Munich Personal RePEc Archive
Stackelberg equilibrium of dynamic
symmetric multi-players zero-sum game with a leader and followers without
differentiability of payoff functions
Tanaka, Yasuhito
2 February 2019
Online at https://mpra.ub.uni-muenchen.de/91898/
MPRA Paper No. 91898, posted 13 Feb 2019 14:45 UTC
Stackelberg equilibrium of dynamic symmetric multi-players zero-sum
game with a leader and followers without differentiability of payoff
functions
Yasuhito Tanaka
†Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto, 602-8580, Japan.
Abstract
This paper studies a Stackelberg type symmetric dynamicn-players zero-sum game.
There is one leader andn 1followers. Players have the symmetric payoff functions.
The game is a two-stages game. In the first stage the leader determines the value of its strategic variable. In the second stage the followers determine the values of their strategic variables given the value of the leader’s strategic variable. In the static game, on the other hand, all players simultaneously determine the values of their strategic variable. We do not assume differentiability of payoff functions. This paper shows that the sub-game perfect equilibrium of the Stackelberg type symmetric dynamic zero-sum game is equivalent to the equilibrium of the static game if and only if the game is fully symmetric.
Keywords: Stackelberg equilibrium, leader, follower, dynamic symmetric zero-sum game.
1 Introduction
We examine the relation between the Stackelberg equilibrium of dynamic game and the equi- librium of the static game in a multi-players zero-sum game, and show that the Stackelberg equilibrium of a dynamic zero-sum game and the equilibrium of the static zero-sum game are equivalent if and only if the game is fully symmetric. The Stackelberg equilibrium of dynamic
This work was supported by Japan Society for the Promotion of Science KAKENHI Grant Number 15K03481 and 18K01594.
†yatanaka@mail.doshisha.ac.jp
game and the equilibrium of the static game are equivalent in a two-person zero-sum game1. We extend this analysis to more general multi-players zero-sum game. We do not assume differentiability of payoff functions. However, we do not assume that the payoff functions are notdifferentiable. We do not use differentiability of payoff functions.
In the next section, using a model of relative profit maximization in an oligopoly with four firms, we show that the Stackelberg equilibrium is not equivalent to the static (Cournot) equilibrium in the following cases which are not fully symmetric.
1. All firms are asymmetric, that is, they have different cost functions.
2. Two followers are symmetric, that is, they have the same cost functions.
3. Three followers are symmetric.
4. The leader and one follower are symmetric.
5. The leader and two followers are symmetric.
The Stackelberg equilibrium is equivalent to the static (Cournot) equilibrium if and only if all firms are symmetric, that is, they have the same cost functions.
In Section 3 we show the main result. All players have symmetric payoff functions. One player is the leader and other players are followers. The game is a two-stages game as follows;
1. In the first stage the leader determines the value of its strategic variable.
2. In the second stage the followers determine the values of their strategic variables given the value of the leader’s strategic variable.
On the other hand, in the static game all players simultaneously determine the values of their strategic variables. We show that the equilibrium of the Stackelberg type dynamic game and the equilibrium of the static game are equivalent if the game is fully symmetric.
2 Example: relative profit maximization in a Stackelberg oligopoly
In the example in this section we consider relative profit maximization in an oligopoly2.
1Please see, for example, Korzhyk et. al. (2014), Ponssard and Zamir (1973), Tanaka (2014) and Yin et. al.
(2010)
2About relative profit maximization in an oligopoly see Matsumura, Matsushima and Cato (2013), Vega- Redondo (1997), Satoh and Tanaka (2014a) and Satoh and Tanaka (2014b). In this example payoff functions are differentiable.
2.1 Case 1: four firms are different each other
Suppose a four firms Stackelberg oligopoly with a homogeneous good. There are Firms A, B, C and D. The outputs of the firms arexA, xB, xC andxD. The price of the good isp. The inverse demand function is
p Da xA xB xC xD; a > 0:
The cost functions of the firms arecAxA2, cBxB2, cCxC2 and cDxD2. cA, cB, cC and cD are positive constants. We assume thatcA, cB, cC and cD are different each other. The relative profit of Firm A is
'ADpxA cAxA2 1
3.pxB cBxB2 CpxC cCx2C CpxD cDxD2/:
The relative profit of Firm B is 'B DpxB cBxB2 1
3.pxA cAxA2 CpxC cCxC2 CpxD2 cDxD2/:
The relative profit of Firm C is 'C DpxC cCxC2 1
3.pxA cAxA2 CpxB cBx2BCpxD cDxD2/;
The relative profit of Firm D is 'D DpxD cDxD2 1
3.pxA cAxA2 CpxB cBxB2 CpxC cCxC2/:
The firms maximize their relative profits. We see
'AC'BC'C C'D D0:
Thus, the game is a zero-sum game. Firm A is the leader and Firms B, C and D are followers.
