Korean J. Com. & Appl. Math. Vol. 3(1996), No. 2, pp. 149 - 156

ON EQUILIBRIUM POINTS IN BIMATRIX GAMES

Hun Kuk

Abstract. We discuss sensitivity of equilibrium points in bimatrix

games depending on small variances (perturbations) of data. Applyingimplicit function theorem to a linear complementarity problem which

is equivalent to the bimatrix game, we investigate sensitivity of equi-librium points with respect to the perturbation of parameters in the

game. Namely, we provide the calculation of equilibrium points deriva-

tives with respect to the parameters.

AMS Mathematics Subject Classification : 90D10, 90C31.

1. Introduction

Extremum problems as well as game usually depend on small vari-ances (perturbations) of data of these problems. Parameters that arenot known exactly appear to be a characteristic feature of mathematicalmedels of complicated systems. Therefore sensitivity analysis, i.e., thequantitative analysis of influence of data perturbations is an importantpart in optimization problems, economics and others.

This paper investigate sensitivity analysis of (Nash) equilibrium poi-nts in bimatrix game (two-person noncooperate game) depend on per-turbations of parameters. Applying implicit function theorem to linearcomplementarity problem which is equivalent to the bimatrix game, weprovide sensitivity of equilibrium points with respect to the perturba-tion of parameters.

2. Bimatrix Games and Linear Complementarity Problems

Received May 24, 1996.c© 1996 Korean SIGCOAM, Korea Information Processing Society

..149..

150 Hun Kuk

Now we introduce concept of equilibrium point in bimatrix gamesand describe well-known correspondence between equilibrium points ofbimatrix games and solutions of linear complementarity problem.

Let A = (aij) and B = (bij) be two m×n matrices. The two-persongame in normal form < S1, S2; EA, EB >, with

S1 =

{s1 ∈ Rm : Si

1 = 0, i = 1, · · · , m,

m∑

i=1

si1 = 1

},

S2 =

s2 ∈ Rn : Sj

2 = 0, j = 1, · · · , n,n∑

j=1

sj2 = 1

,

EA(s1, s2) = s1AsT2 ,

EB(s1, s2) = s1BsT2 ,

for each s1 ∈ S1, s2 ∈ S2, is called the m × n-bimatrix game cor-responding to the ordered pair of matrices A and B and this game isdenoted by (A, B). The matrices A and B are called the payoff matricesof player 1 and player 2, respectively. The class of all m × n-bimatrixgames is denoted by M2

m×n.

Definition 2.1. A pair (s1, s2) ∈ S1 × S2 is called the (Nash) equilib-rium point of the bimatrix game (A, B) if

s1AsT1 = max

s1∈S1s1AsT

2 ,

s1BsT2 = max

s2∈S2s2BsT

2 .

The set of all equilibrium points of (A, B) is nonempty by Theorem1 of Nash. Let (A, B) ∈ M2

m×n. From the strategic equivalence ofgames, we may consider, without loss of generality, that all entries ofmatrices A and B are negative, i.e., A < 0, B < 0. Then there existsa correspondence between the bimatrix game (A, B) and the linear

On Equilibrium Points in Bimatrix Games 151

complementarity problem (LCP):

ui = −n∑

j=1

aijyj − 1, i = 1, · · · , m, (1)

vj = −m∑

i=1

bijxi − 1, j = 1, · · · , n, (2)

uixi = 0, i = 1, · · · , m, (3)vjyj = 0, j = 1, · · · , n, (4)xi = 0, ui = 0, i = 1, · · · , m, (5)yj = 0, vj = 0, j = 1, · · · , n. (6)

If (u, v, x, y) is a solution of the linear complementarity problem (1)-(6), then the equilibrium points of game (A, B) are

si1 = xi/

m∑

i=1

xi, i = 1, · · · , m, (7)

sj2 = yj/

n∑

j=1

yj , j = 1, · · · , n. (8)

Thus equilibrium points of game (A, B) can be computed by solvingthe LCP.

