Equilibrium Existence Issues

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Auctions with multi-dimensional signals are recognized to be important, yet, unless private information is independent, there are no general existence results in such situations. A new single-crossing result is expected to establish the existence of a monotone pure strategy equilibrium in an m+1st price auction with interdependent values and multi-dimensional private information. In particular, this result will supply the missing existence result in the information aggregation result in Pesendorfer and Swinkels (2000). It will also mark an important first step in the theory of auctions with multi-dimensional signals.A more basic issue is the existence of Nash equilibria in games with discontinuous payoffs (auctions are one class of such games). Reny (1999) contains the most general result of this kind to date. Even so, the main result there does not cover some important games from political science (e.g., Downsian electoral competition models). This current study will develop generalization of the pure strategy existence result. Relating these new results to the Downsian model of electoral competition is not at all straightforward and thus will be a significant contribution of this research.Finally, the classic question of the existence of a rational expectations equilibrium in an exchange economy with differential information is analyzed. The new insight will permit a general existence result centers around the use of demand correspondences. Since Kreps (1977) it was thought that restricting attention to strictly concave utilities avoids the need to permit consumers from submitting demand correspondences. The lack of indifference, it was thought, meant that consumers would only want to submit demand functions. This is not the case and permitting consumers to submit correspondences resolves the existence problem in the classic counter examples. The existence problem is similarly resolved much more generally in this study.