Uniqueness of the index for Nash equilibria of two-player games 3

Uniqueness of the index for Nash equilibria of two-player games 3 Srihari Govindan 1 and Robert Wilson 2 1 Economics Department, University of Western Ontario, London, Ontario N6A 5C2, Canada 2 Business School, Stanford University, Stanford, CA 94305-5015, USA Received: ; revised version. Summary: GivenamapwhoserootsaretheNashequilibriaofagame,eachcomponent of the equilibrium set has an associated index, defined as the local degree of the map. This note shows that for a two-player game, every map with the same roots induces the same index. Moreover, this index agrees with the Shapley index constructed from the Lemke-Howson algorithm. JEL Classification Number: C72. 1 Introduction A basic problem of game theory is equilibrium selection; that is, identifying those equilibria that satisfy plausible axioms of the sort postulated by Kohlberg and Mertens (1986) and Mertens (1989), and variants proposed by Banks and Sobel (1987), Cho and Kreps (1987), and others. The main axioms require properties such as Invariance (with respect to addition or deletion of redundant pure strategies), Backwards Induction (subgame perfection), and Admissibility (exclusion of dominated strategies). The importance of 3 Govindan acknowledges research support from the Social Science and Humanities Research Council of Canada; and Wilson, from U.S. National Science Foundation grant SBR9511209. 1

axiomatic criteria became evident in the early 1980s with the demonstration that popular solution concepts fail to satisfy them; e.g., concepts such as sequential equilibrium motivated by Backward Induction need not satisfy Admissibility or Invariance. Kohlberg and Mertens (1986) provided two insights motivating subsequent work. One is that a solution should be set-valued, since for generic extensive-form games all equilibria in the same connected component of the equilibrium set have the same outcomes (i.e., equilibria in the same component agree along the path of equilibrium play). The second is that many in a long list of proposed axioms are satisfied if (the subset of) the component selected is strategically essential; that is, every nearby game (in the space of payoffs) has a nearby equilibrium. Unfortunately, the key axioms of Admissibility and Backwards Induction are not satisfied simultaneously by any of the variants of this selection criterion proposed by Kohlberg and Mertens. On the other hand, Mertens (1989) showed (in effect) that strengthening the criterion to selection of a component that is topologically essential [defined below] with respect to the subspace of strategy perturbations resolves the difficulty. Alternatively, Hillas (1990, Definition 9) obtained a different resolution by allowing as perturbations all (upper hemicontinuous compactand convex-valued) correspondences suitably close to the best-reply correspondence of the game. The work of Mertens and Hillas crystalized the technical issues involved in establishing a decision-theoretic axiomization of equilibrium selection: what in fact is the source of the gap, if any, between topological and strategic essentiality? Actually, Mertens cast his formulation in terms of topological nontriviality, which says that the local projection map from a neighborhood (in the graph of equilibria) of the component into a neighborhood of the game cannot be deformed continuously to a map into the boundary of the neighborhood. When the neighborhood is defined in the full-dimensional space of games (i.e., with respect to payoff perturbations), Kohlberg and Mertens (1986) showed that the global projection map is homotopic to a homeomorphism between oriented manifolds. In this case, nontriviality is equivalent to the requirement that the local projection map has nonzero degree; that is, a cycle around the component s neighborhood in the graph maps onto some nonzero multiple of a cycle around the game s neighborhood. In turn, Govindan and Wilson (1996) showed that games have the remarkable property that this 2

degree is the same as the index of the component when one represents equilibria as the fixed points of the map used by Gül, Pearce, and Stacchetti (1993). For the index, one fixes the game and computes the index as the degree of the local displacement map that carries each nearby strategy profile into the difference between it and its image under the fixed-point map. Because the index is defined for a fixed game, and for generic games it has a simple formula (the sign of the determinant of the associated Jacobian matrix), it is much easier to compute and to analyze. The final link is O Neill s (1953, Theorem 5.2 and Corollary) demonstration that the index is nonzero if and only if the component is topologically essential; i.e., every fixed-point map nearby (in the compact-open topology) hasafixedpointnearby,whichisthekeypropertythatunderlieshillas construction. These technical developments show that, to construct a decision-theoretic axiomization, the remaining piece of the puzzle is intimately connected with the properties of the index. For instance, if there is in fact a gap between topological and strategic essentiality then it will be demonstrated by finding an example of a game for which some component has zero index and yet the local projection map is surjective onto a neighborhood of the game. 3 No such example is known, but if it were to be a two-player game then, in view of our theorem below and O Neill s theorem, it would have the property that every fixed-point map identifying its equilibria can be perturbed to exclude fixed-points near the component, even though every perturbation of the game yields equilibria near the component that is, no nearby game is represented by the nearby map obtained from O Neill s construction. The resolution of this puzzle is important for axiomatic studies because in a decision-theoretic development it would be implausible to impose topological essentiality as an axiom unless it is provable that the space of games is rich enough to obtain equivalence between strategic and topological essentiality; if this is not provable, then the very rich space of nearby games (i.e., behaviors) envisioned by Hillas may be as close as one can come to a purely decision-theoretic axiomization. Our contribution in this article is to clarify whether this formulation of the issues 3 In Govindan and Wilson (1996) we provide such an example for the case considered by Mertens (1989) in which the allowed perturbations of the game are restricted to perturbations of the strategy space. This shows that the issue centers on whether the finite-dimensional space of allowed perturbations of a game is rich enough to obtain an analog of O Neill s theorem in which this space has infinite dimension. 3