3. Sensitivity of Equilibrium Points in Bimatrix Games

In this section we investigate sensitivity of equilibrium point of bi-matrix game through linear complementarity problem. Let (u, v, x, y)be a solution of linear complementarity problem corresponding to thebimatrix game (A, B). Let I and I be a partition of pure strategy set{1, · · · , m} of player 1 satisfying

ui = 0, for all i ∈ I; xi = 0, for all i ∈ I , (9)

and let J and J be a partition of pure strategy set {1, · · · , n} ofplayer 2 satisfying

vj = 0, for all j ∈ J ; yj = 0, for all j ∈ J , (10)

152 Hun Kuk

Replacing entries of u, x, v, y according to the partitions I, I, J, J ,we let

u =(

uI

uI

), x =

(xI

xI

), v =

(vJ

vJ

), y =

(yJ

yJ

).

Now we represent perturbations in bimatrix games as two matricesP and Q. Consider the following equations:

(AIJ + PIJ )yJ + 1 = 0, (11)uI + (AIJ + PIJ )yJ + 1 = 0, (12)

(BTIJ + QT

IJ )xI + 1 = 0, (13)

vJ + (BTIJ + QT

IJ )xI + 1 = 0, (14)uI = 0, vJ = 0, xI = 0, yJ = 0, (15)

where AIJ = (aij)i∈I,j∈J and 0 and 1 are vectors whose entries areall 0 and 1, respectively, with appropriate dimension.

We consider application of implicit function theorem to left-handsides of equations (11)-(15) with independent variables P, Q and de-pendent variables u, v, x, y. Since, from equation (15), we have

uI = 0, vJ = 0, xI = 0, yJ = 0,

we consider only equations (11)-(14) and variables uI , vJ , xJ , yJ .Differentiating the left-hand sides of equations (11)-(14) with respectto uI , vJ , xJ , yJ at P = Q = 0, we obtain the Jacobian matrix

M =

0 0 0 AIJ

E 0 0 AIJ

0 0 BTIJ 0

0 E BTIJ

0

,

where E is an identity matrix with appropriate dimension. If M isnonsingular matrix, then the implicit function theorem is applicable.So we need the following theorem.

Theorem 3.1. Jacobian matrix M is nonsingular if and only if |I| =|J | and AIJ and BIJ are nonsingular, where |I| is cardinal of I.

Proof. It is trivial from the definition of M .

On Equilibrium Points in Bimatrix Games 153

If the conditions of Theorem 3.1 are satisfied, then, by the im-plicit function theorem, there exist C1-class functions u(P, Q), v(P, Q),x(P, Q), y(P, Q) depend on some neighborhood of P = Q = 0 so that

u(0, 0) = u, v(0, 0) = v, x(0, 0) = x, y(0, 0) = y (16)

and satisfy equations (11)-(15). By using of the inverse matrix M−1

of M , where

M−1 =

−AIJA−1IJ E 0 0

0 0 −BTIJ

B−TIJ E

0 0 B−TIJ 0

A−1IJ 0 0 0

,

we obtain the partial differential coefficients of functions u, v, x, yat P = Q = 0 as follows:

∂xI

∂pij= 0, for all i, j, (17)

∂xI

∂qij= −B−T

IJ

∂QTIJ

∂qijxI

=

−B−TIJ

0xi

0

, i ∈ I, j ∈ J,

0, eleswhere,(18)

∂yJ

∂pij= −A−1

IJ

∂PIJ

∂pijyJ

=

−A−1IJ

0yj

0

, i ∈ I, j ∈ J,

0, eleswhere,(19)

154 Hun Kuk

∂yJ

∂qij= 0, for all i, j, (20)

∂uI

∂pij=

[−∂PIJ

∂pij+ AIJA−1

IJ

∂PIJ

∂pij

]yJ

=

0, j ∈ J

−

0yj

0

, i ∈ I, j ∈ J,

AIJA−1IJ

0yj

0

, i ∈ I, j ∈ J,

(21)

