Sion’smini-maxtheoremandNashequilibriuminamulti-playersgamewithtwogroupswhichiszero-sumandsymmetricineachgroup Satoh,AtsuhiroandTanaka,Yasuhito MunichPersonalRePEcArchive

Munich Personal RePEc Archive

Sion’s mini-max theorem and Nash

equilibrium in a multi-players game with two groups which is zero-sum and

symmetric in each group

Satoh, Atsuhiro and Tanaka, Yasuhito

13 September 2018

Online at https://mpra.ub.uni-muenchen.de/88977/

MPRA Paper No. 88977, posted 15 Sep 2018 07:20 UTC

Sion’s mini-max theorem and Nash equilibrium in a multi-players game with

two groups which is zero-sum and symmetric in each group ^∗

Atsuhiro Satoh

Faculty of Economics, Hokkai-Gakuen University, Toyohira-ku, Sapporo, Hokkaido, 062-8605, Japan,

and

Yasuhito Tanaka

Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto, 602-8580, Japan.

Abstract

We consider the relation between Sion’s minimax theorem for a continuous function and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group. We will show the following results.

1. The existence of Nash equilibrium which is symmetric in each group implies a modified version of Sion’s minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group.

2. A modified version of Sion’s minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group implies the existence of Nash equilibrium which is symmetric in each group.

Thus, they are equivalent. An example of such a game is a relative profit maximization game in each group under oligopoly with two groups such that firms in each group have the same cost functions and maximize their relative profits in each group, and the demand functions are symmetric for the firms in each group.

Keywords: multi-players zero-sum game, two groups, Nash equilibrium, Sion’s minimax theorem JEL Classification: C72

∗This work was supported by Japan Society for the Promotion of Science KAKENHI Grant Number 15K03481 and 18K01594.

†atsatoh@hgu.jp

‡yasuhito@mail.doshisha.ac.jp

1 Introduction

We consider the relation between Sion’s minimax theorem for a continuous function and the existence of Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group. There are𝑛players. Players1,2,. . . ,𝑚are in one group, and Players𝑚+1,𝑚+2,. . . ,𝑛are in the other group. We assume𝑛 ≥ 4and2 ≤ 𝑚 ≤ 𝑛−2. Thus, each group has at least two players.

Players1,2,. . . ,𝑚have the same payoff functions and strategy spaces, and they play a game which is zero-sum in this group, that is, the sum of the payoffs of Players1,2,. . . ,𝑚is zero. Similarly, Players

𝑚+1,𝑚+2,. . . ,𝑛have the same payoff functions and strategy spaces, and they play a game which is

zero-sum in this group, that is, the sum of the payoffs of Players𝑚+1,𝑚+2,. . . ,𝑛is zero.

We will show the following results.

1. The existence of Nash equilibrium which is symmetric in each group implies a modified version of Sion’s minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group.

2. A modified version of Sion’s minimax theorem for players with the coincidence of the maximin strategy and the minimax strategy in each group implies the existence of Nash equilibrium which is symmetric in each group.

Thus, they are equivalent.

An example of such a game is a relative profit maximization game in each group under oligopoly with two groups such that firms in each group have the same cost functions and maximize their relative profits in each group, and demand functions are symmetric for the firms in each group. Assume that there are six firms, A, B, C, D, E and F. Let𝜋̄_𝐴, 𝜋̄_𝐵,𝜋̄_𝐶, 𝜋̄_𝐷, 𝜋̄_𝐸 and𝜋̄_𝐹 be the absolute profits of, respectively, Firms A, B, C, D, E and F. Firms A, B and E have the same cost function, and the demand functions are symmetric for them. Firms C, D and F have the same cost function, and the demand functions are symmetric for them. However, the firms in different groups have different cost functions, and the demand functions are not symmetric for firms in different groups.

The relative profits of Firms A, B and E are 𝜋𝐴 =𝜋̄𝐴− 1

2(𝜋̄𝐵+𝜋̄𝐸), 𝜋𝐵 =𝜋̄𝐵− 1

2(𝜋̄𝐴+𝜋̄𝐸), 𝜋_𝐸 =𝜋̄_𝐸− 1

2(𝜋̄_𝐴+𝜋̄_𝐵).

