Regular equilibrium points of n-person games in normal form

Citation for published version (APA):

Damme, van, E. E. C. (1981). Regular equilibrium points of n-person games in normal form. (Memorandum COSOR; Vol. 8112). Technische Hogeschool Eindhoven.

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

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at:

openaccess@tue.nl

providing details and we will investigate your claim.

STATISTICS AND OPERATIONS RESEARCH GROUP

Hemorandum CaSaR 81-12

Regular equilibrium points of n-person games in normal form

by

I

E.E.C. van Damme

)

Eindhoven, October 1981 The Netherlands

Eric van Damme

Abstract

Harsanyi introduced ~~gular equilibrium points in [3J and proved that for almost all noncooperative n-person games in normal form all equili-briUm points are regular. In this paper it is shown that regular equi-librium points are essential, quasi-strong, proper and perfect. Hence, for almost all games in normal form all equilibrium points are quasi-strong,essential, proper and perfect.

1. Introduction

It is well-known that for noncooperative n-person 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 equilibrium concept have been introduced.

Among these refinements there are:

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

-Jiang Jia-he [8J,

- the quasi-strong equilibrium point, introduced by Harsanyi [2J, - the regular equilibrium point, introduced by Harsanyi [3J, - the perfect equilibrium point, introduced by Selten [7J, and - the proper equilibrium point, introduced by Myerson [5J.

Loosely speaking we may say that for a general n-person game

r

in normal form:

an equilibrium point p is essential, if any game near to

r

has an equilibrium point near to p,

- an equilibrium point is quasi-strong, if each player uses each pure best rep'ly (against the strategy combination played by the others) with a positive probability,

- an equilibrium point p is regular, if the Jacobian of a certain mapping, associated with the game, evaluated atp is nonsingular,

- an equilibrium point is perfect, if each player plays a strategy, which is a best reply against small perturbations of the strategy combination used by the others, and

- an equilibrium point is proper, if each player plays a strategy, which is a, best reply against special small perturbations of the strategy combination used by the others.

In the literature the following results can be found with respect to these refinements:

(1;1) the set of games for which all equilibrium points are essential is dense in the set of all games; i f a game has finitely many equilibrium points, then it has at least one essential equilibrium point [8J,

(1.2) for almost all games all equilibrium points are quasi-strong and regular [3J,

(1.3) any game possesses at least one perfect equilibrium point [7J, and (1.4) any game possesses at least one proper equilibrium point; if an

In this paper we concentrate on regular equilibrium points. However, in this paper a slightly different definition of regularity is given. The reason for this fact is, that (1.2) is incorrect when regularity is de-fined as in [3J (example 3.6), whereas (1.2) is correct, when regularity is defined as in this paper. The main results of the paper are that regular equilibrium points are quasi-strong, that regular equilibrium points are essential and that regular equilibrium points are proper (and hence also perfect). Hence, we can conclude from (1.2) that for almost all games all equilibrium points are essential, proper and perfect. The organization of the paper is as follows: In section 2 the notation and some basic concepts are introduced. In section 3 regular equilibrium points are defined and some properties of these equilibria are derived. The main theorems are proved in section 4.

2. Preliminaries

A finite noncooperative n-person game in normal form is a 2n-tuple

r

=

_{(AI"" ,A}

n ,U 1,·· .Un) where (for any i E {1, ••• ,n})

A.

is a finite

n ~

set and U. is a function, U.

.

.

.IT

A.

-+ ]R •

A.

is the set of all pure

~ ~

i=l ~ ~

strategies of player i and U. is the payoff function of this player. It

~

will be convenient to have the sets A. disjoint, so we will assume (with~

~

out loss of generality) that n

n

i=1

A. =

0.

In the following, we will

intro-~

duce some notation with respect to such a game

r.

Mostly we will follow Harsanyi [3J.

Let N = {1, ••• ,n}. For a finite set

S

we denote by

lSi

the number of

n n

elements of this set. For i E N, let K. =

IA. I.

