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Determinacy of Equilibrium Outcome Distributions for Zero Sum and Common Utility Games✩,✩✩

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Determinacy of Equilibrium Outcome Distributions for Zero Sum and Common Utility Games✩,✩✩ Determinacy of equilibrium outcome distributions for zero sum and common utility gamesI,II Cristian M. Litan a, Francisco Marhuenda b,∗ a University Babe³-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 determinacy 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 I We thank an anonymous referee for his/her insightful comments. and elementary proof. II 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 D jS1j ≥ 1; m D jS2j ≥ 1 and let S D S1 × S2. 1 t Consider a finite set of outcomes Ω D f1;:::; lg. We denote by Let k1 D rank .A.u/jdn/ and k2 D rank A .u/jdm , for any ∆.Ω/ the set of probability measures on Ω. An outcome game form u 2 G. Note that k1; k2 2 fk; k C 1g. The argument will be carried is a function θ V S ! ∆.Ω/. out by considering the possible values of k1 and k2. D For each outcome j 1;:::; l, the mapping θ defines a function Case 1: k D k D k. j V ! j 1 2 M S R such that M .sa; sb/ is the probability that θ.sa; sb/ −1 2 j First, remark that B .u/ is defined for all u G. In all the assigns to the outcome j 2 Ω. We identify each M with an n × m D D l Lemmas that follow, it is assumed that k1 k2 k. Note also matrix with real entries. Given u D .u1;:::; ul/ 2 R , we interpret l that the matrices below depend on u. uj as the utility assigned to outcome j. To each u 2 R we assign the matrix Remark 3.1. Since, l X j BC ! A.u/ D ujM : B C dk jD1 k D rank B D rank D rank DE D E dn−k dk dm−k 1 2 l Given two profiles of utilities on outcomes u ; u 2 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 2 ∆.S1/ and y 2 ∆.S2/ of the players induce . / k a probability distribution on Ω. The probability that outcome j a vector z 2 R such that HB D D; HC D E; Hdk D dn−k; zB D dk j −1 −1 occurs is given by x M y. and zC D dm−k. Hence, H D DB ; z D dkB . And we have that 1 1 2 A common utility game is a game in which u D u D u . −1 −1 DB C D E; DB d 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 D dm−k; for every u 2 G: Govindan and McLennan (1998). Fix now uN 1; uN 2 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 uN and A uN has a completely mixed every u 2 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, uN 1 D uN 2 D uN. 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 uN 2 − p U ; uN 2 2 U and d B 1.u/d 6D 0, for every u 2 U [U . Furthermore, The notation dp represents the vector .1;:::; 1/ 2 . The 1 2 k k 1 2 R 1 2 1 n for u 2 U and u 2 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 2 R as an m×1 matrix. 2 p Similarly, for x A u . The scalar product of z; t 2 R is written as 1 1 2 1 α u D ; β u D : (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 uN and A uN . Since Eq. (1) has a solution, we have profiles u ; u 2 R , if a pair of completely mixed strategies, x D 2 1 that x u 2 ∆.S1/ and y D y u 2 ∆.S2/, is Nash equilibrium (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 D rank D k: dm dk dm−k 1 A u1 y D αd ; y · d D 1 (1) n m Therefore, the last row of the above augmented matrix is a linear 2 2 k xA u D βdm; x · dn D 1 (2) combination of its first k rows. There is a vector z R such that −1 z B D dk and αz · dk D 1. Hence, z D dkB and αz · dk D D 1 2 D 2 2 −1 −1 1 for some α α u R (the payoff of player 1) and β β u αdkB dk D 1. It follows that dkB uN dk 6D 0 and the payoff R (the payoff of player 2). The proof is based on a detailed analysis of player 1 is α uN 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 D maxfrank A.u/ V u 2 R g. There is an open, dense subset G p.u/ D det .B.u// dkB .u/dk : of l such that the following hold. R 1 1 Since, p uN 6D 0, there is and open set U1 ⊂ G such that uN 2 U1 (a) rank A.u/ D k, for every u 2 G.
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