The relative profits of Firms C, D and F are

𝜋_𝐶 =𝜋̄_𝐶− 1

2(𝜋̄_𝐷+𝜋̄_𝐹), 𝜋𝐷 =𝜋̄𝐷− 1

2(𝜋̄𝐶+𝜋̄𝐹), 𝜋_𝐹 =𝜋̄_𝐹− 1

2(𝜋̄_𝐶+𝜋̄_𝐷).

We see

𝜋_𝐴+𝜋_𝐵+𝜋_𝐸 =0,

𝜋𝐶+𝜋𝐷+𝜋𝐹 =0.

Firms A, B, C, D, E and F maximize, respectively,𝜋_𝐴,𝜋_𝐵,𝜋_𝐶,𝜋_𝐷,𝜋_𝐸and𝜋_𝐹. Thus, the relative profit maximization game in each group is a zero-sum game¹. In Section 4 we present an example of relative profit maximization in each group under oligopoly with two groups.

2 The model and Sion’s minimax theorem

Consider a multi-players game with two groups which is zero-sum and symmetric in each group. Our analysis can be easily extended to a case with more than two groups. However, since notation is very complicated, we will present arguments of a two groups case. There are𝑛players. Players1,2,. . . ,𝑚 are in one group, and Players𝑚+1,𝑚+2,. . . ,𝑛 are in the other group. We assume 𝑛 ≥ 4 and 2 ≤ 𝑚 ≤ 𝑛−2. Thus, each group has at least two players. Players1,2,. . . ,𝑚have the same payoff functions and strategy spaces, and they play a game which is zero-sum in this group, that is, the sum of the payoffs of Players1,2,. . . ,𝑚is zero. Similarly, Players𝑚+1,𝑚+2,. . . ,𝑛have the same payoff functions and strategy spaces, and they play a game which is zero-sum in this group, that is, the sum of the payoffs of Players𝑚+1,𝑚+2,. . . ,𝑛is zero. The strategic variables for the players are𝑠1,𝑠2, . . . ,𝑠𝑛, and (𝑠₁,𝑠₂,. . . ,𝑠𝑛) ∈𝑆₁×𝑆₂× · · · ×𝑆𝑛. 𝑆₁,𝑆₂, . . . , 𝑆𝑛are convex and compact sets in linear topological spaces.

The payoff function of each player is𝑢_𝑖(𝑠₁,𝑠₂,. . . ,𝑠_𝑛),𝑖 =1,2,. . . ,𝑛. We assume

𝑢𝑖’s for𝑖 =1,2,. . . ,𝑛are continuous real-valued functions on𝑆1×𝑆2× · · · ×𝑆𝑛, quasi- concave on𝑆𝑖for each𝑠𝑗 ∈𝑆𝑗, 𝑗<𝑖, and quasi-convex on𝑆𝑗 for𝑗<𝑖for each𝑠𝑖 ∈𝑆𝑖. Since the game is zero-sum in each group, we have

𝑢₁(𝑠₁,𝑠₂,. . . ,𝑠_𝑛)+𝑢₂(𝑠₁,𝑠₂,. . . ,𝑠_𝑛)+. . . ,𝑢_𝑚(𝑠₁,𝑠₂,. . . ,𝑠_𝑛)=0, (1) 𝑢𝑚+1(𝑠₁,𝑠₂,. . . ,𝑠𝑛)+𝑢𝑚+2(𝑠₁,𝑠₂,. . . ,𝑠𝑛)+. . . ,𝑢𝑛(𝑠₁,𝑠₂,. . . ,𝑠𝑛)=0, (2) for given(𝑠₁,𝑠₂,. . . ,𝑠_𝑛).

Sion’s minimax theorem (Sion (1958), Komiya (1988), Kindler (2005)) for a continuous function is stated as follows.

Lemma 1. Let𝑋 and 𝑌 be non-void convex and compact subsets of two linear topological spaces, and let𝑓 :𝑋 ×𝑌 → ℝbe a function, that is continuous and quasi-concave in the first variable and continuous and quasi-convex in the second variable. Then

max𝑥∈𝑋 min

𝑦∈𝑌𝑓(𝑥,𝑦)=min

𝑦∈𝑌max

𝑥∈𝑋 𝑓(𝑥,𝑦).

We follow the description of this theorem in Kindler (2005).

