Refinements of the Nash equilibrium concept for bimatrix games

Refinements of the Nash equilibrium concept for bimatrix

games

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Damme, van, E. E. C. (1981). Refinements of the Nash equilibrium concept for bimatrix games. (Memorandum COSOR; Vol. 8108). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1981

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Department of Mathematics and Computer Science

STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 81-08

Refinements of the Nash equilibrium concept for bimatrix games

by

E.E.C. van Damme

Eindhoven, May 1981

BIMATRIX GAMES

by

E.E.C. van Damme

Abstract

In the literature several refinements of the Nash equilibrium concept have been introduced. Among these there are: the essential equilibrium

point [16], the quasi-strong equilibrium point [2], the perfect equi-librium point [15], and the proper equiequi-librium point [11]. In this paper bimatrix games are considered. For this class of games a new refinement of the equilibrium concept, called nondegenerate equilibrium point, is introduced. It is proved that nondegenerate equilibrium points are essential, quasi-strong, proper and perfect. Furthermore, it is shown that for almost all bimatrix games all equilibrium points are nondegenerate.

I. Introduction

It is well-known that for noncooperative n-person non-zero sum games in normal form not all Nash equilibrium points are equally suited to be chosen as the solution. The reason for this fact is, that (in general) some equilibrium points are more stable than others. Therefore, in the literature various refinements of the Nash equilibrium concept have been introduced. Each of these refinements requires the equilibrium point to satisfy a particular stability condition.

Among these refinements there are:

- the essential equilibrium point, introduced by Wu Wen-tsun and

Jiang Jia-he ~16J,

- the quasi-strong equilibrium point, introduced by Harsanyi [2J, - the perfect equilibrium point, introduced by Selten [ISJ, and - the proper equilibrium point, introduced by Myerson [IIJ.

Loosely speaking we may say that for a general n-person game G in normal form

an equilibrium point ~ is essential if each game near to G has an

equilibrium point near to ~,

an equilibrium point is quasi-strong if each player uses each pure strategy, which is a best reply against the combination of strate-gies played by the other players, with a positive probability,

- an equilibrium point is perfect i f each player plays a strategy, which is a best reply against small perturbations of the combination of strategies, played by the other players, and

- an equilibrium point is proper if each player plays a strategy, which is a best reply against particular small perturbations of

the combination of strategies, played by the other players. In this paper, we consider bimatrix games (noncooperative 2-person general sum games) and derive relations between the refinements introduced above for this special class of games. The results we obtain are related tot the results obtained by Jansen ([6J and [7J). However, he only considered essential equilibrium points (which he calls stable equilibrium points) and quasi-strong equilibrium points

(which he calls regular equilibrium points). In this paper we restate some of his results. Other results of him are provided with a different proof.

The organisation of the paper is as follows.

In section 2 the formal definitions of the refinements introduced above are given for the special case of a bimatrix game.

In section 3 isolated equilibrium points are introduced (as equilibrium points which are such that there are no equilibrium points close to them) and a characterization of these equilibria is derived. Next, nondegenerate equilibrium points are introduced. An equilibrium point is nondegenerate if it is both isolated and quasi-strong. A useful characterization of nondegenerate equilibrium points is derived.

In section 4 we prove that nondegenerate equilibrium points are essen-tial and that nondegenerate equilibrium points are proper (and hence also perfect). Unfortunately, not all bimatrix games possess nondegene-rate equilibrium points. However, we prove that for almost all bimatrix games all equilibrium points are nondegenerate. More precisely, we show that the set of all m-by-n bimatrix games, for which at least one equi-librium point is degenerate, is a closed set, which has tebesque measure zero, within the class of all m-by-n bimatrix games.

Finally, in section 5, we derive a characterization of perfect equilibrium

points of bimatrix games. Furthermore, three characterizations of non-degenerate equilibrium points are derived.

2. Preliminaries

An

m-by-n bimatrix game consists of two m-by-n matrices

m n

A =[ a .. ]. 1 • 1 and B

1J 1= ,J""

m n

= [b - . _{1J 1= ,J=} J. 1 • 1

where a .. (resp. b .. ) represents the payoff for player 1 (resp. player 2)

1.J 1.J

when player 1 uses his pure strategy i and player 2 uses his pure strategy j. G _mxn is the set of all m-by-n bimatrix games.

I f (A,B) , (A' ,B') €' G we define the distance between(A,B) and (A' ,B') by:

mxn

d«A,B);(A',B'» "" max{ la .. - a:.1 ,

1J 1J

i € {I, .•. ,m}

Iboo - b!.1 ;

1.J 1J

, j E {1, •••

,n}}

Furthermore, we define for e: > 0 and (A,B) E G

mxn

Be-(A,B) "" {(A' ,B') E G ; d«A,B)j(A' ,B'» < e:}

<;. mXn

In this paper vectors will always be column vectors. If x is a vector, th _{en x 1.S t e transpose a x an x.} T. h f d _1S . _{t e} h ₁ . th _{coor 1.nate a} d . f _{t 1.S vector.} h'

1.

k k S := {x If. 1R ; k

I

i=l x. = I}. ~

If (A,B) is an m-by-n bimatrix game, then Sm(Sn) is the set of all mixed strategies of player 1 (player 2). Elements of Sm will be denoted by p, elements of Sn will be denoted by q. e. is the element of Sm

~

(or Sn, no confusion will result) with ith coordinate one.

