nash equilibria for noncooperative n -person games in normal form

SIAM REVIEWVol. 23, No. 2, April 1981

(C) 1981 Society for Industrial and Applied Mathematics

0036-1445/81/2302-0003 $01.00/0

NASH EQUILIBRIA FOR NONCOOPERATIVEn-PERSON GAMES IN NORMAL FORM*

STEVE H. TIJS"

Abstract. This paper concentrates on the problem of the existence of e-equilibrium points in the Nashsense for noncooperative n-person games in normal form. Firstly, a survey of known e-equilibrium pointtheorems and proof techniques is given. Then various new e-equilibrium point theorems are derived, in whichone of the players has a big strategy space and where nonbounded, payoff functions are allowed. To provethese theorems, results of Arzela-Ascoli are extended.

1. Introduction. Many social situations have in common that there are a number ofdecision makers with various objectives involved, who for some reason do notcooperate and where the final outcome of the situation depends only on the indepen-dently chosen actions or strategies of the different decision makers. When, moreover,the number of decision makers is n and when the rewards for the players, correspondingwith an outcome, can be expressed in real numbers, then such a situation can bemodeled as a noncooperative n-person game in normal form.

A very important role in noncooperative game theory is played by equilibria ande-equilibria, introduced by John Nash, which are stable outcomes in the sense that aunilateral deviation from an equilibrium point (e-equilibrium point) by one of theplayers does not increase (increases at most e) the payoff of that player.

In this paper the problem of the existence of equilibria and e-equilibria of n-persongames is central, especially for situations where one of the players has a "big" strategyspace and where unbounded payoff functions are allowed.

As a possible application we note that many oligopoly situations can be trans-formed into n-person games in normal form and that the problem of the existence ofe-Cournot-equilibria is equivalent to the problem of the existence of e-Nash-equilibria(cf. Friedman [10, pp. 168-172]).

For work relating economic situations and noncooperative games we refer to thepapers of Lee and Teo [20], Levitan and Shubik [22], Weddepohl [49] and the books ofFriedman [10], Marschak and Selten [25] and Farquharson [9].

The organization of this paper is as follows. In 2 we give formal definitions andintroduce the notation, used in this paper. Section 3 is devoted to a survey of known(e-)equilibrium point theorems, where in the discussion the emphasis is laid on usedproof techniques. Also, the relation with minimax theorems for two-person games isdiscussed.

In 4, a number of new e-equilibrium point theorems are derived. To prove someof these theorems we also generalize in 4 results of Arzela-Ascoli.

2. Definitions and notation. Let n be a natural number. In the following, N{1, 2,..., n} will denote the set of players in an n-person game.

An n-person game (in normal form) can be described by an ordered 2n-tuple(X1, X2, , Xn, K1, K2, , Kn), where for each player s N the nonempty set Xi ishis (pure) strategy space and where Ki is his payoff function, which assigns to eachelement (x,x,...,xn) of the outcome space X=I-IkNXk a real numberKi(xl, x2,

Such a game is played as follows.

* Received by the editors March 17, 1980.Department of Mathematics, Catholic University, 65.25 ED Nijmegen, the Netherlands.

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226 STEVE H. TIJS

(1) Each of the players chooses a strategy Xi E X which results in the point(xl, x2,’’’, xn) of the outcome space.

(2) Subsequently, each player E N obtains a payoff Ki(xl, x2, , xn).Corresponding to a fixed n-person game F (XI, X2, , X, K1, K2, ’, K) we

will use the following notation.(a) For each N, the Cartesian product IqklV-i Xk will be denoted by X_g and

elements of X_ will often be denoted by x_g. If x- (x l, x2,’", x)X, then theelement (xl, x2,’", xg-1, xi/l,"’, x)X-i will also be denoted by x_i.

(b) For iN, aiEXi and x_i=(xl, xE,...,Xg_l, Xg/, ,xn)X-i, the element(x, x2,"’, Xi-l, ai, xi/l,’" ,xn) will be denoted by (ai’x-i). Hence (ai’x-i) is theelement of X which can be constructed from ai Xg and x_i X_ in the obvious way.Note that for xX we have Ki(xix-i)= Ki(x).

(c) Let N, ei >-O, z X-i. The set of e-best replies to z for player is the set.{a Xi: Ki(a’z)>-_supx,xiKi(xiz)-ei}, which will be denoted by Ri(ei; z).

Now we define the important concept for n-person games in normal form,introduced by J. F. Nash.

