Minimax theorem and Nash equilibrium of symmetric multi-players zero-sum game with two strategic variables
aa r X i v : . [ q -f i n . M F ] J un Minimax theorem and Nash equilibriumof symmetric multi-players zero-sumgame with two strategic variables
Masahiko Hattori ∗ Faculty of Economics, Doshisha University,Kamigyo-ku, Kyoto, 602-8580, Japan,Atsuhiro Satoh † Faculty of Economics, Hokkai-Gakuen University,Toyohira-ku, Sapporo, Hokkaido, 062-8605, Japan,andYasuhito Tanaka ‡ Faculty of Economics, Doshisha University,Kamigyo-ku, Kyoto, 602-8580, Japan.
Abstract
We consider a symmetric multi-players zero-sum game with two strategic variables.There are n players, n ≥
3. Each player is denoted by i . Two strategic variables are t i and s i , i ∈ { , . . . , n } . They are related by invertible functions. Using the minimax theorem bySion (1958) we will show that Nash equilibria in the following states are equivalent.1. All players choose t i , i ∈ { , . . . , n } , (as their strategic variables).2. Some players choose t i ’s and the other players choose s i ’s.3. All players choose s i , i ∈ { , . . . , n } . Keywords: symmetric multi-players zero-sum game, Nash equilibrium, two strategic vari-ables
JEL Classification:
C72 ∗ [email protected] † [email protected] ‡ [email protected] Introduction
We consider a symmetric multi-players zero-sum game with two strategic variables. Thereare n players, n ≥
3. Each player is denoted by i . Two strategic variables are t i and s i , i ∈{ , . . . , n } . They are related by invertible functions. Using the minimax theorem by Sion(1958) we will show that Nash equilibria in the following states are equivalent.1. All players choose t i , i ∈ { , . . . , n } , (as their strategic variables).2. Some players choose t i ’s and the other players choose s i ’s.3. All players choose s i , i ∈ { , . . . , n } .In the next section we present a model of this paper and prove some preliminary resultswhich are variations of Sion’s minimax theorem. In Section 3 we will show the main results.An example of a multi-players zero-sum game with two strategic variables is a relative profitmaximization game in an oligopoly with differentiated goods. See Section 4. We consider a symmetric multi-players zero-sum game with two strategic variables. Thereare n players, n ≥
3. Two strategic variables are t i and s i , i ∈ { , . . . , n } . t i is chosen from T i and s i is chosen from S i . T i and S i are convex and compact sets in linear topological spaces,respectively, for each i ∈ { , . . . , n } . We denote N = { , . . . , n } . The relations of the strategicvariables are represented by s i = f i ( t , . . . , t n ) , i ∈ N , and t i = g i ( s , . . . , s n ) , i ∈ N . f i ( t , . . . , t n ) and g i ( s , . . . , s n ) are continuous invertible functions, and so they are one-to-one and onto functions. Let M = { , . . . , m } , ≤ m ≤ n , be a subset of N , and denote N − M = { m + , . . . , n } . When n − m players in N − M choose s i ’s, t i ’s for them are de-termined according to t m + = g m + ( f ( t , . . . , t m , t m + , . . . , t n ) , . . . , f m ( t , . . . , t m , t m + , . . . , t n ) , s m + , . . . , s n ) . . . t n = g n ( f ( t , . . . , t m , t m + , . . . , t n ) , . . . , f m ( t , . . . , t m , t m + , . . . , t n ) , s m + , . . . , s n ) . We denote these t i ’s by t i ( t , . . . , t m , s m + , . . . , s n ) .When all players choose s i ’s, i ∈ N , t i ’s for them are determined according to t = g ( s , . . . , s n ) . . . t n = g n ( s , . . . , s n ) . Denote these t i ’s by t i ( s , . . . , s n ) . 2he payoff function of Player i is u i , i ∈ N . It is written as u i ( t , . . . , t n ) . We assume u i : T × · · · × T n ⇒ R for each i ∈ N is continuous on T × · · · × T n . Thus, it iscontinuous on S × · · · × S n through f i , i ∈ N . It is quasi-concave on T i and S i fora strategy of each other player, and quasi-convex on T j , j = i and S j , j = i foreach t i and s i .We do not assume differentiability of the payoff functions.Symmetry of the game means that the payoff functions of all players are symmetric and inthe payoff function of each Player i , Players j and k , j , k = i , are interchangeable. f i ’s and g i ’s are symmetric. Since the game is a zero-sum game, the sum of the values of the payofffunctions of the players is zero. All T i ’s are identical, and all S i ’s are identical. Denote themby T and S .Sion’s minimax theorem (Sion (1958), Komiya (1988), Kindler (2005)) for a continuousfunction is stated as follows. Lemma 1.