depends on the special role of the Gül, Pearce, Stacchetti map to define the index of a component. The standard axiomatic characterizations of the Lefschetz index are all expressed as properties of some given fixed-point map; e.g., Dold (1972). The metatheoretic question we address is whether different fixed-point maps might assign different indices to the same fixed point. We show that for two-player games the answer is No, that in fact the index of a component is independent of the fixed-point map used to characterize equilibria. We prove this result by showing that every fixed-point map assigns indices that agree with the index devised by Shapley (1974) as part of his analysis of the Lemke-Howson algorithm for computing equilibria. Our chief mathematical tool is the well-known property that, in the space of k 2 k real matrices, the set of those with positive determinants is path connected. This result has independent interest in view of the many different fixed-point maps that have been invoked since the original formulation of equilibrium points by Nash (1950). It is also relevant to computations, since the implications of the Poincaré-Hopf Theorem (e.g., the sum of the indices of a game s components is the same as the Lefschetz number of the strategy space, namely +1 ) are important tools in identifying the existence, number, and dynamical stability of equilibria. Our result demonstrates that none of thepropertiesoftheindexareaffectedbythechoiceofthefixed-pointmapusedto characterize equilibria. 2 Statement and Proof of the Theorem Let S and T be the finite sets of pure strategies for players 1 and 2, and let S and let T be the corresponding simplices of mixed strategies. The space of games with these strategies is the space G of all pairs G =(A; B) of appropriately dimensioned payoff matrices for players 1 and 2 respectively. If the Nash equilibria of G are the fixed points of the map f : S2T!S2T then the index Ind f (C) of a component C is defined as the integer that is the local degree of the displacement map { 0 f,where { is the identity (or inclusion) map; cf. Dold (1972, IV and VII). Theorem: Let f; g :(S2T) 2G!S2T be two continuous functions such that, for each game G 2G, the restricted maps f = f ( 1 ; G) and g = g( 1 ; G) have as their fixed points the Nash equilibria of G. Then Ind f =Ind g. 4

Proof. Itsufficestoshowthattheindicesunder f and g agree on the isolated equilibria of generic games. There is also no loss of generality in assuming that each game has only positive payoffs, as explained in a footnote later. The proof shows that every such map f assigns indices that agree with the indices assigned by the Lemke-Howson algorithm to the isolated equilibria of a generic game with positive payoffs; cf. Shapley (1974). The index generated by the algorithm is canonical in the sense that it is independent of any fixed-point map used to characterize equilibria. We call it the Shapley index and use the unqualified term index to refer to the index assigned by Let (; ) be an isolated equilibrium of a generic game G =(A; B) with positive payoffs. Then the supports of the strategies have cardinality jsupp()j = jsupp( )j = k for some integer k 1. Considerthe k2k subgame G 0 =(A 0 ;B 0 ), with pure strategies S 0 and T 0 obtained by deleting unused strategies from S and T.Theprojection( 0 ; 0 ) f. of (;)ontos 0 2T 0 is a completely mixed (or pure if k = 1 ) equilibrium in the smaller strategy space. For the game G the Shapley index of (; ) is 6 sgn[det A 0 ][det B 0 ], where the initial sign (6) depends only on the number of strategies. In particular, for a pure-strategy equilibrium the Shapley index is +1, and indeed this is also true of every fixed-point index. 4 To focus on the nontrivial case we therefore assume hereafter that k>1. Our method of proof is to show that there exists a homotopy that deforms the subgame G 0 =(A 0 ;B 0 ) into another k 2 k subgame G 00 =(A 00 ;B 00 ) and that carries ( 0 ; 0 ) into an equilibrium ( 00 ; 00 ) of G 00 whose index is determined solely by the standard axiomatic characterization of indices (e.g., Dold 1972, VII.5.8; O Neill 1953). Because this characterization requires also that the index is a homotopy invariant, it follows that the index of the equilibrium ( 00 ; 00 ) in the game G 00 fullydeterminesthe 4 In Shapley s article the index is so defined that the sum of the indices is 01, but here we reverse Shapley s choice of signs to conform to the standard convention that it is +1, to agree with the Lefschetz number for maps on the strategy space, which is contractible. With this convention, and because the axiomatic characterization of an index requires that it is invariant under homotopies, every index must assign +1 to a purestrategy equilibrium: the homotopy parameterized by t that adds t only to the payoffs from profiles using at least one of the pure strategies used in this equilibrium eventually reaches a game for which this profile is the unique equilibrium, which therefore has index +1. The homotopy axiom also justifies the restriction to games with positive payoffs: the homotopy parameterized by t that adds t to all payoffs leaves all equilibria unchanged, so for each equilibrium its index must be the same all along the path. 5

