Existence of Equilibria in Bayesian Games, Strategic-Form Games, and Extensive-Form Games with Infinite Action Spaces

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Auctions are a ubiquitous means for allocating resources, both in the private and in the public sectors. A key question of interest is whether auctions are able to allocate resources effciently. This question has been studied by many authors since at least the early 1960s. Not surprisingly, auctions work well only when the bidders competing within them bid carefully, taking into account not only their own private information about the value of the object(s) for sale, but also the information they would infer if they were to win one or more objects. Previous research has shown that auctions can produce effcient outcomes when better information leads to higher bids. When this occurs, wesay that bidders employ monotone bidding strategies. The idea that monotonicity of strategies leads to more effcient outcomes can be applied to other settings as well. For example, one might hope that in an oligopoly setting, the competition of the market will lead firms with lower costs to produce more output. And indeed, this will be the case when firms employ monotone strategies. Thus, monotonicity of strategies in economic settings provide important information about how well markets work. One of several goals of the present research is to understand, quite generally, when economic settings haracterized by strategic interaction possesses Bayesian Nash equilibriain which the participants employ monotone pure strategies. The results we expect to obtain will provide the most general answer to this question to date, and so will provide useful information about the efficiency properties of a large class of economic environments.