If (p,q) , (p',q') E Sm x Sn we define the distance between (p,q) and (pl,q') by

d«p,q);(p',q'» := max{lp. - p!1 , Iq. - q!l; i ED , j E:N}

~ ~ J J m n

Furthermore, for e > 0 and (p,q) E Sm x 8n we define

If (A,B) E G and (p,q) E 8m x 8n we define mxn

C

1 (p) := {i EO:N • m' p. ~ > O}

M] (A,q) := {i E:N • _m' e.Aq T = max ekAq} T ~ kiN _m C 2(q) := {j EO 'N • n' q. _J > O} and, M 2(B,p) := {j ED' T T

P Be. max p BeR,}

n' _J

R,6N n

We call Ct(p) the carrier of p. p is completely mixed if Ct(p) ='N m•

M](A,q) is (in the bimatrix game (A,B» the set of all pure best replies

of player I against q. The set of all best replies against

q

is

m

T

{p EO 8 ; P Aq =

By IC 1(p)1 and 1Mt(A,q)I we denote the number of elements of C1(p) and

M₁(A,q), respectively. Of course, similar definitions are given for

C

2(q) and M2(B,p). A strategy pair (p,q) EO Sm x Sn is a Nash equilibrium point of the bimatrix game (A,B) if p is a best reply against q and q is a best reply against p (hence C1(p) c M] (A,q) and C

Nash [12] has shown that any bimatrix game has at least one (Nash) equilibrium point. By E(A,B) we denote the set of all equilibrium points of the bimatrix game (A,B).

In the following we will give the definitions of some refinements of the equilibrium point concept for the special case of a bimatrix game (A,B). The first refinement is a concept called essential equi-librium point. This concept was introduced by Wu Wen-tsun and Jiang Jia-he [I 6J.

Definition 2.1.

(p,q) E E(A,B) is an essential equilibrium point of the bimatrix game

(A,B) if for all g > 0 there exists a 6 > 0 such that for all

(A',B') E Be(A,B) there exists (p',q') E E(A',B') n Bg(p,q).

By Ee(A,B) we denote the set of all essential equilibrium points of the

bimatrix game (A,B). Ge is the set of all (A,B) E G ,which have

mxn mxn

the property that Ee(A,B}

=

E(A,B).

Remark 2.2.

i) the set Ee(A,B) may be empty as we see by considering the bimatrix game (A,B) with A

=

B - (1 1).

ii)in [I6J it is proved that Ge is dense in G and that Ee(A,B)<;:Qj

mXn mXn

whenever (A,B) is a bimatrix game which has a finite number of equilibrium points.

Next, we consider strong equilibrium points. The concept of quasi-strong equilibrium points has been introduced by Harsanyi [2J.

Definition 2.3.

(p,q) E E(A,B) is a quasi-strong equilibrium point of the bimatrix game

(A,B) if C1(p)

=

M)(A,q) and CZ(q)

=

M

2(B,p).

By Eqs(A,B) we denote the set of all quasi-strong equilibrium points

of the bimatrix game (A,B). Gqs is the set of all ~by-n bimatrix games

mxn

Remark 2.4

(i) it is still an open question, whether there exists a bimatrix game (A,B) with Eqs(A,B)

=

0.

(ii) Harsanyi [3J has proved that the complement of G!:n is

a

closed set with (Lebesque) measure zero.

Ne~t, we consider the perfectness concept, introduced by Selten [15J.

Definition 2.5.

(p,q) E E(A,B) is a perfect equilibrium point of the bimatrix game

(A,B) if there exists a sequence {(p(k),q(k»}kiN of elements of

Sm x Sn having the following properties:

(i) p(k) and q(k) are completely mixed, for all k E E,

(ii) p is a best reply against q(k) and q is a best reply against p(k), for all k E E,

(iii) lim (p(k),q(k»

=

(p,q).

k...,

Epe(A,B) will be used to denote the set of all perfect equilibrium points of the bimatrix game (A,B). Selten [15J has shown that Epe(A,B) ~

0

for any bimatrix game (A,B). G_mXn pe is the set of all m-by-n bimatrix games for which all equilibrium points are perfect.

Finally, we consider the properness concept, introduced by Myerson [11].

Defini tion 2.6.

