Symmetricmulti-personzero-sumgamewithtwosetsofstrategicvariables Satoh,AtsuhiroandTanaka,Yasuhito MunichPersonalRePEcArchive

Munich Personal RePEc Archive

Symmetric multi-person zero-sum game with two sets of strategic variables

Satoh, Atsuhiro and Tanaka, Yasuhito

28 December 2016

Online at https://mpra.ub.uni-muenchen.de/75838/

Symmetric multi-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.

December 27, 2016

Abstract

We consider a symmetric multi-person zero-sum game with two sets of alternative strategic variables which are related by invertible functions. They are denoted by.s1; s2; : : : ; sn/ and.t1; t2; : : : ; tn/for players 1, 2,: : :,n. The number of players is larger than two. We consider a symmetric game in the sense that all players have the same payoff functions.

We do not postulate differentiability of the payoff functions of players. We will show that the following patterns of competition, 1) all players choosesi, 2) all players choose tiand 3)mplayers chooseti; i D1; : : : ; mandn mplayers choosesj; j DmC1; : : : ; n where1 m n 1, are equivalent, that is, they yield the same outcome. However, in an asymmetric zero-sum game with more than two players the equivalence does not hold.

keywords multi-person zero-sum game, two strategic variables JEL Classification: C72, D43.

∗atsato@mail.doshisha.ac.jp

†yasuhito@mail.doshisha.ac.jp (corresponding author)

1 Introduction

We consider ann-person symmetric zero-sum game with two sets of strategic variables which are related by invertible functions. They are denoted by.s1; s2; : : : ; sn/and.t1; t2; : : : ; tn/ for players 1, 2,: : : ; n. nis an integer number which is larger than 2. We do not postulate differentiability of the payoff functions of players.

We will show that the following patterns of competition are equivalent, that is, they yield the same outcome.

(1) All players choosesi,i 2N. We call this competitionsi competition.

(2) All players chooseti,i 2N. We call this competitionti competition.

(3) Some players chooseti and other players choose sj. Specifically, mplayers choose ti,i D1; 2; : : : ; m, andn mplayers choosesj, j D mC1; mC2; : : : ; n, where 1mn 1. We call this competitionti sj competition.

We assume that the game is symmetric in the sense that all players have the same payoff functions, and consider symmetric equilibria where all players, whose strategic variables are si’s, choose the same values, and also all players, whose strategic variables areti’s, choose the same values.

Relative profit maximization in a symmetric oligopoly with differentiated goods is an ex- ample of symmetricn-person zero-sum game with two alternative strategic variables. Each firm chooses its output or price. The results of this paper imply that when firms in a symmet- ric oligopoly maximize their relative profits, Cournot and Bertrand equilibria are equivalent, and price-setting behavior and output-setting behavior are equivalent¹.

However, in an asymmetric n-person zero-sum game with more than two players the equivalence does not hold. In Section 7 we present an example that shows the non-equivalence of Cournot and Bertrand equilibria in an asymmetric oligopoly.

2 The model

Consider ann-person zero-sum game withn3as follows. There arenplayers, 1, 2,: : : ; n.

The set of players is denoted byN. They have two sets of alternative strategic variables, (s1; s2; : : : ; sn/ 2S1S2 Snand.t1; t2; : : : ; tn/2 T1T2 Tn. Si andTi for i 2N are compact sets in metric spaces. The relations of them are represented by

si Dfi.t1; t2; : : : ; tn/; i 2N:

.f1; f2; : : : ; fn/is a continuous invertible function, and so it is one-to-one and onto func- tion. We denote

ti Dgi.s1; s2; : : : ; sn/; i 2N:

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). An oligopoly is symmetric when demand functions are symmetric and all firms have the same cost functions.

.g1; g2; : : : ; gn/is also a continuous invertible function. The payoff functions of the players areui.s1; s2; : : : ; sn/fori 2N. They are continuous and quasi-concave. We do not postulate differentiability of the payoff functions². All players have the same payoff functions. Since the game is zero-sum, we have

Xn

iD1

u_i.s₁; s₂; : : : ; s_n/D0; (1)

for given.s1; s2; : : : ; sn/.