In the first stage of the game Firm A determinesxA, and in the second stage Firms B, C and D determinexB,xC andxD givenxA.
Nash equilibrium of the static game
The equilibrium outputs are
xA D acB.cC.27cDC18/C18cDC12/Ca.cC.18cDC12/C12cDC8/
1
; xB D acC.cA.27cD C18/C18cD C12/Ca.cA.18cD C12/C12cD C8/
1
; xC D acB.cA.27cD C18/C18cD C12/Ca.cA.18cD C12/C12cD C8/
1
; xD D acB..27cAC18/cC C18cAC12/Ca..18cAC12/cC C12cAC8/
1
; where
1DcB.cC.cA.54cDC54/C54cD C48/CcA.54cD C48/C48cD C40/
CcC.cA.54cDC48/C48cDC40/CcA.48cDC40/C40cD C32:
Sub-game perfect equilibrium of the dynamic game
The equilibrium outputs are xA D 1
1s
Œ3a.3cBC2/.3cC C2/.3cDC2/.27cBcCcDC27cCcDC27cBcDC20cD
C27cBcC C20cC C20cBC12/;
xB D 1
1s
Œ3a.3cC C2/.3cD C2/.81cAcBcCcDC54cBcCcDC81cAcCcD
C42cCcD C81cAcBcD C42cBcDC72cAcD C32cDC81cAcBcC C42cBcC C72cAcC
C32cC C72cAcBC32cBC60cAC24/;
xC D 1
1s
Œ3a.3cBC2/.3cD C2/.81cAcBcCcDC54cBcCcDC81cAcCcD
C42cCcD C81cAcBcDC42cBcD C72cAcDC32cD C81cAcBcC C42cBcC C72cAcC
C32cC C72cAcBC32cB C60cAC24/;
xD D 1
1s
Œ3a.3cB C2/.3cC C2/.81cAcBcCcDC54cBcCcDC81cAcCcD C42cCcD C81cAcBcD
C42cBcDC72cAcDC32cDC81cAcBcC C42cBcC C72cAcC C32cC C72cAcBC32cB
C60cAC24/;
where
1s D2.2187cAcB2cC2cD2 C2187cB2cC2cD2 C4374cAcBcC2cD2 C3807cBcC2cD2
C2187cAcC2cD2 C1620cC2cD2 C4374cAcB2cCcD2 C3807cB2cCcD2 C8262cAcBcCcD2
C6264cBcCcD2 C3888cAcCcD2 C2556cCcD2 C2187cAcB2cD2 C1620cB2cD2 C3888cAcBcD2 C2556cBcD2 C1728cAcD2 C1008cD2 C4374cAcB2cC2cDC3807cB2cC2cD C8262cAcBcC2cD
C6264cBcC2cD C3888cAcC2cD C2556cC2cD C8262cAcB2cCcDC6264cB2cCcD
C14904cAcBcCcDC9936cBcCcD C6696cAcCcDC3936cCcD C3888cAcB2cD C2556cB2cD
C6696cAcBcD C3936cBcD C2880cAcD C1520cD C2187cAcB2cC2 C1620cB2cC2 C3888cAcBcC2 C2556cBcC2 C1728cAcC2 C1008cC2 C3888cAcB2cC C2556cB2cC
C6696cAcBcC C3936cBcC C2880cAcC C1520cC C1728cAcB2 C1008cB2 C2880cAcB C1520cB C1200cAC576/:
The Nash equilibrium of the static game and the sub-game perfect equilibrium of the dynamic game are not equivalent.
2.2 Case 2: the leader and one follower are symmetric
AssumecD DcA.
