Munich Personal RePEc Archive
Two person zero-sum game with two sets of strategic variables
Satoh, Atsuhiro and Tanaka, Yasuhito
Faculty of Economics, Doshisha Unuversity
20 August 2016
Online at https://mpra.ub.uni-muenchen.de/73704/
MPRA Paper No. 73704, posted 17 Sep 2016 10:39 UTC
Two person zero-sum game with two sets of strategic variables
Atsuhiro Satoh
*Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto, 602-8580, Japan.
and
Yasuhito Tanaka
†Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto, 602-8580, Japan.
Abstract
We consider a two-person zero-sum game with two sets of strategic variables which are related by invertible functions. They are denoted by (𝑠𝐴, 𝑠𝐵) ∈ (𝑆𝐴, 𝑆𝐵) and (𝑡𝐴, 𝑡𝐵) ∈ (𝑇𝐴, 𝑇𝐵)for players A and B. The payoff function of Player A is𝑢𝐴. Then, the payoff function of Player B is−𝑢𝐴.𝑢𝐴is upper semi-continuous and quasi-concave on𝑆𝐴for each𝑠𝐵 ∈ 𝑆𝐵(or each𝑡𝐵 ∈ 𝑇𝐵), upper semi-continuous and quasi-concave on𝑇𝐴for each 𝑡𝐵 ∈ 𝑇𝐵 (or each𝑠𝐵 ∈ 𝑆𝐵), and lower semi-continuous and quasi- convex on𝑆𝐵 for each𝑠𝐴 ∈ 𝑆𝐴(or each𝑡𝐴∈ 𝑇𝐴), lower semi-continuous and quasi- convex on𝑇𝐵for each𝑡𝐴∈ 𝑇𝐴(or each𝑠𝐴∈ 𝑆𝐴). We do not postulate differentiability of payoff functions.
We will show that the following four patterns of competition are equivalent, that is, they yield the same outcome.
1. Player A and B choose𝑠𝐴and𝑠𝐵 (competition by(𝑠𝐴, 𝑠𝐵)).
2. Player A and B choose𝑡𝐴and𝑡𝐵 (competition by(𝑡𝐴, 𝑡𝐵)).
3. Player A and B choose𝑡𝐴and𝑠𝐵 (competition by(𝑡𝐴, 𝑠𝐵)).
4. Player A and B choose𝑠𝐴and𝑡𝐵 (competition by(𝑠𝐴, 𝑡𝐵)).
Keywords.zero-sum game, two strategic variables, equivalence of outcome.
JEL Classification code. C72, L13.
*atsato@mail.doshisha.ac.jp
†yasuhito@mail.doshisha.ac.jp
1 Introduction
We consider a two-person zero-sum game with two sets of strategic variables which are related by invertible functions. They are denoted by (𝑠𝐴, 𝑠𝐵) ∈ (𝑆𝐴, 𝑆𝐵) and(𝑡𝐴, 𝑡𝐵) ∈ (𝑇𝐴, 𝑇𝐵) for players A and B. The payoff function of Player A is 𝑢𝐴. Then, the payoff function of Player B is−𝑢𝐴. 𝑢𝐴 is upper semi-continuous and quasi-concave on 𝑆𝐴 for each𝑠𝐵 ∈ 𝑆𝐵 (or each 𝑡𝐵 ∈ 𝑇𝐵), upper semi-continuous and quasi-concave on 𝑇𝐴 for each𝑡𝐵 ∈ 𝑇𝐵 (or each𝑠𝐵 ∈ 𝑆𝐵), and lower semi-continuous and quasi-convex on𝑆𝐵 for each𝑠𝐴∈ 𝑆𝐴(or each𝑡𝐴 ∈ 𝑇𝐴), lower semi-continuous and quasi-convex on𝑇𝐵 for each 𝑡𝐴 ∈ 𝑇𝐴(or each𝑠𝐴∈ 𝑆𝐴). We do not postulate differentiability of payoff functions.
We will show that the following four patterns of competition are equivalent, that is, they yield the same outcome.
(1) Player A and B choose𝑠𝐴and𝑠𝐵 (competition by(𝑠𝐴, 𝑠𝐵)).
(2) Player A and B choose𝑡𝐴and𝑡𝐵 (competition by(𝑡𝐴, 𝑡𝐵)).
(3) Player A and B choose𝑡𝐴and𝑠𝐵 (competition by(𝑡𝐴, 𝑠𝐵)).