DEFINITION. Let e,eE,’’’,en be nonnegative real numbers and let F=(X1, XE,’",X,K1, KE,’",K) be an n-person game. Then we call a point(:1, 2, ’, ) X an (e 1, e2, , e)-equilibrium point of the game F, if for each Nand each x EX we have

K(x_) <- K() +,or equivalently, if for each N, i R(e;-i). A (0, 0,. , 0)-equilibrium point isalso called an equilibrium point or a Nash equilibrium for the game.

For a nonempty setX let us denote the probability measure (on the o-algebra of allsubsets of X) with mass 1 in a X by e(a). Let be the convex hull of {e(a)" a X};i.e.,/z E " iff there is an s M and there are al, a2, as X and nonnegative realnumbers pl, p2," ", ps with sum Y’.--I Pk 1 such that/x --1 pke(ak).

DEFINITION. Let F (X1, X2, , Xn, K1, K2, , Kn) be an n-person game. Let[" be the n-person game (.., ’2,"’", ,/1,/2,""",/), with

Ki(/.L1, tL2, /tLn)= Ki(X1, X2,""’, Xn) d/.Ll(X1) d/z2(x2)

for all iN and (/Zl,/z2,’’’ ,tx,)f(=l-IilVf(i.Then is called the (finite) mixed extension of the game F and the elements of .i

are called mixed strategies for player i.We note that in the strategy spaces of F are extended by allowing the players to

use mixtures of a finite number of pure strategies. We note further that we have here nomeasurability problems; the integral in the above definition is, essentially, a finite sum.If the players 1, 2,..., n use the mixed strategies /-/,1, /-/,2,""’, /-n, respectively, thenKi(lZl, 2,"’ ",/z,) is the e.xpected payoff for pla.yer i. It will be obvious, what themeaning of X_g and tx-i X-i is, if N and/x X.

Much attention in the literature is paid to an important subclass of n-person gamesin normal form, namely the class of two-person zero-sum games, consisting of thosetwo-person games (X1, X2, KI, Kz) for which K1 /K2 is the zero-function on theoutcome space Xa X2. For a two-person zero-sum game (X1, Xz, K1, K2), importantnotions are the lower value _V(Xl, X2, K1, g2) supxlx inf,x KI(X1, X2) and theupper value O(X, 22, gl, K2) =inf.x SUpx,x, Kl(Xa, x2). If for a game(X1, 22, K1, K2) the lower value is equal to the upper value, then we say that the game

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NASH EQUILIBRIA FOR NONCOOPERATIVE n-PERSON GAMES 227

has a value, and then the elements of

and of

O(X,Xa, K,Kz)={ eX" inf K(:,Xz)>=v_ (X,XE, K,Kz)-e}x2X2

O (Xl, Xz, K1, Kz) {z e X." sup Kl(xl, z) <- f(Xl, Xz, K1, Kz) + e},xlXl

where e >_-0, are called e-optimal strategies for player 1 and for player 2, respectively.Furthermore, 0-optimal strategies are also called optimal strategies.

3. Minimax theorems and _e-equilibrium point theorems. In minimax theoremsone deals with the existence of values and optimal strategies for classes of two-personzero-sum games. The first minimax theorem was proved in 1928 by John von Neumann[29], and this theorem states that the mixed extension of a zero-sum two-person gamewith finite strategy spaces (also called a matrix game) possesses a value and optimalmixed strategies for both players. Since that time, many minimax theorems have beenderived with a great variety of proof techniques and also under a variety of conditions ofa topological and algebraic kind, put on strategy spaces and payoff functions. We do notwant to go into details here but we invite the reader to look at the papers of Fan [8],Hirschfeld [15], K6nig [18], Owen [31], Sion [40], Teh [42], Terkelsen [43] and Tijs[44] to get an impression of sufficient conditions for the existence of a value and of prooftechniques available. For a systematic treatment of minimax theorems, we also refer tothe survey paper of Yanovskaya [52], to Chapter 5 in the book [33] of Parthasarathyand Raghavan and to Chapter 6 in the book [41] of Stoer and Witzgall.

In _e-equilibrium point theorems, sufficient conditions for strategy spaces andpayoff functions are given to guarantee the existence of _e-equilibria for all _e > 0 or forthe existence of equilibrium points.

The first equilibrium point theorem was due to J. F. Nash (see Theorem 3.1). In therest of this section we will try to give an impression of the state of the art of this part ofgame theory by discussing proof techniques used by various authors, and by recallingsome of the equilibrium point theorems which will be used in 4. In that section somenew _e-equilibrium point theorems will be derived.

First of all let us note that the equilibrium problem is a natural extension of theminimax problem, as the following proposition shows. We leave the proof of thisproposition to the reader (cf. Tijs [45, pp. 756-757]).