Nash equilibrium of the static game
The equilibrium outputs are
xA D a.3cB C2/.3cC C2/
2.9cAcBcC C9cBcC C9cAcC C8cC C12cAcBC10cBC10cAC8/;
xB D a.3cAC2/.3cC C2/
2.9cAcBcC C9cBcC C9cAcC C8cC C12cAcB C10cBC10cAC8/;
xC D a.3cAC2/.3cB C2/
2.9cAcBcC C9cBcC C9cAcC C8cC C12cAcBC10cBC10cAC8/;
xD D a.3cAC2/.3cBC2/
2.9cAcBcC C9cBcC C9cAcC C8cC C12cAcBC10cB C10cAC8/:
Sub-game perfect equilibrium of the dynamic game
The equilibrium outputs are
xA D 3a.3cBC2/.27cBcC2 C27cC2 C54cBcC C40cC C20cBC12/
2
;
xB D 3a.3cC C2/.27cAcBcC C18cBcC C27cAcC C14cC C36cAcBC16cB C30cAC12/
2
; xC D 3a.3cB C2/.27cAcBcC C18cBcC C27cAcC C14cC C36cAcBC16cB C30cAC12/
2
; xD D 3a.3cB C2/.27cAcBcC C18cBcC C27cAcC C14cC C36cAcBC16cB C30cAC12/
2
; where
2 D2.243cAcB2cC2 C243cB2cC2 C486cAcBcC2 C423cBcC2 C243cAcC2
C180cC2 C648cAcB2cC C522cB2cC C1188cAcBcC C828cBcC C540cAcC C328cC
C432cAcB2 C252cB2 C720cAcB C380cB C300cAC144/:
The Nash equilibrium of the static game and the sub-game perfect equilibrium of the dynamic game are not equivalent.
2.3 Case 3: two followers are symmetric
AssumecD DcC.
Nash equilibrium of the static game
The equilibrium outputs are
xA D a.3cB C2/.3cC C2/
2.9cAcBcC C12cBcC C9cAcC C10cC C9cAcBC10cBC8cAC8/;
xB D a.3cAC2/.3cC C2/
2.9cAcBcC C12cBcC C9cAcC C10cC C9cAcB C10cBC8cAC8/;
xC D a.3cAC2/.3cB C2/
2.9cAcBcC C12cBcC C9cAcC C10cC C9cAcBC10cBC8cAC8/;
xD D a.3cBC2/.3cC C2/
2.9cAcBcC C12cBcC C9cAcC C10cC C9cAcBC10cB C8cAC8/:
Sub-game perfect equilibrium of the dynamic game
The equilibrium outputs are
xAD 1
3
Œ3a.3cAC2/.3cBC2/.3cC C2/.27cAcBcC C27cBcC C27cAcC
C20cC C27cAcBC20cBC20cAC12/;
xB D 1
3
Œ3a.3cAC2/.3cC C2/.81cA2cBcC C135cAcBcC C42cBcC C81cA2cC C114cAcC C32cC
C81cA2cBC114cAcB C32cB C72cA2 C92cAC24/;
xC D 1
3
Œ3a.3cAC2/.3cB C2/.81cA2cBcC C135cAcBcC C42cBcC C81cA2cC C114cAcC C32cC
C81cA2cB C114cAcBC32cBC72cA2 C92cAC24/;
xD D 1
3
Œ3a.3cB C2/.3cC C2/.81cA2cBcC C135cAcBcC C42cBcC C81cA2cC C114cAcC C32cC
C81cA2cB C114cAcBC32cBC72cA2 C92cAC24/;
where
3 D2.2187cA3cB2cC2 C6561cA2cB2cC2 C5994cAcB2cC2 C1620cB2cC2
C4374cA3cBcC2 C12069cA2cBcC2 C10152cAcBcC2 C2556cBcC2 C2187cA3cC2
C5508cA2cC2 C4284cAcC2 C1008cC2 C4374cA3cB2cC C12069cA2cB2cC C10152cAcB2cC
C2556cB2cC C8262cA3cBcC C21168cA2cBcC C16632cAcBcC C3936cBcC C3888cA3cC
C9252cA2cC C6816cAcC C1520cC C2187cA3cB2 C5508c2AcB2 C4284cAcB2 C1008cB2 C3888cA3cBC9252cA2cBC6816cAcBC1520cBC1728cA3 C3888cA2 C2720cAC576/:
The Nash equilibrium of the static game and the sub-game perfect equilibrium of the dynamic game are not equivalent.
2.4 Case 4: the leader and two followers are symmetric
AssumecD DcC DcA.
