Sion s minimax theorem and Nash equilibrium of symmetric multi-person zero-sum game

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1 MPRA Munich Personal RePEc Archive Sion theorem and Nash equilibrium of symmetric multi-person zero-sum game Atsuhiro Satoh and Yasuhito Tanaka 24 October 2017 Online at MPRA Paper No posted UNSPECIFIED

2 Sion theorem and Nash equilibrium of symmetric multi-person zero-sum game Atsuhiro Satoh a Yasuhito Tanaka b a Faculty of Economics Hokkai-Gakuen University Toyohira-ku Sapporo Hokkaido Japan. b Faculty of Economics Doshisha University Kamigyo-ku Kyoto Japan. Abstract We will show that Sion theorem is equivalent to the existence of Nash equilibrium in a symmetric multi-person zero-sum game. If a zero-sum game is asymmetric i strategies and i strategies of players do not correspond to Nash equilibrium strategies. However if it is symmetric the i strategy and the i strategy constitute a Nash equilibrium. Keywords: multi-person zero-sum game Nash equilibrium Sion theorem. atsatoh@hgu.jp yasuhito@mail.doshisha.ac.jp. Preprint submitted to Elsevier October

3 1. Introduction We consider the relation between Sion theorem and the existence of Nash equilibrium in a symmetric multi-person zero-sum game. We will show that they are equivalent. An example of such a game is a relative profit imization game in a Cournot oligopoly. Suppose that there are n 3 firmn an oligopolistic industry. Let π i be the absolute profit of the i-th firm. Then its relative profit is π i = π i 1 n 1 n j=1 j i π j. We see n π i = i=1 n i=1 π i 1 n 1 (n 1) n j=1 π j = 0. Thus the relative profit imization game in a Cournot oligopoly is a zerosum game 2. If the oligopoly is asymmetric because the demand function is not symmetric or firms have different cost function strategies and i strategies of firms do not correspond to Nash equilibrium strategies. However if the demand function is symmetric and the firms have the same cost function the i strategy and the i strategy constitute a Nash equilibrium. 2. The model Consider a symmetric n-person zero-sum game with n 3 as follows. There are n players n. The set of players denoted by N. A vector of strategic variables (... s n ) S 1 S 2 S n. S i is a convex and compact set in a linear topological space for each i N. The payoff functions of the players are u i (... s n ) for i N. We assume u i for each i N is upper semi-continuous and quasi-concave on S i for each S j j N j i. It is lower semi-continuous and quasi-convex on S j for j N j i for each S i. 2 About relative profit imization under imperfect competition please see Matsumura Matsushima and Cato (2013) Satoh and Tanaka (2013) Satoh and Tanaka (2014a) Satoh and Tanaka (2014b) Tanaka (2013a) Tanaka (2013b) and Vega-Redondo (1997) 2

4 Since the game is symmetric all players have the same payoff function and we have n u i (... s n ) = 0 (1) i=1 for given (... s n ) because we consider a zero-sum game. Also all S i s are identical. Denote them by S. 3. The main results Sion theorem (Sion (1958) Komiya (1988) Kindler (2005)) is stated as follows. Lemma 1 (Sion theorem). Let X and Y be non-void convex and compact subsets of two linear topological spaces and let f : X Y R be a function that is upper semi-continuous and quasi-concave in the first variable and lower semi-continuous and quasi-convex in the second variable. Then f (x y) = f (x y). x X y Y y Y x X We follow the description of this theorem in Kindler (2005). Suppose that s k S k for all k N other than i and j j i are given. Denote a vector of such s k s by s i j. Then u i (... s n ) is written as u i ( s i j ) and it is a function of and. We can apply Lemma 1 to such a situation and get the following lemma. Lemma 2. Let j i and S i and S j be non-void convex and compact subsets of two linear topological spaces and let u i : S i S j R given s i j be a function that is upper semi-continuous and quasi-concave on S i and lower semi-continuous and quasi-convex on S j. Then S i S j u i ( s i j ) = S j S i u i ( s i j ). We assume that arg si S i S j u i ( s i j ) and arg S j si S i u i ( s i j ) are single-valued for any pair of i and j. By the imum theorem they are continuous in s i j. 3