Let

A

= IT

A.

and K =

IAI

= IT K ..

We assume that the elements of A. are numbered consecutively as

J.

1 k Ki

a., ••• ,a., ••• ,a. " If no confusion can result we will write k instead

J. J. J.

k

of a_i " A mixed strategy of player i is a probability distribution on Ai' Hence, any mixed strategy of player i can be identified with a probability

1 k Ki

vector p. ,. (p., ••• ,p., ••• ,p. ).We write:

1. J. 1. J. (2.1) We will write P ,.

*

K -n

I

i=1

K •• If x 1

*

K -K. K. JR 1.

,

,

K.

1

I

k=l k p.

=

J. 1 ,

P~ ~

0 for all k € {l, ••• ,Ki }}. n *

II P. and P. == II P •• P is a subset of JRK , where

• 1 1. J. . ../.. J K

J.=

*

Jr 1. .

€ JRK , then we will write x

=

(x.

,i.),

where x. E JR J.

1. 1. J.

ana

i.

E

1 JR J.. Hence, if p E: P, then p = (p.,p.) with p. € 1 1 1. P. J. and

Pi € Pi' The set C(Pi) of all pure strategies to which Pi assigns a

positive probability is called the carrier of Pi' Pi is completely mixed if C(Pi)

=

Ai' If P

=

(Pl, ••• ,Pi, ••• ,Pn)

n E P, then C(p) ;:::: n

U

C(p.) and • 1 J. 1= p is completely mixed if C(p)

=

U A •• I:::: 1 1. 1 k Ki K.

Let a = (a1, ••• ,a., ••• ,a) € A and x.

=

(x., ••• ,x., •• "x.)€ JRl., We

1. n J. 1. 1. 1

d e · . f1.ne q. x . . a'(,) != 'x. k . y where k J.S, suchtha~ a1.' =:: a .• Hence, J.f p. € P., k .

, 1 . 1 . J. 1. 1. 1.

then q~(Pi) is the probability that Pi assigns to ai' So, in particular, 'f k _{E A., we have} 1. a. 1. J. a k if k q. (a.) ,. a.

₌

a. l. 1. 1. J. a k if

f

k q. (a. )

₌

0 a. a.

.

1. l. ~-1 1.

then the payoff to player i will be:

(2.2) U. (p)

1.

=2

_aeA

[~ q~(P')]

_{j=l J J} U.(a) • _1.

Formula (2.2) can be interpreted in terms of payoffs ever, since q~(p.) is well

J J K. for any p.

=

JR. J J only i f p € P. How-we actually have that by (2.2) a mapping U i defined K*

JR. + JR. is defined, It will be clear that U_i is a polynomial and hence that U

i is infinitely often differentiable. A strategy Pi € Pi is a best reply against a strategy combination Pi € Pi

i f

(2.3) U. (p.

,p.)

1. 1. 1.

A pure best reply against Pi is a best reply, which is an element of Ai' The set of all pure best replies against

p.

will be denoted by

B(p.).

A

1. 1.

strategy combination p

=

(PI, ••• ,Pi, ••• ,Pn) is an equilibrium point ([6J) of

r

if p. is a best reply against

p.

for all i E N. It is easily seen,

1. 1.

that p is an equilibrium point if and orily if

(2.4) C(p.) C

B(p.)

for all i € N. 1. 1.

We will denote the set of equilibrium points of r by E(r). We have E(r)

:f:

~ ([6J).

Let

G

=

G(n,K1, ••• ,K

n) be the set of all games in which player i has exactly K. pure strategies (for any i € N). If we impose a fixed ordering

1 .

on N x A then each specific game

r

€ G is completely characterized by its uK-dimensional payoff vector u

=

<u.(a»(. ) N

A'

and we will sometimes

1 1,a EO: x

write

r

_u for the game which has u as its payoff vector. Hence, the set of all games of a particular size can be viewed as an Euclidian space. For any m EO: E , we will use p to denote the Euclidean distance on JR m

and if x EO: JR m and e: > 0, then p (x) = {y € JR m; p (x,y) < e:}. We say that

e:

a certain mathematical statement S is true for almost all games r € G, if the set of all games for which

S

is false is (in JRnK) a closed set with Lebesgue-measure zero.