Let𝑠ℎ’s forℎ<𝑖,𝑗; 𝑖,𝑗 ∈ {1,2,. . . ,𝑚}be given; then,𝑢𝑖(𝑠₁,𝑠₂,. . . ,𝑠𝑛)is a function of𝑠𝑖 and𝑠𝑗. We can apply Lemma 1 to such a situation, and get the following equation.

max𝑠_𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠₁,𝑠₂,. . . ,𝑠_𝑛)= min

𝑠_𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖 𝑢_𝑖(𝑠₁,𝑠₂,. . . ,𝑠_𝑛). (3)

1About 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)

By symmetry

𝑠max_𝑗∈𝑆_𝑗min

𝑠_𝑖∈𝑆_𝑖𝑢𝑗(𝑠₁,𝑠₂,. . . ,𝑠𝑛)=min

𝑠_𝑖∈𝑆_𝑖max

𝑠_𝑗∈𝑆_𝑗𝑢𝑗(𝑠₁,𝑠₂,. . . ,𝑠𝑛).

Similarly, let𝑠ℎ’s forℎ<𝑘,𝑙; 𝑘,𝑙∈ {𝑚+1,𝑚+2,. . . ,𝑛}be given; then we obtain

𝑠max_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠₁,𝑠₂,. . . ,𝑠𝑛)=min

𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢𝑖(𝑠₁,𝑠₂,. . . ,𝑠𝑛). (4)

By symmetry

max𝑠_𝑙∈𝑆_𝑙 min

𝑠_𝑘∈𝑆_𝑘𝑢𝑙(𝑠₁,𝑠2,. . . ,𝑠𝑛)= min

𝑠_𝑘∈𝑆_𝑘max

𝑠_𝑙∈𝑆_𝑙𝑢𝑗(𝑠₁,𝑠2,. . . ,𝑠𝑛).

We assume thatarg max𝑠_𝑖∈𝑆_𝑖min𝑠_𝑗∈𝑆_𝑗𝑢𝑖(𝑠₁,𝑠₂,. . . ,𝑠𝑛),arg min𝑠_𝑗∈𝑆_𝑗max𝑠_𝑖∈𝑆_𝑖𝑢𝑖(𝑠₁,𝑠₂,. . . ,𝑠𝑛)and so on are unique, that is, single-valued. By the maximum theorem they are continuous in𝑠_ℎ’s,ℎ<𝑖,𝑗or in𝑠_ℎ’s,ℎ < 𝑘,𝑙. Also, throughout this paper we assume that the maximin strategy and the minimax strategy of players in any situation are unique, and the best responses of players in any situation are unique.

Let us consider a point such that𝑠_𝑖 =𝑠for𝑖 ∈ {1,2,. . . ,𝑚}and𝑠_𝑘 =𝑠^′for𝑘∈ {𝑚+1,𝑚+2,. . . ,𝑛}, and consider the following function.

(𝑠 𝑠^′

)

→

(arg max_𝑠𝑖∈𝑆_𝑖min_𝑠𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠,. . . ,𝑠,𝑠^′,. . . ,𝑠^′) arg max𝑠_𝑘∈𝑆_𝑘min𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠,. . . ,𝑠,𝑠_𝑘,𝑠_𝑙,𝑠^′,. . . ,𝑠^′)

) ,

for 𝑖 ∈ {1,2,. . . ,𝑚},𝑘 ∈ {𝑚+1,𝑚+2,. . . ,𝑛}. Since𝑢_𝑖 and𝑢_𝑘 are continuous,𝑆_𝑖 =𝑆_𝑗 is compact

and𝑆_𝑘 =𝑆_𝑙 is compact, these functions are also continuous. Thus, there exists a fixed point of(𝑠,𝑠^′).