Nash equilibrium of the static game
The equilibrium outputs are
xAD a.3cBC2/
2.3cAcBC5cB C3cAC4/; xB D a.3cAC2/
2.3cAcBC5cBC3cAC4/;
xC D a.3cBC2/
2.3cAcB C5cBC3cAC4/; xD D a.3cBC2/
2.3cAcBC5cB C3cAC4/:
Sub-game perfect equilibrium of the dynamic game
The equilibrium outputs are
xAD 3a.3cBC2/.27cA2cBC54cAcBC20cB C27cA2 C40cAC12/
4
; xB D 3a.3cAC2/.27cA2cB C54cAcBC16cBC27cA2 C44cAC12/
4
; xC D 3a.3cBC2/.27cA2cBC54cAcB C16cBC27cA2 C44cAC12/
4
; xD D 3a.3cBC2/.27cA2cBC54cAcBC16cBC27cA2 C44cAC12/
4
; where
4 D2.243cA3cB2 C891cA2cB2 C954cAcB2 C252cB2 C486cA3cBC1611cA2cB C1548cAcB
C380cBC243cA3 C720cA2 C628cAC144/:
The Nash equilibrium of the static game and the sub-game perfect equilibrium of the dynamic game are not equivalent.
2.5 Case 5: three followers are symmetric
AssumecD DcC DcB.
Nash equilibrium of the static game
The equilibrium outputs are
xAD a.3cBC2/
2.3cAcBC3cB C5cAC4/; xB D a.3cAC2/
2.3cAcBC3cBC5cAC4/;
xC D a.3cAC2/
2.3cAcB C3cBC5cAC4/; xD D a.3cAC2/
2.3cAcBC3cB C5cAC4/:
Sub-game perfect equilibrium of the dynamic game
The equilibrium outputs are
xA D 3a.cBC2/.3cBC1/
2.9cAcB2 C9cB2 C30cAcB C23cBC25cAC12/; xB D 3a.3cAcB C2cB C5cAC2/
2.9cAcB2 C9cB2 C30cAcBC23cBC25cAC12/; xC D 3a.3cAcBC2cBC5cAC2/
2.9cAcB2 C9cB2 C30cAcBC23cBC25cAC12/; xD D 3a.3cAcBC2cBC5cAC2/
2.9cAcB2 C9cB2 C30cAcBC23cBC25cAC12/:
The Nash equilibrium of the static game and the sub-game perfect equilibrium of the dynamic game are not equivalent.
2.6 Case 6: all firms are symmetric
AssumecB DcC DcD DcA.
Nash equilibrium of the static game
The equilibrium outputs are
xAD a
2.cAC2/; xB D a
2.cAC2/; xC D a
2.cAC2/; xD D a 2.cAC2/:
Sub-game perfect equilibrium of the dynamic game
The equilibrium outputs are
xAD a
2.cAC2/; xB D a
2.cAC2/; xC D a
2.cAC2/; xD D a 2.cAC2/:
The Nash equilibrium of the static game and the sub-game perfect equilibrium of the dynamic game are equivalent.
3 Symmetric dynamic zero-sum game
There is ann-players and two-stages game. Players are called Player 1, 2,: : :,n. The strategic variable of Player i is si; i 2 f1; 2; : : : ; ng . The set of strategic variable of Player i is Si; i 2 f1; 2; : : : ; ng, which is a convex and compact set of a linear topological space. One of players is the leader and other players are followers.
The structure of the game is as follows.
1. The first stage
The leader determines the value of its strategic variable.
2. The second stage
Followers determine the values of their strategic variables given the value of the leader’s strategic variable.
Thus, the game is a Stackelberg type dynamic game. We investigate a sub-game perfect equilibrium of this game.
On the other hand, there is a static game in which all players simultaneously determine the values of their strategic variables.
The payoff of Playeriis denoted byui.s1; s2; : : : ; sn/. ui is jointly continuous insi and all sj; j ¤i. We assume
n
X
iD1
ui.s1; s2; : : : ; sn/D0given.s1; s2; : : : ; sn/:
Therefore, the game is a zero-sum game.
We assume that the game is symmetric in the sense that the payoff functions of all players are symmetric, and assume that the sets of strategic variables for all players are the same. Denote them byS. We do not assume differentiability of players’ payoff functions3.
We show the following theorem
Theorem 1. The sub-game perfect equilibrium of the symmetric Stackelberg type dynamic zero-sum game is equivalent to the equilibrium of the static game.