We close this section by giving the formal definitions of the refinements of the equilibrium concept, which were mentioned in the introduction. Let r e:

G

be fixed.

P € P is an essential equilibrium point of

r

if for any e: > 0, there exists a 0 >

°

such that E(r') n Pe:(p)

+

~ for any r' € po(r). There

exist games without essential equilibrium points, as we see by considering k t

the two person game

r

with

IAII

=

IA21

=

2 and U_i(a₁,a

₂₎

=

1, for all i,k,t € {.t,2}.

P EO: P is a guasi·strong equilibrium point of

r,

if C(p.)

=

B(p.) for all

1 1

i € N. It is still an open question whether any game possesses at least one

quasi-strong equilibrium point.

P € P is a perfect equilibrium point of

r,

if there exists a sequence

{p(k)}kEEOf completely mixed strategy conbinations, which converges to p, such that p. is a best reply against p.(k) for any i € Nand k € E.

1 1

PEP is an e-proper equilibrium point.(E > 0) if i) p is completely mixed, and

1.'1.') 1. ' f U. a.,p, (k - ) < U. a.,Pl· , ten p, ( t - ) h k ~ sp.; or a j/, f 11' 1. E N an d k

,x.

n E {I , ••• " . K }

1. 1. 1. 1. 1. 1. 1. 1.

p e: P is a proper equilibrium point of

r,

i f there exists a sequence {ek,p(k)}kEE such that Ek converges to 0, p(k) converges to p and p(k) is an Ek-proper equilibrium point of

r.

Any game possesses at least one proper equilibrium point and any proper equilibrium point is perfect ([5J).

3. Regular equilibrium points

In this section the concept of regular equilibrium points is introduced. The definition of regularity given in this paper is slightly different

from the definition given in [3J. The reason for such a different defini-tion is illustrated in example 3.6. In this example it is shown that (1.2) is not correct if regularity is defined as in [3JO' However, from Harsanyi's analysis ([3J) it easily follows that (1.2) is indeed correct if regularity is defined as in this paper. Furthermore, we derive in this section some elementary properties of regular equilibrium points and investigate what regularity means for bimatrix games.

We consider the set G

=

G(n,K₁, ••• ,Kn) of all games of a particular size .•.

n nK

Let K

=

IT K. and K*

=

I

K .• Let u E JR be the payoff vector of some

1.'=} 1. _k '1 L

1.= k. k

1 1. n

game rEG. Let a :r:: (&1 , ••• ,a.

,.> ••

,a ) € A. We define a mapping 1. n

(3.1) ~.(x;u)

k

=

x.~-x.

~.

.1.

k[' k

U.(a.,x.) - U.(a. ,x.) -

k.]

1. - ,

1. 1. 1. 1. 1. 1. 1. 1. 1. i ~ N, and for each i:

(3.2) K. 1.

=

L

k=l

x~

- 1 1. ' W e W1. '11 somet:unes wr1.te . . v~k (- ) k • x.;u 1. 1. i ~ N. k _ k. 1.

-for U.(a.,x.) - U.(a. ,x.). 1. 1. 1. 1. 1. 1. we deduce that for any a E A and u E JR nK:

(3.3) E(r ) c {x

~

:m.

K

u

*

_.

,

~(x;u)

=

a}.

From (2.4)

The inclusion in (3.3) may be strict,. since ~(a';u) = 0 for all a,a leA and u

~

JRnK. By Ja(p;u) we will denote the Jacobian of

~(.;u)

evaluated at p. Hence,

(3.4) J (p; u) a = --:?-....::...:...., a~(x;u)

ax

_x=p

Futhermore, if

M

is a matrix, then

IMI

is its determinant.

nK

Definition 3.5. Let

r

EO G, with payoff vector u ~:m.