(4) Player A and B choose𝑠𝐴and𝑡𝐵 (competition by(𝑠𝐴, 𝑡𝐵)).
Relative profit maximization in duopoly with differentiated goods is an example of zero- sum game with two alternative strategic variables1. Each firm chooses its output or price.
The results of this paper imply that when firms in duopoly maximize their relative profits, Cournot and Bertrand equilibria are equivalent, and price-setting behavior and output- setting behavior are equivalent2.
The key to our results is Lemma 4 in Section 6. This lemma implies that the maximin strategies in four patterns of competition are equivalent, and the minimax strategies in four patterns of competition are equivalent.
2 The model
Consider a two-person zero-sum game as follows. There are two players, A and B. They have two sets of alternative strategic variables, (𝑠𝐴, 𝑠𝐵) ∈ 𝑆𝐴× 𝑆𝐵 and(𝑡𝐴, 𝑡𝐵) ∈ 𝑇𝐴× 𝑇𝐵. 𝑆𝐴,𝑆𝐵,𝑇𝐴and𝑇𝐵are compact sets in metric spaces. The relations of them are represented by 𝑠𝐴= 𝑓𝐴(𝑡𝐴, 𝑡𝐵), and𝑠𝐵 = 𝑓𝐵(𝑡𝐴, 𝑡𝐵).
(𝑓𝐴, 𝑓𝐵)is a continuous invertible function, and so it is a one-to-one and onto function.
We denote
𝑡𝐴 = 𝑔𝐴(𝑠𝐴, 𝑠𝐵), and𝑡𝐵 = 𝑔𝐵(𝑠𝐴, 𝑠𝐵).
1A game of relative profit maximization in duopoly is a zero-sum game because the sum of the relative profits of firms is zero.
2About 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).
(𝑔𝐴, 𝑔𝐵)is also a continuous invertible function. The payoff function of Player A is𝑢𝐴(𝑠𝐴, 𝑠𝐵) and the payoff function of Player B is𝑢𝐵(𝑠𝐴, 𝑠𝐵). Since the game is zero-sum, we have 𝑢𝐵(𝑠𝐴, 𝑠𝐵) = −𝑢𝐴(𝑠𝐴, 𝑠𝐵).𝑢𝐴is upper semi-continuous and quasi-concave on𝑆𝐴for each 𝑠𝐵 ∈ 𝑆𝐵 (or each 𝑡𝐵 ∈ 𝑇𝐵), upper semi-continuous and quasi-concave on 𝑇𝐴 for each 𝑡𝐵 ∈ 𝑇𝐵 (or each𝑠𝐵 ∈ 𝑆𝐵), and lower semi-continuous and quasi-convex on𝑆𝐵 for each 𝑠𝐴 ∈ 𝑆𝐴 (or each 𝑡𝐴 ∈ 𝑇𝐴), lower semi-continuous and quasi-convex on 𝑇𝐵 for each 𝑡𝐴 ∈ 𝑇𝐴(or each𝑠𝐴∈ 𝑆𝐴). We do not postulate differentiability of payoff functions3.
3 Competition by (𝑠
𝐴, 𝑠
𝐵)
First consider competition by(𝑠𝐴, 𝑠𝐵). Let𝑠∗𝐴and𝑠∗𝐵 be the values of𝑠𝐴 and𝑠𝐵 which, respectively, (locally) maximizes 𝑢𝐴(𝑠𝐴, 𝑠𝐵) given 𝑠∗𝐵 and (locally) maximizes 𝑢𝐵(𝑠𝐴, 𝑠𝐵) given𝑠∗𝐴in a neighborhood around(𝑠∗𝐴, 𝑠∗𝐵)in𝑆𝐴× 𝑆𝐵. Then,
𝑢𝐴(𝑠∗𝐴, 𝑠∗𝐵) ≥ 𝑢𝐴(𝑠𝐴, 𝑠∗𝐵)for all𝑠𝐴 ≠ 𝑠∗𝐴, and 𝑢𝐵(𝑠∗𝐴, 𝑠∗𝐵) ≥ 𝑢𝐵(𝑠∗𝐴, 𝑠𝐵)for all𝑠𝐵 ≠ 𝑠∗𝐵. Since𝑢𝐵 = −𝑢𝐴, this is rewritten as
𝑢𝐴(𝑠∗𝐴, 𝑠𝐵) ≥ 𝑢𝐴(𝑠∗𝐴, 𝑠∗𝐵), for all𝑠𝐵 ≠ 𝑠∗𝐵. Thus, we obtain
𝑢𝐴(𝑠∗𝐴, 𝑠𝐵) ≥ 𝑢𝐴(𝑠∗𝐴, 𝑠∗𝐵) ≥ 𝑢𝐴(𝑠𝐴, 𝑠∗𝐵)for all𝑠𝐴≠ 𝑠∗𝐴, and all𝑠𝐵 ≠ 𝑠∗𝐵. This is equivalent to
𝑢𝐴(𝑠∗𝐴, 𝑠∗𝐵) = max
𝑠𝐴 𝑢𝐴(𝑠𝐴, 𝑠∗𝐵) = min
𝑠𝐵 𝑢𝐴(𝑠∗𝐴, 𝑠𝐵).
