Distortion of Utilities and the Bargaining Problem
研究多商品讨价还价中，代理人通过扭曲报告效用函数来影响结果的非合作博弈，证明在纳什等解下，真实效用的约束平等竞争均衡是纳什均衡结果。
Given two agents with vonNeumann-Morgenstern utilities who wish to divide n commodities, consider the two-person noncooperative game with strategies consisting of concave, increasing von Neumann-Morgenstern utility functions as well as rules to break ties and whose outcomes are some solution to the bargaining game determined by the strategies used.It is shown that, for a class of bargaining solutions which includes those of Nash and Raiffa, Kalai and Smorodinsky, any constrained equal-income competitive equilibrium allocation for the true utilities is a Nash equilibrium outcome for the noncooperative game.credible utilities for an agent.Kurz takes 1Q to be the set of all von Neumann-Morgenstern utility functions that are increasing, concave, and continuously differentiable.In [11] he shows that for the one-commodity Aumann-Kurz model [1], reporting any linear function in QI is a dominant strategy for each agent.The marginal tax rate implied by the use of linear strategies is 50 per cent.Kurz generalizes this result to the n-commodity case in [121.Once again, players have dominant strategies that lead to a marginal tax rate of 50 per cent.In this case, however, the dominant strategy reported utility functions need not be linear.The significance of Kurz's results is that, regardless of the true preferences of the agents, the distortion game has a dominant strategy equilibrium that yields a Pareto-efficient outcome.Crawford and Varian [31 use the methods of Kurz to analyze the effect that distortion of utilities has on the solutions to bargaining games.Assuming that agents may report any concave, increasing utility function, they find that in Nash [15] or Raiffa [17]-Kalai-Smorodinsky [8] bargaining over the division of a single good reporting linear utility functions constitutes a dominant strategy equilibrium.The allocation implied by the equilibrium reports is equal division.The purpose of this paper is to generalize this result to include bargaining over more than one commodity.The main link between the one-commodity bargaining game and its multicommodity generalization has to do with the effect a player's attitude towards risk has on the utility he receives at the solution.A utility function U is said to be more risk averse than V if there is an increasing concave function k with U = k(V); an agent is more risk averse than another agent if his utility function is more risk averse than the other agent's.Kihlstrom, Roth, and Schmeidler [1012 show that for a class of bargaining solutions that include the Nash [151 and the Raiffa [17]-Kalai-Smorodinsky [8] solution, a player's utility increases as his opponent becomes more risk averse.This result, which is related to a theorem of Kannai [9], makes it possible to deduce that players will report linear utilities in the one-commodity distortion game.This follows because all monotonic preferences defined over one commodity are (ordinally) equivalent.The Kihlstrom-Roth-Schmeidler results thus imply that the players will select the least risk averse representation of these preferences.In the one-commodity case this will be a linear function.As long as the solution for the bargaining problem satisfies the axioms of Pareto optimality, symmetry, and invariance with respect to affine transformations of utility, the linear strategies give rise to equal division.The situation is made more complicated in the n-commodity case because there are many possible ordinal rankings of the outcomes.While the Kihlstrom, Roth, and Schmeidler result restricts the possible strategies that the players will find advantageous to report, a broad class of possible distortions (including any increasing linear function) cannot be excluded on the basis of their theorem.Consequently, it is not surprising that the characterization of equilibria for the distortion game is less satisfactory for multi-commodity bargaining than for 2This result is presented in Roth [18, pp.38-48, 104-105].