index of ( 0 ; 0 ) in G 0, and therefore by the formula for the Shapley index, also the index of (;) in the original game G. In principle, this method has two halves. The first half supposes that the Shapley index of (; ) in G is positive and we construct a homotopy between G 0 and a game G 00 withauniqueequilibriumthatmusthaveindex +1.Thesecondhalfsupposesthat the Shapley index is negative and we construct a homotopy to a game G 00 that has three equilibria, of which one with index 01 is completely mixed. For this it suffices to select a game with a completely mixed equilibrium whose Shapley index is 01: fromthefirst half we know that each positive index is +1, and the sum of all three indices must be the Lefschetz number +1, so the other two equilibria have indices +1, implying that the completely mixed equilibrium has index 01, in agreement with the Shapley index. We omit the details of the construction of the games G 00 used in the two halves. Briefly: In the first half, G 00 is the higher-dimensional analog of matching pennies, or paper-scissors-rock, depending on whether k is even or odd. In the second half, G 00 is obtained by embedding the appropriate one of these two games as a (k 0 1) 2 (k 0 1) subgame whose unique equilibrium becomes a positive-index incompletely-mixed equilibriumofthefull k 2 k game; the other payoffs are chosen to produce a pure-strategy equilibrium in which both players use the residual strategy k : the third equilibrium is then completely mixed and has Shapley index 01. The exposition of these two halves is exactly parallel, so we present only the first half, inwhichitissupposedthattheshapleyindexof (;) for the game G is positive. As mentioned, there exists a game G 00 =(A 00 ;B 00 ) with the same pure strategies S 0 and T 0 that has a unique equilibrium, and such that sgn det A 0 =sgndeta 00 and sgn det B 0 = sgn det B 00. To construct the homotopy between G 0 and G 00 we use the key fact that the group of k 2 k real matrices with determinants having a specified sign (positive or negative) is path connected; cf. Osborn (1982, 2.6.19). This implies that there exists a one-dimensional connected path of k 2 k real matrices (A t ;B t ) such that sgndeta t = sgn det A 0 and sgn det B t =sgndetb 0 for all t 2 [0; 1], and (A 0 ;B 0 )=(A 0 ;B 0 ) and (A 1 ;B 1 )=(A 00 ;B 00 ). Moreover, because these matrices are all nonsingular, for each t 6

