DIRECT PROOFS OF GENERIC FINITENESS OF NASH EQUILIBRIUM OUTCOMES SRIHARI GOVINDAN AND ROBERT WILSON Abst r act . Using elementary techniques from semi-algebraic geometry, we give short proofs of two generic ¯ niteness results for equilibria of ¯ nite games. For each assignment of generic payo®s to a ¯ xed normal form or to an extensive form with perfect recall, the Nash equilibria induce a ¯ nite number of distributions over t he possible out comes. 1. Int r oduct ion Harsanyi (1973), RosenmÄ uller (1971), and Wilson (1971) showed t hat for each assignment of generic payo®s to t he normal form of a game, the number of Nash equilibria is ¯ nit e and odd. Nowadays, this result is seen as an immediate corollary of Kohlberg and Mertens' (1986, Theorem 1) structure theorem, which st ates that the Nash graph is homeomorphic to the space of normal-form payo®s. For games in ext ensive form wit h perfect recall, K reps and Wilson (1982, T heorem 2) proved t he st ronger result that for each assignment of generic payo®s to the terminal nodes, the Nash equilibria induce a ¯ nite number of probability distributions over the terminal nodes; in particular, each component of equilibria has a unique payo®distribution. Kohlberg and Mertens (1986, Appendix C) sketched a shorter proof but Elmes (1990) ident i¯ ed a basic ° aw and suggest ed a possible ¯ x. An extensive form with simultaneous moves represents a normal form, so the ¯ niteness result for normal-form games is a special case. As with most genericity theorems in economics, these proofs rely on Sard's theorem from di®erential topology, and as usual the manipulations required to adapt the formulation to stringent smoothness and compactness conditions account for long complicated constructions such as the Appendix in Kreps and Wilson (1982). In this note we exploit the fact that in ¯ nite games an equilibrium payo® is a polynomial funct ion of t he mixed-st rat egy probabilit ies and t he payo®s from pure st rat egies. T his enables proofs derived from stronger results in the theory of semi-algebraic sets. Blume and Zame (1994, Theorems 3 and 4) also use t his approach t o obt ain weaker analogs of K ohlberg and Mert en' s const ruct ions for subgame-perfect , sequential, and perfect equilibria of ext ensive-form games, but t hey do not address the generic-¯ niteness t heorem for such games t hat follows easily, as we show below. A semi-algebraic set is a subset of a Euclidean space de¯ ned by a ¯ nite number of polynomial equalities and inequalities. The classic reference is van den Waerden (1939) who established the key property that a semi-algebraic set can be triangulated. The following Lemma is the only result about semi-algebraic sets needed here. It follows readily from the Generic Local Triviality Theorem (Bochnak, Coste, and Roy, 1987; Blume and Zame, 1994). Genericity is used here in the sense that a subset is generic if its complement is a closed set of smaller dimension, and we say that a point is generic if it resides in a generic set. Lemma 1.1. Let f :X ! Y be a continuous function whose graph is a semi-algebraic subset of X £ Y. If dim(X ) 6 dim(Y) then for generic y 2 Y; f ¡1 (y) is a ¯ nite set (possibly empty). If f :X ! Y is a su± ciently smoot h map, not necessarily semi-algebraic, and X is compact t hen an analog of this lemma follows from Sard's Theorem. Compactness is essential, however: consider the map f :R! S 1 given by f (μ) = (cos(μ); sin(μ)), for which the inverse image of a point in S 1 is an in¯ nite set of isolat ed point s. T his rest rict ion prevent s any easy or direct applicat ion of Sard' s T heorem t o games. For Date: October, 1999. This work was funded in part by grants from the Social Sciences and Humanities Research Council of Canada and the Nat ional Science Foundation of t he Unit ed St ates, SBR9511209. 1