3 s

competition

First, consider competition bysi; i 2 N;for all players. Let s_i; i 2 N;be the values of si’s which, respectively, maximizesui; i 2 N;givens_j; j ¤ i, in a neighborhood around .s₁; s₂; : : : ; s_n/inS1S2 Sn. Then,

ui.s₁; : : : ; s_i; : : : ; s_n/ui.s₁; : : : ; si; : : : ; s_n/for allsi ¤s_i; i 2N: (2) We assume that alls_i’s are equal at equilibria. Thus, ui.s₁; : : : ; s_i; : : : ; s_n/’s for alli are equal, and by the property of zero-sum game they are zero. By symmetry of the game we have

uj.s₁; : : : ; si; : : : ; s_n/Du_k.s₁; : : : ; si; : : : ; s_n/forj ¤i; k ¤i; j ¤k:

From this and (1) X

jD1;j¤i

uj.s₁; : : : ; si; : : : ; s_n/D .n 1/uj.s₁; : : : ; si; : : : ; s_n/Dui.s₁; : : : ; si; : : : ; s_n/:

Therefore, from (2)

uj.s₁; : : : ; si; : : : ; s_n/uj.s₁; : : : ; s_i; : : : ; s_n/forj ¤i:

By symmetry

ui.s₁; : : : ; sj; : : : ; s_n/ui.s₁; : : : ; s_i; : : : ; s_n/forj ¤i:

Combining this and (2)

ui.s₁; : : : ; si; : : : ; s_n/ui.s₁; : : : ; s_i; : : : ; s_n/ui.s₁; : : : ; sj; : : : ; s_n/ for allsi ¤s_i; and allsj ¤s_j; j ¤i; i 2N:

This is equivalent to

ui.s₁; : : : ; s_i; : : : ; s_n/Dmax

si

ui.s₁; : : : ; si; : : : ; s_n/Dmin

s_j ui.s₁; : : : ; sj; : : : ; s_n/;

j ¤igivens_k; k¤i; j;

2In Satoh and Tanaka (2016) we analyze maximin and minimax strategies in oligopoly when payoff functions of firms are differentiable.

.s₁; s₂; : : : ; s_n/is a Nash equilibrium of thesi competition game. By the Glicksberg’s the- orem (Glicksberg (1952)) there exists a Nash equilibrium.

Lets_i;j be a vector ofs_kfork¤i; j. We can show the following lemma.

Lemma 1. The following three statements are equivalent.

(1) There exists a Nash equilibrium in thesi competition game.

(2) Givens_kfor allk¤i; j, the following relation holds.

v^s_i max

s_i min

sj

ui.si; sj; s_i;j/Dmin

sj

maxs_i ui.si; sj; s_i;j/v^s_jfor any pair ofiandj:

(3) There exists a real numbervs,s_i^mands_j^msuch that

ui.s_i^m; sj; s_i;j/vsfor anysj; andui.si; s_j^m; s_i;j/vsfor anysi; (3) for any pair ofi andj:

Proof. (1!2)

Lets_iands_j be the equilibrium strategies of Playeri andj. Then, v^s_j Dmin

sj

maxsi

ui.si; sj; s_i;j/max

si

ui.si; s_j; s_i;j/Dui.s_i; s_j; s_i;j/ Dmin

sj

ui.s_i; sj; s_i;j/max

s_i min

sj

ui.si; sj; s_i;j/Dv^s_i:

On the other hand, mins_jui.si; sj; s_i;j/ui.si; sj; s_i;j/, then maxs_imins_jui.si; sj; s_i;j/ maxsiui.si; sj; s_i;j/, and so maxsiminsjui.si; sj; s_i;j/ minsjmaxsiui.si; sj; s_i;j/.

Thus,v^s_i v^s_j, and we havev^s_i Dv^s_j. (2!3)

Lets_i^mDarg maxs_imins_jui.si; sj; s_i;j/(the maximin strategy),s^m_j Darg mins_jmaxs_iui.si; sj; s_i;j/ (the minimax strategy), and letvsDv^s_i Dv^s_j. Then, we have

ui.s_i^m; sj; s_i;j/min

sj

ui.s_i^m; sj; s_i;j/Dmax

si

minsj

ui.si; sj; s_i;j/Dvs

Dmin

sj

maxs_i u_i.s_i; s_j; s_i;j/Dmax

s_i u_i.s_i; s^m_j ; s_i;j/u_i.s_i; s_j^m; s_i;j/:

(3!1) From (3)

ui.s_i^m; sj; s_i;j/vsui.si; s_j^m; s_i;j/for allsi 2Si; sj 2Sj:

Puttingsi Ds_i^mandsj Ds_j^m, we seevs Dui.s^m_i ; s^m_j ; s_i;j/and.s_i^m; s^m_j ; s_i;j/is an equi- librium. Thus,s_i^mDs_iands^m_j Ds_j.

Since at equilibria allui’s are zero, we havev^s_i D v^s_j D vs D 0. Denote the values of ti; i 2N, which are derived from the following equation;

.t1; t2; : : : ; tn/D.gi.s₁; s₂; : : : ; s_n/; g2.s₁; s₂; : : : ; s_n/; : : : ; gn.s₁; s₂; : : : ; s_n//;

byt_i; i 2N.