Denote it by(̃𝑠,𝑠). It satisfieŝ 𝑠̃=arg max

𝑠_𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠,̃ . . . ,̃𝑠,𝑠,̂ . . . ,𝑠),̂ 𝑖 ∈ {1,2,. . . ,𝑚}, (5) 𝑠̂=arg max

𝑠_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠,̃ . . . ,𝑠,̃ 𝑠_𝑘,𝑠_𝑙,𝑠,̂ . . . ,𝑠),̂ 𝑘∈ {𝑚+1,𝑚+2,. . . ,𝑛}. (6) Now we assume

Assumption 1. About𝑠̃and𝑠̂which satisfy (5) and (6), arg max

𝑠_𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠,̃ . . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠)̂ =arg min

𝑠_𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖 𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠,̃ . . . ,̃𝑠,𝑠,̂ . . . ,𝑠),̂ arg max

𝑠_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠,̃ . . . ,𝑠,̃ 𝑠_𝑘,𝑠_𝑙,𝑠,̂ . . . ,𝑠)̂ =arg min

𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠,̃ . . . ,̃𝑠,𝑠_𝑘,𝑠_𝑙,𝑠,̂ . . . ,𝑠),̂

for 𝑖 ∈ {1,2,. . . ,𝑚},𝑘∈ {𝑚+1,𝑚+2,. . . ,𝑛}, that is, the maximin strategy and the minimax strategy coincide.

Based on Assumption 1 we present a modified version of Sion’s minimax theorem.

Lemma 2(Modified version of Sion’s minimax theorem). Let𝑗 <𝑖, 𝑖,𝑗 ∈ {1,2,. . . ,𝑚}, and𝑆_𝑖 and 𝑆𝑗 be non-void convex and compact subsets of two linear topological spaces, and Let𝑙 < 𝑘, 𝑘,𝑙 ∈

{𝑚+1,𝑚+2,. . . ,𝑛}, and𝑆_𝑘and𝑆_𝑙be non-void convex and compact subsets of two linear topological

spaces. Let𝑢_𝑖 :𝑆_𝑖 ×𝑆_𝑗 →ℝgiven the strategies of all other players and𝑢_𝑘 :𝑆_𝑘×𝑆_𝑙 →ℝgiven the strategies of all other players be functions that is continuous on𝑆₁×𝑆₂× · · · ×𝑆𝑛, quasi-concave on 𝑆_𝑖 (or𝑆_𝑘) and quasi-convex on𝑆_𝑗 (or𝑆_𝑙). Then, there exist𝑠̃and𝑠̂which satisfy (3), (4), (5), (6) and Assumption 1.

As we will show in the Appendix, without Assumption 1 we may have a Nash equilibrium which is asymmetric in each group.

3 The main results

Consider a Nash equilibrium which is symmetric in each group. Let𝑠^∗_𝑖’s and𝑠^∗_𝑘’s be the values of𝑠𝑖’s

for𝑖 ∈ {1,2,. . . ,𝑚} and𝑠_𝑘’s for𝑘 ∈ {𝑚+1,𝑚+2,. . . ,𝑛}which, respectively, maximize𝑢_𝑖’s and

𝑢_𝑘’s, that is,

𝑢𝑖(𝑠^∗₁,𝑠^∗₂,. . . ,𝑠^∗_𝑖,. . . ,𝑠^∗_𝑛) ≥𝑢𝑖(𝑠^∗₁,𝑠^∗₂,. . . ,𝑠𝑖,. . . ,𝑠^∗_𝑛)for any𝑠𝑖 ∈𝑆𝑖, and

𝑢_𝑘(𝑠^∗₁,𝑠^∗₂,. . . ,𝑠^∗_𝑘,. . . ,𝑠^∗_𝑛) ≥𝑢_𝑘(𝑠^∗₁,𝑠^∗₂,. . . ,𝑠_𝑘,. . . ,𝑠^∗_𝑛)for any𝑠_𝑘 ∈𝑆_𝑘,

If the Nash equilibrium is symmetric in each group,𝑠^∗₁’s for all𝑖 ∈ {1,2,. . . ,𝑚}are equal, and𝑠^∗_𝑘’s for

all𝑘∈ {𝑚+1,𝑚+2,. . . ,𝑛}are equal.

Notations of strategy choice by players are as follows.

(𝑠_𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)is a vector of strategy choice by players such that Players 1,. . ., 𝑚other than𝑖choose𝑠^∗and Players𝑚+1,. . .,𝑛choose𝑠^∗∗.(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗) is a vector such that Players 1,. . .,𝑚choose𝑠^∗and Players𝑚+1, . . .,𝑛other than𝑘 choose𝑠^∗∗. (𝑠_𝑖,𝑠𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)is a vector such that Players 1,. . .,𝑚other than 𝑖and𝑗choose𝑠^∗and Players𝑚+1,. . .,𝑛choose𝑠^∗∗. (𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗)is a vector such that Players 1,. . .,𝑚choose𝑠^∗and Players𝑚+1,. . .,𝑛other than𝑘and𝑙 choose𝑠^∗∗.