(𝑠∗𝐴, 𝑠∗𝐵)is a Nash equilibrium of competition by(𝑠𝐴, 𝑠𝐵)game.
On the other hand, by the Sion’s minimax theorem (Sion (1958), Komiya (1988), Kindler (2005)) we have
𝑣𝑠𝐴 ≡max
𝑠𝐴 min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵) =min
𝑠𝐵 max
𝑠𝐴 𝑢𝐴(𝑠𝐴, 𝑠𝐵) ≡ 𝑣𝑠𝐵. We can show the following lemma.
Lemma 1. The following three statements are equivalent.
(1) There exists a Nash equilibrium in competition by(𝑠𝐴, 𝑠𝐵)game.
3In Satoh and Tanaka (2016) we analyze maximin and minimax strategies in duopoly when payoff func- tions of firms are differentiable.
(2) The following relation holds.
𝑣𝑠𝐴≡ max
𝑠𝐴 min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵) ≡min
𝑠𝐵 max
𝑠𝐴 𝑢𝐴(𝑠𝐴, 𝑠𝐵) = 𝑣𝑠𝐵,
in a neighborhood around(arg max𝑠𝐴min𝑠𝐵𝑢𝐴(𝑠𝐴, 𝑠𝐵),arg min𝑠𝐵max𝑠𝐴𝑢𝐴(𝑠𝐴, 𝑠𝐵))in 𝑆𝐴× 𝑆𝐵.
(3) There exists a real numberv𝑠,𝑠𝑚𝐴and𝑠𝑚𝐵 such that
𝑢𝐴(𝑠𝑚𝐴, 𝑠𝐵) ≥v𝑠for any𝑠𝐵, and𝑢𝐴(𝑠𝐴, 𝑠𝑚𝐵) ≤v𝑠for any𝑠𝐴 (1) in a neighborhood around(𝑠𝑚𝐴, 𝑠𝑚𝐵)in𝑆𝐴× 𝑆𝐵.
Proof. (1 → 2)
Let𝑠∗𝐴and𝑠∗𝐵 be the equilibrium strategies. Then, 𝑣𝑠𝐵 = min
𝑠𝐵 max
𝑠𝐴 𝑢𝐴(𝑠𝐴, 𝑠𝐵) ≤max
𝑠𝐴 𝑢𝐴(𝑠𝐴, 𝑠∗𝐵) = 𝑢𝐴(𝑠∗𝐴, 𝑠∗𝐵)
= min
𝑠𝐵 𝑢𝐴(𝑠∗𝐴, 𝑠𝐵) ≤max
𝑠𝐴 min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵) = 𝑣𝑠𝐴.
