arxiv:10607253v1 [econem] 17 Jun 201 On the relation between Sion s i theorem and existence of Nash equilibrium in asymmetric multi-players zero-sum game with only one alien Atsuhiro Satoh Faculty of Economics Hokkai-Gakuen University Toyohira-ku Sapporo Hokkaido 062-605 Japan and Yasuhito Tanaka Faculty of Economics Doshisha University Kamigyo-ku Kyoto 602-50 Japan Abstract We consider the relation between Sion s i theorem for a continuous function and a Nash equilibrium in an asymmetric multi-players zero-sum game in which only one player is different from other players and the game is symmetric for the other players Then 1 The existence of a Nash equilibrium which is symmetric for players other than one player implies Sion s i theorem for pairs of this player and one of other players with symmetry for the other players 2 Sion s i theorem for pairs of one player and one of other players with symmetry for the other players implies the existence of a Nash equilibrium which is symmetric for the other players Thus they are equivalent This work was supported by Japan Society for the Promotion of Science KAKENHI Grant Number 15K0341 and 1K01594 atsatoh@hgujp yasuhito@maildoshishaacjp
Keywords: multi-players zero-sum game one alien Nash equilibrium Sion s i theorem JEL Classification: C72 1 Introduction We consider the relation between Sion s i theorem for a continuous function and a Nash equilibrium in an asymmetric multi-players zero-sum game in which only one player is different from other players and the game is symmetric for the other players We will show the following results 1 The existence of a Nash equilibrium which is symmetric for players other than one player implies Sion s i theorem for pairs of this player and one of other players with symmetry for the other players 2 Sion s i theorem for pairs of one player and one of other players with symmetry for the other players implies the existence of a Nash equilibrium which is symmetric for the other players Thus they are equivalent Symmetry for the other players means that those players (players other than one player) have the same payoff function and strategy space and so their equilibrium strategies i strategies and i strategies are the same An example of such a game is a relative profit imization game in a Cournot oligopoly Suppose that there are four firms A B C and D in an oligopolistic industry Let π A π B π C and π D be the absolute profits of the firms Then their relative profits are We see π A = π A 1 3 ( π B+ π C + π D ) π B = π B 1 3 ( π A+ π C + π D ) π C = π C 1 3 ( π A+ π B + π D ) π D = π D 1 3 ( π A+ π B + π C ) π A + π B + π C + π D = π A + π B + π C + π D ( π A + π B + π C + π C )=0 Thus the relative profit imization game in a Cournot oligopoly is a zero-sum game 1 If the oligopoly is fully asymmetric because the demand function is not symmetric (in a case of differentiated goods) or firms have different cost functions (in both homogeneous and differentiated goods cases) i strategies and i strategies of firms do not correspond to Nash equilibrium strategies However if the oligopoly is symmetric for three firms in the sense that the demand function is symmetric and those firms have the same cost function the i strategies of those firms with the corresponding i strategy of one firm (for the other players) constitute a Nash equilibrium which is symmetric for the three firms In Appendix we present an example of a four-firms relative profit imizing oligopoly We see from this example that with two aliens the equivalence result does not hold 1 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
2 The model and Sion s i theorem Consider a multi-players zero-sum game with only one alien There are n players i=1n n 3 The strategic variables for the players are s 1 s 2 s n and (s 1 s 2 s n ) S 1 S 2 S n S 1 S 2 S n are convex and compact sets in linear topological spaces The payoff function of each player is u i (s 1 s 2 s n )i=12n We assume u i s for i = 12n are continuous real-valued functions on S 1 S 2 S n quasi-concave on S i for each s j S j j i and quasi-convex on S j for j i for each s i S i n players are partitioned into two groups Group 1 and Group n Group 1 includes n 1 players Players 1 2 n 1 and Group n includes only Player n In Group 1 n 1 players are symmetric in the sense that they have the same payoff function and strategy space Thus their equilibrium strategies i strategies and i strategies are the same Only Player n has a different payoff function and a strategy space Its equilibrium strategy may be different from those for the other players Since the game is a zero-sum game