(𝑠_𝑖,𝑠,̃ . . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠)̂ is a vector of strategy choice by players such that Players 1,. . .,𝑚 other than𝑖choose𝑠̃and Players𝑚+1,. . .,𝑛choose𝑠.̂ (𝑠,̃ . . . ,𝑠,̃ 𝑠_𝑘,𝑠,̂ . . . ,𝑠)̂ is a vector such that Players 1, . . ., 𝑚choose 𝑠̃and Players 𝑚+1, . . ., 𝑛 other than𝑘 choose 𝑠.̂ (𝑠_𝑖,𝑠𝑗,𝑠,̃ . . . ,̃𝑠,𝑠,̂ . . . ,𝑠)̂ is a vector such that Players 1,. . .,𝑚other than𝑖and𝑗choose𝑠̃ and Players𝑚+1,. . .,𝑛choose𝑠.̂ (̃𝑠,. . . ,𝑠,̃ 𝑠_𝑘,𝑠_𝑙,𝑠,̂ . . . ,𝑠)̂ is a vector such that Players

1,. . .,𝑚choose𝑠̃and Players𝑚+1,. . .,𝑛other than𝑘and𝑙choose𝑠.̂

The same applies to other similar notations.

We show the following theorem.

Theorem 1. The existence of Nash equilibrium which is symmetric in each group implies Sion’s minimax theorem with the coincidence of the maximin strategy and the minimax strategy.

Proof. Let(𝑠₁,. . . ,𝑠𝑚,𝑠𝑚+1,. . . ,𝑠𝑛) =(𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) be a Nash equilibrium which is sym- metric in each group. Since the game is zero-sum in each group.

𝑢_𝑖(𝑠_𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)+

𝑚

∑

𝑗=1,𝑗<𝑖

𝑢_𝑗(𝑠_𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=0,

and

𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗)+

∑𝑛

𝑙=𝑚+1,𝑘<𝑘

𝑢_𝑙(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗)=0 imply

𝑢𝑖(𝑠𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=−(𝑚−1)𝑢𝑗(𝑠𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗), and

𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗)=−(𝑛−𝑚−1)𝑢_𝑙(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗).

These equations hold for any𝑠𝑖and𝑠𝑘. Therefore, arg max

𝑠_𝑖∈𝑆_𝑖𝑢_𝑖(𝑠_𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=arg min

𝑠_𝑖∈𝑆_𝑖𝑢_𝑗(𝑠_𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗), arg max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗)=arg min

𝑠_𝑘∈𝑆_𝑘𝑢_𝑙(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗).

By the assumption of uniqueness of the best responses, they are unique. By symmetry for each group arg max

𝑠_𝑖∈𝑆_𝑖 𝑢𝑖(𝑠𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=arg min

𝑠𝑗∈𝑆𝑗

𝑢𝑖(𝑠𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗),

arg max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗)=arg min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗).

Therefore,

𝑢_𝑖(𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)= min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) ≤𝑢_𝑖(𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗), 𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗) ≤𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗).

We get

max𝑠_𝑖∈𝑆_𝑖 𝑢_𝑖(𝑠_𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=𝑢_𝑖(𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)= min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗),

𝑠max_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗)=𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗), They mean

𝑠min_𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖 𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) ≤max

𝑠_𝑖∈𝑆_𝑖𝑢_𝑖(𝑠_𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) (7)

=min

𝑠_𝑗∈𝑆_𝑗𝑢𝑖(𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) ≤max

𝑠𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢𝑖(𝑠_𝑖,𝑠𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗).

and

min𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗) ≤ max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗) (8)

=min

𝑠_𝑙∈𝑆_𝑙𝑢𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠𝑙,𝑠^∗∗,. . . ,𝑠^∗∗) ≤ max

𝑠_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠𝑘,𝑠𝑙,𝑠^∗∗,. . . ,𝑠^∗∗).

On the other hand, since

𝑠min_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) ≤𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗),

𝑠min_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗) ≤𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗),

we have

max𝑠_𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) ≤max

𝑠_𝑖∈𝑆_𝑖 𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗),

𝑠max_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗) ≤max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗).