4 t

competition

Next consider competition byti; i 2N;for all players. In this section we use the following notation.

vi.t1; : : : ; tn/Dui.f1.t1; : : : ; tn/; : : : ; fn.t1; : : : ; tn//for eachi 2N:

LettQi; i 2N, be the values ofti’s which, respectively, maximizesvi; i 2N;giventQj; j ¤i, in a neighborhood around.tQ1;tQ2; : : : ;tQn/inT1T2 Tn. Then,

vi.tQ1; : : : ;tQi; : : : ;tQn/vi.tQ1; : : : ; ti; : : : ;tQn/for allti ¤ Qti; i 2N; (4) We assume that alltQi’s are equal at equilibria. Thus,vi.Qt1; : : : ;tQi; : : : ;tQn/for alli are equal, and by the property of zero-sum game allvi’s are zero. By symmetry of the model

vj.tQ1; : : : ; ti; : : : ;tQn/Dv_k.Qt1; : : : ; ti; : : : ;tQn/forj ¤i; k¤i; j ¤k:

From this and (1) X

jD1;j¤i

vj.tQ1; : : : ; ti; : : : ;tQn/D .n 1/vj.tQ1; : : : ; ti; : : : ;tQn/Dvi.tQ1; : : : ; ti; : : : ;tQn/:

Therefore, from (4)

vj.tQ1; : : : ; ti; : : : ;tQn/vj.tQ1; : : : ;tQi; : : : ;tQn/forj ¤i:

By symmetry we get

vi.tQ1; : : : ; tj; : : : ;tQn/vi.tQ1; : : : ;tQi; : : : ;tQn/forj ¤i:

Combining this and (4)

v_i.tQ₁; : : : ; t_i; : : : ;tQ_n/v_i.Qt₁; : : : ;tQ_i; : : : ;tQ_n/v_i.tQ₁; : : : ; t_j; : : : ;tQ_n/ for allti ¤ Qti; and alltj ¤ Qtj; j ¤i; i 2N:

This is equivalent to

vi.tQ1; : : : ;tQi; : : : ;tQn/Dmax

ti

vi.tQ1; : : : ; ti; : : : ;tQn/Dmin

tj

vi.tQ1; : : : ; tj; : : : ;tQn/;

j ¤i giventQ_k; k ¤i; j:

LettQ i;j be a vector oftQ_kfork¤i; j. Similarly to Lemma 1 we can show the following lemma.

Lemma 2. The following three statements are equivalent.

(1) There exists a Nash equilibrium in theti competition game.

(2) GiventQ_k for allk¤i; j, the following relation holds.

v^t_i max

ti

mint_j vi.ti; tj;tQ i;j/Dmin

t_j max

ti

vi.ti; tj;tQ i;j/v^t_j for any pair ofi andj:

(3) There exists a real numbervt,t_i^mandt_j^msuch that

vi.t_i^m; tj;tQ i;j/vt for anytj; andvi.ti; t_j^m;tQ i;j/vt for anyti

for any pair ofi andj:

Thus,t_i^mD Qti andt_j^mD Qtj. Since at equilibria allvi’s are zero, we havev^t_i Dv^t_j Dvt D0.

Denote the values ofsi; i 2N, which are derived from the following equation;

.s1; s2; : : : ; sn/D.fi.tQ1;tQ2; : : : ;tQn/; f2.tQ1;tQ2; : : : ;tQn/; : : : ; fn.tQ1;tQ2; : : : ;tQn//;

bysQi; i 2N.

5 t

s

competition

Next, considerti sj competition. Assume thatmplayers chooseti; i D1; 2 : : : ; m, and the remainingn mplayers choosesj; j DmC1; mC2; : : : ; n. mis an integer such that 1 m n 1. At least one player choosesti, and at least one player chooses sj. In this section we use the following notation.

wi.t1; : : : ; tm; smC1; : : : ; sn/

Dui.f1.t1; : : : ; tm; gmC1.: : : /; : : : ; gn.: : : //; : : : ; fm.t1; : : : ; tm; gmC1.: : : /; : : : ; gn.: : : //; smC1; : : : ; sn/ for eachi 2N; where

gj.: : : /Dgj.s1; : : : ; sm; smC1; : : : ; sn/forj 2 fmC1; : : : ; n;g with

si Dfi.t1; : : : ; tm; gmC1.: : : /; : : : ; gn.: : : //fori 2 f1; : : : ; mg:

LettNi; i D1; 2; : : : ; m, andsNj; j DmC1; : : : ; n, be the values oftiandsjwhich maximizes, respectively,wi andwj, in a neighborhood around the equilibrium point. Then,

wi.tN1; : : : ;tNi; : : : ;tNm;sNmC1; : : : ;sNn/ (5) wi.tN1; : : : ; ti; : : : ;tNm;sNmC1; : : : ;sNn/for allti ¤ Nti; i D1; 2; : : : ; m;

and

wj.tN1; : : : ;tNm;sNmC1; : : : ;sNj; : : : ;sNn/ (6) wj.tN1; : : : ;tNm;sNmC1; : : : ; sj; : : : ;sNn/for allsj ¤ Nsj; j DmC1; mC2; : : : ; n;