Let X and Y be non-void convex and compact subsets of two linear topologicalspaces, and let f : X × Y → R be a function that is continuous and quasi-concave in the firstvariable and continuous and quasi-convex in the second variable. Then max x ∈ X min y ∈ Y f ( x , y ) = min y ∈ Y max x ∈ X f ( x , y ) . We follow the description of Sion’s theorem in Kindler (2005).Applying this lemma to the situation of this paper such that m players choose t i ’s and n − m players choose s i ’s as their strategic variables, we have the following relations.max t i ∈ T min t j ∈ T u i ( t i , t j , t k , t l ) = min t j ∈ T max t i ∈ T u i ( t i , t j , t k , t l ) . max t i ∈ T min s j ∈ S u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = min s j ∈ S max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) , where t k is a vector of t k , k ∈ M , of the players other than Players i and j who choose t k ’s astheir strategic variables. On the other hand, t l is a vector of t l , l ∈ N − M , of the players otherthan Player j who choose s l ’s as their strategic variables. Also, relations which are symmetricto them hold. u i ( t i , t j , t k , t l ) is the payoff of Player i when Players i and j choose t i and t j . Onthe other hand, u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) means the payoff of Player i when he chooses t i andPlayer j chooses s j .Further we show the following results. 3 emma 2. max t j ∈ T min t i ∈ T u j ( t i , t j , t k , t l ) = max s j ∈ S min t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l )= min t i ∈ T max s j ∈ S u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = min t i ∈ T max t j ∈ T u j ( t i , t j , t k , t l ) , u j ( t i , t j , t k , t l ) is the payoff of Player j when Players i and j choose t i and t j . On the otherhand, u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) means the payoff of Player j when he chooses s j and Player i chooses t i . Proof. min t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) is the minimum of u j with respect to t i given s j . Let˜ t i ( s j ) = arg min t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) , and fix the value of t j at t j = g j ( f i ( ˜ t i ( s j ) , t j , t k , t l ) , s j , f k , s l ) , (1)where f k denotes a vector of the values of s k ’s of players who choose t k ’s, and s l denotes avector of the values of s l ’s of players who choose s l ’s. Then, we havemin t i ∈ T u j ( t i , t j , t k , t l ) ≤ u j ( ˜ t i ( s j ) , t j , t k , t l ) = min t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) , where min t i ∈ T u j ( t i , t j , t k , t l ) is the minimum of u j with respect to t i given the value of t j at t j . We assume that ˜ t i ( s j ) = arg min t i ∈ T u j ( u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l )) is single-valued. By themaximum theorem and continuity of u j , ˜ t i ( s j ) is continuous. Then, any value of t j can berealized by appropriately choosing s j according to (1). Therefore,max t j ∈ T min t i ∈ T u j ( t i , t j , t k , t l ) ≤ max s j ∈ S min t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) . (2)On the other hand, min t i ∈ T u j ( t i , t j , t k , t l ) is the minimum of u j with respect to t i given t j .Let ˜ t i ( t j ) = arg min t i ∈ T u j ( t i , t j , t k , t l ) , and fix the value of s j at s j = f j ( ˜ t i ( t j ) , t j , t k , t l ) . (3)Then, we havemin t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) ≤ u j ( ˜ t i ( t j ) , t j ( ˜ t i ( t j ) , s j , t k , t l ) , t k , t l ) = min t i ∈ T u j ( t i , t j , t k , t l ) , where min t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) is the minimum of u j with respect to t i given the valueof s j at s j . We assume that ˜ t i ( t j ) = arg min t i ∈ T u j ( t i , t j , t k , t l ) is single-valued. By the maximumtheorem and continuity of u j , ˜ t i ( t j ) is continuous. Then, any value of s j can be realized byappropriately choosing t j according to (3). Therefore,max s j ∈ S min t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) t k , t l ) ≤ max t j ∈ T min t i ∈ T u