PROPOSITION 3.1. Let F (Xx, X2, K1, K2) be a two-person zero-sum game. Thenwe have"

(1) F possesses a value and optimal strategies for both players iff F possesses anequilibrium point.

(2) F possesses a real value (i.e., _v(F) 7(F) [) iff F possesses (e , e2)-equilibriumpoints for all e > O, e2 > O.

(3) If F possesses a real value and el and e2 are nonnegative real numbers, then(1, 2) is an (el + e2, el + e2)-equilibrium point of F, if 1 is el-optimal and 2 ise2-0ptimal.

(4) ff F possesses a real value and if (1, 2) is an (e 1, e2)-equilibrium point, then 1is (el + e2)-optimal for player 1 and 2 is (el + e2)-optimal for player 2.

In view of Proposition 3.1 it is obvious that zero-sum game theory is a source ofinspiration for the general noncooperative theory. However, the equilibrium problem ismuch more complicated and many proof techniques in minimax theorems cannot be

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228 STEVE H. TIJS

used. Also stronger conditions are often needed in equilibrium point theorems. Thefollowing two theorems will be needed later on, but at this moment they can serve togive the reader an impression of the conditions which appear in equilibrium pointtheorems.

THEOREM 3.1. (Equilibrium point theorem of J. Nash [27], [28]). Let F=(X, X2, , X,, K, K., ., K,) be an n-person game, where the strategy spacesX,X2,..., X, are finite sets. Then the mixed extension [" possesses at least oneequilibrium point.

We note that the strategy spaces of F in Theorem 3.1 can be identified withcompact convex subsets of finite dimensional spaces (simplices), and that the payofffunctions of [" are continuous and multilinear.

THEOREM 3.2. (Equilibrium point theorem of Nikaido-Isoda [30]). Let F=(Xx, Xz, ., X, K, K, ., K,) be an n-person game such that

(1) X1, X, , X, are compact and convex subsets of topological vector spaces,(2) Kx, K., ., K, are continuous functions,(3) for all e N and all x_iX_i the function on Xi defined by xi--gi(xix-i) is a

concave function.Then F possesses at least one equilibrium point.

Now we want to discuss some proof techniques.1. In many proofs the e_-best reply multifunction R (e_,.) plays a role. This map

assigns to an x X the subset 1-I,rR(e x_) ofX (_e -> (0, , 0)). The usefulness ofthis function follows from the following proposition.

PROPOSITION 3.2. Let F (Xx, Xz, , X,, Kx, Kz, , K, be an n-person gameand e_ (e 1, F_, 2, In) -. Then we have

X is an e_-equilibrium point iff is a fixed pointof the multifunction R (e, (i.e., R (e, )).

In view of this proposition, the equilibrium point existence problem can bereplaced by the problem of the existence of fixed points of certain multifunctions.

The first proof of Theorem 3.1 in [27] used the best reply multifunction and thefixed-point theorem of Kakutani [17] for multifunctions. Other equilibrium pointtheorems with proofs based on the (_e-) best reply multifunction were given byGlicksberg [11], Browder [4], Rupp [37] and Bloemberg [2]. The first three mentionedauthors and also Halpern [12] derived new fixed-point theorems extending Kakutani’stheorem. Bloemberg gave an elementary proof for the existence of fixed points of the_e-best reply multifunction, when N {1, 2}, X1 [0, 1] and X2 [0, oo).

Because Bloemerg’s master’s thesis [2] is not published, we will recall here hismain result.

THEOREM 3.3. Let F=([0, 1], [0, oo),K1, Kz) be a two-person game with thefollowing properties"

(1) K2 is an upper bounded function.(2) For each x26[0, oe) the function Xl--Kx(xx, x2) is quasi-concave; for each

xa [0, 1] the function xz-- K2(xa, x2) is quasi-concave.(3) The family {xx--Kx(xx, x2): x26[0, oo)} of functions on [0, 1] is an equicon-

tinuous [amily.Then the game F has e_-equilibrium points [or each e_ > (0, 0).Ponstein [34] derived the existence of equilibrium points of constrained games

with the same technique, using the Glicksberg-Fan extension [11], [8] of Kakutani’stheorem. Debreu [5] also derived a very general existence theorem for constrainedn-person games using the fixed point theorem of Eilenberg-Montgomery [7].

