The Return Function: A Tool for Computing Bayesian-Nash Equilibria in Mechanism Design

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In many situations, such as art auctions, privatization of public assets and allocation of television airwaves to wireless carriers, the value of the object on sale (product, service or asset) is not known beforehand, and a market has to be designed to determine its price. The market (or mechanism) designer has to set the rules of the game in a context where typically none of the parties has complete information about the preferences of the others. Finding a solution amounts at determining some Bayesian-Nash equilibria to that game, given that the designer has an interest in the outcome. In this article, we introduce the idea of return function, and use it to compute Bayesian-Nash equilibria in mechanism design. In a nutshell, given a player's choice of action, the other players' strategies and the mechanism chosen by the market designer, the return function of that given player is the density of the means of the probability distribution function of the outcome. Further, we define and consider optimality concepts for general forms of the principal's objective function. We also introduce the ideality gap function to assess the difference between the optimality of a mechanism with perfect information and its implementability constraint. Finally, we give a method for computing Bayesian-Nash equilibria and optimizing mechanisms, which is based on the return function.

, 25 pages

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