Let E > O. (p(E),q(E» E Sm x Sn is an E-proper equilibrium point of the bimatrix game (A,B) if

(i) p(E) and q(E) are completely mixed,

(ii) for all i,k E Em: eIAq(E) <

e~Aq(E)

implies Pi CE)

~EPk(E),

(iii) for al1 j,R. EE : _{p(E.:)TBe .} < _{p(E)TBeo implies q.{E)}

~

Eqo(E),

n J " J N

(p,q) is a proper equilibrium point of (A,B), if there exists a sequence

{(~, p (~), q (~»}k E E which satisfies:

(i) Ek (ii)

> 0 for all k E E, lim ~

=

₀

lim (peEk) ,q(E k

»

== k...,

(iii) _{(peEk) ,q(E}

k

»

is an k E E.

k...,

(p ,q)

We will use Epr(A,B) to denote the set of all proper equilibrium points

of the bimatrix game (A,B). Myerson [11J has shown that

0

~ Epr(A~B) c EpeA,B)

for any bimatrix game (A,B). Moreover, it is possible that Epr(A,B) ~ Epe(A,B)

(see e.g. example 5.4). We will use Gpr to denote the set of all ~by-n mxn

bimatrix games for which all equilibrium points are proper.

3. Nondegenerate equilibrium points

In the sequel relations between the concepts introduced in section 2 will be derived. Some of these relations can already be obtained by examining a simple example.

Example 3.J.

Let the bimatrix game (A,B) be given by:

We have 1 2 (0,0) (0,1) 2 (1 ,0) (0,0) E(A,B)= {(pe

1+ (l-p)e2 ,e1); p€[O,lJ}U {eel ,qel + (l-q)e2.); qdO,I]L

Hence, we see that

- a proper equilibrium point need not be quas i -s trang, - a proper equilibrium point need not be essential,

- a quasi-strong equilibrium point need not be perfect, and - a quasi-strong equilibrium point need not be essential.

For a concept to be useful as a refinement of the equilibrium concept, it should be easy to check whether an equilibrium point satisfies the requirements of this concept. Of the concepts introduced in section 2 only the requirements of the quasi-strongnessconcept are trivial to check.

In example 3.1 we have however seen, that quasi-strong equilibrium points do not possess all nice properties we want equilibrium points to have. In the sequel we will see that this is caused by the fact that a quasi-strong equilibrium point need not be isolated. On the other hand, it will turn out, that equilibrium points, which are both quasi-strong and isolated, possess very nice properties (also see [6J and [7J). Therefore, we will introduce a new name for these

"l"b . . "11 11 "l"b . " 1 )

equ1 1 r1um p01nts; we W1 ca them nondegenerate equ1 1 r1um p01nts •

We will show that it is easy to check whether an equilibrium point is nondegenerate.

Definition 3.2.

Let (A,B) € G _mXn and (p,q) € E(A,B).

(p,q) is an isolated equilibrium point of (A,B) if there exists an

e; > 0 such that B (p,q) (l E(A,B)\ {(p,q)}

=

0.

e:

Ei(~B)is

the set of all isolated equilibrium points of (A,B) and G_mxn i is the set of all m-by-n bimatrix games for which all equilibrium _. points are isolated. In example 3.1 we see that E1(A,B)

=

0

for some (A,B).

Definition 3.3.

Let (A,B) € G and (p,q) € E(A,B).

mXn

(p,q) is a nondegenerate equilibrium point of (A,B) if (p,q) is both a quasi-strong and an isolated equilibrium point of (A,B).

End(A,B) is the set of all nondegenerate equilibrium points of (A,B) and Gnd _mxn is the set of all m-by-n bimatrix games for which all equilibrium points are nondegenerate.

1) We have chosen for this terminology, since a bimatrix game which is

nondegenerate in the sense of Lemke and Howson [19J is a game for which all equilibrium points are nondegenerate in this sense.

In order to obtain a characterization of nondegenerate equilibrium points,we will first derive a characterization of isolated equilibrium points. This latter characterization is obtained via the so called maximal Nash subsets ([4J,[10J).

Defini don 3.

Let (A,B) E G • Assume (p,q),(pl,ql) E E(A,B), S c E(A,B).

mxn

(p,q) and (pl,ql) are S-interchangeable if (pl,q) and (p,ql) are also elements of S. S is a Nash subset of (A,B) if any two elements of S are S-interchangeable. S is a maximal Nash subset of (A,B) if S is a Nash subset of (A,B) and if there does not exist a Nash subset of

(A,B), which properly contains S.

In [5J the proof of the following theorem can be found.

Theorem 3.5 Let (A,B) E G

mxn

(i) if S is a maximal Nash subset of (A,B), then S is a closed and convex set,

(ii) there are only finitely many maximal Nash subsets of (A,B), (iii) E(A,B) is the union of all maximal Nash subsets of (A,B).

Theorem 3.6

Let (A,B) E G and (p,q) € E(A,B).

mXn

(p,q) is an isolated equilibrium point of (A,B) if and only if {(p,q)} is a maximal Nash subset of (A,B).