Proof. (1) Suppose that the leader is Player 1. Let.s2.s1/; s3.s1/; : : : ; sn.s1//be a solution of the following equation;
8 ˆˆ ˆ<
ˆˆ ˆ:
s2.s1/Darg maxs22Su2.s1; s2; s3.s1/; : : : sn.s1//
s3.s1/Darg maxs32Su3.s1; s2.s1/; s3; : : : sn.s1//
: : :
sn.s1/Darg maxsn2Sun.s1; s2.s1/; s3.s1/; : : : sn/
3As we said in the introduction, we do not assume that the payoff function isnotdifferentiable. We do not use differentiability of payoff functions.
givens1. Assume that arg maxsi2Sui.s1; s2.s1/; : : : ; si; : : : ; sn.s1//fori 2 f2; 3; : : : ; ng are unique. SinceS is compact, ui.s1; s2; : : : ; sn/ for alli 2 f1; 2; : : : ; ng are jointly continuous, by the maximum theorem s2.s1/, s3.s1/, : : :, sn.s1/ are continuous. We have
maxsi2Sui.s1; s2.s1/; : : : ; si; : : : ; sn.s1// Dui.s1; s2.s1/; : : : ; si.s1/; : : : ; sn.s1//; i 2 f2; : : : ; ng:
By symmetry of the game
s2.s1/Ds3.s1/D Dsn.s1/;
and
u2.s1; s2.s1/; : : : ; sn.s1// Du3.s1; s2.s1/; : : : ; sn.s1//D Dun.s1; s2.s1/; : : : ; sn.s1//;
given s1. s1.s2/, s3.s2/, : : :, sn.s2/, s2.s3/, : : :, sn.s3/, : : :, s1.sn/, : : :, sn 1.sn/are similarly defined. By symmetry of the game we have
s1.s2/Ds3.s2/D Dsn.s2/; s2.s3/D Dsn.s3/; : : : ; s1.sn/D Dsn 1.sn/:
s2.s1/is also obtained as a fixed point of the following function max
s2S u2.s1; s; s2.s1/; : : : ; s2.s1//:
(2) The Nash equilibrium of the static game is obtained as a fixed point of a function from SntoSn;
0 B B
@
arg maxs12Su1.s1; s.1/; s.2/; : : : ; s.n// arg maxs22Su2.s.1/; s2; : : : ; s.n//
: : :
arg maxsn2Sun.s.1/; s.2/; : : : ; sn/ 1 C C A :
By symmetry of the game for all players we assume that s1 D s2 D D sn at the equilibrium. Denote the equilibrium by.Qs;s; : : : ;Q s/.Q sQis also obtained as a fixed point of the following function.
maxs2S u1.s;s;Q s; : : : ; ;Q s/:Q
We assume uniqueness of the Nash equilibrium of the static game. At the equilibrium of the static game.Qs;s; : : : ;Q s/, we haveQ
u1.s;Q s; : : : ;Q s/ > uQ 1.s;s; : : : ;Q s/Q for anys 2S; s ¤ Qs; (1) and
u1.Qs;s; : : : ;Q s/Q D0:
Similarly,
ui.Qs; : : : ;s; : : : ;Q s/ > uQ i.Qs; : : : ;s; : : : ; s; : : : ;Q s; : : : ;Q s/Q for anys 2S; s ¤ Qs; si Ds;
ui.Qs; : : : ;s; : : :Q s/Q D0 fori 2 f2; : : : ; ng. Note that
s2.Qs/Darg max
s22Su2.Qs; s2;s; : : : ;Q s/Q D Qs;
and so on. Since the game is zero-sum and symmetric, we have u1.s;s; : : : ;Q s/Q D .n 1/u2.s;s; : : : ;Q s/:Q Thus, (1) means
u2.s;s; : : : ;Q s/ > 0:Q By symmetry, we get
u1.s; s;Q s; : : : ;Q s/ > 0:Q Therefore,
u1.s;s; : : : ;Q s/ < 0 < uQ 1.Qs; s;s; : : : ;Q s/:Q (2) Similarly,
u1.s;s; : : : ;Q s/ < 0 < uQ 1.Qs; : : : ;s; s;Q s; : : : ;Q s/; sQ i Ds; i 2 f3; 4; : : : ; ng: (3) Also we have
ju1.s;s; : : : ;Q s/j DQ .n 1/ju1.s; s;Q s; : : : ;Q s/j:Q (4) (3) The equilibrium strategy of Player 1 in the dynamic game is written as
arg max
s12Su1.s1; s2.s1/; : : : ; sn.s1//:
Let
s1Darg max
s12Su1.s1; s2.s1/; : : : ; sn.s1//:
.s1; s2.s1/; : : : ; sn.s1// is a Stackelberg equilibrium of the dynamic game when Player 1 is the leader. We assume uniqueness of the Stackelberg equilibrium. Similarly, we get sisuch that
siDarg max
si2S ui.s1.si/; : : : ; si 1.si/; si; siC1.si/; : : : ; sn.si//:
By symmetry of the game
s1Ds2D Dsn: Denote them bys.
