MATHEMATICS OF OPERATIONS RESEARCH Vol. 18, No. 3, August 1993 Primed in U.S.A.

A BOUND ON THE PROPORTION OF PURE STRATEGY EOUILIBRIA IN GENERIC GAMES FARUK GUL, DAVID PEARCE AND E N N I O

STACCHETTI

In a generic finite normal form game with 2a + 1 Nash equilibria, at least a of the equilibria are nondegenerate mixed strategy equilibria (that is, they involve randomization by some players).

1. Introduction. In this paper we prove the following result: if a generic n-person game T has a > 1 Nash equilibria in pure strategies, then the total number of Nash equilibria of F is at least 2a - 1. In other words, at least a - 1 equilibria involve some randomization by some players. The proof of this result relies on a characterization of Nash equilibria as the fixed points of a continuous function g on a compact convex set. It is known that the sum of the Leftchetz indices of the fixed points of a Leftchetz function equals + 1 (see Guillemin and Pollack (1974)). We establish our main result by showing that generically g is a Leftchetz function and that pure strategy Nash equilibria are fixed points of g with Leftchetz index + 1 . 2. Characterization of equilibria. The conditions characterizing a Nash Equilibrium (NE) of an ^-person game are a straightforward extension of the necessary and sufficient conditions introduced by Lemke and Howson (1964) for 2-pIayer games. Consider a (finite) /i-player game U = ( 1 1 ' , . . . , 11") with payoff matrices 11' = [•n-'(;,,..., ;„)], / = 1 , . . . , n. Player / (/ = 1 , . . . , n) has action set ^4, — { 1 , . . . , m,}, and Tr'(j\,..., ;„) is his payoff when the players select the action profile ( y , , . . . , y'^) G A••=Aj X ••• XA^. The set of (mixed) strategies for player i is the simplex 5 , ••= { A e R ' " ' i A > 0 , e • \ = 1 ) , where e^ — ( 1 , . . . , 1). Below, e will always denote a column vector with all its entries equal to 1; we will not specify its dimension since it will be clear from the context. Let K ••= Si X • • • X S^ and m — m , + • • • +m„. Note that K is a. convex compact set. Also, for each player i, let A_i — Xj^.Aj, and for each k e A^ and (;',, . . . , ; „ ) G A, let y _ , := ( ; , , . . . , y,._,, y , . ^ , , . . . , y j G ^ _ , . and (j_i, k) ••= ( ^ l ) • • • , Ji-li

l^t Ji+v

• • • •> in'-

A Strategy A' G 5, is a pure strategy for player i if A is an extreme point of 5,. (That is. A' = e^ for some k G y4,. Here e^ denotes a vector with component k equal to 1 and all other components equal to 0.) A' G 5, is a mixed strategy if it is not a pure strategy. An NE A = ( A ' , . . . , A") G /C is a pure strategy NE if it is an extreme point of K; otherwise, A is a mixed strategy NE. Our use of the term "strategy" will include both pure and mixed strategies. Received August 20, 1990; revised August 29, 1991. AMS 1980 subject classification. Primary: 90D10, 90D12. IAOR 1973 subject classification. Main: Games. OR/MS Index 1978 subject classification. Primary: 231 Games. Key words, n-person games, Nash equilibria, number of pure strategies. 548 0364-765X/93/1803/0548/$01.25 Copyright © 1993. The In.stitute of Management Sciences/Operations Research Society of America

PROPORTION OF PURE STRATEGY EQUILIBRIA IN GENERIC GAMES DEFINITION. Let C c R ' be a convex set and x defined by N^-ix) •= 0 when x € C, and by

G R'. The

549

normal cone to C at x is

Ncix) ••= {q e R'\(q,c - x ) < 0 for all c ^ C) when jc e C. DEFINITION. For any strategy profile A = (A',..., A") e R'"'"^ action k e /I,, let

"^'"", player /, and

and let n'(A) denote the vector (n'CA),,..., n'(A),^ )'. When \ ^ K, n'(A)^ is the expected value for player / of action k, when his opponents follow the strategy profile A"'. L E M M A 1. A = ( A ' , . . . , A") e K is an NE with value u = iu^,..., player i, there exists x' e R'"' such that

uj

iff for each

n'(A) + x' = t;,e, jc' > 0, and A'' • x' = 0. The condition of Lemma 1 requires that player / give strictly positive weight to a pure strategy only if it is a best response to the profile A. The proof of Lemma 1 is immediate, and is omitted. COROLLARY L

\ e

K

is an NE iff for each player i, U'i\) e N^i\).