Proof

Assume {(p,q)} is not a maximal Nash subset. Assume (pl,q') is such that q' ~ q and that (p,q) and (p',q') are E(A,B)-interchangeable.

Define q(A) : = Aq + (J-A) q' for A € [0,1].

We have (p,q(A» E E(A,B) \ {(p,q)} for A E [0,1).

Moreover

ltf

(p,q(A»

=

(p,q). Hence (p,q) is not an isolated equilibrium point of (A,B).

Assume (p,q) is not an isolated equilibrium point of (A,B).

For n €:N, let (p(n),q(n» € B1/n(p,q) n E(A,B)\{(p,q)}.

By (ii) and (iii) of theorem 3.5 we may assume that there exists a

maximal Nash subset S of (A,B) such that (p(n),q(n»ES for all n€ IN.

Since S

is

closed (p,q) € S.

Hence, Hp,q)} is not a maximal Nash subset of (A,B).

0

In theorem 3.9 we give a characterization of nondegenerate equilibrium points of bimatrix games (A,B) € G _mxn ,which satisfy

(3.7) a .. > 0 and b .. > 0 for all i € IN , j € IN

1J 1J m n

This is no restriction, since we have

Lennna 3.8 Let (A,B) € G mxn • Define (A+,B+) € G by mXn + 1 + max{

I

a •. I a .. '" a .. + 1J 1J 1J + + max{ lb .. I b .. '" b .. + 1 1J Then Proof 1J E(A,B)

=

E(A+,B+) Eqs(A,B)

=

Eqs(A+,B+) Epr(A,B)

=

Epr(A+,B+) Elementary. 1.J ;iElN ,jElN }

.

. m n ;iEJN ,jElN}. m n Ee(A,B)

=

Ee(A+,B+) Epe(A,B)= Epe(A+,B+) . . + + E1(A,B) = E1(A ,B )

o

Theorem 3.9

Let (A,B) E G _mxn such that (3.7) is satisfied. Let (p,q) E E(A,B). Define matrices A and

B

by

,..., A

=

[a .. ] ~ B = [b •. ] 1J iEM t(A,q),jEC2(q) 1J iEC}(p),jEM2(B,p) •

The following two statements are equivalent

(

~) _... ( _p,q ) E E nd ( A,B, )

(ii) IM}(A,q)1

=

ICt(p)1

=

IC₂(q)1

=

IM₂(B,p)I and the matrices

"" ,...

A and Bare nonsingular.

Proof

'

-Assume (p,q) E End(A,B). -Assume y ;.: 0 is such that Ay = O. Define YE:JRn by

For e: E:R, define q (e:) lE:Rn by

""

Since Ay

=

0, we have

Hence, if e: is sufficiently close to zero (say e: E (-e:O,e:_O

»

we have

Hence, (p~q(e:» E E(A,B), if E E (-EO,E

O)' This in in contradiction with

1 ""

-(p,q) E E (A,B). Hence, if y ;.: 0, we have Ay ;.: 0 and so the columns of A constitute a set of independent vectors. Therefore we must have

Applying a similar reasoning as above, we can prove

T""

if x ~ 0 then x B ~ 0

Hence

(3. 11)

From (3.10), (3.11) and (p,q) E Eqs(A,B) we may conclude

~ '"

Hence, the matrices

A

and B are square. The above reasoning shows that

both matrices are nonsingular.

Assume that IM)(A,q)1

=

IC

1(p)1

=

IC

2(q)1

=

IM2(B,p)1 and that the

matrices

A

and Bare nonsingular. It suffices to prove that

i

(p,q) E E (A,B).

Assume q' ~ q is such that (p,q') E E(A,B). We have C

2(q') c C2(q).

Let

q :

=

[q.] and

q' :

=

[q!J. C ( )

J jEC

2(q) J JE 2 q

Since :t1.} (A,q ') ::> ~(A,q). there exists a A E R such that

Aq'

=

hAc!

Since

q'

~

q

and

L.q!

=

I

=

Lq .•

we have

q'

~

Aq.

"" '" J J "" '" l J

Hence, q' - Aq ~ 0 and A(q' - q)

=

O. Which is a contradiction. Similarly, we can prove that there does not exist a p' ~ p such that

i

(p' ,q) E E(A,B). By theorem 3.6 we have (p,q) E E (A,B).

4. Properties of nondegenerate equilibrium points

o

In this section we will prove that nondegenerate equilibrium points have very nice properties. Namely: nondegenerate equilibrium points are essential (theorem 4.S) and nondegenerate equilibrium points are proper (theorem 4.9) and hence also perfect. Furthermore, we show that for almost all games all equilibrium points are nondegenerate

(theorem 4.11). The proof that nondegenerate equilibrium points are

essential is only slightly different from the proof in [7J (thm.7.S). Our proof is splitted into two lemmas, each of them interesting in his own right.