•

Let p be an

equilibrium point of

r.

p is a regular equilibrium point if iJa(p;u)I ~ 0 for some a EO C(p).

Harsanyi [3J defined regular equilibrium points as equilibrium points p for which IJa(p;u) I

~

0, where a = (all, •••

,a~,

••• ,al). The difference

~ n

between these two regularity concepts is illustrated in example 3.6.

Example 3.6. Consider the set G(l,3) of 1 person games in which player

1 2 3 k

has exactly 3 pure strategies a

l ,al ,al • We will write ~ for Ul (al ) (k € {l,2,3}). Let U c G(I,3) be defined by:

u

l,u2 € [O,lJ u3 € [2,3J}'

Each element of U has exactly one equilibrium point, namely p

=

(0,0,1). If u € U, a

=

a! and a

=

ai we have: u l-u3 0

...

_{a a}

°

J (p ;u) = 0 0

°

J (p;u) = u

_Z

-u 3 u 3-ul

° °

Hence, for any u € U we have IJa(p;u)I

=

°

and IJa(p;u)I

~

0.

In example 3.6 we see that (1.2) is not 'correct if regularity is defined as in [3J (the set U has positive Lebesgue-measure in G(1,3». However, in his analysis Harsanyi everywhere implicitly assumes that any equili-brium point p is such that

p~

>

°

for all i € N, and so he actually

~

works with our definition of regularity. Since Harsanyi's arguments can be easily adjusted to our definition of regularity, we can conclude:

Theorem 3.7. For almost all games all equilibrium points are regular.

°

°

To check whether an equilibrium point is regular one has to find some a € C(p) with IJa(p;u)I ~ O. In theorem 3.8 we show that IJa(p;u)I ~ 0

for all a € C(p), if p is regular. Let a

=

(al, ••• tai, ••• ,an) € A and

... a

p

=

(p}, ••• ,p., ••• ,p ) € P. In Theorem 3.8 we write J (p;u) for the 1. n

matrix that results from Ja(p;u) when we cross out the rows and columns corresponding to the pure strategies not belonging to C(p) •

Theorem 3.8. Let rEG with payoff vector u € nnK. Let

1} 1n

and a'

=

(a

l , ••• ,an ) E A. Let p be an equilibrium point of

r.

We have: i) If a € C(p), ·then

t

ii) If a,a' E C(p), then IJa(p;u)j

=

0 if and only if IJa (p;u) I

= o.

iii) If a E.C(p), and p is. not quasi-strong, then IJa(p;u)I

=

0; hence

any regular equilibrium point is quasi-strong.

Proof. i) If a E C(p) and

a~

¢

C(p.), then 1. 1. aj.:,~ J x=p k

ax.

1. x=p

=

0 , i f x. .2, _J ~ x., and k 1. k., k 1. -= p.

ok

(p, ;u). 1. • 1. 1.

From this the statement follows immediately. ii) Let a,a' € C(p). It is clear that

n k _ IT IT ok (p.;u) ,.. 0 i=1 kiC(Pi) i ~ if and only if n k _ IT IT o~. (Pi;u) ,.. 0 • i-I kiC(p.) ~ l. ~

-a'

Hence, we have to show that

IJ

(p;U)

I

=

0 if and only if

IJ

(p;u)1

=

o.

This follows from the following observations:

k

ax.

]. k

ax.

]. :x=p .x=p

=

x=p

=

1

=

::: 0

=

, 'Z. a l.

a

F. (x; u) ]. k

ax.

_]. a' mt ,

a

.F i . (x;U) k

ax.

_]. , k. a ].

.a .

Fi(x;u) t

ax.

J x=p :x=p

for all i € Nand k € {1, ••• ,K.}

].

for all i c N; k,m,m' c C(Pi); m ;' k.;m' " _].

t:.

