Determinacy of equilibrium distributions for zero sum and common utility games✩,✩✩

Cristian M. Litan a, Francisco Marhuenda b,∗ a University Babeş-Bolyai, Faculty of Economics and Business Administration, Romania b Department of Economics, University Carlos III, C/ Madrid 126, 28903-Getafe (Madrid), Spain

a b s t r a c t

We show the generic finiteness of probability distributions induced on outcomes by the Nash equilibria in two player zero sum and common interest outcome games.

JEL classification: C70 C72

Keywords: Generic finiteness Outcome zero sum games Outcome common interest games

1. Introduction which outcome games a similar result to the one obtained for normal and extensive form games continues to be valid. It has been established by Harsanyi (1973) that for normal Progress towards answering the above question has been form games with an arbitrary number of players, if the payoffs made by Govindan and McLennan (2001) who have proved the of the players can be perturbed independently, generically there generic finiteness of equilibrium distributions for games with two is a finite number of equilibria. Kreps and Wilson (1982) proved outcomes and any number of players. González-Pimienta (2010) the generic finiteness of equilibrium probability distributions on argues that the same result holds for games with two players and the end nodes of an extensive form game. In games arising from three outcomes. Park (1997) has shown the generic of economic models, it may happen that different actions of the equilibria for sender–receiver cheap-talk games. players yield the same outcome. Thus, the scope of the above Mas-Colell (2010) has shown that the equilibrium payoffs are results is limited to the extent that it does not generally hold for generically finite for two player game forms. In this note, we outcome games. combine Mas-Colell’s approach with elementary linear algebra to show the generic finiteness of equilibrium distributions on Govindan and McLennan (2001) and Kukushkin et al. (2008) outcomes when the associated game is either a two player zero have shown examples for which there is a continuum of probability sum or a common interest game. This result has already been distributions on outcomes induced by the Nash equilibria of the obtained, in a so far unpublished manuscript, by Govindan and associated game. A natural question that arises is to find out for McLennan (1998) using semi-algebraic geometry techniques. The interest of the present work lies on the fact that by relying only on straightforward linear algebra, we are able to provide a fairly short ✩ We thank an anonymous referee for his/her insightful comments. and elementary proof. ✩✩ The authors acknowledge financial support from project No. ECO2010- 19596 from the Spanish Ministry of Science and Innovation. Cristian Litan also 2. Common utility and zero sum outcome games with two acknowledges financial support by CNCSIS-UEFISCSU, project number PN II-RU players 415/2010. ∗ Corresponding author. Tel.: +34 91 624 93 66; fax: +34 91 624 93 29. E-mail addresses: [email protected] (C.M. Litan), There are two players. Let S1, S2 be their sets of pure strategies. [email protected], [email protected] (F. Marhuenda). We assume n = |S1| ≥ 1, m = |S2| ≥ 1 and let S = S1 × S2.