Lemma 4. I

Let (A,B) E G and (p,q) € E(A,B). Define matrices

A

and

B

and vectors

~ mxn

p and q by

A = [a .. ]

Ii'

=

[b .. ]

1J iEM

1(A,q),jEM2(B,p) 1J iEHl (A,q) ,j EM2 (B,p)

p

=

[po ] 1 i€M t (A,q) q

=

[q.} J jEM 2(B,p) Proof

This follows almost immediately from the fact that there exists an e: >

a

such that for all (A'.B') E B (A,B) and all (p',q') E B (p,q):

€ E

M1(A',q') c M

1(A,q) and M2(B',p') c M2(B,p). IJ

Lemma 4.2 (cf [14J)

Let (A,B) E G be such that m

=

n and that A and Bare nonsingular.

mxn

Define F(B) E~n and G(A) E ~n by

n

0.

n

(4.3) F. (B)

₌

_I

B..

I

B .. (i E 'N

n

)

1 j=l 1J i,j=l l.J

n 0.n

(4.4) G. (A) =

L

A..

I

A .. (j> E :N ), J _{i=1 1J i ,j=l} 1J n

where A.: represents the co-factor of a .. in the matrix A.

1J 1J

I f F.{B) > 0 for all i E:N and G.(A) > 0 for all j E:N , then

1 n J n

(F{B),G(A» is the unique completely mixed equilibrium point of (A,B).

I f F.{B) :S 0 for some i E:N or G.(A) s

a

for some j E:IN , then

1 n J n

there does not exist a completely mixed equilibrium point of (A,B).

Proof

This immediately follows from the fact that if (p,q) is a completely mixed equilibrium point of (A,B), then there exist A,p E ~ such that

(p,q,A,~) is(the unique) solution of the system:

T T T

P B

=

Ae; p e

=

I; Aq

=

~e; q e

=

I.

and Cramerts rule (e is the vector with all coefficients equal to one).

Theorem 4.5 (cf [7], thm. 7.5)

nd e

E (A,B) c E (A,B).

Proof

Let (A,B) E G and (p,q) E End(A,B). By theorem 3.9 and lemma 4.1

mXn

we may assume that the matrices A and B are square and nonsingular and that p and q are completely mixed. Let 6> 0 be such that C and Dare

nonsingular, whenever (C,D) E B

6(A,B). By (4.3) and (4.4) a continuous

mapping (F,G): Bo(A,B) ~~n x~n is defined. We have (F,G)«A,B»

=

(p,q).

Let e:: E (0, !min{p. ,q. ;i,j E:N })

1 J n

Since (F,G) is continuous there exists a n E (0,0) such that

(F,G)«C,D» E B (p,q) whenever (C,D) E B (A,B).

e:: e n

By lemma 4.2: (p,q) E E (A,B).

0

Next, we will prove that nondegenerate equilibrium points are proper. Again, the proof is splitted into several lemmas.

Lemma 4.6

nd

Let (A,B) E G • Assume (p,q) E E (A,B). There exists a 0

0 > 0 such mxn

that for all ~ E (0,0

0) and ~or all i E:Nm \C1 (p) and all j E:Nn \C

z

(q)

there exist f1(0) E Sm and gJ(6) E Sn such that

i £.(0)

=

0 1 C 2(gJ(0»

=

C2(q) u {j}

g~

(0) = 0 J M}(A,gj(o»

=

C1(p).

Proof

We will only indicate how one can construct fiCo). In a similar fashion one can

construc~

gj(o). By theorem 3.9 we may assume C

1{p) =Ek ... C2{q). Define matrices Band B by:

"'" k k 13 ... [b •• J. 1 • 1 1.J 1.= ,J-m k B

=

[b •. J. I • I 1.J 1.= ,J=

Since

B

is nonsingular there exists for all

that (xi)T'i

=

e: B.

1.

X

i(~~; m i

For i E Em\Ek and 0 E lR+, define u. E lR and Yo E lR by

i x. (0) == 0 1. i x. (0) J i x. (c) J i p. - ox. J J

=

0 k = 1 - 0(1 -

I

j=l if j else i x.) J E Ek i

For 0 sufficiently close to zero we have Yo > O. In this case we define

i i i i f (0) == Yo

x

(o/yo)

If 0 > 0 is sufficiently small, we have

i

£. (6) = 6 •

1.

Moreover, we have lim fiCo)

=

0+0

p •

Hence, if 0 is sufficiently close to zero: M 2(B,f

i

(o» c M

i T k Furthermore: (f (0)) B

=

L

~j=1 i T T f.(o)e. B + oe. B J J ~ . k T • k

=

Y~{

L

PJ' e J•

B -

(o/Y~)

L

i T - T x. e. B} + oe. B j=1 j=1 J J ~

Hence, if 0 is sufficiently small: MZ(B,fi(o))

=

MZ(B,p).

o

The idea of the proof that a nondegenerate equilibrium point (p,q) is proper is the following: instead of the game in which the pure strategy sets of the players areE and E , we look at a game in which the pure

m n

strategy sets are

We then define a generalization of the e:-proper equilibrium point concept

(definition 4.7) and show that a certain sequence of strategy pairs in

this new game, which satisfy the requirements of this new concept, induces in the original game a sequence of e:-proper equilibrium points, converging to (p,q).