the equation system A t 1 t = u t 1 ; 1 > 1 t =1; (?) B t 1 t = v t 1 ; 1 > 1 t =1; where 1 is a k -vector of ones, has either no solution or a unique solution ( t ; t ; u t ;v t ). Here, u t and v t are the players payoffs, which must be nonzero. Thus, if a solution exists for every t 2 [0; 1] then the locus ( t ; t ) of strategy pairs is also a one-dimensional connected path with endpoints that are the equilibria of (A 0 ;B 0 ) and (A 00 ;B 00 ). We divide the remainder of the proof into two parts. Part 1 assumes the existence of solutions all along the path and addresses the main argument, which is to show that the indices must agree at the two ends of the path. Part 2 addresses the more technical argument, showing that there is a path of payoff matrices for which a solution exists at every point. Part 1: Agreement of the Indices As mentioned, this part assumes that a solution ( t ; t ; u t ;v t ) to the system (?) exists for each t 2 [0; 1]. It could happen that the locus of solutions does not lie entirely in the restricted mixed strategy space S 0 2T 0 because some coordinates are negative. However, this is immaterial for the calculation of indices: because the locus starts and ends in S 0 2T 0, it suffices to expand the simplices of mixed strategies within the affine spaces generated by S 0 and T 0 so that the entire locus lies in the interior of this expanded strategy space, say S 3 2T 3. Alternatively, which is the convention we shall use, the payoffs can be expanded. Given S 3 2T 3, there is a unique representation of ( t ; t ) as t = ˆ + [ t 0 ˆ] 2S 3 ; t =ˆ + [ t 0 ˆ] 2T 3 ; where t 2S 0, ˆ 2S 0 is the uniform mixture, and 1 is the coefficient that expands S 0 radially (outward from ˆ )to S 3 ( =1 if S 0 = S 3 ); and analogously t 2T 0,etc. Then ( t ; t ) is a (completely mixed) equilibrium of the game (Āt; B t ), where Ā t = A t +[10](A t 1 ˆ ) 1 1 > ; B t = B t +[10 ](B t 1 ˆ) 1 1 > ; 7

where the second dot-products are outer products. Hence, the replacement (A t ;B t ; t ; t ) (Āt; B t ; t ; t ) ; which is continuous and leaves the signs of the determinants unchanged, ensures that the locus of equilibria resides entirely in the interior of S 0 2T 0. To include the strategies in S or T that are not among those in S 0 the locus in S2T and T 0, embed by specifying zero probabilities for these additional strategies in S n S 0 and T n T 0. Further, construct a connected one-dimensional path of payoffs for these strategies such that, with all strategies now included, the equilibria of (A t ;B t ) include ( t ; t ); that is, for all t along the path, the additional strategies have payoffs against t and t, respectively, that are strictly inferior to u t and v t respectively. From this construction we obtain a locus of solutions ( t ; t ; t) that is a subset of S2T 2[0; 1]. Moreover, there exists a neighborhood of this one-dimensional locus that does not contain any other equilibria in its closure. By the homotopy axiom for indices, the same index must be assigned to the equilibria at the two ends of this locus. With the standard orientation, f 1 = f ( 1; A 1 ;B 1 ) can assign only the index +1 to the unique equilibrium ( 1 ; 1 ) of (A 1 ;B 1 ) for t = 1 ; that is, its Lefschetz number is L(f 1 )=+1 because the strategy space is contractible. Consequently, f 0 = f ( 1; A 0 ;B 0 ) must assign the same index +1 to the other endpoint ( 0 ; 0 ) that is the equilibrium for (A 0 ;B 0 ). This equilibrium is the same as the original equilibrium of the original game (A; B), independently of any rescaling (; ). Consequently, f 0 agrees with the Shapley index. Part 2: Existence of Solutions In this part we show that the path (A t ;B t ) of payoff matrices can be selected to ensure that for each t asolution ( t ; t ; u t ;v t ) to (?) exists. The construction of such a path proceeds in three steps. Step 1: Normalize the Equilibria at the Endpoints The first step is to transform the two games G 0 =(A 0 ;B 0 ) and G 00 =(A 00 ;B 00 ) at the endpoints t =0 and t = 1 so that for each player the equilibrium strategy is the uniform mixture ˆ or ˆ over the k pure strategies, and the equilibrium payoff for each player is 1=k. We illustrate only the transformation of (A 0 ;B 0 ). Select k completely mixed 8