On the other hand, min𝑠𝐵𝑢𝐴(𝑠𝐴, 𝑠𝐵) ≤ 𝑢𝐴(𝑠𝐴, 𝑠𝐵), then max𝑠𝐴min𝑠𝐵𝑢𝐴(𝑠𝐴, 𝑠𝐵) ≤max𝑠𝐴𝑢𝐴(𝑠𝐴, 𝑠𝐵), and so max𝑠𝐴min𝑠𝐵𝑢𝐴(𝑠𝐴, 𝑠𝐵) ≤ min𝑠𝐵max𝑠𝐴𝑢𝐴(𝑠𝐴, 𝑠𝐵). Thus, 𝑣𝑠𝐴 ≤ 𝑣𝑠𝐵, and we have
𝑣𝑠𝐴= 𝑣𝑠𝐵. (2 → 3)
Let𝑠𝑚𝐴 =arg max𝑠𝐴min𝑠𝐵𝑢𝐴(𝑠𝐴, 𝑠𝐵)(the maximin strategy),𝑠𝑚𝐵 =arg min𝑠𝐵max𝑠𝐴𝑢𝐴(𝑠𝐴, 𝑠𝐵) (the minimax strategy), and letv𝑠= 𝑣𝑠𝐴= 𝑣𝑠𝐵. Then, we have
𝑢𝐴(𝑠𝑚𝐴, 𝑠𝐵) ≥min
𝑠𝐵 𝑢𝐴(𝑠𝑚𝐴, 𝑠𝐵) =max
𝑠𝐴 min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵) =v𝑠
=min
𝑠𝐵 max
𝑠𝐴 𝑢𝐴(𝑠𝐴, 𝑠𝐵) = max
𝑠𝐴 𝑢𝐴(𝑠𝐴, 𝑠𝑚𝐵) ≥ 𝑢𝐴(𝑠𝐴, 𝑠𝑚𝐵).
(3 → 1) From (1)
𝑢𝐴(𝑠𝑚𝐴, 𝑠𝐵) ≥v𝑠≥ 𝑢𝐴(𝑠𝐴, 𝑠𝑚𝐵)for all𝑠𝐴 ∈ 𝑆𝐴, 𝑠𝐵 ∈ 𝑆𝐵.
Putting𝑠𝐴 = 𝑠𝑚𝐴and𝑠𝐵 = 𝑠𝑚𝐵, we seev𝑠 = 𝑢𝐴(𝑠𝑚𝐴, 𝑠𝑚𝐵)and(𝑠𝑚𝐴, 𝑠𝑚𝐵)is an equilibrium.
We write(𝑠𝑚𝐴, 𝑠𝑚𝐵) = (𝑠∗𝐴, 𝑠∗𝐵). Denote the value of𝑡𝐴which is derived from𝑡𝐴= 𝑔𝐴(𝑠∗𝐴, 𝑠∗𝐵) by𝑡∗𝐴, and denote the value of𝑡𝐵 which is derived from𝑡𝐵 = 𝑔𝐵(𝑠∗𝐴, 𝑠∗𝐵)by𝑡∗𝐵.
4 Competition by (𝑡
𝐴, 𝑡
𝐵)
Next consider competition by(𝑡𝐴, 𝑡𝐵). Substituting𝑓𝐴and𝑓𝐵 into𝑢𝐴and𝑢𝐵 yields 𝑢𝐴= 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)), 𝑢𝐵 = 𝑢𝐵(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)).
Let 𝐴̃𝑡 and 𝐵̃𝑡 be the values of𝑡𝐴and𝑡𝐵which, respectively, (locally) maximizes𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)) given 𝐵̃𝑡 and (locally) maximizes 𝑢𝐵(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)) given 𝐴̃𝑡 in a neighborhood
around( ̃𝑡𝐴, ̃𝑡𝐵)in𝑇𝐴× 𝑇𝐵. Then,
𝑢𝐴(𝑓𝐴( ̃𝑡𝐴, ̃𝑡𝐵), 𝑓𝐵( ̃𝑡𝐴, ̃𝑡𝐵)) ≥ 𝑢𝐴(𝑓𝐴(𝑡𝐴, ̃𝑡𝐵), 𝑓𝐵(𝑡𝐴, ̃𝑡𝐵))for all𝑡𝐴≠ ̃𝑡𝐴, and
𝑢𝐵(𝑓𝐴( ̃𝑡𝐴, ̃𝑡𝐵), 𝑓𝐵( ̃𝑡𝐴, ̃𝑡𝐵)) ≥ 𝑢𝐵(𝑓𝐴( ̃𝑡𝐴, 𝑡𝐵), 𝑓𝐵( ̃𝑡𝐴, 𝑡𝐵))for all𝑡𝐵 ≠ ̃𝑡𝐵. Since𝑢𝐵 = −𝑢𝐴, this is rewritten as
𝑢𝐴(𝑓𝐴( ̃𝑡𝐴, 𝑡𝐵), 𝑓𝐵( ̃𝑡𝐴, 𝑡𝐵)) ≥ 𝑢𝐴(𝑓𝐴( ̃𝑡𝐴, ̃𝑡𝐵), 𝑓𝐵( ̃𝑡𝐴, ̃𝑡𝐵))for all𝑡𝐵 ≠ ̃𝑡𝐵. Thus, we obtain
𝑢𝐴(𝑓𝐴( ̃𝑡𝐴, 𝑡𝐵), 𝑓𝐵( ̃𝑡𝐴, 𝑡𝐵)) ≥ 𝑢𝐴(𝑓𝐴( ̃𝑡𝐴, ̃𝑡𝐵), 𝑓𝐵( ̃𝑡𝐴, ̃𝑡𝐵)) ≥ 𝑢𝐴(𝑓𝐴(𝑡𝐴, ̃𝑡𝐵), 𝑓𝐵(𝑡𝐴, ̃𝑡𝐵)) for all𝑡𝐴 ≠ ̃𝑡𝐴, and all𝑡𝐵 ≠ ̃𝑡𝐵.