we have u 1 (s 1 s 2 s n )+u 2 (s 1 s 2 s n )+u n (s 1 s 2 s n )=0 (1) for given(s 1 s 2 s n ) Sion s i theorem (Sion (195) Komiya (19) Kindler (2005)) for a continuous function is stated as follows Lemma 1 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 continuous and quasi-concave in the first variable and continuous and quasi-convex in the second variable Then x X y Y f(xy)= y Y x X f(xy) We follow the description of this theorem in Kindler (2005) Let s j s for j in; i j {12n 1} be given Then u i (s 1 s 2 s n ) is a function of s i and s n We can apply Lemma 1 to such a situation and get the following equation Note that we do not require u i (s 1 s 2 s n )= u i (s 1 s 2 s n ) (2) nor u n (s 1 s 2 s n )= u n (s 1 s 2 s n ) s j S j u i (s 1 s 2 s n )= s j S j u i (s 1 s 2 s n ) j i; i j {12n 1} 3
We assume that arg si S i sn S n u i (s 1 s 2 s n ) and arg sn S n si S i u i (s 1 s 2 s n ) are unique that is single-valued By the imum theorem they are continuous in s j s j i n Also throughout this paper we assume that the i strategy and the i strategy of players in any situation are unique and the best responses of players in any situation are unique Let s j = s for all j i j {12n 1} Consider the following function s arg u i (ss i ss n ) Since u i is continuous S i and S n are compact and all S i s are the same this function is also continuous with respect to s Thus there exists a fixed point Denote it by s s satisfies From (2) we have arg u i ( ss i ss n )= s (3) u i ( ss i ss n )= u i ( ss i ss n ) (4) From symmetry for Players 1 2 n 1 s satisfies (3) and (4) for all i {12n 1} 3 The main results Consider a Nash equilibrium of an n-players zero-sum game Let s i s i {12n 1} and s n be the values of s i s which respectively imize u i s Then They mean and u i (s 1 s i s n ) u i(s 1 s is n ) for all s i S i i {12n 1} u n (s 1s 2s n 1s n) u n (s 1s 2s n 1s n ) for all s n S n arg u i (s 1 s is n )=s i i {12n 1} arg u n (s 1 s i s n)=s n We assume that the Nash equilibrium is symmetric in Group 1 that is it is symmetric for Players 1 2 n 1 Then s i s are the same and u i(s 1 s i s n) s are equal for all i {12n 1} Also we have u i (s 1s is js n )=u j (s 1s is js n ) j i; i j {12n 1} Since the game is zero-sum n 1 u i (s 1 s i s j s n)=(n 1)u i (s 1 s i s j s n)= u n (s 1 s i s j s n) i=1 4
Thus This implies arg u i (s 1 s i s j s n)=arg u n (s 1 s i s j s n)=s n u i (s 1 s n S s i s j s n)=u i (s 1 s i s j s n ) n = u i (s 1 s is j s n ) First we show the following theorem Theorem 1 The existence of a Nash equilibrium which is symmetric in Group 1 implies Sion s i theorem for pairs of a player in Group 1 and Player n with symmetry in Group 1 Proof 1 Let(s 1 s 2 s n ) be a Nash equilibrium of a multi-players zero-sum game This means u i (s 1s i s n ) u i (s 1s i s n) (5) = u i (s 1s is n ) u i (s 1s i s n ) for Player i i {12n 1} On the other hand since we have u i (s 1 s is n ) u i (s 1 s is n ) u i (s 1 s is n ) u i (s 1 s is n ) This inequality holds for any s n Thus With (5) we obtain (5) and (6) imply u i (s 1s i s n ) u i (s 1s i s n ) u i (s 1 s is n )= u i (s 1 s is n ) (6) u i (s 1s i s n )=u i (s 1s i s n) u i (s 1 s is n )= u i (s 1 s n S s i s n) n 5
From and we have u i (s 1s i s n ) u i (s 1s i s n) u i (s 1 s is n )=u i (s 1 s is n ) arg u i (s 1 s is n )=arg u i (s 1 s is n)=s i for all i {12n 1} s i s are equal for all i {12n 1} Also from and we get u i (s 1 s is n ) u i (s 1 s i S s i s n) i u i (s 1s i s n )= u i (s 1s is n ) arg u i (s 1 s is n )=arg u i (s 1 s i s n)=s n for all i {12n 1} Next we show the following theorem Theorem 2 Sion s i theorem with symmetry in Group 1 implies the existence of a Nash equilibrium which is symmetric in Group 1 Proof Let s be a value of s j s j i j {12n 1} such that Then we have s=arg u i ( ss i ss n ) u i ( ss i ss n )= u i ( s s ss n ) (7) = u i ( ss i ss n ) Since and u i ( s s ss n ) u i ( ss i ss n ) u i ( s s ss n )= u i ( ss i ss n ) 6
we get arg u i ( s s ss n )=arg u i ( ss i ss n ) () Since the game is zero-sum n 1 u i ( s s ss n )=(n 1)u i ( s s ss n )= u n ( s s ss n ) i=1 Therefore Let arg u i ( s s ss n )=arg u n ( s s ss n ) ŝ n = arg u i ( s s ss n )=arg u n ( s s ss n ) (9) Then from (7) and () Thus u i ( ss i ss n )=u i ( ss i sŝ n ) = u i ( s s ss n )=u i ( s s sŝ n ) arg u i ( ss i sŝ n )= s for all i {12n 1} (10) (9) and (10) mean that (s 1 s 2 s n 1 s n ) = ( s s sŝ n ) is a Nash equilibrium in which only Player n may choose a different strategy 4 Example of relative profit imizing four-firms oligopoly Consider a four-players game Suppose that the payoff functions of the players are π A =(a x A x B x C )x A c A x A 1 3 [(a x A x B x C )x B c B x B +(a x A x B x C )x C c C x C +(a x A x B x C ) c D ] π B =(a x A x B x C )x B c B x B 1 3 [(a x A x B x C )x A c A x A +(a x A x B x C )x C c C x C +(a x A x B x C ) c D ] π C =(a x A x B x C )x C c C x C 1 3 [(a x A x B x C )x A c A x A +(a x A x B x C )x B c B x B +(a x A x B x C ) c D ] 7