For any closed convex set C c R' and z e R', P^iz) will denote the projection of z onto C. A well-known characterization result states that Pciz) is the unique point in C that satisfies (z - Pciz),c - Pciz)) ^0

forallceC.

Let / : R ' -> R ' and define g: R' ^ C by giz) •= Pj^iz + fiz)), z G R'. It is easy to check that z* is a fixed point of g iff /(z*) e Nciz*). DEFINITION. Let / : R™ -^ R'" and g: R'" -> K be, respectively, the functions

n'(A) /(A) -

COROLLARY 2.

, and g(A) :=P^(

A e /C is an NE iff it is a fixed point of g.

The idea behind Corollary 2 can be expressed as follows. Consider any convex set C c R ' and a function /^: C -> R; the point z e C is a local maximizer of F in K iff z is the projection onto C of the point obtained by moving away from z along the gradient of F at z. That is, Pciz + F'iz)) = z. The construction' preceding Corollary 2 utilizes this observation to simultaneously solve the interdependent maximization problems of the players in order to obtain a Nash equilibrium. 'This construction was introduced by Hartman and Stampacchia (1966) to provide an existence result for variational inequalities. The Lemke-Howson (1964) characterization of Nash equilibria is a special case of a variational inequality. A similar construction has been used by Kehoe (1980, 1983) to study properties of competitive equilibria.

550

FARUK GUL, DAVID PEARCE & ENNIO STACCHETTI

3. Number of pure strategy equilibria. dimensional subspace in R"" defined by

The tangent space to K is the (m - n)-

T--= {x = where 0 0

A :=

e R" 0

Let A, be a smooth deformation of the unit sphere in R'"', such that 5, c A,, and let A := Al X ••• X A^. Clearly A is a compact manifold without boundary and K d A. Suppose h: R^ -^ ^ A '^ a continuous function. A fixed point z of /z is a Leftchetz fixed point if h is continuously differentiable in a neighborhood of z and / - h'iz) is an isomorphism from the tangent space to A at z into itself. Further, h is a Leftchetz map if each of its fixed points is a Leftchetz point. If /; is a Leftchetz map, it admits in the compact set A only a finite number of fixed points, and if all its fixed points are in K, its Leftchetz number can be computed^ by

i(h;z),

where/(/;; z) = sgndet

h(z)

A'' 0

is the index of h at z. Because the Leftchetz number is a homotopy invariant, L(h) = L{h) for any h homotopic to h, and since A is convex, all maps h: A -^ K are in the same homotopy class^In particular, if z G A; and Ji: A -* K is the constant map h(z) = z, z e A, then h is a Leftchetz map, z is its only fixed point, and L{h) = i{h;z)

= sgndet

-A

0

= sgndet A A = 1,

because det AA^ = X"^^m^. DEFINITION. An NE A of 11 is strongly nondegenerate if (i) for each player, every best response is used with positive probability. (ii) det(/ - g'ik)) + 0. The game 11 is regular if each of its NE is strongly nondegenerate; otherwise, II is said to be singular. Condition (i) guarantees that g'{X) exists for each NE A of II. Suppose g is differentiable at A e A:. Since g'(A)f G T for all ^ e R™, condition (ii) is equivalent to requiring that / - g'iX): J ^ T be an isomorphism. This is also equivalent to

(ii') -A

0.

The Leftchetz number is a topologieal concept and is defined for any continuous function on K. Since g is defined everwhere In R'", Lin Zhou has suggested that using Brower's degree function deg(/ - g, R'", 0) instead of the Leftchetz number L(.g) could simplify the analysis below.