This is equivalent to
𝑢𝐴(𝑓𝐴( ̃𝑡𝐴, ̃𝑡𝐵), 𝑓𝐴( ̃𝑡𝐴, ̃𝑡𝐵)) = max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, ̃𝑡𝐵), 𝑓𝐵(𝑡𝐴, ̃𝑡𝐵))
= min
𝑡𝐵 𝑢𝐴(𝑓𝐴( ̃𝑡𝐴, 𝑡𝐵), 𝑓𝐵( ̃𝑡𝐴, 𝑡𝐵)).
Similarly to Lemma 1 we can show.
Lemma 2. The following three statements are equivalent.
(1) There exists a Nash equilibrium in competition by(𝑡𝐴, 𝑡𝐵)game.
(2) The following relation holds.
𝑣𝑡𝐴≡max
𝑡𝐴 min
𝑡𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)) =min
𝑡𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)) ≡ 𝑣𝑡𝐵, in a neighborhood around
(arg max
𝑡𝐴 min
𝑡𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)),arg min
𝑡𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵))) in𝑇𝐴× 𝑇𝐵.
(3) There exists a real numberv𝑡,𝑡𝑚𝐴 ∈ 𝑇𝐴and𝑡𝑚𝐵 ∈ 𝑇𝐵 such that
𝑢𝐴(𝑓𝐴(𝑡𝑚𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝑚𝐴, 𝑡𝐵)) ≥ v𝑡 for any𝑡𝐵 ∈ 𝑇𝐵, and𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝑚𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝑚𝐵)) ≤v𝑡 for any𝑡𝐴∈ 𝑇𝐴in a neighborhood around(𝑡𝑚𝐴, 𝑡𝑚𝐵)in𝑇𝐴× 𝑇𝐵.
We write(𝑡𝑚𝐴, 𝑡𝑚𝐵) = ( ̃𝑡𝐴, ̃𝑡𝐵). Denote the value of𝑠𝐴which is derived from𝑠𝐴= 𝑓𝐴( ̃𝑡𝐴, ̃𝑡𝐵) by 𝐴̃𝑠 , and denote the value of𝑠𝐵 which is derived from𝑠𝐵 = 𝑓𝐵( ̃𝑡𝐴, ̃𝑡𝐵)by 𝐵̃𝑠 .
5 Competition by (𝑡
𝐴, 𝑠
𝐵)
Next consider competition by(𝑡𝐴, 𝑠𝐵). we have
𝑠𝐴 = 𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑠𝐴, 𝑠𝐵)), 𝑡𝐵 = 𝑔𝐵(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵).
The payoffs of Player A and B are written as
𝑢𝐴(𝑠𝐴, 𝑠𝐵) = 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵), 𝑢𝐵(𝑠𝐴, 𝑠𝐵) = 𝑢𝐵(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵).
Let 𝐴̄𝑡 and 𝐵̄𝑠 be the values of𝑡𝐴and𝑠𝐵 which, respectively, (locally) maximizes𝑢𝐴given
̄𝑠𝐵 and (locally) maximizes 𝑢𝐵 given ̄𝑡𝐴 in a neighborhood around ( ̄𝑡𝐴, ̄𝑠𝐵) in𝑇𝐴 × 𝑆𝐵. Then,
𝑢𝐴(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵), ̄𝑠𝐵) ≥ 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), ̄𝑠𝐵)for all𝑡𝐴≠ ̄𝑡𝐴, and
𝑢𝐵(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵), ̄𝑠𝐵)) ≥ 𝑢𝐵(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵), 𝑠𝐵))for all𝑠𝐵 ≠ ̄𝑠𝐵. Since𝑢𝐵 = −𝑢𝐴, this is rewritten as
𝑢𝐴(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵), 𝑠𝐵)) ≥ 𝑢𝐴(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵), ̄𝑠𝐵))for all𝑠𝐵 ≠ ̄𝑠𝐵. Thus, we obtain
𝑢𝐴(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵), 𝑠𝐵)) ≥ 𝑢𝐴(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵), ̄𝑠𝐵)) ≥ 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), ̄𝑠𝐵)) for all𝑡𝐴≠ ̄𝑡𝐴, and all𝑠𝐵 ≠ ̄𝑠𝐵.