5 Consider the following function; arg s1 S 1 s2 S 2 u 1 ( s 12 ) arg s2 S 2 s3 S 3 u 2 ( s 23 ) arg sn S n s1 S 1 u n (s n s 1n ) s n given (... s n ). This function is continuous and each S i is convex and compact. Therefore there exists a fixed point (... s n ) (by Glicksberg s fixed point theorem (Glicksberg (1952))). Similarly we can consider the following function; arg s1 S 1 s2 S 1 u 2 ( s 12 ) arg s2 S 2 s3 S 3 u 3 ( s 23 ) arg sn S n s1 S 1 u 1 (s n s 1n ) s n given (... s n ). This function also has a fixed point ( s 1 s 2... s n). Since we consider a symmetric game in which all players have the same payoff function we can assume that when s i j = s kl and S u i( s i j ) = s k S s l S u k(s k s l s kl ) = s l S s k S u k(s k s l s kl ) = S S u i( s i j ) arg S u i( s i j ) = arg s k S s l S u k(s k s l s kl ) = arg s l S s k S u k(s k s l s kl ) = arg S S u i( s i j ) for i j k l N. They mean and S u i( s i j ) = S S u j( s i j ) = S S u j( s i j ) = S S u i( s i j ) arg S u i( s i j ) = arg S S u j( s i j ) = arg S S u j( s i j ) = arg S S u i( s i j ) for any i j. 4

6 Then we find ( s 1 s 2... s n) = (... s n ). Let s = (s s... s). If (... s n ) = s all arg si S S u i ( s... s) s and all arg si S S u j ( s... s) s are the same. Thus the fixed point obtained from above two functions symmetric. Denote it by s = ( s s... s). These arguments ensure the existence of symmetric i and i strategies. Summarizing the results Sion theorem for a symmetric multiperson zero-sum game is stated as follows. Theorem 1. Let S i s for i N be non-void convex and compact subsets of linear topological spaces let u i : S i S j R given s i j be a function that is upper semi-continuous and quasi-concave on S i and lower semi-continuous and quasiconvex on S j for all j i and i N and S i = S for all i N. Then there exists s = ( s s... s) such that and S u i( s i j ) = S S u j( s i j ) = S S u j( s i j ) = S S u i( s i j ) arg S u i( s i j ) = arg S S u j( s i j ) = arg S S u j( s i j ) = arg S S u i( s i j ) for any i j where s i j = ( s s... s) for k N k i j. Now we consider a Nash equilibrium of a symmetric multi-person zero-sum game. Let s i i N be the values of s which respectively imize u i i N given s j j i in a neighborhood around (s 1 s 2... s n) in S 1 S 2 S n. Then u i (s 1... s i... s n) u i (s s n) for all s i i N. (2) Since the game is symmetric we consider a symmetric equilibrium such that all s i s are equal at equilibria. Thus u i(s 1... s i... s n) s for all i are equal and by the property of zero-sum game they are zero. By symmetry of the game we have u j (s s n) = u k (s s n) for j i k i j k. From this and (1) n u j (s s n) = (n 1)u j (s s n) = u i (s s n). j=1 j i 5

7 Therefore from (2) By symmetry Combining this and (2) u j (s s n) u j (s 1... s i... s n) for j i. u i (s s n) u i (s 1... s i... s n) for j i. u i (s s n) u i (s 1... s i... s n) u i (s s n) for all s i and all s j j i i N. This equivalent to u i (s 1... s i... s n) = j i given s k k i j u i (s s n) = u i (s s n) Denote the symmetric Nash equilibrium of the zero-sum game by s = (s s... s ). Let s i j = ( s s... s ) for k N k i j and s i j = ( s s... s) for k N k i j. We can show the following result. Theorem 2. The following three statements are equivalent. (1) There exists a symmetric Nash equilibrium in a symmetric multi-person zero-sum game. (2) There exists s = ( s s... s) such that the following relation holds. v u i ( s i j ) = u i ( s i j ) v for any pair of i and j. (3) There exists a real number v s s m i and s m j such that u i (s m i s i j ) v s for any and u i ( s m j s i j) v s for any (3) for any pair of i and j. Proof. (1 2) Set s = s. Then v = = u i ( s i j) u i ( s j s i j) = u i (s i s j s i j) u i (s i s i j) u i ( s i j) = v. 6

8 On the other hand u i ( s i j ) u i( s i j ) then u i ( s i j ) si u i ( s i j ) and so u i ( s i j ) si u i ( s i j ). Thus v v and we have v = v. (2 3) Set s = s. Let s m i = arg si u i ( s i j ) (the i strategy) s m j = arg si u i ( s i j ) (the i strategy) and let v s = v = v. Then we have u i (s m i s i j) = u i (s m i s i j) = u i ( s i j) = v s u i ( s i j) = u i ( s m j s s i i j) u i ( s m j s i j). By Theorem 1 s m i = s = s and s m j = s = s. (3 1) Set s = s. Since s m i = s m j = s from (3) we get u i (s s i j) v s u i ( s s i j) for all S i S j. Putting = s i and = s j we see v s = u i (s s s i j ) and s = (s s... s ) is an equilibrium. Therefore Sion theorem is equivalent to the existence of Nash equilibrium of a symmetric multi-person zero-sum game. 4. Example of asymmetric multi-person zero-sum game Consider a three-person game. Suppose that the payoff functions of players are π 1 = (a ( + ) c [(a ( + ) c 2 +(a ( + ) c 3 ] π 2 = (a ( + ) c [(a ( + ) c 1 +(a ( + ) c 3 ] and π 3 = (a ( + ) c [(a ( + ) c 1 +(a ( + ) c 2 ]. This a model of relative profit imization in a three firms Cournot oligopoly with constant marginal cost and zero fixed cost producing a homogeneous good. 7