1  t  Consider a finite set of outcomes Ω = {1,..., l}. We denote by Let k1 = rank (A(u)|dn) and k2 = rank A (u)|dm , for any ∆(Ω) the set of probability measures on Ω. An outcome game form u ∈ G. Note that k1, k2 ∈ {k, k + 1}. The argument will be carried is a function θ : S → ∆(Ω). out by considering the possible values of k1 and k2. = For each outcome j 1,..., l, the mapping θ defines a function Case 1: k = k = k. j : → j 1 2 M S R such that M (sa, sb) is the probability that θ(sa, sb) −1 ∈ j First, remark that B (u) is defined for all u G. In all the assigns to the outcome j ∈ Ω. We identify each M with an n × m = = l Lemmas that follow, it is assumed that k1 k2 k. Note also matrix with real entries. Given u = (u1,..., ul) ∈ R , we interpret l that the matrices below depend on u. uj as the utility assigned to outcome j. To each u ∈ R we assign the matrix Remark 3.1. Since, l  j  BC  A(u) = ujM .   B C dk j=1 k = rank B = rank = rank DE D E dn−k dk dm−k 1 2 l Given two profiles of utilities on outcomes u , u ∈ R for the players, the matrices A u1 and A u2 define a two-person game. the remaining rows of the above matrices are a linear combination of its first k rows. Therefore, there is an n − k × k matrix H and The strategies x ∈ ∆(S1) and y ∈ ∆(S2) of the players induce ( ) k a probability distribution on Ω. The probability that outcome j a vector z ∈ R such that HB = D, HC = E, Hdk = dn−k, zB = dk j −1 −1 occurs is given by x M y. and zC = dm−k. Hence, H = DB , z = dkB . And we have that 1 1 2 A common utility game is a game in which u = u = u . −1 −1 DB C = E, DB d = d − , We will write A(u) for common utility games. The purpose of this k n k −1 work is to provide an elementary proof of the following result in dkB C = dm−k, for every u ∈ G. Govindan and McLennan (1998). Fix now u¯ 1, u¯ 2 ∈ G. From now on, we assume that the game l  1  2 Proposition 2.1. There is an open dense subset G ⊂ R such that, for defined by the matrices A u¯ and A u¯ has a completely mixed every u ∈ G, the set of probability distributions on outcomes induced NE. Later, we will consider the particular case of a common utility by the Nash equilibria of the game A(u) is finite. game, u¯ 1 = u¯ 2 = u¯. However, this last assumption is not needed in the lemmas below. 3. Proof of Proposition 2.1 1 Lemma 3.2. There are open subsets U1, U2 of G such that u¯ ∈ − p U , u¯ 2 ∈ U and d B 1(u)d ̸= 0, for every u ∈ U ∪U . Furthermore, The notation dp represents the vector (1,..., 1) ∈ . The 1 2 k k 1 2 R 1 2  1 n for u ∈ U and u ∈ U , the systems of linear equations (1) and n × 1 matrix A u y is also interpreted as a vector in R . In the 1 2  1 m (2) have a solution only for the following payoffs expression A u y, we regard the vector y ∈ R as an m×1 matrix.  2 p Similarly, for x A u . The scalar product of z, t ∈ R is written as 1 1 2 1 α u  = , β u  = . (3) · t −1  1 −1  2 z t. By A we denote the transpose of the matrix A. dkB u dk dkB u dk The number of pure NE of a finite game is finite. By eliminating those strategies that are played with zero probability, it is enough Proof. Let α be the payoff of player 1 in the completely mixed NE to prove Proposition 2.1 for completely mixed NE. Given two utility  1  2 1 2 l of the game A u¯ and A u¯ . Since Eq. (1) has a solution, we have profiles u , u ∈ R , if a pair of completely mixed strategies, x =  2  1 that x u ∈ ∆(S1) and y = y u ∈ ∆(S2), is (NE)  1  2  BC  αd  of the game defined by A u and A u , then, they constitute a  k  A  solution of the following systems of linear equations rank DE  αdn−k = rank = k.  dm dk dm−k 1 A u1 y = αd , y · d = 1 (1) n m Therefore, the last row of the