PROPORTION OF PURE STRATEGY EOUILIBRIA IN GENERIC GAMES THEOREM

551

1. Suppose U is regular and J e K is a pure NE. Then iig; A) = 1.

PROOF. Without loss of generality assume that 11' > 0 for each player /, and A = (A', ..., A") = ( e , , . . . , e,). One can show that ^'(A) = 0: since A is a pure NE A is an extreme point of_K and int 7\^(A) # 0. Condition (i) of strong nondegeneracy then implies that /(A) e int N^(A). Hence_for all A in a neighborhood of A, A -t- /(A) G A -f N^iJ), and therefore g(A) = A. That is, g is constant on a neighborhood of A and ^'(A) must be 0. Thus,

THEOREM 2. Assume Yl is regular and has a > 1 pure NE. Then 11 must have at least 2a - 1 NE in total. PROOF.

i = ^(^) =

E

'(5;A) = « +

A=i'(A)

Z

'(g;A)

A=i'(A) A is not pure

and since iig\X) is either -)-1 or - 1 for each NE, there must be at least (a - 1) mixed strategy NE A with iig; A) = - 1 . Q.E.D. While it is the case that each pure strategy equilibrium must have Leftchetz index -1-1, mixed strategy equilibria can have Leftchetz indices +1 or - 1 . To see this consider any game with a unique mixed strategy NE and no pure strategy NE (such as the "matching pennies" game). It follows from the discussion above that the unique equilibrium must have index -I-1. On the other hand the "battle of the sexes" game has 3 equilibria, 2 of which are pure. Thus, in this case it follows that the mixed strategy equilibrium must have Leftchetz index - 1 . In general it is quite possible to have a game with 1 pure strategy NE (Leftchetz index -(-1 by the above theorem), two mixed strategy NE each with Leftchetz indices -)-1 and two other mixed strategy NE with Leftchetz indices - 1 . 4. Generic games. In this section we show that the set of regular games is generic. More precisely, we show that the set of regular games is open and its complement has measure 0. Let M ••= R'"'""

"""", P ••= M",

and f: R"

X P -> R'" be the function

/(A;n):= n"(A) and /n(A; 11) denote the Jacobian of / with respect to n. We have 0 if a =^ /, X A',, if a = /. For each A G /C and player /, there exists a tuple ( i , , . . . , ;„) such that [ X ,^ .AyJ ^ 0. It is easy then to see that the rows of the Jacobian /n(A; 11) are linearly independent, and /||(A; Yl) is of rank m for each A e K.

552

FARUK GUL, DAVID PEARCE & ENNIO STACCHETTI

A Sard Theorem (see Theorem 3 in Stacchetti (1987) and Theorem 4.1 in Reinoza (1983)^) establishes the following result. THEOREM 3. The set PQ a P of regular games H = ( n ' , . . . , n") is open and its complement has Lebesgue measure 0 in P.

Acknowledgement. We are grateful to Robert Wilson for discussions and encouragement and, in particular, for suggesting the use of Leftchetz fixed point theory. We also received valuable suggestions from John Geanakoylos and Lin Zhou. GUI and Pearce gratefully acknowledge support from the Alfred P. Sloan Foundation. Pearce and Stacchetti also wish to thank the National Science Foundation for its financial support. References Guillemin, Y, and Pollack, A, (1974), Differenlial Topology. Prentice-Hall, New Jersey, Hartman, P, and Stampacchia, G, (1966), On Some Nonlinear Elliptic Differential Functional Equations Ada Math. 115 271-310, Kehoe, T, (1980), An Index Theorem for General Equilibrium Models with Production, Econometrica 48 1211-1132, (1983), Regularity and Index Theory for Economies with Smooth Production Technologies, Econometrica 51 895-917, Lemke, C, E, and Howson, J, T, (1964), Equilibrium Points of Bimatrix Games, SIAM J. Appl Malh 12 413-423, Reinoza, A, (1983), Global Behavior of Generalized Equations: A Sard Theorem, SIAM J Control Optim 21 443-450, Stacchetti, E, (1987), Variational Inequalities and General Equilibrium, Mimeo, F, GUI: Stanford University, Stanford, California 94305 D, Pearce: Yale University, New Haven, Connecticut 06520 E, Stacchetti: Department of Economics, University of Michigan, Ann Arbor, Michigan 48109

Reinoza (1983) does not prove that P^ is open. This is a consequence of the Implicit Function Theorem (see Theorem 2 in Stacchetti (1987)),