Defini tion 4. 7

Let (A,B) E G • Let pI

=

{PI 1 , •••

,P~}

be a partition of Em and let

Z Z _mxnZ 1\

p

=

{PI"'" P~} be a partition of En' Let (p,q) E 8m x 8n and let e: > O. (p,q) is an e:-proper equilibrium point of (A,B) with respect to

(pl,pZ) i f

(i) p and q are completely mixed , (ii) for all i,k EE and all C/. ,/3 E E

A:

i f C/. > ₁₃ and m i E P , 1 j _{E P 13' then P i} I ~ _{e:Pk '} if i ,k _{E P ,} I then _e.Aq TC/. _{< ~} TA _{q ~mp ~es} ' I ' _Pi ~ _e:Pk

(iii) for all j,~ EN and all a,S E'N :

n 2 2 ~

if a > 13 and j

E P

_{a ,} ~

EPa'

then qj ~ €qt'

' f ' n p2 h TB T ' 1 ' ~ J,N E , t en p e, < p Ben 1mp 1es q.

a J N J

( _{p,q 1S a proper} ) ' _equ~11br1um . . . , f ( _{p01nt 0 A,B W1t} ) ' h _{respect to} (pI ,p2) _{_}

if there exists a sequence {(€k,P(€k),q(€k»}kaN such that a) €k > 0 (k € N); lim €k

=

0,

k-+<»

b) lim (P(€k)' q(€k»

=

(p,q), k-+<»

c) (P(€k),q(€k» is an €k-proper equilibrium point of (A,B) with respect

1 2

to (P ,P ), for all k-t:"N.

Lemma 4.8

Let (A,B) E G x • Let pI

=

{PII, ••• ,F!} be a partition of N and let

2 2 m n 1\ m

p

=

{Pl, ••• ,p2} be a partition of N . Then

~ n

(i) if € E (0,1), then there exists an €-proper equilibrium point of (A,B) with respect to (pI ,p2) ,

(ii) there exists a proper equilibrium point of (A,B) with respect to

(pI ,p2) • Froof (i) Let F (p,q) := m , 1 m i n € € (0,1). Define 0 := m1n{m€ , ~€ } k k S (0) := {x E S ; xi ~ 0 for all i EN k} (k E {m,n})

for al1 i,k E N

m and all

a,a

E NA: m

X E S (0) _~ ' f 0 ' pI , pI h <

a > p ; 1 E a' J E 13' t en Xi - €~,

and define a point-to-set map F : Sm(o) x Sn(o) + Sn(o) in a similar n

way (cf. definition 4.7 iii). Let Fe.)

=

F (.) x F (.). Then F is a

m n

point-to-set map from Sm(o) x Sn(o) to Sm(o) x Sn(o). It is not dif-ficult to see that F satisfies the conditions of the Kakutani fixed point theorem [8]. Hence, there exists a fixed point (p,q) of F. In this case (p,q) is an €-proper equilibrium point of (A,B) with

I

Z

respect to

(P ,P ).

(ii) Follows immediately from (i) and the fact that Sm x Sn is a compact set.

0

Theorem 4.9

End(A,B) C Epr(A,B).

Proof

Let (A,B) E G x • Assume (p,q) E End(A,B). For i E:N \ e

1 (p), j EN \ eZ(q)

m n m . n.

and a certain fixed 0, with 0 sufficiently small, let f1(0) and gJ(o) be defined as in lemma 4.6. To simplify the notation, we will write fi and gj for fiCo) and gj(o), respectively.

v

and (as long as U pI

~N

\el(p» a.=1 a. m V E N m \ (el (p) U U a.=1

T

T }

e.Aq

=

max aAq

1 v k

keN \(e

1(p)u U pI)

m a.=1 a.

v

and (as long as U p2

~~

\C

2(q»: a=t a n p2 ₌

{j

_{E 'N} v _U p2) . T _{P Be. = max} n \(C2(q) u v+1 a=] a ' J v _p2} 9..E~n \ (C₂ (q) u U a We define a bimatrix game

a .. 1J '" _{b ..} 1J = (fi)TAgj = (fi}TBgj a=1

(ASh

by:

For s E (O,t), let (~(s),y(s» be an s-proper equilibrium point of

(A,B) with respect to (pt ,p2). We define a probability distribution

on {fi; i E'N \Ct(p)} by xes) =

I.

~.(s)fi.

m 1 1

pTBeR,}

Since fi E Sm, for all i, we have xes) E Sm. In a similar way yes) E sn

is defined. We define

pes) = (J + s)-l(p + sx(s»

-I

q(s) = (J + s) (q + sy(s».