π D =(a x A x B x C ) c D 1 3 [(a x A x B x C )x A c A x A +(a x A x B x C )x B c B x B +(a x A x B x C )x C c C x C ] This is a model of relative profit imization in a four firms Cournot oligopoly with constant marginal costs and zero fixed cost producing a homogeneous good x i i=abcd are the outputs of the firms The conditions for imization of π i i=abcd are π A x A = a 2x A (x B + x C + ) c A + 1 3 (x B+ x C + )=0 π B = a 2x B (x A + x C + ) c B + 1 x B 3 (x A+ x C + )=0 π C = a 2x C (x A + x B + ) c C + 1 x C 3 (x A+ x B + )=0 π D = a 2x C (x A + x B + x C ) c D + 1 3 (x A+ x B + x C )=0 The Nash equilibrium strategies are x A = 2a 5c A+c B +c C +c D x B = 2a 5c B+c A +c C +c D x C = 2a 5c C+c A +c B +c D = 2a 5c D+c A +c B +c C Next consider i and i strategies about Player A and Player D The condition for imization of π A with respect to is π A = 0 Denote which satisfies this condition by (x A x B x C ) and substitute it into π A Then the condition for imization of π A with respect to x A given (x A x B x C ) x B and x C is π A x A + π A x A = 0 It is denoted by arg xa xd π A The condition for imization of π A with respect to x A is π A x = 0 Denote x A A which satisfies this condition by x A (x B x C ) and substitute it into π A Then the condition for imization of π A with respect to given x A (x B x C ) is π A + π A x A x A = 0 It is denoted by arg xd xa π A In our example we obtain arg x A π A = 2a 3c A+ c D Similarly we get the following results arg x B arg x A π A = 6a 3c A 3c D x B x C π B = 2a 3c B+ c D (11)
arg x B arg x C π B = 6a 3c B 3c D x A x C π C = 2a 3c C+ c D If c C = c B = c A arg x C π C = 6a 3c C 3c D x A x B arg x A π A = arg x B π B = arg x C π C = 2a 3c A+ c D These are equal to the Nash equilibrium strategies for Firms A B and C with c C = c B = c A and c D c A When x A = x B = x C = 2a 3c A+c D we have arg x A π A = arg x B π B = arg x C π C = 2a 5c D+ 3c A These are equal to the Nash equilibrium strategy for Firm D with c C = c B = c A On the other hand if c B = c A and c C = c D we have arg x A π A = arg x B π B = 2a 3c A+ c D This is not equal to the Nash equilibrium strategies for Firms A and B with c B = c A and c C = c D c A which are x A = x B = a 2c A+ c D 4 2a 3c A+ c D 5 Concluding Remark In this paper we have exaed the relation between Sion s i theorem for a continuous function and a Nash equilibrium in an asymmetric multi-players zero-sum game in which only one player is different from other players We have shown that the following two statements are equivalent 1 The existence of a Nash equilibrium which is symmetric for players other than one player implies Sion s i theorem for pairs of this player and one of other players with symmetry for the other players 2 Sion s i theorem for pairs of one player and one of other players with symmetry for the other players implies the existence of a Nash equilibrium which is symmetric for the other players As we have shown in Appendix if there are two aliens this equivalence does not hold 9
References Kindler J (2005) A simple proof of Sion s i theorem American Mathematical Monthly 112 pp 356-35 Komiya H (19) Elementary proof for Sion s i theorem Kodai Mathematical Journal 11 pp 5-7 Matsumura T N Matsushima and S Cato (2013) Competitiveness and R&D competition revisited Economic Modelling 31 pp 541-547 Satoh A and Y Tanaka (2013) Relative profit imization and Bertrand equilibrium with quadratic cost functions Economics and Business Letters 2 pp 134-139 2013 Satoh A and Y Tanaka (2014a) Relative profit imization and equivalence of Cournot and Bertrand equilibria in asymmetric duopoly Economics Bulletin 34 pp 19-27 2014 Satoh A and Y Tanaka (2014b) Relative profit imization in asymmetric oligopoly Economics Bulletin 34 pp 1653-1664 Sion M (195) On general i theorems Pacific Journal of Mathematics pp 171-176 Tanaka Y (2013a) Equivalence of Cournot and Bertrand equilibria in differentiated duopoly under relative profit imization with linear demand Economics Bulletin 33 pp 1479-146 Tanaka Y (2013b) Irrelevance of the choice of strategic variables in duopoly under relative profit imization Economics and Business Letters 2 pp 75-3 2013 Vega-Redondo F (1997) The evolution of Walrasian behavior Econometrica 65 pp 375-34 10