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2. In the papers of Nikaido and Isoda [30] and of Rosen [35] a role is played by thefunction L"X xX R and the multifunction M"X-X defined by

L(x, y)= EiN

M(x) {y s X’L(x, y) max L(x, z)}.zX

This multifunction M has the following interesting property.PROPOSITION 3.3.. Let F, L andMbe as above. Then Xis an equilibrium point of

the game F iff is a fixed point of the multifunction M.Hence Proposition 3.3 also reduces the equilibrium problem to a fixed-point

problem.3. In his second proof in [28], J. F. Nash reduced the problem of the existence of

equilibrium points to the problem of the existence of a fixed point of a (univalent)function. The ingenious function f’. , which he constructed, we like to call thecorrection function of Nash. Extensions of Nash’s result, also using a correctionfunction, were given by Borges [3], Mahn [24], Marchi [26] and Jiang Jia-He [16].Fixed-point theorems of Brouwer or of Schauder-Tykhonov play a role. Vrieze alsoused, in [47], a correction function (and the fixed-point theorem of Schauder-Tykh-onov) to show that a class of games with a countable infinite number of players has anequilibrium point.

4. Another approach to the existence problem was followed in the papers of Tijs[44, pp. 95-96], Hazen [14] and Owen [32], where a sequence of finite subgames of theoriginal game was considered and where an (e-)equilibrium point of the original gamewas found, using the sequence of equilibrium points of the subgames considered. In[44], this approach was used to prove that tn oo-bimatrix games (A, B) with (upper)bounded B have e-equilibrium points for each e >0, and in [14] and [32] mixedextensions of noncooperative games were considered, where the strategy spaces wereequal to the closed interval [0, 1].

5. In Tijs [45], e-equilibrium points for two-person games F were derived bylooking at subgames F’ of F for which e-equilibria exist and where the strategy spaces ofF’ arise from the corresponding strategy spaces of F by the removal of e-dominatedstrategies. With the aid of this technique some of the results of [45] will be extended in4 of this paper to games with more than two persons.

6. In the papers [37] and [38] Rupp reduces the existence problem to the existenceof e-equilibria for certain subgames called A-marginal games (A-Rand Spiele). LetF (X1, X2, Xn, K1, K2, ",Kn) and A > 0. Then the A-marginal game of F is thegame

F(A) (XI(A), X2(A),..., X, (A), K1, K2, , K,),

where Xi(A) is the subset LI{Ri(A x-i)" x-i X-i} of Xi consisting of A-best replies forplayer to some x-i in X-i, and where the payoff functions of F(A) are restrictions of thepayoff functions of F. We note that A-marginal games were introduced by Bierlein [1]for two-person games. The connection between F and F(A) with respect to _e-equili-brium points is described in the following

PROPOSITION 3.4. Let A > 0 and e 1, e2, en [0, A ]. Then"(1) If is an (el, e:,’’’, e,)-equilibrium point of F, then s IIXi(A) and is an

(el,’"’, e,)-equilibrium point of F(A).(2) If is an _e-equilibrium point of F(A), then is also an e_-equilibrium point of F.

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230 STEVE H. TIJS

7. An elementary algebraic proof of the two-person version of the equilibriumpoint theorem of Nash is included in the paper of Lemke-Howson [21], where theexistence problem is transformed into a linear complementarity problem (cf. Shapley[39]). Also a method is described there (the Lemke algorithm) of finding equilibriumpoints for bimatrix games. For a quite different proof of the Nash theorem, usingalgebraic geometry, we refer to Harsanyi [13].

We conclude "this section with two remarks.(i) In the foregoing, the emphasis was on proof techniques and we paid less

attention to the precise conditions in the various mentioned e-equilibrium pointtheorems. But, roughly speaking, we can say that in the proofs using fixed-pointtheorems, the strategy spaces of the considered games are small (compact, precompactmetric) sets and the payoff functions satisfy heavy continuity and concavity conditions.The methods, where a game is approximated by subgames make it possible to derive_e-equilibrium point theorems for games where not all strategy spaces need to be small.

(ii) For works in which computational aspects of equilibrium points are consideredwe refer to Rosen [35], Kuhn [19], Lemke-Howson [21], Rosenmiiller [36], Liithi [23],Wilson [50] and Winkels [51]. Much research can still be done here; promising arefixed-point algorithms for functions and multifunctions (cf. [23]).

4. Some new f-equilibrium point theorems for n-person games in normal form.When the author was presenting in Heidelberg the paper [45] concerning the existenceof e-equilibrium points for two-person games in normal form, the question arose,whether results in that paper could be extended to games with more than two persons.In this section we show that most of the theorems in [45] can be extended in a certainway to n-person games. We derive, also, some other e-equilibrium point theorems forn-person games and mixed extensions with big outcome sets and unbounded payofffunctions. In the proofs the games will be compared with suitable small subgames forwhich one already knows that e-equilibrium points exist.