We assume that at equilibria alltNi; i D1; 2; : : : ; m, are equal, and allsNj; j DmC1; mC 2; : : : ; n, are equal. Since all players have the same payoff functions, allwi; i D1; 2; : : : ; m, are equal, and allwj; j DmC1; mC2; : : : ; n, are equal. Then, from (1) we obtain

mwi.tN1; : : : ;tNm;sNmC1; : : : ;sNj; : : : ;sNn/C.n m/wj.tN1; : : : ;tNm;sNmC1; : : : ;sNj; : : : ;sNn/D0;

and so

wj.tN1; : : : ;tNm;sNmC1; : : : ;sNj; : : : ;sNn/D m

n mwi.tN1; : : : ;tNm;sNmC1; : : : ;sNj; : : : ;sNn/:

Ifwi D0(orwj D0), thenwj D0(orwi D0). (6) is rewritten as wi.tN1; : : : ;tNm;sNmC1; : : : ;sNj; : : : ;sNn/

wi.tN1; : : : ;tNm;sNmC1; : : : ; sj; : : : ;sNn/for allsj ¤ Nsj; j DmC1; mC2; : : : ; n;

Combining this and (5),

wi.tN1; : : : ; ti; : : : ;tNm;sNmC1; : : : ;sNj; : : : ;sNn/wi.tN1; : : : ;tNi; : : : ;tNm;sNmC1; : : : ;sNj; : : : ;sNn/ wi.Nt1; : : : ;tNm;sNmC1; : : : ; sj; : : : ;sNn/

for allti ¤ Nti; i D1; 2; : : : ; m; and allsj ¤ Nsj; j DmC1; mC2; : : : ; n:

This is equivalent to

wi.tN1; : : : ;tNi; : : : ;tNm;sNmC1; : : : ;sNj; : : : ;sNn/Dmax

ti

wi.tN1; : : : ; ti; : : : ;tNm;sNmC1; : : : ;sNn/ Dmin

sj

wi.tN1; : : : ;tNm;sNmC1; : : : ; sj; : : : ;sNn/for any pair ofi; j:

LettN i be a vector oftN_k for k 2 f1; : : : ; mg; k ¤ i and sN j be a vector ofsN_l forl 2 fmC1; : : : ; ng; l ¤j. Similarly to Lemma 1 we can show the following lemma.

Lemma 3. The following three statements are equivalent.

(1) There exists a Nash equilibrium in theti sj competition game.

(2) GiventN_k,k¤i; k 2 f1; : : : ; mgandsN_l,l ¤j; l 2 fmC1; : : : ; ng, the following relation holds.

v^{t s}_i max

t_i min

sj

wi.ti;tN i; sj;sN j/Dmin

sj

maxt_i wi.ti;tN i; sj;sN j/v^{t s}_j for any pair ofi andj:

(3) There exists a real numbervt s,t_i^{t s} ands_j^{t s}such that

wi.t_i^{t s};tN i; sj;sN j/vt sfor anysj; andwi.ti;tN i; s_j^{t s};sN j/vt s for anyti

for any pair ofi andj;

Thus,t_i^{t s} D Nti ands_j^{t s} D Nsj. Denote the values ofsi; i 2 f1; 2; : : : ; mgand the values of tj; j 2 fmC1; mC2; : : : ; ng, which are derived from the following equation;

.s1; s2; : : : ; sm/D.f1.tN1;tN2; : : : ;tNm; tmC1; tmC2; : : : ; tn/; f2.: : : /; : : : ; fm.: : : //;

.tmC1; tmC2; : : : ; tn/D.g1.s1; s2; : : : ; sm;sNmC1;sNmC2; : : : ; sn/; g2.: : : /; : : : ; gm.: : : //;

bysNi; i 2 f1; 2; : : : ; mgandtNj; j 2 fmC1; mC2; : : : ; ng.

6 Equivalence of three patterns of competition

First we show the following proposition.

Proposition 1. si competition andti sj competition where one player, Player 1, choosest1

are equivalent.

Each playerj inf2; : : : ; ngchoosessj as his/her strategic variable. To prove this propo- sition we need the following lemma.

Lemma 4.

maxt1

minsj

w1.t1; sj;sN j/Dmax

s1

minsj

u1.s1; sj;sN j/:

N

s j is a vector ofsN_l forl 2 f2; : : : ; ng; l ¤j.

