1126 East 59th Street Chicago, Illinois 60637 T: 773.702.7587 F: 773.795.6891 [email protected]THE BECKER FRIEDMAN INSTITUTE FOR RESEARCH IN ECONOMICS BFI Working Paper Series No. 2013-004 Nash Equilibrium in Discontinuous Games Philip J. Reny University of Chicago September 3, 2013
Nash Equilibrium in Discontinuous Games ° Philip J. Reny Department of Economics University of Chicago September 3, 2013 Abstract We provide several generalizations of the various equilibrium existence results in Reny (1999), Barelli and Meneghel (2013), and McLennan, Monteiro, and Tourky (2011). We also provide an example demonstrating that a natural additional general- ization is not possible.All of the theorems yielding existence of pure strategy Nash equilibria here are stated in terms of the players°preference relations over joint strate- gies. Hence, in contrast to virtually all of the previous work in the area the present results for pure strategy equilibria are entirely ordinal, as they should be. 1. Introduction A primary objective here is to resolve a nagging problem in the literature on the existence of Nash equilibrium in discontinuous games.1 Because pure strategy equilibria are invariant to ordinal transformations of payo/s, the ±right²pure strategy equilibrium existence result should be stated in purely ordinal terms. Yet, virtually all of the existence theorems in the literature rely on non-ordinal properties of the players°utility functions in the sense that their hypotheses, when satis³ed in one game, need not be satis³ed in an ordinally equivalent game.2 All of the conditions introduced here are entirely ordinal and are stated in terms of players°preference relations over the joint strategy space. A second objective is to better connect the existence results for discontinuous games with the more standard existence results for continuous games that are based upon well-behaved ° I am grateful to Andy McLennan, Roger Myerson and Guilherme Carmona for helpful comments and discussions.Financial support from the National Science Foundation (SES-1227506, SES-0922535, SES- 0617884) is gratefully acknowledged. 1This literature has grown substantially since the seminal contribution of Dasgupta and Maskin (1986). A sample of papers is Simon (1987), Simon and Zame (1990), Baye, Tian, and Zhou (1993), Reny (1999, 2009, 2011), Jackson, Simon, Swinkels and Zame (2002), Carmona (2005, 2009, 2011), Bagh and Jofre (2006), Monteiro and Page (2007, 2008), Barelli and Soza (2009), Bich (2009), Carbonell-Nicolau (2011), Prokopovych (2011, 2013), De Castro (2011), McLennan, Monteiro, and Tourky (2011), Barelli and Meneghel (2013), Barelli, Govindan, and Wilson (2013), Bich and Laraki (2013), He and Yannelis (2013), Nessah (2013). 2Recent exceptions are Barelli and Soza (2009) and Propkopovych (2013). An important practical feature of the hypotheses we shall introduce here is their local nature, which is in keeping with the conditions in most of the literature. This is in contrast to the hypotheses of Barelli and Soza°s (2009) Theorem 2.2 and Prokopovych°s (2013) Theorem 2, which, because of their global nature, are likely to be rather more di¢ cult to verify in practice.
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best-reply correspondences (e.g., Nash (1950, 1951), Glicksberg (1952)).To this end, we introduce the concept ofpoint-security with respect to a subset of players, where only the preferences of players within the subset are restricted because players outside the subset are presumed to have well-behaved best-reply correspondences. This idea not only leads to more powerful results, it helps to better connect the ideas introduced by McLennan, Monteiro and Tourky (2011) and Barelli and Meneghel (2013) with those of Reny (1999). The paper proceeds as follows. Section 2 provides notation and some basic de³nitions. Section 3 provides a new and ordinal ±point security²condition as well as our main result, Theorem 3.4. Section 4 shows how Theorem 3.4 can be used to derive various results from the literature. Section 5 shows a variety of ways that the results obtained in previous sections can be straightforwardly extended and re³ned, and also contains a related result on the existence of mixed strategy equilibria. Section 6 discusses a natural weakening of the ±security²part of the assumptions from previous sections and provides an example showing that, under this weakening, existence of a Nash equilibrium cannot be assured. Section 7 provides an application to abstract games. 2. Preliminaries LetNbe a ³nite set of players. For eachi2N;letXi be a set of pure strategies for player iand let±ibe a binary relation onX=²i2NXi. This de³nes a gameG= (Xi;±i)i2N : The symbol³idenotes ±all players buti.²In particular,X°i=²j6 =iXj;andx°i denotes an element ofX°i :The product of any number of sets is endowed with the product topology and unless otherwise speci³ed, we restrict attention to the topology relative toX: A strategyx±2Xis a (pure strategy)Nash equilibriumofGifx±±i(xi ; x±°i )for every playeri2Nand everyxi2Xi :3 Consider the following assumptions onG= (Xi;±i)i2N :For everyi2N; A.1Xiis a nonempty, compact, subset of a Hausdor/ topological vector space, and±i is complete, re´exive, and transitive. A.2Xi is a convex set. A.3For everyx2X;fx0i2Xi: (x0i ; x°i)±i xgis a convex set. When A.3 holds, we will say that the preference realtions±i areconvex,4and when A.2 and A.3 hold we say that the gameGisconvex. 3Nash equilibrium will always mean pure strategy Nash equilibrium, although we will include ±pure strategy²for emphasis from time to time. We will always say mixed strategy Nash equilibrium when mixed strategies are introduced. 4Together with A.1, convexity implies also thatfx0i2Xi: (x0 i ; x±i)>i xgis a convex set. Indeed, suppose that(x1i; x±i)>ixand(x2i; x±i)>ix:By completeness, assume without loss that(x1i; x±i)±i(x2i ; x±i ): Then(°x1i+ (1³°)x2i; x±i)±i(x2 i ; x±i)>i x;where the ³rst inequality follows from convexity. The desired result follows from transitivity. 2
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