j ( t i , t j , t k , t l ) . (4)4ombining (2) and (4), we getmax s j ∈ S min t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) t k , t l ) = max t j ∈ T min t i ∈ T u j ( t i , t j , t k , t l ) . Since any value of s j can be realized by appropriately choosing t j , we havemax s j ∈ S u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = max t j ∈ T u j ( t i , t j , t k , t l ) . Thus, min t i ∈ T max s j ∈ S u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = min t i ∈ T max t j ∈ T u j ( t i , t j , t k , t l ) . Therefore, max t j ∈ T min t i ∈ T u j ( t i , t j , t k , t l ) = max s j ∈ S min t i ∈ T u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l )= min t i ∈ T max s j ∈ S u j ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = min t i ∈ T max t j ∈ T u j ( t i , t j , t k , t l ) . Lemma 3. min t j ∈ T max t i ∈ T u i ( t i , t j , t k , t l ) = min s j ∈ S max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l )= max t i ∈ T min s j ∈ S u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = max t i ∈ T min t j ∈ T u i ( t i , t j , t k , t l ) , Proof. max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) is the maximum of u i with respect to t i given s j . Let¯ t i ( s j ) = arg max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) , and fix the value of t j at t j = g j ( f i ( ¯ t i ( s j ) , t j , t k , t l ) , s j , f k , s l ) , (5)where f k denotes a vector of the values of s k ’s of players who choose t k ’s, and s l denotes avector of the values of s l ’s of players who choose s l ’s. Then, we havemax t i ∈ T u i ( t i , t j , t k , t l ) ≥ u i ( ¯ t i ( s j ) , t j , t k , t l ) = max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) , where max t i ∈ T u i ( t i , t j , t k , t l ) is the maximum of u i with respect to t i given the value of t j at t j . We assume that ¯ t i ( s j ) = arg max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) is single-valued. By themaximum theorem and continuity of u i , ¯ t i ( s j ) is continuous. Then, any value of t j can berealized by appropriately choosing s j according to (5). Therefore,min t j ∈ T max t i ∈ T u i ( t i , t j , t k , t l ) ≥ min s j ∈ S max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) . (6)On the other hand, max t i ∈ T u i ( t i , t j , t k , t l ) is the maximum of u i with respect to t i given t j .Let ¯ t i ( t j ) = arg max t i ∈ T u i ( t i , t j , t k , t l ) , and fix the value of s j at s j = f j ( ¯ t i ( t j ) , t j , t k , t l ) . (7)5hen, we havemax t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) ≥ u i ( ¯ t i ( s j ) , t j ( t i , s j , t k , t l ) , t k , t l ) = max t i ∈ T u i ( t i , t j , t k , t l ) , where max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) is the maximum of u i with respect to t i given the valueof s j at s j . We assume that ¯ t i ( t j ) = arg max t i ∈ T u i ( t i , t j , t k , t l ) is single-valued. By the maximumtheorem and continuity of u i , ¯ t i ( t j ) is continuous. Then, any value of s j can be realized byappropriately choosing t j according to (7). Therefore,min s j ∈ S max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) ≥ min t j ∈ T max t i ∈ T u i ( t i , t j , t k , t l ) . (8)Combining (6) and (8), we getmin s j ∈ S max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = min t j ∈ T max t i ∈ T u i ( t i , t j , t k , t l ) . Since any value of s j can be realized by appropriately choosing t j , we havemin s j ∈ S u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = min t j ∈ T u i ( t i , t j , t k , t l ) . Thus, max t i ∈ T min s j ∈ S u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = max t i ∈ T min t j ∈ T u i ( t i , t j , t k , t l ) . Therefore, min t j ∈ T max t i ∈ T u i ( t i , t j , t k , t l ) = min s j ∈ S max t i ∈ T u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) , = max t i ∈ T min s j ∈ S u i ( t i , t j ( t i , s j , t k , t l ) , t k , t l ) = max t i ∈ T min t j ∈ T u i ( t i , t j , t k , t l ) . In this section we present the main results of this paper. First we show
Theorem 1.