above augmented matrix is a linear  2 ∈ k xA u = βdm, x · dn = 1 (2) combination of its first k rows. There is a vector z R such that −1 z B = dk and αz · dk = 1. Hence, z = dkB and αz · dk = =  1 ∈ =  2 ∈ −1 −1  1 for some α α u R (the payoff of player 1) and β β u αdkB dk = 1. It follows that dkB u¯ dk ̸= 0 and the payoff R (the payoff of player 2). The proof is based on a detailed analysis of player 1 is α u¯ 1, where α is defined in Eq. (3) Consider the of the solutions of the above equations. polynomial We will rely on the following fact shown by Mas-Colell (2010). l  −1  Let k = max{rank A(u) : u ∈ R }. There is an open, dense subset G p(u) = det (B(u)) dkB (u)dk . of l such that the following hold. R  1 1 Since, p u¯ ̸= 0, there is and open set U1 ⊂ G such that u¯ ∈ U1 (a) rank A(u) = k, for every u ∈ G. After reordering, if necessary, and p(u) ̸= 0 for every u ∈ U . In particular, d B−1(u)d ̸= 0 for the strategies of the players, we may write A = A(u) as 1 k k every u ∈ U1. Moreover, the above argument also shows that, for BC any u ∈ U , the only possible value of α for which the system of Eqs. A = 1 DE (1) has a solution is α(u), as given in Eq. (3). A similar argument, using the payoff of player 2, determines the set U2 and the value of where B = B(u) is a k × k matrix with det B ̸= 0.  t  β, the payoff of player 2.  (b) The functions rank (A(u)|dn) and rank A (u)|dm are constant on G. The following two lemma s restate the familiar procedure of writing the general solution of a linear system as a particular solution of the complete system plus the general solution of 1 = 1 = 2 ∈ \{ } the associated homogeneous system. The novelty here is in Our method applies as well to games for which u u tu , with t R 0 . 1 2 In particular, taking t = −1, Proposition 2.1 holds also for two-person zero sum showing that, near u¯ and u¯ , we can express those solutions in games. a differentiable manner. 2 − x (u¯)Mjy (u¯, v) = 0, for every v ∈ m k. Lemma 3.3. There are differentiable functions α : U1 → R, yp : p h R U → m and y : U × m−k → m such that for each u ∈ U , 1 R h 1 R R 1 On the other hand, differentiating A(u)yh(u, v) = 0 with respect to u and evaluating at u¯, we see that (i) A(u)yp(u) = α(u)dn, dm · yp(u) = 1. j (ii) A(u)y (u, v) = 0, d · y (u, v) = 0, for every v ∈ m−k. h m h R j ∂ 0 = M yh(u¯, v) + A(u¯) yh(u¯, v). p j p−k Proof. We split R as R × R . Accordingly, we will explicitly ∂uj write the action of each of the block matrices B,..., E of A, when ¯ n m Multiplying on the left by xh(u, w), the second term vanishes. Thus, it acts on a vector in R . For example, given z = (zk, zm−k) ∈ R = m m−k = ¯ j ¯ ∈ m−k ∈ n−k R × R , we will write Az = (Bzk + Czm−k, Dzk + Ezm−k) ∈ 0 xh(u, w)M yh(u, v) for every v R and w R . n × n−k = n. Let α : U → be given by Eq. (3). Define R R R 1 R Now, when player 1 follows the x(u¯, w) and player 2 y : U → m and y : U × m−k → m as follows p 1 R h 1 R R follows the strategy y(u¯, v), the probability that outcome j occurs is  −1  y (u) = α(u) B (u)d , 0 j j j p k x(u¯, w)M y(u¯, v) = xp(u¯)M yp(u¯) + xp(u¯)M yh(u¯, v) − = − 1  j j yh(u, v) B (u)C(u)v, v . + xh(u¯, w)M yp(u¯) + xh(u¯, w)M yh(u¯, v) j Since, det(B(u)) ̸= 0 on G, all the above functions α, yp and yh = xp(u¯)M yp(u¯) are differentiable. By Remark 3.1, it follows that which, for a fixed u¯ ∈ G, is independent of v and w.  −1 −1  Ayp = α BB dk, DB dk = α(dk, dn−k) = αdn Case 2: ki = k + 1 for some i = 1, 2. −1 Consider first the possibility that k = k + 1. The case k = dm · yp = αdkB dk = 1 1 2 k + 1 is similar. Assume that for some u1, u2 ∈ G, the game and A u1 , A u2 has a NE in which player 1 uses completely mixed  −1 −1  strategy. This player must be indifferent among all his strategies. Ayh = −BB Cv + Cv, −DB Cv + Ev = 0 Thus, the linear system (1) has a solution, for some α ∈ R. If · = − −1 + · = − · + · = dm yh dkB Cv dm−k v dm−k v dm−k v 0.  