This is equivalent to
𝑢𝐴(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵), ̄𝑠𝐵) =max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), ̄𝑠𝐵) =min
𝑠𝐵 𝑢𝐴(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵), 𝑠𝐵).
Similarly to Lemma 1 we can show.
Lemma 3. The following three statements are equivalent.
(1) There exists a Nash equilibrium in competition by(𝑡𝐴, 𝑠𝐵)game.
(2) The following relation holds.
𝑣𝑡𝑠𝐴 ≡max
𝑡𝐴 min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) =min
𝑠𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) ≡ 𝑣𝑡𝑠𝐵, in a neighborhood around
(arg max
𝑡𝐴 min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵),arg min
𝑠𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵)) in𝑇𝐴× 𝑆𝐵.
(3) There exists a real numberv𝑡𝑠,𝑡𝑡𝑠𝐴 ∈ 𝑇𝐴and𝑠𝑡𝑠𝐵 ∈ 𝑆𝐵 such that
𝑢𝐴(𝑓𝐴(𝑡𝑡𝑠𝐴, 𝑡𝐵), 𝑠𝐵) ≥v𝑡𝑠for any𝑠𝐵 ∈ 𝑆𝐵, and𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝑡𝑠𝐵) ≤v𝑡𝑠for any𝑡𝐴 ∈ 𝑇𝐴 in a neighborhood around(𝑡𝑡𝑠𝐴, 𝑠𝑡𝑠𝐵)in𝑇𝐴× 𝑆𝐵.
We write(𝑡𝑡𝑠𝐴, 𝑠𝑡𝑠𝐵) = ( ̄𝑡𝐴, ̄𝑠𝐵). Denote the value of𝑠𝐴which is derived from𝑠𝐴 = 𝑓𝐴( ̄𝑡𝐴, 𝑔𝐵(𝑠𝐴, ̄𝑠𝐵)) by 𝐴̄𝑠 , and denote the value of𝑡𝐵which is derived from𝑡𝐵 = 𝑔𝐵(𝑓𝐴( ̄𝑡𝐴, 𝑡𝐵) ̄𝑠𝐵), by 𝐵̄𝑡 . Then,
̄𝑡𝐴and ̄𝑠𝐵 are written as
̄𝑡𝐴= 𝑔𝐴( ̄𝑠𝐴, ̄𝑠𝐵), and ̄𝑠𝐵 = 𝑓𝐵( ̄𝑡𝐴, ̄𝑡𝐵).
6 Equivalence of four patterns of competition
In this section we show the equivalence of four patterns of competition. First we show the following lemma which is key to our results.
Lemma 4. The following relations hold.
(1) max𝑡𝐴min𝑠𝐵𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) =max𝑠𝐴min𝑠𝐵𝑢𝐴(𝑠𝐴, 𝑠𝐵).
(2) min𝑠𝐵max𝑡𝐴𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) =min𝑡𝐵max𝑡𝐴𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)).
Proof. (1) min𝑠𝐵𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵)is the minimum of𝑢𝐴with respect to𝑠𝐵given𝑡𝐴. Let 𝑠𝐵(𝑡𝐴) =arg min𝑠𝐵𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵), and fix the value of𝑠𝐴at𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑡𝐴))). Then, we have
min𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑡𝐴))), 𝑠𝐵)
≤ 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑡𝐴))), 𝑠𝐵(𝑡𝐴)) =min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵),
where min𝑠𝐵𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑡𝐴))), 𝑠𝐵) is the minimum of𝑢𝐴 with respect to𝑠𝐵 given the value of𝑠𝐴at𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑡𝐴))). This holds for any𝑡𝐴. Thus,
𝑓𝐴(𝑡𝐴,𝑔𝐵max(𝑠𝐴,𝑠𝐵(𝑡𝐴)))min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑡𝐴))), 𝑠𝐵) ≤max
𝑡𝐴 min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵).