We will prove that (p(s),q(s» is a VE-proper equilibrium point of (A,B) if E is sufficiently small. The proof is divided into several steps.

Step

If i E Ct(p) and j E'N \C

2(q), then

e~Agj

= pTAgj and

e~AY(s)

=

pTAy(s).

n 1 . 1

This immediately follows from the way in which gJ was constructed and the definition of yes).

Step 2 If i E]>lm \C 1 (p) and j E En \C2 (q), then T ' T ' - (I - 0) P AgJ + oe.AgJ • ~ ~tep 3 If i E Em \C 1 (p), then T e.Ay(e:)

=

~ Step 4 I f i E ~m \C 1 (p), then T j""

e.Ag y_ (E) - (step 2) =

~ J

-1 ~ i T -} '"

== ( 1 + E:) e: x. (d (f ) e. == (1 + d E: x. (d IS •

St~p 5

T T

Assume i,k E:N are such that e:-Aq(E) < ekAq(E).

m 1.

If E is sufficiently small, it follows that:

T T

Case (i): eiAq < ekAq.

Then i i C

1(p). If k E CJ(p), we have for E sufficiently small:

-1'" -J -1

p. (e:) ,. (1 + e:) x. (e:) 0 :s; (1 == E) E:S; 5,1 + e) ve Pk:S; ve P'k(€:)'

1. 1. -1 ._ -I '""

If k i C₁(p), then PiCe:) == ( 1+ e::) e xi(e) 0 and Pk(e::) == (1 + e) e: ~(e:) 0 Since (i(e:),y(e» is an e-proper equilibrium point of

(A,E)

with respect

1 2 '" '"

to

(P ,P ),

we have xi(e:) :s; e: ~(e:).

Hence, Pi (E) :s; e: Pk (e:).

T T T T

Case (ii): eiAq

=

ekAq and eiAy(e) < ekAy(e).

By step 1 and the fact that (p,q) is a quasi-strong equilibrium point,

T- T~

we have i,k i C](p). By step 3: eiAY(E) < ekAy(e).

Since (i(E),y(e:» is an E-proper equilibrium point of (A,B) with respect

] 2 '" '"

to (P ,P): xi(e):s;e ~(e:). By step 4: PiCe:) :s; e:Pk(E).

In a similar way as above we can prove that, qj(e) :s;

VE

qR-(e:) , whenever p(e:)TBe . < p(e:)TBen (j,t E:N ).

J ~ n

Hence, (p(e),q(E» is a ve-proper equilibrium point of (A,B) if E > 0 is sufficiently small. Since lim (p(E),q(e» ,. (p,q) we have that (p,q)

E:-tO

is a proper equilibrium point of (A,B).

Corollary 4.10 End(A,B) c Epe(A,B).

Theorem 4. 1 ]

For almost all games (A,B) E G we have End(A,B)

=

E(A,B).

Proof

Jansen ([7], thIn ) proved that if (p,q) € Eqs(A,B), then (p,q) is an

element of the relative interior of some maximal Nash subset. Consequently, if all equilibrium points are quasi-strong, then all equilibrium points

are isolated. Hence, if E(A,B)

=

Eqs(A,B), then E(A,B)

=

End(A,B). Since,

for almost all games (A,B) € G we have that Eqs (A,B) = E(A,B) ([3] thm.2) mxn

the proof is complete.

0

5. Some characterizations

In this section we will derive a characterization of perfect equilibrium points of bimatrix games. Furthermore, we will derive a property of proper equilibrium points of bimatrix games. Finally, we will prove two theorems, which are more or less converses to the theorems 4.5 and 4.9.

Defini tion 5. 1

Let (A,B) € G and let (p,q),(~,o) € Sm x Sn.

mxn

p is dominated by ~ if:

T T T T

P Ae. ::;; p Ae. for all j €::N and p Ae. < p Ae. for some j €:N •

J J n J J n

q is dominated by 0 if:

T

't

T T

e.Aq ::;; e.Ars for all i € Nand e.Aq < e.Acr for some i €::N •

3. 3. m 3. 3. m

p (q) is dominated if there exists a ~ € Sm (0 E Sn) such that

p (q) is dominated by ~ (0).

In [1] one can find the proof of the following theorem:

Theorem 5.2

Let (A,B) € G and let (p,q) E E(A,B).

mXn

(p,q) E Epe(A,B) if and only if p and q are not dominated.

Theorem 5.3

"" '" ""

Let (A,B) E G and let (p,q) € E(A,B). Define A,B,p and q as in lemma

mxn

. pr "" '" pe '" '" 4.1. I f (p,q) € E (A,B), then (p,q) € E (A,B).