Proof. mins_jw1.t1; sj;sN j/is the minimum ofw1.D u1/with respect to sj givent1 and N

s j. Letsj.t1/Darg minsj w1.t1; sj;sN j/, and fix the value ofs1at

s₁⁰Df₁.t₁; g₂.s₁⁰; s₂; : : : ; s_j.t₁/; : : : ; s_n/; : : : ; g_n.s₁⁰; s₂; : : : ; s_j.t₁/; : : : ; s_n//:

Then, we have minsj

u1.s₁⁰; sj;sN j/u1.s⁰₁; sj.t1/;sN j/Dmin

sj

w1.t1; sj;sN j/;

where minsj u1.s₁⁰; sj;sN j/is the minimum of u1 with respect tosj given the value ofs1

ats₁⁰. We assume thatsj.t1/Darg mins_jw1.t1; sj;sN j/is single-valued. By the maximum theorem and continuity ofw1,sj.t1/is continuous. Then, any value ofs₁⁰in some neighbor- hood around.Ns1;sN2; : : : ;sNn/can be realized by appropriately choosingt1givensj andsN j

ass₁⁰Df1.t1; g2.s⁰₁; s2; : : : ; sj.t1/; : : : ; sn/; : : : ; gn.s₁⁰; s2; : : : ; sj.t1/; : : : ; sn//. Therefore, maxs1

minsj

u1.s1; sj;sN j/max

t1

minsj

w1.t1; sj;sN j/: (7)

On the other hand, mins_ju1.s1; sj;sN j/is the minimum ofu1with respect tosj givens1

andsN j. Letsj.s1/Darg minsj u1.s1; sj;sN j/, and fix the value oft1atg1.s1; sj.s1/;sN j/.

Then, we have

mins_j w1.t1; sj;sN j/Dmin

s_j w1.g1.s1; sj.s1/;sN j/; sj;sN j/u1.s1; sj.s1/;sN j/Dmin

s_j u1.s1; sj;sN j/;

where mins_j w1.g1.s1; sj.s1/;sN j/; sj;sN j/is the minimum ofw1.Du1/with respect tosj

given the value oft1atg1.s1; sj.s1/;sN j/. We assume thatsj.s1/Darg mins_j u1.s1; sj;sN j/ is single-valued. By the maximum theorem and continuity ofu1,sj.s1/is continuous. Then, any value oft1in some neighborhood around.tN1;sN2; : : : ;sNn/can be realized by appropriately choosings1givensj andsN j ast1Dg1.s1; sj.s1/;sN j/. Therefore,

maxt₁ min

sj

w1.t1; sj;sN j/max

s1

minsj

u1.s1; sj;sN j/: (8)

Combining (7) and (8), we get maxt1

minsj

w1.t1; sj;sN j/Dmax

s1

minsj

u1.s1; sj;sN j/:

Proof of Proposition 1. We show that the condition for .sN1; : : : ;sNn/ and the condition for .s₁; : : : s_n/are the same. From Lemma 3

maxt₁ min

sj

w1.t1; sj;sN j/Dmin

sj

maxt₁ w1.t1; sj;sN j/:

Since any value ofs1can be realized by appropriately choosingt1givensj; j ¤1, andsN j, we have maxs₁u1.s1; sj;sN j/Dmaxt₁w1.t1; sj;sN j/for anysj. Thus,

mins_j max

s1

u1.s1; sj;sN j/Dmin

s_j max

t1

w1.t1; sj;sN j/:

With Lemma 4 we conclude maxs1

minsj

u1.s1; sj;sN j/Dmin

sj

maxs1

u1.s1; sj;sN j/Du1.Ns1;sN2; : : : ;sNn/D0:

This is 2 of Lemma 1. The result of this proposition means thatw1.tN1;sNj;sN j/Dwj.tN1;sNj;sN j/D 0.

Next we show the following proposition.

Proposition 2. ti competition andti sj competition where one player, Playern, choosessn, are equivalent.

To prove this proposition we need the following lemma.

Lemma 5.

minsn

maxti

wi.ti;tN i; sn/Dmin

tn

maxti

vi.ti;tN i; tn/:

tN _i is a vector oftN_k fork2 f1; : : : ; n 1g; k ¤i.

Proof. maxt_iwi.ti;tN i; sn/is the maximum ofwi.Dvi/with respect toti givensnandtN i. Letti.sn/Darg max_tiwi.ti;tN i; sn/, and fix the value oftnat

t_n⁰Dgn.fi.ti.sn/;tN i; tn/; f i..ti.sn/;tN i; tn//; sn/;

wheref i is a vector off_kfork2 f2; : : : ; n 1g; k¤i. Then, we have maxt_i vi.ti;tN i; t_n⁰/Dmax

t_i vi.ti;tN i; gn.fi.ti.sn/;tN i; tn/; f i..ti.sn/;tN i; tn//; sn//

wi.ti.sn/;tN i; sn/Dmax

ti

wi.ti;tN i; sn/;

where max_tivi.ti;tN i; t_n⁰/is the maximum ofvi with respect toti given the value oftnat gn.fi.ti.sn/;tN i; tn/; f i..ti.sn/;tN i; tn//; sn/:

We assume thatti.sn/Darg maxt_iwi.ti;tN i; sn/is single-valued. By the maximum theorem and continuity ofwi, ti.sn/is continuous. Then, any value of t_n⁰ in some neighborhood around.tN1;tN2; : : : ;tNn/can be realized by appropriately choosingsngiventi andtN i as

t_n⁰Dgn.fi.ti.sn/;tN i; tn/; f i..ti.sn/;tN i; tn//; sn/:

Therefore,

mintn

maxt_i vi.ti;tN i; tn/min

sn

maxt_i wi.ti;tN i; sn/: (9) On the other hand, maxtivi.ti;tN i; tn/is the maximum ofviwith respect totigiventnand tN i. Letti.tn/ D arg maxt_ivi.ti;tN i; tn/, and fix the value ofsnatfn.ti.tn/;tN i; tn/. Then, we have

maxti

wi.ti;tN i; sn/Dmax

ti

wi.ti;tN i; fn.ti.tn/;tN i; tn//vi.ti.tn/;tN i; tn/Dmax

ti

vi.ti;tN i; tn/;

where max_tiwi.ti;tN i; sn/ is the maximum ofwi.D vi/ with respect toti given the value of sn atfn.ti.tn/;tN i; tn/.We assume thatti.tn/ D arg maxtivi.ti;tN i; tn/is single-valued.

By the maximum theorem and continuity ofvi,ti.tn/is continuous. Then, any value ofsn

in some neighborhood around.Nt₁;tN₂; : : : ;sN_n/can be realized by appropriately choosingt_n giventNi andtN i assnDfn.ti.tn/;tN i; tn/. Therefore,

minsn

maxt_i wi.ti;tN i; sn/min

tn

maxt_i vi.ti;tN i; tn/: (10) Combining (9) and (10), we get

minsn

maxti

wi.ti;tN i; sn/Dmin

t_n max

ti

vi.ti;tN i; tn/:

Proof of Proposition 2. We show that the condition for .Nt1; : : : ;tNn/ and the condition for .tQ1; : : : ;tQn/are the same. From Lemma 3

maxt_i min

sn

wi.ti;tN i; sn/Dmin

sn

maxt_i wi.ti;tN i; sn/:

Since any value oftncan be realized by appropriately choosingsngiventi,i ¤n, andtN i, we have minsnwi.ti;tN i; sn/Dmintnvi.ti;tN i; tn/for anyti. Thus,

maxti

mintn

vi.ti;tN i; tn/Dmax

ti

minsn

wi.ti;tN i; sn/:

From Lemma 5 we have minsnmaxt_iwi.ti;tN i; sn/ Dmintnmaxt_ivi.ti;tN i; tn/. Therefore, we obtain

maxti

mintn

vi.ti;tN i; tn/Dmin

tn

maxti

vi.ti;tN i; tn/Dvi.tN1;tN2; : : : ;tNn/D0:

This is 2 of Lemma 2. The result of this proposition means thatwi.tNi;tN i;sNn/Dwn.tNi;tN i;sNn/D 0.

Finally we show the following proposition.

Proposition 3. ti sj competition in whichmplayers chooseti’s as their strategic variables, and ti sj competition in whichm 1players chooseti’s as their strategic variables are equivalent, where2mn 1.

To prove this proposition we need the following lemma.

Lemma 6.

maxti

minsj

wi.ti;tN i; sj;sN j/Dmax

s_i min

sj

wi.si;tN i; sj;sN j/:

tN i is a vector of tNk for k 2 f1; : : : ; mg; k ¤ i. sN j is a vector of sNl for l 2 fmC 1; : : : ; ng; l ¤j.

Proof. minsjwi.ti;tN i; sj;sN j/is the minimum ofwi.Dui/with respect tosj giventi,tN i

andsN j. Letsj.ti/ D arg mins_jwi.ti;tN i; sj;sN j/. The values of variables other thanti, sj.ti/,tN i andsN j are determined by the following equations;

s1Df1.tN1; : : : ; ti; : : : ;tNm; tmC1; : : : ; tj; : : : ; tn/;

: : : ;

si Dfi.tN1; : : : ; ti; : : : ;tNm; tmC1; : : : ; tj; : : : ; tn/;

: : : ;

smDfm.tN1; : : : ; ti; : : : ;tNm; tmC1; : : : ; tj; : : : ; tn/;

tmC1DgmC1.s1; : : : ; si; : : : ; sm;sNmC1; : : : ; sj.ti/; : : : ;sNn/;