The equilibrium where all players choose t i ’s is equivalent to the equilibriumwhere one player (Player j) chooses s j and all other players choose t i ’s as their strategicvariables.Proof.
1. Consider a situation ( t , . . . , t n ) = ( t , . . . , t ) , that is, all players choose the samevalue of t i . Let s ( t ) = f i ( t , . . . , t ) , i ∈ N . By symmetry of the gamemax t ∈ T u ( t , t , . . . , t ) = · · · = max t n ∈ T u n ( t , . . . , t n ) , t ∈ T u ( t , t , . . . , t ) = · · · = arg max t n ∈ T u n ( t , . . . , t n ) . Consider the following function. t → arg max t i ∈ T u i ( t i , t , . . . , t ) , i ∈ N . Since this function is continuous and T is compact, there exists a fixed point. Denote itby t ∗ . Then, t ∗ → arg max t i ∈ T u i ( t i , t ∗ , . . . , t ∗ ) . We have max t i ∈ T u i ( t i , t ∗ , . . . , t ∗ ) = , for all i ∈ N .
2. Because the game is zero-sum, u i ( t i , t ∗ , . . . , t ∗ ) + n ∑ j = , j = i u j ( t i , t ∗ , . . . , t ∗ ) = . By symmetry u i ( t i , t ∗ , . . . , t ∗ ) + ( n − ) u j ( t i , t ∗ , . . . , t ∗ ) = . This means u i ( t i , t ∗ , . . . , t ∗ ) = − ( n − ) u j ( t i , t ∗ , . . . , t ∗ ) . and max t i ∈ T u i ( t i , t ∗ , . . . , t ∗ ) = − ( n − ) min t i ∈ T u j ( t i , t ∗ , . . . , t ∗ ) . From this we getarg max t i ∈ T u i ( t i , t ∗ , . . . , t ∗ ) = arg min t i ∈ T u j ( t i , t ∗ , . . . , t ∗ ) = t ∗ . We have max t i ∈ T u i ( t i , t ∗ , . . . , t ∗ ) = min t i ∈ T u j ( t i , t ∗ , . . . , t ∗ ) = u i ( t ∗ , . . . , t ∗ ) = . By symmetry max t i ∈ T u i ( t i , t ∗ , . . . , t ∗ ) = min t j ∈ T u i ( t j , t ∗ , . . . , t ∗ ) = . Then, min t j ∈ T max t i ∈ T u i ( t i , t j , t ∗ , . . . , t ∗ ) ≤ max t i ∈ T u i ( t i , t ∗ , . . . , t ∗ )= min t j ∈ T u i ( t j , t ∗ , . . . , t ∗ ) ≤ max t i ∈ T min t j ∈ T u i ( t i , t j , t ∗ , . . . , t ∗ ) . t j ∈ T max t i ∈ T u i ( t i , t j , t ∗ , . . . , t ∗ ) = max t i ∈ T u i ( t i , t ∗ , . . . , t ∗ ) = min t j ∈ T u i ( t j , t ∗ , . . . , t ∗ ) (9) = max t i ∈ T min t j ∈ T u i ( t i , t j , t ∗ , . . . , t ∗ ) = min s j ∈ S max t i ∈ T u i ( t i , t j ( t i , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ )= max t i ∈ T min s j ∈ S u i ( t i , t j ( t i , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) = .