9 i = are the outputs of the firms. The conditions for imization of π i i = are π 1 = a 2 ( + ) c ( + ) = 0 π 2 = a 2 ( + ) c ( + ) = 0 and π 3 = a 2 ( + ) c ( + ) = 0. The Nash equilibrium strategies are = 3a 5c 1 + c 2 + c 3 9 = 3a 5c 2 + c 1 + c 3 9 = 3a 5c 3 + c 2 + c 1. (4) 9 We consider i and i strategy about Player 1 and 2. The condition for imization of π 1 with respect to is π 1 = 0. Denote which satisfies this condition by ( ) and substitute it into π 1. Then the condition for imization of π 1 with respect to given ( ) and is π 1 + π 1 d d = 0. We call the strategy of Player 1 obtained from these conditions the i strategy of Player 1 to Player 2. It is denoted by arg s1 s2 π 1. The condition for imization of π 1 with respect to is π 1 = 0. Denote which satisfies this condition by ( ) and substitute it into π 1. Then the condition for imization of π 1 with respect to given ( ) is π 1 + π 1 d d 2 = 0. We call the strategy of Player 2 obtained from these conditions the i strategy of Player 2 to Player 1. It is denoted by arg s2 s1 π 1. In our example we obtain arg π 1 = 3a 4c 1 + c 2 9 Similarly we get the following results. arg π 2 = 3a 4c 2 + c 1 9 arg arg 8 π 1 = 6a 9 2c 1 4c 2. 9 π 2 = 6a 9 2c 2 4c 1 9

10 arg arg arg π 1 = 3a 4c 1 + c 3 9 π 3 = 3a 4c 3 + c 1 9 π 2 = 3a 4c 2 + c 3 9 arg arg arg π 1 = 6a 9 2c 1 4c 3 9 π 3 = 6a 9 2c 3 4c 1 9 π 2 = 6a 9 2c 2 4c 3 9 arg π 3 = 3a 4c 3 + c 2 arg π 3 = 6a 9 2c 3 4c If the game is asymmetric for example c 2 c 3 arg s1 s2 π 1 arg s1 s3 π 1 arg s2 s3 π 2 arg s3 s2 π 3 arg s3 s2 π 2 arg s2 s3 π 3 and so on. However if the game is symmetric we have c 2 = c 3 = c 1 and arg = arg π 1 = arg π 2 = arg π 2 = arg π 3 = a c 1. 3 π 1 = arg π 3 All of the Nash equilibrium strategies of the playern (4) are also equal to a c 1 3. Assume = = as well as c 2 = c 3 = c 1. Then Further if we obtain arg = arg arg = arg π 1 = arg π 2 = arg = arg π 1 = arg π 2 = arg π 3 = 2a 3 2c 1. 3 π 2 = arg π 2 = arg π 3 π 2 = arg π 3 = a c 1. 3 π 1 = arg π 3 π 1 = arg π 3 Therefore the i strategy the i strategy and the Nash equilibrium strategy for all players are equal. Acknowledgment This work was supported by Japan Society for the Promotion of Science KAK- ENHI Grant Number 15K

11 References Glicksberg I.L. (1952) A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points. Proceedings of the American Mathematical Society 3 pp Kindler J. (2005) A simple proof of Sion theorem American Mathematical Monthly 112 pp Komiya H. (1988) Elementary proof for Sion theorem Kodai Mathematical Journal 11 pp Matsumura T. N. Matsushima and S. Cato (2013) Competitiveness and R&D competition revisited Economic Modelling Satoh A. and Y. Tanaka (2013) Relative profit imization and Bertrand equilibrium with quadratic cost functions Economics and Business Letters 2 pp Satoh A. and Y. Tanaka (2014a) Relative profit imization and equivalence of Cournot and Bertrand equilibria in asymmetric duopoly Economics Bulletin 34 pp Satoh A. and Y. Tanaka (2014b) Relative profit imization in asymmetric oligopoly Economics Bulletin Sion M. (1958) On general i theorems Pacific Journal of Mathematics 8 pp Tanaka Y. (2013a) Equivalence of Cournot and Bertrand equilibria in differentiated duopoly under relative profit imization with linear demand Economics Bulletin Tanaka Y. (2013b) Irrelevance of the choice of strategic variablen duopoly under relative profit imization Economics and Business Letters 2 pp Vega-Redondo F. (1997) The evolution of Walrasian behavior Econometrica