Proof

For k E

E,

let (p(€k),q(E

k

»

be an Ek-proper equilibrium point of (A,B)

such that lim Ek

= a

and lim (p(Ek),q(£k»

=

(p,q).

k-!><X> k'"""" If j

t

M

2(B,p), 9, E M2(B,p), we have for k sufficiently large:

T T

p(ek) Bej < peek) Be£. Hence qj(£k) ~ _Ekq9,(ek)·

Assume

p

is (in

A)

dominated by ~. Let jo E M

2(B,p) and 0 >

a

be such that

~T"" P Ae. J

_O

m n Le t II A II :. =

I I

i=1 j= 1 I a .. 1 • 1J

For k sufficiently large, we have

T T

Hence, there exist i E C1(p) and 9, E Cl(~) such that eiAq(£k) < e9,Aq(£k) (if k is sufficiently large).

Therefore p.

=

lim P.(E

k) $ lim £k

P

(£k)

=

0,

which is a contradiction.

1. k-+«> 1. k-+«> 9,

Similarly we can prove that

q

cannot be dominated in

B.

0

A similar result as in theorem 5.3 is not true for perfect equilibrium points of (A,B) as we see in example 5.4. In example 5.5 we see that the converse of theorem 5.3 is false in general.

Example 5.4

Let the bimatrix game (A,B) be defined by:

(1,1) (0,0) (-2,-2) (0,0) (0,0) (-1,-1)

(-2,-2) (-1,-1) (-1,-1).

Example 5.5

Let the bimatrix game (A,B) be defined by:

Theorem 5.6 (1 , l) (1,1) (I,D) (1,1) (0,0) (0,0)

End(A,B)

=

Ei(A,B) n Epr(A,B).

Proof

(0,1) (0,0) (1,1).

We have already shown that a nondegenerate equilibrium point is isolated and proper. Assume (p,q) is an isolated and proper equilibrium point of

(A,B). It is sufficient to show that (p,q) is quasi-strong. Now by theorem 5.3 there exists a ~ € Sn such that C2(~)

=

M2(~'P) and

Cl(p) c M

1(A,a). Hence, (p,a) € E{A,B). Since (p,q) E E~(A,B), we have

q

=

a. Hence, C

2_qs (q)

=

M2(B,p). Similarly, we have C1(p)

=

M1(A,q).

Remark 5.7

In example 5.4 we see that a perfect equilibrium point, which is isolated

need not be nondegenerate.

Theorem 5.8 (cf [7], theorem 6.2)

-nd i e

E (A,B)

=

E (A,B) n E (A,B).

Proof

We have already seen that nondegenerate equilibrium points are isolated

and essential. Assume (p,q) E Ei(A,B) n Ee(A,B). We have to prove

(p,q) E Eqs(A,B). Assume C

2(q)

~

M2(B,p). Since (p,q) E Ei(A,B) and because of theorem 5.2 we have: there exists

a

~ E sm such that

pTAe .

~ ~TAe.

for all j E M

2(B,p) and pTAe . <

~TAe.

for some j E M2(B,p).

J ] J J

Choose iO E ~ such that p. <~ • • For k E~, define an m-by-m matrix

m 10 10 A(k) by: a .. (k)

=

a .. 1J 1J a. . (k) = a. . + 11k. 1 0J 10J For j E M 2(B,p), we have pTA(k)e. < 1T T_{A(k)e .•} J J

For k E~, let (p(k),q(k» E E(A(k) ,B) such that lim (p(k),q(k» - (p,q).

k~

Then, for k sufficiently large: C

2(q(k» c M2(B,p(k» c M2(B,p).

Hence, pTA(k)q(k) < 1TTA(k)q(k). Hence, there exists an i E Cl(p), such

that i i M1(A(k),q(k» for infinitely many k. But this is a contradiction,

since for k sufficiently large: Ct(p) c Cl(p(k» C M](A(k),q(k».

Similarly, we can prove Ct(p)

=

M}(A,q).

Hence, (p,q) E Eqs(A,B).

0

To complete the picture, we mention the following theorem. For a proof, see [7], theorem 7.4.

Theorem 5.9

Remark 5.10

We have proved that an equilibrium point which is essential and isolated is nondegenerate. Moreover, we have that an equilibrium point which is essential and quasi-strong is nondegenerate. By looking at simple examples it seems that only nondegenerate equilibrium points can be essential. Although we cannot prove it, we conjecture that the assertion "an equi-librium point is nondegenerate if and only if it is essential" is true.

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[9J Lemke, C.E. and J.T. Howson, Equilibrium points of bimatrix games.J.

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[IOJ Millham, C.B., On Nash subsets of bimatrix games. Naval Res. Logistics

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[11J Myerson, R.B., Refinements of the Nash equilibrium concept. Int. J. Game Theory

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(1978) 73 - 80.

[12J Nash, J.F., Equilibrium points in n-person games. Proc. Nat. Acad. Sci. U.S.A., 36 (1950) 48 - 49.

[13J Parthasarathy, T. and T.E.S. Raghavan. Some Topics in two-person games. American Elsevier Publ. Camp •• New York, 1971.

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