DEFINITIONS. Let F (X1, Xz, , X,, K1, K2, K,) be an n-person game andlet Y1, Y2,’ ", Y, be nonempty subsets of X1, X2,’’’, X,, respectively. Then thegame F’ (Y1, Y2, , Y,, K1, , K,), in which the payoff functions are restrictionsof the payoff functions in F to the set Y l-Ii= Y (and in which restrictions are denotedby the same symbol), is called a subgame of F.

Let 61, 62, , 6, be n nonnegative real numbers. We shall say that the subgame F’of the game F is a (61, 62, , 6,)-approximation of F if for each player the followingholds’

APP(i)’ For each xi X there exists a y Y, such that for all z X-i we haveK(x’z <= Ki(yi’z + 6.

Much use will be made of the following two lemmas, describing a relation betweene-equilibrium points of a game and of its approximations.

LEMMA 4.1. Let _e (el, 62," en) and _6 (81, t2,’ 8n) be nonnegativevectors in . Let F and F’ be as above and suppose that the n-person game F’ is a6_-approximation of the game F. Then each e_-equilibrium point of F’ is also an (e_ + 6_ )-equilibrium point of F.

Proof. Let (:91, 2,’" ", ,) be an _e-equilibrium point of F’. Take eN andx e X. Since APP(i) holds, there is a y e Y/such that

(1)Furthermore,(2) gi(yi/:-i) Ki(:) + Ei,

because 3 is an _e-equilibrium point of F’.

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Combining (1) and (2), we may conclude that

Ki(xi_i)<--Ki()-.l-i+Ei for all xiXi and sN.

Hence is an (_e +_)-equilibrium point of F. [3

LEMMA 4.2. Let e_,

_ff_ and let F’ be a _-approximation of the n-person game F.

For the mixed extensions and ’ o[ F and F’, respectively, we have"(1) ts a _-approximation of F.(2) An _e-equilibrium point of ’ is an (_e + 8_)-equilibrium point of .Proof. In view of Lemma 4.1, we need only to prove conclusion (1), because (2) is a

direct consequence of (1).Let s N and lzi Xi. We have to prove that there is a ui Yi such that

(3) i([Jli[)<:Ii(l’il,)’-i for allki

Suppose that i =,/=lpe(xe(])), where sN and p,..., p are nonnegative realnumbers with sum 1 and x(1),..., x(s)X. It follows from APP(i) that there existyi(1),..., yi(s) Yi such that for j 1,..., s we have

(4) Ii(e(xi(j))t.,)<=Ii(e(yi(]))l)+i for all x (X-i.

Take b’ 2j=1 pje(yi(])). Then it follows immediately from (4), that (3) holds. 71DEFINITIONS. Let e > 0 and let and be two families of real-valued functions on

a set K. We will say that e-dominates , if for each f s there is a g e ( such thatf(x)<-_g(x)+e for all xK. The family @ will be called an upper bounded family,if there is a c s N such that f(x) <= c for all x K and f .

The following proposition was proved in [45, pp. 759-760].PROPOSITION 4.1. Let e > O. Let Vbe a nonempty upper bounded subset of’ (i.e.,

there is a c such that for all (xl, , xn) V we have Xl <-_ c, x2 <-_ c, , x, <- c).Then there exists a finite subset W of V such that W e-dominates V; i.e., for each v Vthere is a w W, such that vi <- wi + e for 1,..., m.

LEMMA 4.3. Let e > O. LetE be a finite set and let be an upper bounded family ofreal-valuedfunctions on E. Then there exists a finite subfamily of;, which e-dominatesthe family

Proof. Let us denote the elements of the finite set E by a 1, a2, , am (m N). Foreach f s , let a(f) be the vector (f(al), f(a2),’", f(a,n))S ". Since is an upperbounded family, the set V {c(f)" f e } is an upper bounded subset of ". In viewof Proposition 4.1, we can take a finite subset W, which e-dominates V. Butthen {f : a(f) W} is a finite subfamily of , which e-dominates

The first e-equilibrium point theorem, which we now are going to prove, deals withmixed extensions of n-person games, where all but one pure strategy space are finitesets and one of them is an arbitrary set.

THEOREM 4.1. Let F (Xl, X2, Xn, K1, g2, Kn) be an n-person game innormal form such that

(1) XI, X2, ", X,_ are finite sets,(2) K is an upper bounded function.on X.

Then for each e > 0 the mixed extension F has (0, 0,...,0., e)-equilibrium points.

