Reducible Equilibrium Properties: Comments on Recent Existence Results ∗ Guilherme Carmona † University of Cambridge October 18, 2011 Abstract We show that each of the properties of equilibrium problems considered in recent Nash equilibrium existence results satisfy the following condition: All the existence problems that satisfy the property on a compact subset K of a fixed strategy space have, on K, an equilibrium set that contains the equilibrium set of an existence problem that is well-behaved on K. The properties that satisfy this condition are called reducible equilibrium properties. This notion is used to obtain an abstract existence result and also as a tool to derive a new reducible equilibrium property. Both the abstract existence result and the new reducible equilibrium property are used to clarify recent existence results and the relationship of some of the properties considered in those results. Journal of Economic Literature Classification Numbers: C72 Keywords: Nash equilibrium; discontinuous games; properties. ∗ I wish to thank Pavlo Prokopovych and Phil Reny for very helpful comments. Financial support fromFunda¸c˜ ao para a Ciˆ encia e Tecnologia is gratefully acknowledged. Any remaining errors are, of course, mine. † Faculty of Economics and King’s College, University of Cambridge and Nova School of Busi- ness and Economics. Address: University of Cambridge, Faculty of Economics, Sidgwick Avenue, Cambridge CB3 9DD, UK; phone: (44) 122333 5225; email: gmc42@cam.ac.uk. 1