3. Since any value of s j can be realized by appropriately choosing t j ,min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , . . . , t ∗ ) = min t j ∈ T u i ( t ∗ , t j , t ∗ . . . , t ∗ ) (10) = u i ( t ∗ , . . . , t ∗ ) = . Then, arg min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) = s ( t ∗ ) . (9) and (10) mean min s j ∈ S max t i ∈ T u i ( t i , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) (11) = min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) = . And we havemax t i ∈ T u i ( t i , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) ≥ u i ( t ∗ , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) . Then, arg min s j ∈ S max t i ∈ T u i ( t i , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ )= arg min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) = s ( t ∗ ) . Note s ( t ∗ ) = f ( t ∗ , t ∗ , . . . , t ∗ ) .Thus, by (11)min s j ∈ S max t i ∈ T u i ( t i , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) = max t i ∈ T u i ( t i , t j ( t i , s ( t ∗ ) , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ )= min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) = u i ( t ∗ , t j ( t ∗ , s ( t ∗ ) , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) = . Therefore, arg max t i ∈ T u i ( t i , t j ( t i , s ( t ∗ ) , t ∗ , . . . , t ∗ ) , t ∗ , . . . , t ∗ ) = t ∗ . (12)This holds for all i ∈ N , i = j . 8n the other hand, because any value of s j is realized by appropriately choosing t j ,max s j ∈ S u j ( t ∗ , t j ( t ∗ , s j , t ∗ , . . . , t ∗ )) = max t j ∈ T u j ( t ∗ , t j , t ∗ , . . . , t ∗ ) = u j ( t ∗ , . . . , t ∗ ) = . Therefore, arg max s j ∈ S u j ( t ∗ , t j ( t ∗ , s j , t ∗ , . . . , t ∗ )) = s ( t ∗ ) . (13)From (12) and (13), ( t ∗ , s ( t ∗ ) , t ∗ , . . . , t ∗ ) is a Nash equilibrium which is equivalent to ( t ∗ , . . . , t ∗ ) . ( t ∗ , s ( t ∗ ) , t ∗ , . . . , t ∗ ) denotes an equilibrium where t i = t ∗ , s j = s ( t ∗ ) and t k = t ∗ for k = i , j .Consider a Nash equilibrium where m players choose t ∗ and n − m players choose s ( t ∗ ) .Let t k be a vector of t k , k ∈ M , of players other than i and j who choose t k ’s as their strategicvariables; t l be a vector of t l , l ∈ N − M , of players who choose s l ’s as their strategic variables.These expressions mean that t i = t j = t ∗ ; each t k = t ∗ and each s l = s ( t ∗ ) . We write suchan equilibrium as ( t i , t j , t k , t l , ) = ( t ∗ , t ∗ , t ∗ k , t ∗ l , ) . In the next theorem, based on Assumption 1,we will show that such a Nash equilibrium is equivalent to a Nash equilibrium where m − t ∗ and n − m + s ( t ∗ ) .Now we assume Assumption 1.
At the equilibrium where m players choose t ∗ and n − m players choose s ( t ∗ ) ,the responses of u k and u l to a small change in t i have the same sign.u k is the payoff of each player, other than i , whose strategic variable is t k , and u l is thepayoff of each player whose strategic variable is s l .When t i = t ∗ and s l = s ( t ∗ ) for i ∈ M , l ∈ N − M , we have t l = t ∗ for all l ∈ N − M . u k , k ∈ M \ i and u l , l ∈ N − M respond to a change in t i , i ∈ M given t k , k ∈ M \ i and s l , l ∈ N − M . Since s k , k ∈ M \ i and t l , l ∈ N − M are not constant, theresponses of u k , k ∈ M \ i and the responses of u l , l ∈ N − M to a change in t i , i ∈ M may be different. However, because all t i ’s are equal and all u i ’s for i ∈ N are equal at the equilibrium, we may assume that the responses of u k , i ∈ M \ i and the responses of u l , l ∈ N − M to a change in t i , i ∈ M have the same sign ina sufficiently small neighborhood of the equilibrium.Using this assumption we show the following result. Theorem 2.