strategies in S 0,andsimilarly k in T 0, such that the given (completely mixed) equilibrium strategies ( 0 ; 0 ) are the barycenters of their convex hulls. These selections can be done by pulling the vertices of S 0 and T 0 towards 0 and 0. Consequently, the transformation is represented by two k 2 k matrices Q and R with positive determinants such that 0 = Q 1 ˆ and 0 = R 1 ˆ,andalso 1 > 1 Q = 1 > and 1 > 1 R = 1 >.Fromthis it follows that the equations A 0 1 0 = u 0 1 and 1 > 1 0 = 1 become [Q > 1 A 0 1 R] 1 ˆ = u 0 1 and 1 > 1 ˆ = 1 and similarly for B 0 and 0. Finally, the transformed payoff matrices A 0 2 Q > 1 A 0 1 R and B 0 2 R > 1 B 0 1 Q include rescaling parameters ; > 0 chosen so that the resulting equilibrium payoffs become u 0 =1=k and v 0 =1=k.Thischoiceisfeasiblesincethepresumedexistenceof asolutionto (?) at (A 0 ;B 0 ) ensures that the players payoffs are nonzero, and hence positive (because the matrices are positive). Because the uniform mixtures are ˆ = ˆ =[1=k]1, the result of this transformation is to ensure that the vector 1 is an eigenvector of both A 0 and B 0,andalsoatthe other, similarly transformed, endpoint (A 00 ;B 00 ). For the subsequent steps, let H be the (k 0 1) -dimensional hyperplane through the origin that has 1 as its normal vector. Step 2: Exclude 1 from the Image of H. Thesecondstepistoensurethateachofthefourpayoffmatricesatthetwoendsof the path define linear maps than carry H into itself. This can be done as follows, for each payoff matrix separately say A 0. Rotate coordinates (again with positive determinant) to map 1 into a vector e k in the positive direction along dimension k. After this rotation, let H 0 be the image of < k01 under the linear map, and let P be the linear map that preserves e k and projects H 0 to < k01. Then Â0 = P A 0 is a linear map that carries < k01 into itself. Consequently, we can follow the path indicated by the linear homotopy that deforms A 0 continuously into Â0 to arrive at a linear map that carries < k01 into itself. (This homotopy merely shrinks to zero the matrix elements in row k except for the 1 in column k, which remains unchanged.) Being linear, this homotopy has full rank so the sign of the determinant of the linear map is preserved throughout. At the end of this step, each of the four payoff matrices at the endpoints has the 9

property that the vector e k (which replaces 1 ) is mapped to itself and its orthogonal hyperplane < k01 (which replaces H ) is mapped to itself. In particular, its determinant is the same as the determinant of the submatrix comprising its initial (k 0 1) rows and columns. Step 3: Select a Path of Submatrices The last step is to choose the path in a way that ensures the existence of solutions to (?) throughout. Select any path of the initial (k 0 1) 2 (k 0 1) submatrices that preserves the signs of their determinants (and therefore the signs of the determinants of the full matrices in which they are embedded) and that connects the pairs of matrices at the two endpoints. One can see as follows that along such a path there exists a solution to the corresponding equations (?). Because e k is never in the image of < k01 in the transformed system, it follows that, after reversing the transformations, 1 is never in the image of H undereitherofthelinearmapsspecifiedby A t and B t. Consequently, Farkas Lemma implies that the nonsingularity of the matrices A t and B t along the path ensures that also the full equation system (?) is nonsingular and therefore has a solution, and this solution is unique. Remarks 1. The theorem can be extended to functions whose fixed points include the Nash equilibria, provided the extra fixed points are disconnected from the equilibria and the sum of their indices is zero. 2. The theorem applies also to vector fields for which the Nash equilibria are zeroes. References Banks, Jeffery, and Joel Sobel (1987), Equilibrium selection in signaling games, Econometrica, 55: 647-662. Cho, In-Koo, and David Kreps (1987), Signaling games and stable equilibria, Quarterly Journal of Economics, 102: 179-221. Dold, Albrecht (1972), Lectures on Algebraic Topology. New York: Springer-Verlag. Govindan, Srihari, and Robert Wilson (1996), Equivalence and invariance of the index and degree of Nash equilibria, Games and Economic Behavior, to appear. 10

Gül, Faruk; David Pearce; and Ennio Stacchetti (1993), A bound on the proportion of pure strategy equilibria in generic games, Mathematics of Operations Research, 18: 548-52. Hillas, John (1990), On the definition of the strategic stability of equilibria, Econometrica, 58: 1365-90. Kohlberg, Elon, and Jean-François Mertens (1986), On the strategic stability of equilibria, Econometrica, 54: 1003-38. Mertens, Jean-François (1989), Stable equilibria: a reformulation, Math. Oper. Res., 14: 575-625; and (1991),...: Part II, 16: 694-753. Nash, John (1950), Equilibrium points in n-person games, Proc. Nat. Acad. Sciences USA, 36: 48-49. O Neill, Barrett (1953), Essential sets and fixed points, Amer. J. Math., 75: 497-509. Osborn, Howard (1982), Vector Bundles, Vol. I: Foundations and the Stiefel-Whitney Classes. New York: Academic Press. Shapley, Lloyd (1974), A note on the Lemke-Howson algorithm, Mathematical Programming Study 1: 175-189. New York: North-Holland. 11

Running Head: Uniqueness of Index Journal of Economic Literature Classification Number: C72. Authors Mailing Address: Professor Robert Wilson Stanford Business School Stanford, CA 94305-5015 12