Ex-post equilibria in combinatorial auctions

Abstract

Given a game with incomplete information, such as a combinatorial auction, one may ask what is a good solution concept for (or predictor for players’ behaviors in) such a game. Most of the literature on combinatorial auctions appeals to a nonBayesian setting, and to the concept of dominant strategy implementation. A nice property of a dominant strategy for a given game is its uniqueness. However, this nice property and the natural appeal of dominant strategies might be misleading. In a general non-Bayesian setting, one can define the natural solution concept of ex-post equilibrium. Roughly speaking, an ex-post equilibrium is a strategy profile in which unilateral deviations are not beneficial regardless of the state of nature. Notice that in difference to a dominant strategy, where arbitrary behaviors of the other players are considered, in an ex-post equilibrium only other players’ behaviors which conform to the prescribed strategy profile are considered. From a purely economic mechanism design perspective one may be tempted to ignore ex-post equilibria: The revelation principle implies that, in a private value setting, if a function is implemented as an ex-post equilibrium, then it is also implementable as a dominant strategy equilibrium of another mechanism. However, in a computational setting the revelation principle may be of little use: the translation from a mechanism to a corresponding revelation mechanism may be exponential. Hence, one should carefully characterize the set of ex-post equilibria of a game with incomplete information even if this game possesses a dominant strategy equilibrium; in fact, in principle, even if a game possesses a dominant strategy equilibrium, there can be another ex-post equilibrium which will be selected due to the fact that this equilibrium is more tractable from the computational perspective than the dominant strategy equilibrium. The above argument is not hypothetical. Indeed, as it turns out, it touches on the most famous mechanism in the context of economic mechanism design in general, and combinatorial auctions in particular: the VCG combinatorial auction. While the literature on the VCG mechanism acknowledged the existence of a variety of ad-hoc equilibria for particular valuations, when considering the general setting the literature refers to the VCG mechanism as possessing a dominant strategy and does not refer to other non-Bayesian equilibria. However, as shown in [1]