_~

for all i,j € N; j ;' i;9, € C(Pj)'

a~~(x;u)

~ a' k k _a~.~(x;u) p.

a

F. (x;u) p. ~ ~ J. + ~ .::'~ =

T-it !t !to it

ax. ~ ax. J. ax.

J _..x=p p.

1. J x=p p. ~ J x=p

for all

i.j

e N; i

+

j; ke C(p.); it e C(~.); k ~ k.,it ••

1. ) 1 . J .

iii) This statement follows immediately from i).

Next, we will invest~gate what regularity means f?r bimatrix games. Let

r

€ G(2,K

1,K2), with payoff vector u. Assume p is a regular equilibrium point of

r.

Then p is a quasi-strong equilibrium point. Without loss of

1 m I n

generality we may assume that C(Pl) = {al, ••• ,a_{1} and C(P2) = {a2, ••• ,a2}·} 1 2 Let a = (a 1,a2). Then e 0 m n ... a J (p;u) =

.,

~1

,

0 e'. m n t;2

.,

where

ek(Ok) is the row vector in R k with all coefficients equal to 1 (0),

o

is a matrix with all entries 0, t;l is an (m - 1)-by-n matrix with

and

t;2 is an (n - l)-by-m matrix with

2

6 .. ==

~J

... a

J (p;u) is nonsingular if and only if the matrices

are nonsingular.

From this it immediately follows that lC(PI)I= m

=

n

=

IC(P2)\'

More-~aT, by applying some elementary linear algebra and using the fact that p ~ E(f) it is, seen that 3.9 is equivalent with:

the matrices:

-(3.10) and U 2 ar~ nonsingular. We can conclude: Theorem 3.11. If

r

=

(A

1,A2,U1,U2) is a bimatrix game and p is an equi-librium point of f, then p is a regular equilibrium point if and only if:

i) p is a quasi-strong equilibrium point, and

ii) [C(pt)1

=

IC(pz)1 and the matrices U

1 andU2 (as in (3.10» are non-singular.

In [lJ equilibrium points which satisfy the conditions i) and ii) of theorem 3.11 are called nondegenerate equilibrium points. In that paper it is proved that (for all bimatrix games):

(3.12) any nondegene~ate equilibrium point ~s essential, and (3.13) any nondegene'ratlhequilibrium point is proper.

heavily use the 2-person character of the game. In section 4 of this paper we will."prove the. analogous statements o'f (3.12) and (3.13) for general n-person games in normal form, by using arguments based on Ja(p;u). The proofs given in this paper are a considerable improvement of those given in [1]. In section 4 we first prove that re-gular equilibrium points are essential. After that, we prove that rere-gular equilibrium points are proper, using the fact that regular equilibrium points are essential.

4. The main theorems

Theorem 4.1. Let

r

E G(n,K

1, ••• ,Kn) with defining vector U E JRnK • Assume

p

is a regular equilibrium point of

r.

Then there exists a neighborhood (= ngbh.)V

_O

of

p,

such that for any ngbh.

V

of p, with

V

c

VO'

there exists a ngbh. U of

u

such that for any U E U we have

IE(r )

n

vi

=

1.

u

Proof. Let a E C(p) be fixed. For any u E JRnK we will write F(. ;u) and J(.;u) for ~(.;u) and Ja(.;u), respectively. Consider the mappings:

*

*

ql JRnK x JRK + JRnK x JRK , defined by ql(u,x)

=

_(u,F(x,u»,

and

*

*

" JRnK JRK JRK _{, defined by ~(u,x)}

₌

_x.