The equilibrium where m, ≤ m ≤ n − , players choose t i ’s and n − m playerschoose s i ’s as their strategic variables is equivalent to the equilibrium where m − playerschoose t i ’s and n − m + players choose s i ’s as their strategic variables.Proof. Suppose that Player i chooses t i in both equilibria, but Player j chooses t j when m players choose t i ’s and he chooses s j when m − t i ’s. Then,arg max t i ∈ T u i ( t i , t ∗ , t ∗ k , t ∗ l ) = arg max t j ∈ T u j ( t ∗ , t j , t ∗ k , t ∗ l ) = t ∗ . t j is realized by appropriately choosing s j , we getmax s j ∈ S u j ( t ∗ , t j ( t ∗ , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = max t j ∈ T u j ( t ∗ , t j , t ∗ k , t ∗ l ) = u j ( t ∗ , t ∗ , t ∗ k , t ∗ l ) , and arg max s j ∈ S u j ( t ∗ , t j ( t ∗ , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = s ( t ∗ ) . (14)Since the game is zero-sum, u i ( t i , t ∗ , t ∗ k , t ∗ l ) + u j ( t i , t ∗ , t ∗ k , t ∗ l ) + ( m − ) u k ( t i , t ∗ , t ∗ k , t ∗ l ) + ( n − m ) u l ( t i , t ∗ , t ∗ k , t ∗ l ) = , and so u i ( t i , t ∗ , t ∗ k , t ∗ l ) = − [ u j ( t i , t ∗ , t ∗ k , t ∗ l ) + ( m − ) u k ( t i , t ∗ , t ∗ k , t ∗ l ) + ( n − m ) u l ( t i , t ∗ , t ∗ k , t ∗ l )] , u k denotes the payoff of each player who chooses t k as its strategic variable. Player j is oneof such players. u l denotes the payoff of each player who chooses s l as its strategic variable.Then, we obtain u i ( t i , t ∗ , t ∗ k , t ∗ l ) = − [( m − ) u j ( t i , t ∗ , t ∗ k , t ∗ l ) + ( n − m ) u l ( t i , t ∗ , t ∗ k , t ∗ l )] . Thus, max t i ∈ T u i ( t i , t ∗ , t ∗ k , t ∗ l ) = − min t i ∈ T [( m − ) u j ( t i , t ∗ , t ∗ k , t ∗ l ) + ( n − m ) u l ( t i , t ∗ , t ∗ k , t ∗ l )] . By Assumption 1 since u i ( t i , t ∗ , t ∗ k , t ∗ l ) ≤ u j ( t i , t ∗ , t ∗ k , t ∗ l ) ≥ , u l ( t i , t ∗ , t ∗ k , t ∗ l ) ≥ , in any neighborhood of ( t ∗ , t ∗ , t ∗ k , t ∗ l ) . Thus, we havemin t i ∈ T u j ( t i , t ∗ , t ∗ k , t ∗ l ) = , arg min t i ∈ T u j ( t i , t ∗ , t ∗ k , t ∗ l ) = t ∗ , (15a)min t i ∈ T u l ( t i , t ∗ , t ∗ k , t ∗ l ) = , and arg min t i ∈ T u l ( t i , t ∗ , t ∗ k , t ∗ l ) = t ∗ . (15b)By symmetry min t j ∈ T u i ( t ∗ , t j , t ∗ k , t ∗ l ) = , arg min t j ∈ T u i ( t ∗ , t j , t ∗ k , t ∗ l ) = t ∗ . Thus, max t i ∈ T u i ( t i , t ∗ , t ∗ k , t ∗ l ) = min t j ∈ T u i ( t ∗ , t j , t ∗ k , t ∗ l ) = u i ( t ∗ , t ∗ , t ∗ k , t ∗ l ) = . t j ∈ T max t i ∈ T u i ( t i , t j , t ∗ k , t ∗ l ) ≤ max t i ∈ T u i ( t i , t ∗ , t ∗ k , t ∗ l ) = min t j ∈ T u i ( t ∗ , t j , t ∗ k , t ∗ l ) ≤ max t i ∈ T min t j ∈ T u i ( t i , t j , t ∗ k , t ∗ l ) . From Lemma 3min t j ∈ T max t i ∈ T u i ( t i , t j , t ∗ k , t ∗ l ) = max t i ∈ T u i ( t i , t ∗ , t ∗ k , t ∗ l ) = min t j ∈ T u i ( t ∗ , t j , t ∗ k , t ∗ l ) (16) = max t i ∈ T min t j ∈ T u i ( t i , t j , t ∗ k , t ∗ l ) = min s j ∈ S max t i ∈ T u i ( t i , t j ( t i , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l )= max t i ∈ T min s j ∈ S u i ( t i , t j ( t i , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) . Since any value of t j is realized by appropriately choosing s j given s i = s ( t ∗ ) for all i = n ,min t j ∈ T u i ( t ∗ , t j , t ∗ k , t ∗ l ) = min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = . (17)Thus, arg min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = s ( t ∗ ) . From (16) and (17)min s j ∈ T max t i ∈ T u i ( t ∗ , t j ( t ∗ , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = . (18)And we have max t i ∈ T u i ( t i , t j ( t i , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) ≥ u i ( t i , t j ( t i , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) . Then,arg min s j ∈ S max t i ∈ T u i ( t i , t j ( t i , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = arg min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ k , t ∗ l )) , t ∗ k , t ∗ l ) = s ( t ∗ ) . By (18) we getmin s j ∈ T max t i ∈ T u i ( t i , t j ( t i , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = max t i ∈ T u i ( t i , t j ( t i , s ( t ∗ ) , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l )= min s j ∈ S u i ( t ∗ , t j ( t ∗ , s j , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = u i ( t ∗ , t j ( t ∗ , s ( t ∗ ) , t ∗ k , t ∗ l )) , t ∗ k , t ∗ l ) = . Therefore, arg max t i ∈ T u i ( t i , t j ( t i , s ( t ∗ ) , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) = t ∗ . (19)This holds for all i ∈ N , i = j .From (14) and (19) ( t ∗ , t j ( t ∗ , s ( t ∗ ) , t ∗ k , t ∗ l ) , t ∗ k , t ∗ l ) is a Nash equilibrium which is equivalentto ( t ∗ , t ∗ , t ∗ k , t ∗ l ) , and hence it is equivalent to ( t ∗ , . . . , t ∗ ) . Note that i and j are arbitrary.11y mathematical induction this theorem means that the Nash equilibrium where one playerchooses t i and n − s i ’s as their strategic variables is equivalent to the Nashequilibrium where all players choose t i ’s as their strategic variables. Suppose that in the for-mer equilibrium only Player n chooses t n and the other players choose s i ’s as their strategicvariables. Then, this equilibrium is denoted by ( s ( t ∗ ) , . . . , s ( t ∗ ) , t ∗ ) , and soarg max s i ∈ S u i ( t i ( s i , t n , s ( t ∗ ) , . . . , s ( t ∗ )) , t n , s ( t ∗ ) , . . . , s ( t ∗ )) = s ( t ∗ ) , for i = n , arg max t n ∈ T u n ( t i ( s i , t n , s ( t ∗ ) , . . . , s ( t ∗ )) , t n , s ( t ∗ ) , . . . , s ( t ∗ )) = t ∗ . Since any value of t n is realized by appropriately choosing s n ,max t n ∈ T u n ( t i ( s i , t n , s ( t ∗ ) , . . . , s ( t ∗ )) , t n , s ( t ∗ ) , . . . , s ( t ∗ ))= max s n ∈ T u n ( t i ( s i , s n , s ( t ∗ ) , . . . , s ( t ∗ )) , t n ( s i , s n , s ( t ∗ ) , . . . , s ( t ∗ )) , s ( t ∗ ) , . . . , s ( t ∗ )) , and arg max s n ∈ T u n ( t i ( s i , s n , t ∗ l ) , t n ( s i , s n , t ∗ l ) , t ∗ l ) = s ( t ∗ ) . Then, ( t i ( s i , s n , t ∗ l ) , t n ( s i , s n , t ∗ l ) , t ∗ l ) is a Nash equilibrium, in which all players choose s ( t ∗ ) . Itis equivalent to ( t ∗ , . . . , t ∗ ) .Summarizing the results we have shown Theorem 3.