We assume 𝑠𝐵(𝑡𝐴) is single-valued. By the maximum theorem and continuity of the functions, 𝑢𝐴 and 𝑓𝐴, 𝑠𝐵(𝑡𝐴) is continuous. The values of 𝑠𝐴 in some neigh- borhood around( ̄𝑠𝐴, ̄𝑠𝐵)can be realized by appropriately choosing𝑡𝐴 given𝑠𝐵 as 𝑠𝐴 = 𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑡𝐴))). Therefore, this can be rewritten as
max𝑠𝐴 min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵) ≤max
𝑡𝐴 min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵). (2)
On the other hand, min𝑠𝐵𝑢𝐴(𝑠𝐴, 𝑠𝐵)is the minimum of𝑢𝐴with respect to𝑠𝐵 given 𝑠𝐴. Let 𝑠𝐵(𝑠𝐴) = arg min𝑠𝐵𝑢𝐴(𝑠𝐴, 𝑠𝐵), and fix the value of 𝑡𝐴 at 𝑔𝐴(𝑠𝐴, 𝑠𝐵(𝑠𝐴)). Then, we have
min𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑔𝐴(𝑠𝐴, 𝑠𝐵(𝑠𝐴)), 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑠𝐴))), 𝑠𝐵)
≤ 𝑢𝐴(𝑓𝐴(𝑔𝐴(𝑠𝐴, 𝑠𝐵(𝑠𝐴)), 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑠𝐴))), 𝑠𝐵(𝑠𝐴)) = 𝑢𝐴(𝑠𝐴, 𝑠𝐵(𝑠𝐴)) =min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵), where min𝑠𝐵𝑢𝐴(𝑓𝐴(𝑔𝐴(𝑠𝐴, 𝑠𝐵(𝑠𝐴)), 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑠𝐴))), 𝑠𝐵)is the minimum of𝑢𝐴with re- spect to𝑠𝐵 given the value of𝑡𝐴at𝑔𝐴(𝑠𝐴, 𝑠𝐵(𝑠𝐴)). This holds for any𝑠𝐴. Thus,
𝑔𝐴(𝑠max𝐴,𝑠𝐵(𝑠𝐴))min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑔𝐴(𝑠𝐴, 𝑠𝐵(𝑠𝐴)), 𝑔𝐵(𝑠𝐴, 𝑠𝐵(𝑠𝐴))), 𝑠𝐵) ≤max
𝑠𝐴 min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵) We assume𝑠𝐵(𝑠𝐴)is single-valued. By the maximum theorem and continuity of𝑢𝐴, 𝑠𝐵(𝑠𝐴)is continuous. The values of𝑡𝐴in some neighborhood around( ̄𝑡𝐴, ̄𝑠𝐵)can be realized by appropriately choosing 𝑠𝐴 given 𝑠𝐵 as 𝑡𝐴 = 𝑔𝐴(𝑠𝐴, 𝑠𝐵(𝑠𝐴)). Therefore, this can be rewritten as
max𝑡𝐴 min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) ≤max
𝑠𝐴 min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵). (3) Combining (2) and (3), we get
max𝑠𝐴 min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵) =max
𝑡𝐴 min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵).
(2) max𝑡𝐴𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) is the maximum of 𝑢𝐴 with respect to 𝑡𝐴 given 𝑠𝐵. Let 𝑡𝐴(𝑠𝐵) =arg max𝑡𝐴𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵), and fix the value of𝑡𝐵at𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵). Then, we have
max𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵)), 𝑠𝐵)
=max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵)), 𝑓𝐵(𝑡𝐴, 𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵)))
≥ 𝑢𝐴(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵)), 𝑠𝐵) =max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵),
where max𝑡𝐴𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵)), 𝑠𝐵)is the maximum of𝑢𝐴with respect to𝑡𝐴given the value of𝑡𝐵 at𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵)). This holds for any𝑠𝐵. Thus,
𝑔𝐵(𝑓𝐴(𝑡𝐴min(𝑠𝐵),𝑡𝐵),𝑠𝐵))max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵)), 𝑓𝐵(𝑡𝐴, 𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵)))
≥min
𝑠𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵).