(4) Since, by symmetry for Players 2 ton,sn.s/Dsn 1.s/ D Ds2.s/for anys, we have sDarg max
s2S u1.s; s2.s/; : : : ; s2.s//:
This is equivalent to
u1.s; s2.s/; : : : ; s2.s// > u1.s; s2.s/; : : : ; s2.s//for anys 2S; s ¤s: Suppose a state such thats1 Ds2 D D Qs. From (2) and (3), fors ¤ Qs,
u1.s;s; : : : ;Q s/ < 0; : : : ; uQ 1.Qs; : : : ;s; s;Q s; : : : ;Q s/ > 0 .sQ i Ds/; : : : ; u1.Qs; : : : ;s; s/ > 0 .sQ nDs/:
Since u1.s1; s2; : : : ; sn/ is jointly continuous, there exists a neighborhood V0.Qs/ of sQ such that, fors02 V0.s/; sQ 0 ¤ Qs
ju1.Qs; s0; : : : ; s0/j <ju1.s; s;Q s; : : : ;Q s/j;Q and
u1.Qs; s0; : : : ; s0/ > 0;
fors which satisfies (2) and (3). Since the game is zero-sum,
u1.Qs; s0; : : : ; s0/Cu2.Qs; s0; : : : ; s0/Cu3.Qs; s0; : : : ; s0/C Cun.Qs; s0; : : : ; s0/D0:
By symmetry
u1.Qs; s0; : : : ; s0/D .n 1/u2.Qs; s0; : : : ; s0/D .n 1/u1.s0;s; sQ 0; : : : ; s0/:
Thus,
u1.s0;s; sQ 0; : : : ; s0/ < 0 .s2 D Qs/; : : : ; u1.s0; s0; : : : ; s0;s/ < 0 .sQ nD Qs/:
Also we have
ju1.Qs; s0; : : : ; s0/j D.n 1/ju1.s0;s; sQ 0; : : : ; s0/j:
Sinceu1.s1; s2; : : : ; sn/is jointly continuous, ifV .Qs/is sufficiently small, we can assume ju1.Qs; s0; : : : ; s0/ u1.Qs;s; : : : ;Q s/j Q .n 1/ju1.Qs; s0;s; : : : ;Q s/Q u1.Qs;s; : : : ;Q s/j:Q or
ju1.Qs; s0; : : : ; s0/j .n 1/ju1.Qs; s0;s; : : : ;Q s/j:Q Consequently, from (4)
ju1.Qs; s0; : : : ; s0/j ju1.s0;s; : : : ;Q s/j:Q There exists a neighborhoodV .Qs/ofsQ such that fors 2V .Qs/
ju1.s; s2.s/; : : : ; s2.s//j<ju1.s0;s;Q s; : : : ;Q s/j;Q fors0 2V0.Qs/:
It seems to be that
js2.s/ sjQ <js sj:Q Since
u1.s;s; : : : ;Q s/ < 0;Q and
u1.Qs; s2.s/; : : : ; s2.s// > 0;
we get
u1.s; s2.s/; : : : ; s2.s// < 0:
This means
u1.Qs;s; : : : ;Q s/ > uQ 1.s; s2.s/; : : : ; s2.s//; fors 2V .s/:
Thus,.Qs;s; : : : ;Q s/Q is the Stackelberg equilibrium.
We have completed the proof.
4 Concluding Remark
As we said in the introduction, the equivalence of the Stackelberg type dynamic game and the static game in a two-players zero-sum game is a widely known result. But, this problem in a multi-players case has not been analyzed. In this paper we have analyzed a generaln-players game.
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