: : : ;

tj Dgj.s1; : : : ; si; : : : ; sm;sNmC1; : : : ; sj.ti/; : : : ;sNn/;

: : : ;

tnDgn.s1; : : : ; si; : : : ; sm;sNmC1; : : : ; sj.ti/; : : : ;sNn/:

Denote thissi bys_i⁰, and fix the value ofsi ats_i⁰. Then, we have mins_j wi.s_i⁰;tN i; sj;sN j/wi.ti;tN i; sj.ti/;sN j/Dmin

s_j wi.ti;tN i; sj;sN j/;

where min_sjw_i.s⁰_i;tN _i; s_j;sN _j/ is the minimum ofw_i.D u_i/ with respect to s_j given the value ofsi ats_i⁰. We assume thatsj.ti/Darg minsjwi.ti;tN i; sj;sN j/is single-valued. By the maximum theorem and continuity ofwi, sj.ti/is continuous. Then, any value of s_i⁰ in some neighborhood around.sNi;tN i;sNj;sN j/can be realized by appropriately choosingti

givensj,tN i andsN j. Therefore, maxsi

mins_j wi.si;tN i; sj;sN j/max

ti

mins_j wi.ti;tN i; sj;sN j/: (11) On the other hand, mins_jwi.si;tN i; sj;sN j/is the minimum ofwi.Dui/with respect to sj givensi,tN i andsN j. Letsj.si/Darg mins_j wi.si;tN i; sj;sN j/. The values of variables other thansi,sj.si/,tN i andsN j are determined by the following equations;

s1Df1.tN1; : : : ; ti; : : : ;tNm; tmC1; : : : ; tj; : : : ; tn/;

: : : ;

smDfm.tN1; : : : ; ti; : : : ;tNm; tmC1; : : : ; tj; : : : ; tn/;

tmC1DgmC1.s1; : : : ; si; : : : ; sm;sNmC1; : : : ; sj.si/; : : : ;sNn/;

: : : ;

ti Dgi.s1; : : : ; si; : : : ; sm;sNmC1; : : : ; sj.si/; : : : ;sNn/;

tj Dgj.s1; : : : ; si; : : : ; sm;sNmC1; : : : ; sj.si/; : : : ;sNn/;

: : : ;

tnDgn.s1; : : : ; si; : : : ; sm;sNmC1; : : : ; sj.si/; : : : ;sNn/:

Denote thisti byt_i⁰, and fix the value ofti att_i⁰. Then, we have mins_j wi.t_i⁰;tN i; sj;sN j/wi.si;tN i; sj.si/;sN j/Dmin

s_j wi.si;tN i; sj;sN j/;

where mins_jwi.t_i⁰;tN i; sj;sN j/ is the minimum of wi.D ui/ with respect tosj given the value ofti att_i⁰. We assume thatsj.si/Darg minsjwi.si;tN i; sj;sN j/is single-valued. By the maximum theorem and continuity ofwi, sj.si/ is continuous. Then, any value oft_i⁰ in some neighborhood around.tNi;tN i;sNj;sN j/can be realized by appropriately choosingsi

givensj,sN j andtN i. Therefore, maxt_i min

sj

wi.ti;tN i; sj;sN j/max

s_i min

sj

wi.si;tN i; sj;sN j/: (12) Combining (11) and (12), we get

maxt_i min

sj

w_i.t_i;tN _i; s_j;sN _j/Dmax

s_i min

sj

w_i.s_i;tN _i; s_j;sN _j/:

Proof of Proposition 3. We show that the condition for.tNi;tN i;sNj;sN j/ and the condition for.sNi;tN i;sNj;sN j/are the same. From Lemma 3

maxti

minsj

wi.ti;tN i; sj;sN j/Dmin

sj

maxti

wi.ti;tN i; sj;sN j/:

Since any value ofsi can be realized by appropriately choosingti givensj,tN i,sN j, we have maxtiwi.ti;tN i; sj;sN j/Dmaxsiwi.si;tN i; sj;sN j/for anysj. Thus,

mins_j max

ti

wi.ti;tN i; sj;sN j/Dmin

s_j max

s_i wi.si;tN i; sj;sN j/:

With Lemma 6 we conclude maxs_i min

s_j wi.si;tN i; sj;sN j/Dmax

s_i min

s_j wi.si;tN i; sj;sN j/:

Summarizing these results we conclude.

Proposition 4. s_icompetition,t_icompetition andt_i s_jcompetition with any number of players whose strategic variables areti’s are equivalent, and payoffs of all players at any equilibrium are zero.