Nash equilibria in the following states are equivalent.1. All players choose t i , i ∈ { , . . . , n } (as their strategic variables).2. Some players choose t i ’s and the other players choose s i ’s.3. All players choose s i , i ∈ { , . . . , n } . Consider a relative profit maximization game in an oligopoly with three firms producing differ-entiated goods . It is an example of multi-players zero-sum game with two strategic variables.The firms are A, B and C. The strategic variables are the outputs and the prices of the goodsof the firms.We consider the following four cases. About relative profit maximization 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)
12. Case 1: All firms determine their outputs.The inverse demand functions are p A = a − x A − bx B − bx C , p B = a − x B − bx A − bx C , and p C = a − x C − bx A − bx B , where 0 < b < p A , p B and p C are the prices of the goods of Firm A, B and C, and x A , x B and x C are the outputs of them.2. Case 2: Firms A and B determine their outputs, and Firm C determines the price of itsgood.From the inverse demand functions, p A = ( − b ) a + b x B − bx B + b x A − x A + bp C , p B = ( − b ) a + b x B − x B + b x A − bx A + bp C , and x C = a − bx B − bx A − p C are derived.3. Case 3: Firms B and C determine the prices of their goods, and Firm A determines itsoutput.Also, from the above inverse demand functions, we obtain p A = ( − b ) a + b x A − bx A − x A + bp C + bp B + b , x B = ( − b ) a + b x A − bx A + bp C − p B ( − b )( + b ) , and x C = ( − b ) a + b x A − bx A − p C + bp B ( − b )( + b .
4. Case 4: All firms determine the prices of their goods.From the inverse demand functions the direct demand functions are derived as follows; x A = ( − b ) a − ( + b ) p A + b ( p A + p C )( − b )( + b ) , x B = ( − b ) a − ( + b ) p B + b ( p B + p C )( − b )( + b ) , and x C = ( − b ) a − ( + b ) p C + b ( p A + p B )( − b )( + b ) . π A = p A x A − c A x A , π B = p B x B − c B x B , and π C = p C x C − c C x C . c A , c B and c C are the constant marginal costs of Firm A, B and C. The relative profits of thefirms are ϕ A = π A − π B + π C , ϕ B = π B − π A + π C , and ϕ C = π C − π A + π B . The firms determine the values of their strategic variables to maximize the relative profits. Wesee ϕ A + ϕ B + ϕ C = , so the game is zero-sum.We compare the equilibrium prices of the good of Firm B in four cases. Denote the valueof p B in each case by p B , p B , p B and p B . Then, we get p B = bc C − b c B + bc B + c B + bc A + ab − ab + a ( − b )( b + ) , p B = A ( − b )( b + )( b + ) , p B = B ( b + )( b + )( b + ) , and p B = b c C + bc C + b c B + bc B + c B + b c A + bc A − ab + ab + a ( b + )( b + ) , where A = b c C + bc C − b c B + b c B + bc B + c B − b c A + b c A + bc A + ab − ab − ab + a , and B = b c C + b c C + bc C + b c B + b c B + bc B + c B + b c A + b c A + bc A − ab − ab + ab + a . c C = c A , they are p B = bc B − b c B + c B + bc A + ab − ab + a ( − b )( b + ) , p B = b c B − b c B + bc B + c B − b c A + b c A + bc A + ab − ab − ab + a ( − b )( b + )( b + ) , p B = b c B + b c B + bc B + c B + b c A + b c A + bc A − ab − ab + ab + a ( b + )( b + )( b + ) , and p B = b c B + bc B + c B + b c A + bc A − ab + ab + a ( b + )( b + ) . Further when c C = c B = c A , we get p B = p B = p B = p B = bc A + c A − ab + ab + . We can show the same result for the equilibrium prices of the goods of the other firms. Thus,in a fully symmetric game the four cases are equivalent.It can be verified that this example with c A = c B = c C satisfies Assumption 1 in the sensethat the argmin (argument of the minimum) of the relative profit of Firm B with respectto the strategy of Firm A is equal to that of Firm C with the Nash equilibriumstrategies of Firms B and C in Case 2 and Case 3. See (15a) and (15b). In this paper we have shown that in a symmetric multi-players zero-sum game with two strate-gic variables, choice of strategic variables is irrelevant to the Nash equilibrium. In an asym-metric situation the Nash equilibrium depends on the choice of strategic variables by playersother than two-players case . Acknowledgment
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