'It x +

By the implicit function theorem and the fact that p is a regular equi-librium point of

r,

_{there exists a ngbh. Ut of} li, a ngbh. VI of p and

a

ngbh. WI of 0, such that ~ is a diffeomorphism (= a differentiable mapping which is bijective) from U1 x VI to UI x WI" Let U

_o

cUI' V

_o

c VI and W

_o

C WI be ngbh's of u,p and 0, respectively, such that

(4.2) ~ is a diffeomorphism from U

_o

x V

o

to U

o

x WO' (4.3) for all i € Nand k € {1, ••• ,K.}:

p~

>

°

implies

l. l. X € V; and k x. > 0, for all l. A k A A ~ ~

(4.4) for all i € N and all k,t € {1, ••• ,K.}: U.(a.;p) < U.(a.jp)

im-l. l. l. l. l.

k - 1

-plies U.(a.;x.) < U.(a.jx.), for all u € U

_o

and x € VO'

l. l. l. l. l. ].

-1

Let u € UO' Define x(u) := ~~ (u,O). Then x(u) € Vo and F(x(u);u)

=

O.

Since ~ is a diffeomorphism we have that x(u) is the only element x of Vo for which F(xju)

=

O. From (4.3) and 4.4) it follows that actually

x(u) €

E(r ).

From (3.3) it follows that {x(u)}

=

E(r )

n VO' The

asser-u u

tion from the theorem now follows from the fact that the mapping· Uo 3 u + x(u) € Vo is continuous.

Corollary 4.5. Any regular equilibrium point is essential.

Theorem 4.6. Any regular equilibrium point is proper.

Proof. Let r,p,u,uO'V

O and x(u) (u € UO) be as in (the proof of) theorem 4.1. For e: € [O,IJ and 'IT

=

('lTl' .. .,'lT

n) € P, define u(e:,'IT) €::nnK by:

(4.7)

where (a~ws.c) is that element of P where any player j in Splays a_j and any player

j

in SC(=N\S) plays

tt.~

The reader can verify that for any

J a E: A:

(4.8)

where (I €)a. + €w. is that element of p~ where any play~r j in N\{i}

~ 1. .6.

plays (I - €)aj + €W

j . Moreover, since for any p

=

(Pl, ••• ,Pn) E P, SeN and i E: N:

(4.9)

(4.10)

U~€,tt)(p)

=

U.(p.,(l - €)p. + €;.).

1. .. 1. 1. 1. 1.

It is clear that for € sufficiently small (say € E [O,e

O

»

u(e,w) will be in U

O' for all W e P. Let E E (O,EO)' For W E P let p(E,tt)

=

x(u(e,rr» be such that {p(E,tt)}

=

E(r

)

n VO' Let

0

=

eK/K and define P.(o) by:

",(e,tt) 1.

u

(4.11) P.(&)

=

{x. E P;

1. ~ 1.

k

x.

~ 0 for all k E {1, •• o,K.}}.

1. 1.

n

Let P(o)

=

II Pi (0). We define a multi valued mapping G€ P(o) ~ P(o) ~-­ i=1

by

(4.12)

' f (e:,IT)( k - (

»

(£,'11')( t - (

»

h k t 1 U. a.,p. e:,rr < U. a.,p. e:,rr , t en x. ~ e:x.

1 1 1 1 1 1 1 1

Then G7("Ii') :F

0

for any rr.,€

pea).

For, if k € {l, ••• ,K.}, let

1 1

(4.13)

and define x. € P.(o) by

1 1 (4.14) x. k

=

1 k.

t

Eg(R.) R.=1 (e:,rr) t - (E,rr) k - .

I .

U. (a.,p.(E,rr» > U. (a.,p.(e,rr»}, 1 1 1 1 1 1

Furthermore, G:(rr) is a closed and convex set, for any rr € P(c). From 1

the proof of theorem 4.1 it follows that the mapping

is continuous, and this implies that GE is upper semi continuous. Hence GE satisfies the conditions of the Kakutani fixed point. theorem [4J; so

For any i E N, k,~ E {1, ••• ,K.}, we have:

1.

(4.15) U.' (a.,p.(e:,1I"» (e: 11") k - . < U. (e:,1I") (a.,p.(e:,1I"»

~-1. 1. 1. 1. 1. 1.

implies

11"~ ~ e:1I"~.

