched his axiomatic bargaining solution. By representing the institutional arrangements and negotiation protocols by means of a non-cooperative game, we get a deeper understanding about how coalitions form and how players interact. In this non cooperative approach, cooperative solutions will be understood as the equilibrium outcome of the strategic problem faced by the individuals. As stated by Nash (1953, p. 129), “both approaches to the bargaining problem, via a negotiation model or via the axioms, are complementary; each helps to justify and clarify the other”. This vision, intended to bridge the gap between the cooperative and non-cooperative approaches, is known in the literature as the Nash program (see Serrano, 2005, for a comprehensive survey). A large body of the analysis of the strategic bargaining in economic situations, however, has adopted the non-cooperative approach. Unfortunately, the non-cooperative analysis depends strongly on the particular form of the negotiation procedure. In general it is uncertain what is the best negotiation protocol for analyzing a particular problem, or which are the communication schemes used by players in a bargaining situation. Moreover, even when the outcome of a bargaining problem is robust to a variety of negotiation procedures, such procedures often do not lead to a unique equilibrium, so that results may not be conclusive.

Given a strategic situation, we can describe it in the most informative way by specifying the timing and set of moves available and the information held by each player at each point in time (extensive form representation). We can also abstract away from this complex representation by suppressing information concerning strategies and focusing only on the outcomes that result when the players come together in different coalitions. By doing that, one lacks the description of the procedures that coalitions follow to achieve particular outcomes. Nevertheless, the analysis becomes robust to irrelevant details of the different negotiation procedures that underlie the same set of feasible outcomes. This idea is better expressed by Aumann (1989, p. 8-9):

“Cooperative theory starts with a formalization of games that abstracts away altogether from procedures and… concentrates, instead, on the possibilities for agreement… There are several reasons that explain why cooperative games came to be treated separately. One is that when one does build negotiation and enforcement procedures explicitly into the model, then the results of a non-cooperative analysis depend very strongly on the precise form of the procedures, on the order of making offers and counter-offers and so on. This may be appropriate in voting situations in which precise rules of parliamentary order prevail, where a good strategist can indeed carry the day. But problems of negotiation are usually more amorphous; it is difficult to pin down just what the procedures are. More fundamentally, there is a feeling that procedures are not really all that relevant; that it is the possibilities for coalition forming, promising and threatening that are decisive, rather than whose turn it is to speak… Detail distracts attention from essentials. Some things are seen better from a distance; the Roman camps around Metzada are indiscernible when one is in them, but easily visible from the top of the mountain.”

This article is an effort to summarize some valuable contributions of the theory of cooperative games to the study of bargaining problems in economic theory. I believe that both the cooperative approach and the non-cooperative approach are complementary: on the one hand, axiomatic cooperative solutions are appropriate for a wider variety of situations than those satisfying only the assumptions we made in a particular non cooperative bargaining procedure. On the other hand, the models in the non-cooperative approach will enhance our understanding of a cooperative solution.

by Andrés Salamanca

References:

- Arrow, K. J. and Kurz, M. (1970). Public Investment, the Rate of Return and Optimal Fiscal Policy, Resources for the Future, The Johns Hopkins Press.
- Aumann, R. (1964). “Markets with a continuum of traders”. Econometrica, 32, pp. 39-50.
- Aumann, R. (1989). Game Theory. In: Eatwell, J., Milgate, M., and P. Newman (Eds.). The New Palgrave, New York, Norton, pp. 8-9.
- Debreu, G. and Scarf, H. (1963). “A limit theorem on the core of an economy”. International Economic Review, 4, pp. 235-246.
- Edgeworth, F. Y. (1881). Mathematical Psychics. London: Kegan Paul.
- Gilles, D. B. (1959). Solutions to general non-zero-sum games. In: Contributions to the Theory of Games IV. H.W. Kuhn y A.W. Tucker (Eds.). Princeton: Princeton University Press, pp. 47-85.
- Lindahl, E. (1958). Just taxation-A positive solution. In: Musgrave, R. A.; Peacock, A. T., Classics in the Theory of Public Finance, London: Macmillan.
- Myerson, R. (1991). Game Theory: Analysis of Conflict, Harvard University Press.
- Nahs, J. (1950). “The bargaining problem”. Econometrica, 18, pp. 155-162.
- Nash, J. (1953). “Two-person cooperative games”. Econometrica, 21, pp. 128-140.
- Osborne, M. J. and Rubinstein, A. (1990). Bargaining and Markets, Academic Press, San Diego.
- Serrano, R. (2005). “Fifty years of the Nash program, 1953-2003”. Investigaciones

Económicas, 29, pp. 219-258.

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