Proof. Let e > 0 and let E be the finite set X_, [-I7; Xg. By (2) the family

c’--{(X1, X2,""", Xn-1)r-->Kn(Xl, x2, Xn-1, Xn)" Xn Xn}

of functions on E is an upper bounded family. In view of Lemma 4.3, there exists a finitesubset Yn of Xn, such that the subfamily

(4__--{(Xl, X2,"" ", Xn-1)-->Kn(Xl, X2, Xn-1, Xn)" Xn Yn}

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232 STEVE H. TIJS

of e-dominates ,. This implies that the finite subgame (Xl, X2, Xn-1, Yn, K1,K2,’ , Kn) is a (0, 0,. ., 0, e)-approximation of F. By Nash’s Theorem 3.1, for ’equilibrium points exist. So it follows from Lemma 4.2 that such an equilibrium point isa (0, 0,..., 0, e)-equilibrium point of .

If two (or more) pure strategy spaces of an n-person game are nonfinite, then such agame may have no _e-equilibrium points for small positive el, e2,’", en, as thefollowing example shows.

Example 4.1. For n _->2 let Fn (X1, X2, Xn, K1, K2,’’’, K,) be the gamewhere Xi {1} for 1, 2,. , n 2 and X,-1 X [, where Ki(x) 0 for all x e Xand 1, 2,. ., n-2 and where

)[=1 ifx_l_->xn,K_l(1,1,’",l,x,_l,x,)=-K,(1,1,’..,1, Xn_l,X

=0 ifx_l<x.

Then it is easy to show that , has no (e 1, e2," , en)-equilibrium points if en-1 + en <1.Note that in this example n 2 of the n pure strategy spaces of F, are finite, and that thepayoff functions are bounded. Note further that for n 2 the corresponding game I2 isthe zero-sum game of A. Wald [48], which he used to show that not all mixed extensionsof oo oo-matrix games have a value.

If we want to derive e-equilibrium point theorems for games, where more than onepure strategy space is infinite, then that is possible if extra topological properties forstrategy spaces and payoff functions hold. We will need some definitions.

DEFINITIONS. Let Z be a metric space with metric d and let be a family ofreal-valued functions on Z.

1. We will say that is an equicontinuous family of functions on Z if

2. We will say that is an equi-upper semicontinuous family on Z if

VzezV>o::la>oV=ezVfe[d(x, z) < a ::> f(x) <f(z)+ e].

3. We will say that is an equi-uniform continuous family if

THEOREM 4.2. Let F (X1, X2, Xn, K1, g2, gn) be an n-person gamewith the following properties"

(1) for each 6 {1, 2,..., n- 1}, there is a metric di on Xi, such that (Xi, di) is acompact metric space and such that the family {xi - Ki(xi’x-i)" x-i X-i} is an equi-upper semicontinuous family on Xg,

(2) for each x X_,, the function xn--K,(x,’x) is an upper bounded function on

x..Then for each e_ (el, e2, en)> (0, 0,’ ", 0) the game has g-equilibrium

points.Proof. Take (el, e2,""", en)>(0, 0,’", 0). Let i6{1,... ,n-l}. Denote the

open ball {x Xi’dg(x, xi)<8} by U(xi, 8). In view of condition (1), for each xi Xithere is a 6(xi)> 0, such that

(5) Ki(xitx_i)<Ki(xitx_i)+ei for all Xi U(xi, a(xi)) and x_iX_i.

Since Xi is a compact space, we can find a finite subset Eg of X/such that

(6) U U(ei, t(ei)) Xi.ei Ei

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It follows from (5) and (6) that the game

(7) F’ (Ea, Ez,""" ,En-I,X,,K,"" ,Kn)

(el, e2,"" ’, en-, 0)-dominates F. Now the first n-1 pure strategy spaces of F’ arefinite, and by condition (2) the nth payoff function of F’ is upper bounded.

Hence the game F’ satisfies the conditions of Theorem 4.1, and this implies thatthere is a (0, 0,. , 0, e, -equilibrium point/2 of ". In view of (7) and Lemma 4.2 wemay conclude that/2 is an (e, e2,""", e, )-equilibrium point of

The proof of the following theorem runs along similar lines as that of the foregoingtheorem and is left for the reader to perform.

THEOREM 4.3. Let F (Xx, X2, , X,, K1, K2, Kn) be an n-person game, inwhich the strategy spaces X, X2, , X-I are precompact metric spaces and where ]:oreach {1, 2,. ., n 1} the family i {xi Ki(xi’x-i)" x-i X-i} is an equi-uniformcontinuous family on Xi and where for each x X_, the function x K, (x,’x) on X, isupper bounded.

Then F has (e, e2, e,)-equilibrium points for all (e, e2, e,) >(o, o,..., o).

The following theorem extends results in [32] and is a simple consequence ofTheorem 4.2. Such games as are in this theorem can be interpreted as games of timing,where the first n 1 players are obliged to act in the time interval [0, 1], and player n hasthe freedom to choose his action moment in A.