α ̸= 0, then dn is a linear combination of the columns of A, but then k1 = rank (A(u)|dn) = k. Hence, α = 0. Similarly, one can prove the following result. Therefore, if k1 = k + 1, the payoff of player 1 vanishes in any completely mixed NE of any game generated by utilities in G. Lemma 3.4. There are differentiable functions β : U2 → R, xp : We claim that no common interest game generated by utilities n n−k n ¯ ∈ U2 → R and xh : U2 × R → R such that for each u ∈ U2 we in G can have a completely mixed NE. Otherwise, there is u G have, such that the game A(u¯) has a completely mixed NE, say x ∈ ∆(S1) and y ∈ ∆(S ). In this equilibrium, the payoff of player 1 is zero. (i) x (u)A(u) = β(u)d d · x (u) = 1. 2 p m n p Since G is open, we can add a small enough, non-zero constant ε to = · = ∈ n−k (ii) xh (u, w) A(u) 0, dn xh (u, w) 0, for every w R . the utilities of the players and obtain a new game, say A(u¯ +ε), also  1   1  1  generated by utilities from G. Since the payoffs of the games A(u¯) Of course, not all the functions y u , v = yp u + yh u , v and A(u¯ + ε) differ only by a fixed constant ε, they have the same and x u2, w = x u2 + x u2, w obtained from the above p h NE. In particular, x ∈ ∆ S and y ∈ ∆ S is also a completely lemma s correspond necessarily to a NE. For this, one would need to ( 1) ( 2) mixed NE in the game A(u¯ + ε), in which player 1 gets the payoff check also that the coordinates of y u1, v and x u2, w are non- , but this contradicts the above remark. negative and that these strategies constitute a to the ε other player’s action. References The proof of Proposition 2.1 for the case k1 = k2 = k is now a straightforward application of Lemmas 3.3 and 3.4. Using the  1 2 González-Pimienta, C., 2010. Generic finiteness of outcome distributions for two notation in those lemma s, we note first that, for u , u ∈ U1 ×U2, person game forms with three outcomes. Mathematical Social Sciences 59,  2   1  1  1 2 364–365. xh u , w A u yp u = α u xh(u , w) · dn = 0. Govindan, S., McLennan, A., 1998. Generic finiteness of outcome distributions for Differentiating this expression with respect to u1 we obtain that two person game forms with zero sum and common interest utilities. Mimeo, j University of Western Ontario.   Govindan, S., McLennan, A., 2001. On the generic finiteness of equilibrium outcome ∂ distributions in game forms. Econometrica 69, 455–471. x u2  Mjy u1 + A u1 y u1 = 0 h , w p 1 p . Harsanyi, J.C., 1973. Oddness of the number of equilibrium points: a new proof. ∂uj International Journal of 2, 235–250. Kreps, D.M., Wilson, R., 1982. Sequential equilibria. Econometrica 50, 863–894. We consider now the particular case of a common utility game in Kukushkin, N.S., Litan, C., Marhuenda, F., 2008. On the generic finiteness of 1 2 which u¯ = u¯ = u¯. Since, xh(u¯, w)A(u¯) = 0, when plugging equilibrium outcome distributions in bimatrix game forms. Journal of Economic u1 = u2 = u¯ in the previous equation, it reduces to Theory 139, 392–395. Mas-Colell, A., 2010. Generic finiteness of equilibrium payoffs for bimatrix games. j n−k xh(u¯, w)M yp(u¯) = 0, for every w ∈ R . Journal of Mathematical Economics 46, 382–383. Park, I.U., 1997. Generic finiteness of equilibrium outcome distributions for Similarly, we see that sender–receiver cheap-talk games. Journal of Economic Theory 76, 431–448.

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