These inequalities hold for any𝑠𝑗 and𝑠𝑙. Thus, max𝑠_𝑖∈𝑆_𝑖 min

𝑠𝑗∈𝑆𝑗

𝑢𝑖(𝑠𝑖,𝑠𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) ≤ min

𝑠𝑗∈𝑆𝑗

max𝑠_𝑖∈𝑆_𝑖𝑢𝑖(𝑠𝑖,𝑠𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗),

𝑠max_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠𝑘,𝑠𝑙,𝑠^∗∗,. . . ,𝑠^∗∗) ≤min

𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠𝑘,𝑠𝑙,𝑠^∗∗,. . . ,𝑠^∗∗), With (7) and (8), we obtain

max𝑠_𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)= min

𝑠_𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖 𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗), (9)

𝑠max_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗)= min

𝑠_𝐷∈𝑆_𝐷 max

𝑠_𝐶∈𝑆_𝐶𝑢_𝐶(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗). (10) From

𝑠min𝑗∈𝑆𝑗

𝑢𝑖(𝑠𝑖,𝑠𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) ≤𝑢𝑖(𝑠𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗), min𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗) ≤𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗), max𝑠_𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢𝑖(𝑠𝑖,𝑠𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=max

𝑠_𝑖∈𝑆_𝑖𝑢𝑖(𝑠𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗), and

𝑠max_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗)= max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗), we have

arg max

𝑠𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=arg max

𝑠𝑖∈𝑆_𝑖 𝑢_𝑖(𝑠_𝑖,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=𝑠^∗, arg max

𝑠_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗)=arg max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠^∗∗,. . . ,𝑠^∗∗)=𝑠^∗∗. From

max𝑠_𝑖∈𝑆_𝑖𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) ≥𝑢_𝑖(𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗),

𝑠max_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗) ≥𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗),

𝑠min_𝑗∈𝑆_𝑗max

𝑠𝑖∈𝑆𝑖

𝑢𝑖(𝑠_𝑖,𝑠𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)= min

𝑠_𝑗∈𝑆_𝑗𝑢𝑖(𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗), and

min𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠𝑘,𝑠𝑙,𝑠^∗∗,. . . ,𝑠^∗∗)=min

𝑠_𝑙∈𝑆_𝑙𝑢𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠𝑙,𝑠^∗∗,. . . ,𝑠^∗∗), we get

arg min

𝑠_𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=arg min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=𝑠^∗, arg min

𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑘,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗)=arg min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠_𝑙,𝑠^∗∗,. . . ,𝑠^∗∗)=𝑠^∗∗. Therefore,

arg max

𝑠_𝑖∈𝑆_𝑖 min

𝑠𝑗∈𝑆𝑗

𝑢𝑖(𝑠𝑖,𝑠𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗) (11)

=arg min

𝑠_𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠^∗,. . . ,𝑠^∗,𝑠^∗∗,. . . ,𝑠^∗∗)=𝑠^∗,

arg max

𝑠_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠𝑘,𝑠𝑙,𝑠^∗∗,. . . ,𝑠^∗∗) (12)

=arg min

𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢𝑘(𝑠^∗,. . . ,𝑠^∗,𝑠𝑘,𝑠𝑙,𝑠^∗∗,. . . ,𝑠^∗∗)=𝑠^∗∗.

□ Next we show the following theorem.

Theorem 2. Sion’s minimax theorem with the coincidence of the maximin strategy and the minimax strategy implies the existence of a Nash equilibrium which is symmetric in each group.

Proof. We denote a state such that Players1,2,. . . ,𝑚choose𝑠, and Players̃ 𝑚+1,𝑚+2,. . . ,𝑛choose

̂

𝑠by(𝑠,̃ . . . ,𝑠,̃ 𝑠̂. . . ,𝑠).̂

Let𝑠̃and𝑠̂be the values of𝑠_𝑖’s for𝑖 ∈ {1,2,. . . ,𝑚}and𝑠_𝑘’s for𝑘∈ {𝑚+1,𝑚+2,. . . ,𝑛}such that

̃

𝑠=arg max

𝑠_𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢𝑖(𝑠𝑖,𝑠𝑗,𝑠,̃ . . . ,̃𝑠,𝑠,̂ . . . ,𝑠)̂ =arg min