THEOREM 4.4. Let F (Xa, X2, ", X,_x, X,, Kx, K2, Kn) be an n-persongame such that

(1) X X2 X,_x [0, 1] and X, A ,(2) Ki" X is a uniformly continuous function for 1, 2, n 1,(3) K, X-g is upper bounded.

Then " has _e-equilibrium points for each _e > (0, O,. , 0).A simple consequence of Theorem 4.3, which will be used later on, is the following

theorem.THEOREM 4.5. Let F (X1, X2, , X,, Ka, , K,) be a game such that"(1) X, X2, ,X are precompact metric spaces,(2) gl, g2," gn-1 are uniformly continuous functions,(3) K is an upper bounded function.Then has e_-equilibrium points for each e_ > (0, 0,. ., 0).In the following theorem we give suffcient conditions for the existence of pure

_e-equilibrium points.THEOREM 4.6. Let F (X1, X2, , X, KI, Kz, , K,) be an n-person game

with the properties (1) and (2) of Theorem 4.2 and with the following additionalproperties:

(3) X, X2, ", X, are convex subsets of certain vector spaces,(4) for each Nand x-i X-i the function xi -Ki(xix-i) is a concave function on

X,(5))’or each Nand each xi Xi the function x-iKi(xix-i) is an affine function

on X_i.

Then the game F has _e-equilibrium points for each _e > (0, O, , 0).Proof. Take _e > 0. In view of the properties (1) and (2) and Theorem 4.2, we can

take an e-equilibrium point/3, (/2 ,/22,. ,/2,) s 3 of the mixed extension " of F. Foreach s N the mixed strategy i is of the form i E pi(s)e(xi(s)), where pi(s) > 0and xi(s) Xi for each s {1, 2,. ., ni} and Y pi(s) 1.

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234 STEVE H. TIJS

Let .i 7’= pi(s)xi(s). Then .i Xi for each N, because Xi is a convex set. Weare going to prove that (1, x2, , n) is an _e-equilibrium point of the game F. Forthat purpose we note that for each N and each xi Xi we have:

(a) Ki(xi’-i)= Ig2i(e(xi)12-i)in view of property (5).(b) Ki(e(xi)’t2_i)<=Ii(fii’fi_i)+ el, since/2 is an _e-equilibrium point of(c) Ii([ffl,i[ffl,-i)"-" i(,spi(s)e(xi(s))tl2-i) 7’= pi(s)Ki(xi(s)t-i) by property

(5).(d) 7’__ pi(s)Ki(xi(s),_i)<-Ki(,i,-i) by property (4).

Combining (a), (b), (c) and (d) yields for each N

Ki(xi_i)Ki(i/_i)-J-Ei Ki().--E for all xisXi.

Hence is an (el, e2,’’ ", en)-equilibrium point of F. [3The following two propositions, 4.2 and 4.3, can be seen as extensions of well-

known results of Arzela-Ascoli (cf. [6, pp. 266-267]) to nonbounded families offunctions. They will be used to find suitable approximations for certain classes of games.

First we recall that a family of real-valued functions on a topological space K issaid to be an equicontinuous lamily, if for each x K and each e > 0, there exists aneighborhood U(x) of x, such that I](x)-’(y)l < e for all y e U(x) and all 1 e .

PROPOSITtON 4.2. LetK be a compact topological space, let be an upper boundedand equicontinuous]amily o] real-valued]’unctions on Kand let e > O. Then there exists a

finite subfamily and r such that e-dominates .Pro@ For each a e K, let U(a) be an open neighborhood of a such that

(8) It’(x)-’(a)l <-- 1/2e for all x g(a) and all " ft.

Since K is compact, we can take a finite subset E of K such that K t.Jz U(a). Foreach 1 , let ’* E - N be the function on E with 1* (x) ’(x) for each x E.

Then the family -*={1* :fe } is an upper bounded family of real-valuedfunctions on the finite set E. In view of Lemma 4.3, we can find a finite subfamily * of

* such that *e-dominates *. Let * ={hi, h2,..., hs}. For each k e{1,..., s}take a gk , such that g hk.

Then the family d {gl, g2, , gs} is a finite subfamily of and we .will show nowthat e-dominates r. Let " e . Then there is a k e {1, , s}, such that

(9) ?[(e)=J*(e)<--g’(e)+1/2e=gk(e)+1/2e for each e E.

We are finished with the proof, if we can show that [(x) <- gk (X) q- e for all x e X.Let x X. Take e e E such that x U(e). Then in view of (8),

(10) f(x)<-]r(e)+1/2e, gk(e)<--gk(X)+1/2e.