∂uI

∂qij= 0, for all i, j, (22)

∂vJ

∂pij= 0, for all i, j, (23)

∂vJ

∂qij=

[−∂QIJ

∂pij+ BT

IJB−TIJ

∂QTIJ

∂qij

]xI

=

0, j ∈ I

−

0

xi

0

, i ∈ I, j ∈ J ,

BIJB−TIJ

0

xj

0

, i ∈ I, j ∈ J,

(24)

Partial differential coefficients of equilibrium points of game (A, B)with respect to parameters P, Q are obtained immediately from the

On Equilibrium Points in Bimatrix Games 155

relationship (7) and (8) between equilibrium points of game (A, B) andsolution of LCP. Namely, we have

∂si1

∂pij= 0, (25)

∂si1

∂qij=

∂xi

∂qij

∑mi=1 xi − xi

∑mi=1

∂xi

∂qij

(∑m

i=1 xi)2, (26)

∂sj2

∂pij=

∂yj

∂pij

∑nj=1 yj − yj

∑nj=1

∂yj

∂pij

(∑n

j=1 yj)2, (27)

∂sj2

∂qij= 0. (28)

Example 3.1. Consider the bimatrix game in which the payoffmatrices are

A =(−2 −3−3 −2

), B =

(−2 −2−2 −3

).

and the perturbation matrices are

P =(

0 00 0

), B =

(0 θ0 0

).

Let E(θ) be the set of equilibrium points of game (A + P, B + Q).Then

E(θ) =

{((1, 0), (1, 0))}, θ < 0,{((1, 0), (α, 1− α)) : 1

2 5 α 5 1}, θ = 0,

{(( 1(1+θ) ,

θ(1+θ) ), (

12 , 1

2))}, θ > 0.

First, we consider the following solutions of LCP corresponding to((1, 0), (1, 0)) ∈ E(0):

x = (12, 0), y = (

12, 0), u = (0,

12), v = (0, 0).

Partition satisfying conditions (9) and (10) and the conditions ofTheorem 3.1 is

I = {1}, I = {2}, J = {1}, J = {2}.

156 Hun Kuk

Hence, from (25)-(28), we obtain∂s1

∂θ= (0, 0),

∂s2

∂θ= (0, 0).

Next, we consider the following solutions of LCP corresponding to((1, 0), (1

2 , 12 )) ∈ E(0):

x = (12, 0), y = (

15,15), u = (0, 0), v = (0, 0).

Partition satisfying conditions (9) and (10) and the conditions ofTheorem 3.1 is

I = {1, 2}, I = ∅, J = {1, 2}.J = ∅.Hence, from (25)-(28), we obtain

∂s1

∂θ= (−1,−1),

∂s2

∂θ= (0, 0).

Remark 3.1. In this paper we have studied sensitivity of Nashequilibria in two-person games. However, there remain some problemswhich should be solved in the future, as follows: (i) extension of theproperties of Nash equilibria in two-person game to n-person game, (ii)development of computer software in order to calculate sensitivity ofNash equilibria of bimatrix game.

References

1. A.V. Fiacco, Introduction to sensitivity and stability analysis in nonlinear pro-

gramming, Academic Press, New York, 1983.2. J. Nash, Non-cooperative games, Annals of Mathematics 54 (1951), 286-295.

3. B.C. Eaves, The linear complementarity problem, Managemant Science 17 (1971), 612-634.

4. M.J.M. Jansen, Regularity and stability equilibrium points of bimatrix games,

Mathematics of Operations Research 6 (1981), 530-550.5. E.V. Damme, Stability and perfection of Nash equilibria, Springer-Verlag, Berlin,

1991.

6. R.W. Cottle, J.S. Pang and R.E. Stone, The linear complementarity problem,Academic Press, Boston, 1992.

Department of Applied MathematicsNational Fisheries University of PusanPusan 608-737, Korea