We assume 𝑡𝐴(𝑠𝐵) is single-valued. By the maximum theorem and continuity of the functions, 𝑢𝐴 and 𝑓𝐴, 𝑡𝐴(𝑠𝐵) is continuous. The values of 𝑡𝐵 in some neigh- borhood around( ̄𝑡𝐴, ̄𝑡𝐵) can be realized by appropriately choosing𝑠𝐵 given 𝑡𝐴 as 𝑡𝐵 = 𝑔𝐵(𝑓𝐴(𝑡𝐴(𝑠𝐵), 𝑡𝐵), 𝑠𝐵). Therefore, this can be rewritten as
min𝑡𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)) ≥min
𝑠𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵). (4)
On the other hand, max𝑡𝐴𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)) is the maximum of𝑢𝐴 with re- spect to 𝑡𝐴 given 𝑡𝐵. Let 𝑡𝐴(𝑡𝐵) = arg max𝑡𝐴𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)), and fix the value of𝑠𝐵 at𝑓𝐵(𝑡𝐴(𝑡𝐵), 𝑡𝐵). Then, we have
max𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴(𝑡𝐵), 𝑡𝐵))
≥ 𝑢𝐴(𝑓𝐴(𝑡𝐴(𝑡𝐵), 𝑡𝐵), 𝑓𝐵(𝑡𝐴(𝑡𝐵), 𝑡𝐵)) =max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)),
where max𝑡𝐴𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴(𝑡𝐵), 𝑡𝐵))is the maximum of 𝑢𝐴 with respect to𝑡𝐴 given the value of𝑠𝐵 at𝑓𝐵(𝑡𝐴(𝑡𝐵), 𝑡𝐵). This holds for any𝑡𝐵. Thus,
𝑓𝐵(𝑡min𝐴(𝑡𝐵),𝑡𝐵)max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴(𝑡𝐵), 𝑡𝐵)) ≥ min
𝑡𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)).
We assume𝑡𝐴(𝑡𝐵)is single-valued. By the maximum theorem and continuity of the functions, 𝑢𝐴, 𝑓𝐴 and 𝑓𝐵, 𝑡𝐴(𝑡𝐵) is continuous. The values of 𝑠𝐵 in some neigh- borhood around( ̄𝑡𝐴, ̄𝑠𝐵) can be realized by appropriately choosing 𝑡𝐵 given 𝑡𝐴 as 𝑠𝐵 = 𝑓𝐵(𝑡𝐴(𝑡𝐵), 𝑡𝐵). Therefore, this can be rewritten as
min𝑠𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) ≥min
𝑡𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)). (5) Combining (4) and (5), we get
min𝑡𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑓𝐵(𝑡𝐴, 𝑡𝐵)) =min
𝑠𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵).
Now we show the following propositions.
Proposition 1. (1) Competition by(𝑠𝐴, 𝑠𝐵)and competition by(𝑡𝐴, 𝑠𝐵)are equivalent.
(2) Competition by(𝑡𝐴, 𝑠𝐵)and competition by(𝑡𝐴, 𝑡𝐵)are equivalent.
Proof. (1) We show that the condition for( ̄𝑠𝐴, ̄𝑠𝐵)and the condition for(𝑠∗𝐴, 𝑠∗𝐵)are the same. From Lemma (3)
max𝑡𝐴 min
𝑠𝐵 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) = min
𝑠𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) = 𝑢𝐴( ̄𝑠𝐴, ̄𝑠𝐵).
Since any value of𝑠𝐴can be realized by appropriately choosing𝑡𝐴given𝑠𝐵, we have max𝑡𝐴𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) = max𝑠𝐴𝑢𝐴(𝑠𝐴, 𝑠𝐵)for any𝑠𝐵. Thus,
min𝑠𝐵 max
𝑠𝐴 𝑢𝐴(𝑠𝐴, 𝑠𝐵) =min
𝑠𝐵 max
𝑡𝐴 𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) = 𝑢𝐴( ̄𝑠𝐴, ̄𝑠𝐵).
From Lemma 4 we have max𝑡𝐴min𝑠𝐵𝑢𝐴(𝑓𝐴(𝑡𝐴, 𝑡𝐵), 𝑠𝐵) = max𝑠𝐴min𝑠𝐵𝑢𝐴(𝑠𝐴, 𝑠𝐵). Therefore, we obtain
max𝑠𝐴 min
𝑠𝐵 𝑢𝐴(𝑠𝐴, 𝑠𝐵) = min
𝑠𝐵 max
𝑠𝐴 𝑢𝐴(𝑠𝐴, 𝑠𝐵) = 𝑢𝐴( ̄𝑠𝐴, ̄𝑠𝐵).
This is 2 of Lemma 1.
