Bargaining and the theory of cooperative games: John Nash and beyond

This essay surveys the literature on the axiomatic model of bargaining formulated by Nash ("The Bargaining Problem," Econometrica 28, 1950, 155-162).Nash's bargaining model, Nash solution, Kalai-Smorodinsky solution, Egalitarian solution

BARGAINING, VOTING, AND VALUE

This paper addresses the following issue: If a set of agents bargain on a set of feasible alternatives 'in the shadow' of a voting rule, that is, any agreement can be enforced if a 'winning coalition' supports it, what general agreements are likely to arise? In other words: What influence can the voting rule used to settle (possibly non-unanimous) agreements have on the outcome of negotiations? To give an answer we model the situation as an extension of the Nash bargaining problem in which an arbitrary voting rule replaces unanimity to settle agreements by n players. This provides a setting in which a natural extension of Nash's solution is obtained axiomatically. Two extensions admitting randomization on voting rules based on two informational scenarios are considered.Bargaining, voting, value, bargaining in committees.

Gradual Negotiations and Proportional Solutions

I characterize the proportional N-person bargaining solutions by individual rationality, translation invariance, feasible set continuity, and a new axiom - interim improvement. The latter says that if the disagreement point d is known, but the feasible set is not - it may be either S or T, where S is a subset of T - then there exists a point d' in S, d' > d, such that replacing d with d' as the disagreement point would not change the final bargaining outcome, no matter which feasible set will be realized, S or T. In words, if there is uncertainty regarding a possible expansion of the feasible set, the players can wait until it is resolved; in the meantime, they can find a Pareto improving interim outcome to commit to - a commitment that has no effect in case negotiations succeed, but promises higher disagreement payoffs to all in case negotiations fail prior to the resolution of uncertainty.Bargaining; Proportional solutions

Lost in Translation? Basis Utility and Proportionality in Games

Cooperative and noncooperative games have no representation of players's basis utilities. Basis utility is the natural reference point on a player's utility scale that enables the determination the marginal utility of any payoff or allocation. A player's basis utility can be determined by an observer and other players under standard rationality assumptions. Basis utility allows interpersonal comparison of proportional utility gains relative to the disagreement outcome. Proportional pure bargaining is the unique solution satisfying efficiency, symmetry, affine transformation invariance and monotonicity in pure bargaining games with basis utility. Characterization of the Nash (1950) bargaining solution requires the assumption of the irrelevance of basis utility in games with basis utility. All existing cooperative solution functions become translation invariant once proper account is taken of basis utility. The noncooperative rationality of these results is demonstrated with a proportional bargaining based on Gul (1988). Further noncooperative application is demonstrated by showing that quantal response equilibria with multiplicative error structures (Goeree, Holt and Palfrey (2004)) become translation invariant with specification of basis utility.Basis utility, equal split, Kalai-Smorodinsky solution, Nash bargaining, quantal response equilibria, proportional bargaining, translation invariance.

Social Contract II: Gauthier and Nash

This is the second of several free-standing papers whose beginnings lie in Rawls' [1958, 1968, 1972] theory of the social contract. The aim of the sequence of papers is to defend a version of Rawls' "egalitarian" conclusion for a world in which agents are assumed to be constrained only by rational self-interest.Center for Research on Economic and Social Theory, Department of Economics, University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/100627/1/ECON104.pd

A theory of unstructured bargaining using distribution-valued solution concepts

In experiments it is typically found that many joint utility outcomes arise in any given unstructured bargaining game. This suggests using a positive unstructured bargaining concept that maps a bargaining game to a probability distribution over outcomes rather than to a single outcome. We show how to "translate" Nash's bargaining axioms to apply to such distributional bargaining concepts. We then prove that a subset of those axioms forces the distribution over outcomes to be a power-law. Unlike Nash's original result, our result holds even if the feasible set is finite. When the feasible set is convex and comprehensive, the mode of the power law distribution is the Harsanyi bargaining solution, and if we require symmetry it is the Nash bargaining solution. However in general these modes of the joint utility distribution are not Bayes-optimal predictions for the joint uitlity, nor are the bargains corresponding to those outcomes the most likely bargains. We then show how an external regulator can use distributional solution concepts to optimally design an unstructured bargaining scenario. Throughout we demonstrate our analysis in computational experiments involving flight rerouting negotiations in the National Airspace System.JEL Codes:

Bargaining Power in Marriage: Earnings, Wage Rates and Household Production

What determines bargaining power in marriage? This paper argues that wage rates, not earnings, determine well-being at the threat point and, hence, determine bargaining power. Observed earnings at the bargaining equilibrium may differ from earnings at the threat point because hours allocated to market work at the bargaining solution may differ from hours allocated to market work at the threat point. In the divorce threat model, for example, a wife who does not work for pay while married might do so following a divorce; hence, her bargaining power would be related to her wage rate, not to her earnings while married. More generally, a spouse whose earnings are high because he or she chooses to allocate more hours to market work, and correspondingly less to household production and leisure, does not have more bargaining power. But a spouse whose earnings are high because of a high wage rate does have more bargaining power. Household production has received little attention in the family bargaining literature. The output of household production is analogous to earnings, and a spouse's productivity in household production is analogous to his or her wage rate. Thus, in a bargaining model with household production, a spouse's productivity in home production is a source of bargaining power.