𝑠_𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖 𝑢𝑖(𝑠𝑖,𝑠𝑗,̃𝑠,. . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠),̂

̂

𝑠=arg max

𝑠_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢𝑘(𝑠,̃ . . . ,̃𝑠,𝑠𝑘,𝑠𝑙,𝑠,̂ . . . ,𝑠)̂ =arg min

𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢𝑘(̃𝑠,. . . ,𝑠,̃ 𝑠𝑘,𝑠𝑙,𝑠,̂ . . . ,𝑠),̂

max𝑠_𝑖∈𝑆_𝑖 min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,̃𝑠,. . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠)̂ = min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑗,̃𝑠,. . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠)̂

= min

𝑠𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖 𝑢𝑖(𝑠𝑖,𝑠𝑗,̃𝑠,. . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠)̂ =max

𝑠_𝑖∈𝑆_𝑖 𝑢𝑖(𝑠𝑖,𝑠,̃ . . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠),̂

and

𝑠max_𝑘∈𝑆_𝑘min

𝑠_𝑙∈𝑆_𝑙𝑢𝑘(𝑠,̃ . . . ,𝑠,̃ 𝑠𝑘,𝑠𝑙,𝑠,̂ . . . ,𝑠)̂ =min

𝑠_𝑙∈𝑆_𝑙𝑢𝑘(̃𝑠,. . . ,𝑠,̃ 𝑠𝑙,𝑠,̂ . . . ,𝑠)̂

=min

𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢𝑘(𝑠,̃ . . . ,𝑠,̃ 𝑠𝑘,𝑠𝑙,𝑠,̂ . . . ,𝑠)̂ =max

𝑠_𝑘∈𝑆_𝑘𝑢𝑘(𝑠,̃ . . . ,𝑠,̃ 𝑠𝑘,𝑠,̂ . . . ,𝑠).̂ Since

𝑢𝑖(𝑠𝑗,𝑠,̃ . . . ,̃𝑠,𝑠,̂ . . . ,𝑠) ≤̂ max

𝑠_𝑖∈𝑆_𝑖𝑢𝑖(𝑠𝑖,𝑠𝑗,̃𝑠,. . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠),̂

𝑠min_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑗,̃𝑠,. . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠)̂ = min

𝑠_𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖 𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠,̃ . . . ,̃𝑠,𝑠,̂ . . . ,𝑠),̂ we get

arg min

𝑠_𝑗∈𝑆_𝑗𝑢_𝑖(𝑠_𝑗,𝑠,̃ . . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠)̂ =arg min

𝑠_𝑗∈𝑆_𝑗max

𝑠_𝑖∈𝑆_𝑖 𝑢_𝑖(𝑠_𝑖,𝑠_𝑗,𝑠,̃ . . . ,̃𝑠,𝑠,̂ . . . ,𝑠)̂ =𝑠.̃ Similarly, from

𝑢_𝑘(𝑠,̃ . . . ,̃𝑠,𝑠_𝑙,𝑠,̂ . . . ,𝑠) ≤̂ max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠,̃ . . . ,̃𝑠,𝑠_𝑘,𝑠_𝑙,𝑠,̂ . . . ,𝑠),̂ min𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(̃𝑠,. . . ,𝑠,̃ 𝑠_𝑙,𝑠,̂ . . . ,𝑠)̂ =min

𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠,̃ . . . ,𝑠,̃ 𝑠_𝑘,𝑠_𝑙,𝑠,̂ . . . ,𝑠),̂ we get

arg min

𝑠_𝑙∈𝑆_𝑙𝑢_𝑘(𝑠,̃ . . . ,𝑠,̃ 𝑠_𝑙,𝑠,̂ . . . ,𝑠)̂ =arg min

𝑠_𝑙∈𝑆_𝑙max

𝑠_𝑘∈𝑆_𝑘𝑢_𝑘(𝑠,̃ . . . ,𝑠,̃ 𝑠_𝑘,𝑠_𝑙,𝑠,̂ . . . ,𝑠)̂ =𝑠.̂ Since

𝑢𝑖(𝑠_𝑖,̃𝑠,. . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠) ≥̂ min

𝑠_𝑗∈𝑆_𝑗𝑢𝑖(𝑠_𝑖,𝑠𝑗,̃𝑠,. . . ,𝑠,̃ 𝑠,̂ . . . ,𝑠),̂