1. 1.

From (4.10) it follows that for any i E N and k,~ E {1, ••• ,K.}:

1.

(4.16) ... U.(a.,(l - e:)p.(e,rr) k + err.) < U.(a.,(l - e:)p.(e:,rr) .... R- + e~.)

1. 1. 1. 1. 1. 1. 1. 1.

k ~

imp· lies rr. _1. ~ _. e:rr .•

= 1.

Define p(e,1I") E P by p(e:,1I")

=

(1 - e)P(e,rr) + e1l"- From (4.16) it follows k t

that, for all i EN and a.,a. E A.\C(p.(e,rr»: 1. 1. 1. 1.

(4.17) U.(a·,P·(e,rr» ... k - < U.(a.,p.(e:'1I"» .... t

-1. 1. 1. 1. 1. 1.

implies

Since p(e,rr) is an equilibrium point of u(e,rr) , we have for any i E N

k and a. E C(p.(e,rr»: 1. 1. (4.18) Hence, if (4.19) A R,-max U.(a"P'(e,rr». 1. 1. 1. R,E{l, ••• ,K.} 1.

then, we have for all i e Nand k,! e {l, ••. ,K_i}:

(4.20) -U.(a.,p.(e:,1T» k - . < U.(a.,p.(e:,1T» -

1-1. 1. 1. 1. 1. 1.

implies

Hence, p(e:,1T) is an n(e:)-proper equilibrium point of

r.

Since, (4.21) p is a quasi-strong equilibrium point of

r(= r_),

u

(4.22) p(e:,1T) is an equilibrium point of

r

and l' (e:,1T)

U(e:,1T)' (4.23) l.Dl u = fi, lim p(e:,1T)

=

p,

e:+0 e:+O

we have that C(p(e:,1T) == C(pL for e: sufficiently small.

Hence,

(4.24) lim min min

e:~0 i€N leC(p.(e:,1T»

1.

and so lim nee:) == O. Since

dO

1 p.(e:,1T) == 1. min min p:' ieN leC(p.) 1-1.

(4.25) p(e:,1T) is completely mixed, for e: >

O.

(4.26) p(e:,1T) is an n(e:)-proper equilibrium point of

r,

and (4.27) lim nee:)

=

0; lim p(e:,1T)

=

p,

e:+0 e:+0

we have that p is a proper equilibrium point of

r.

> 0

h h 3 7 ... C! 4' 1 d 4.6 a"nd the fact that any prope.r. equilibrium

. From t e t eorems .)'. ~. w ,,~,~ an

Corollary 4.28. For almost all finite noncooperative n-person games in normal form, we have

i) a11 equilibrium. points are quasi-strong, ii) all equilibrium. points are essential, iii) a11 equilibrium. points are proper, and

iv) a11 equilibrium. points are perfect.

References

[IJ Damme, E.E.C., van, Refinements of the Nash equilibrium concept

for bimatrix games. Memorandum. COSOR M 81-08, Dept. of Mathematics, Eindhoven University of Technology, Eindhoven, The Netherlands, 1981.

[2J Harsanyi, J.C., Games with randomly disturbed payoffs: a new

rationale for mixed strategy equilibrium points. Int. J. Game Theory

l

(1973)

1-23.-[3J Harsanyi, J.C., Oddness of the number of equilibrium points: a

new proof. Int. J. Game Theory

l

(1973) 235-250.

[4J Kakutani, S., A generalization of Brouwer's fixed point theorem.

Duke Math. J.,

!

(1941) 457-458.

[5J Myerson, R.B., Refinements of the Nash equilibrium concept.

Int. J. Game Theory

Z

(1978) 73-80.

[6J Nash, J.F., Equilibrium points in n-person games. Proc. Nat.

librium points in extensive games. Int. J. Game Theory

i

(1975) 25-55.

[8J Wu Wen-tsUn and Jiang Jia-he, Essential equilibrium points of