In view of (9) and (10) we obtain

f(x) (f(x)- f(e)) + (f(e)- gk(e)) + (gk(e)- gk(X)) + gk(X) <---- e + gk(X).

Hence we have proved that e-dominates 5g. [3PROPOSITION 4.3. LetKbe a precompact metric space with metric d, let be an upper

bounded equi-uniform continuous family of real-valued functions on K and let e > O.Then there exists a finite subfamily of ;such that f e-dominates .

Proof. Take a 6 > 0 such that for all x, y K with d(x, y) < 6 we have ]/(x)-/’(y)l <--3X-e. Since K is precompact, we can find a finite subset E of K, such that K t.JaE U(a),where U(a) is the ball {x e K: d(x, a)< 6}. Now similarly as in the proof of Proposition4.2, we can introduce a family 5* of functions on E and find a ge-domlnatmg subfamily

* of * and construct with the aid of c.g. an e-dominating subfamily of . 71

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THEOREM 4.7. Let F (X1, Xz, , Xn, K1, K2, Kn) be an n-person gamewith the following properties:

(1) XI, X., , Xn are convex subsets of linear topololgical spaces.(2) Xx, X2, , Xn-x are compact sets.(3) Ka, K2, Kn are continuous functions on X.(4) For each N and x-i X-i the function xi - Ki(xix-i) is a concave function

OH Xi.(5) Thefamily ; {x-n -Kn (Xn’X-n)" Xn Xn} is an upper bounded and equicon

tinuous family offunctions on X_,.Then F has (0, 0,. , 0, e)-equilibrium points for each e > O.Proof. Take e > 0. It follows from property (2), that the set X-n is compact. So, in

view of property (5) and Proposition 4.2, we can take a finite subset E, of X, such thatthe finite family

e-dominates . This implies that the game F’ (X1, X2, , Xn-1,(0, 0,.. , 0, e)-dominates F. Then also F"(X, X2,’ , Xn-, eonv (En), K1,.. ,(0,..., 0, e)-dominates F, where conv (En) is the convex hull of En. Since conv (En) isconvex and compact, the game F" satisfies all conditions in the equilibrium pointTheorem 3.2 of Nikaido-Isoda. Hence F" has equilibrium points and then F has(0, 0,..., e)-equilibrium points in view of Lemma 4.1.

THEOREM 4.8. Let F <X, 22, Xn, Kx, K2, Kn) be an n-person gamewith the following properties"

(1) Xx, X2, ", Xn- are precompact metric spaces with metrics d, dE, dn-1,respectively.

(2) For each {1, 2,..., n 1} and each x, Xn, the function on

(Xl, X2, Xn-1)>Ki(xl, X2, Xn-1, Xn)

is uniformly continuous.(3) The famity Y; {x-n --Kn(xnx-n)" xn Xn} is an upper bounded equi-.uniform

continuous family offunctions onThen " has (el, e2, en)-equilibrium points ]’or each (el, e2,’’’,

(o, o,..., o).Proof. Take _e > 0. In view of (1), the set X-n is a precompact metric space with

metric d(x_,, y-n) maxk=.....n-1 dk(xk, Yk), where x_, (xa, x2, , xn-1) and y-n(ya, y2, , yn-a). Hence, it follows from (3) and Proposition 4.3 that there is a finitesubset Y, of Xn such that

F’- (X1, X2,’ ’, Xn-1, Yn, K, ", Kn)

(0; 0, .., 0, e)-dominates F. It follows from (2) and the finiteness of Y, that Ki’XIX2 " X,,_ Yn is uniformly continuous for 1,.. , n 1. Hence F’ satisfiesthe conditions in Theorem 4.5. This implies that " has (el, e2,’’" ,en-,1/2en)-equilibrium points. By Lemma 4.2, " (0,..., 0, 1/2en)-dominates ’. Hence " has(e,..., en)-equilibrium points .

It will be obvious now, that with the aid of other well-known e-equilibrium pointtheorems and the technique of approximation, many other new _e-equilibrium pointtheorems may be derived.

5. Final comments. In 4 we have seen that the proof technique using _6-approximations of games yields _e-equilibrium point theorems, where rather weak

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236 STEVE H. TIJS

conditions are posed on strategy spaces and payoff functions. In a forthcoming paper[46], we will use a similar technique to derive _e-equilibrium point theorems forstochastic games, where one of the players at each stage may have a big action space.

In this paper we considered only finite mixed extensions of n-person games, toavoid measurability problems. But in a standard manner many of the results in thispaper for finite mixed extensions can be generalized to other mixed extensions.

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nash equilibria for noncooperative n -person games in normal form

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