Proof. From Proposition 1

w1.tN1;sNj;sN j/Dwj.tN1;sNj;sN j/D0; j 2 f2; 3; : : : ; ng:

This means that the payoffs of all players when only one player chooses ti and all other players choosesj’s are zero. From Proposition 2

wn.tNi;tN i;sNn/Dwi.tNi;tN i;sNn/D0; i 2 f1; 2; : : : ; n 1g:

This means that the payoffs of all players when only one player choosessj and all other players chooseti’s are zero. From the result of Proposition 3

wi.tN1;tN2;sNj;sN j/Dwj.tN1;tN2;sNj;sN j/D0; i 2 f1; 2g; j 2 f3; 4; : : : ; ng:

This means that the payoffs of all players when two players chooseti’s and all other players choosesj’s are zero. Then, inductively we conclude that

wi.tNi;tN i;sNj;sN j/Dwj.tNi;tN i;sNj;sN j/D0;

in the game wheremplayers chooseti’s as their strategic variables for anymsuch that2 m n 1. i denotes a player whose strategic variable isti, andj denotes a player whose strategic variable issj. Thus, payoffs of all players in anyti sj competition are zero. By the definitions ofsi competition andti competition payoffs of all players in thesi competition and theti competition are zero.

7 Example: relative profit maximization in oligopoly with differentiated goods

Consider an oligopoly with three firms producing differentiated goods. The firms are A, B and C. The inverse demand functions are

pADa xA bxB bxC;

pB Da xB bxA bxC;

and

p_C Da x_C bx_A bx_B;

where0 < b < 1. p_A, p_B and p_C are the prices of the goods of Firm A, B and C, and xA, xB andxC are the outputs of them. From these inverse demand functions the direct demand functions are derived as follows;

xAD .1 b/a .1Cb/pACb.pACpC/

.1 b/.1C2b/ ;

xB D .1 b/a .1Cb/p_BCb.p_BCp_C/

.1 b/.1C2b/ ;

and

x_C D .1 b/a .1Cb/pC Cb.pACpB/

.1 b/.1C2b/ :

The (absolute) profits of the firms are

ADpAxA cAxA; B DpBxB cBxB; and

C DpCxC cCxC:

c_A,c_B andc_C are the constant marginal costs of Firm A, B and C. The relative profits of the firms are

'_AD_A BCC

2 ;

'B DB

_AC_C

2 ;

and

'C DC

_AC_B

2 :

We see

'AC'BC'C D0;

so . the game is zero-sum. In a Cournot model the firms determine their outputs to maxi- mize their relative profits. In a Bertrand model they determine the prices of their goods to maximize their relative profits. The Cournot equilibrium outputs are

x_A^C D bc_C Cbc_B bc_A 4c_A abC4a

.4 b/.2Cb/ ;

x_B^C D bcC bcB 4cBCbcA abC4a

.4 b/.2Cb/ ;

x^C_C D bcB bcC 4cC CbcA abC4a

.4 b/.2Cb/ :

The Bertrand equilibrium prices are

p^B_A D 3b²cC C3bcC C3b²cBC3bcBC4b²cAC7bcAC4cA 5ab²CabC4a

.2Cb/.4C5b/ ;

p_B^B D 3b²cC C3bcC C4b²cBC7bcBC4cBC3b²cAC3bcA 5ab²CabC4a

.2Cb/.4C5b/ ;

p_C^B D 4b²cC C7bcC C4cC C3b²cBC3bcBC3b²cAC3bcA 5ab²CabC4a

.2Cb/.4C5b/ ;

The difference between the relative profit in the Bertrand equilibrium and that in the Cournot equilibrium for each of Firm A, B, C is

'_A^B '_A^C D 9b³.bC2/.c_C² 4cBcC C2cAcC Cc²_BC2cAcB 2c_A²/

.b 4/².b 1/.5bC4/² ;

'_B^B '_B^C D 9b³.bC2/.c_C² C2c_Bc_C 4c_Ac_C 2c²_BC2c_Ac_BCc_A²/

.b 4/².b 1/.5bC4/² ;

and

'_C^B '^C_C D 9b³.bC2/.2c_C² 2cBcC 2cAcC c²_BC4cAcB c_A²/

.b 4/².b 1/.5bC4/² :

If and only ifcADcB DcC, we have'_A^C D'_A^B,'_B^C D'_B^B,'_C^C D'_C^B. Thus, in a symmetric oligopoly Cournot and Bertrand equilibria coincide. However, in an asymmetric oligopoly they do not coincide. For example, ifcB D cC butcA > cB, the difference between the relative profit in the Bertrand equilibrium and the relative profit in the Cournot equilibrium for each firm is

'_A^B '_A^C D 18b³.bC2/.cB cA/² .b 4/².b 1/.5bC4/² < 0;

'_B^B '_B^C D 9b³.bC2/.cB cA/² .b 4/².b 1/.5bC4/² > 0;

and

'_C^B '_C^C D 9b³.bC2/.cB cA/² .b 4/².b 1/.5bC4/² > 0:

References

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