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Perfection of Nash equilibria in continuous games
mathematical social sciences ELSEVIER
Mathematical Social Sciences 29 (1995) 225-237
Perfection of Nash equilibria in continuous games L. M 6 n d e z - N a y a a'*, I. G a r c i a - J u r a d o b, J . C . C e s c o c "Departamento de Econometria y Metodos Cuantitativos, Universidad de Santiago de Compostela, 15771 Santiago de Compostela, Spain bDepartarnento de Estadistica e Investigacion Operativa, Universidad de Santiago de Compostela, 15771 Santiago de Compostela, Spain CInstituto de Matematica Aplicada San Luis, Universidad Nacional de San Luis, E]ercito de Los Andes 950, 5700 San Luis, Argentina
Received October 1993 Revised April 1994
Abstract
In this paper we introduce a generalization of Selten's perfect equilibrium for continuous n-person games in normal form. We also study some properties of our new concept as well as its relationship with other equilibria. Some facts that we show contrast significantly with the finite theory. Keywords: Nash equilibrium; Perfect equilibrium; Normal form game; Infinite games
1. Introduction
In this p a p e r we define a notion of perfectness in a class of normal form games with uncountable sets of strategies (generalizing the concept introduced in Selten, 1975, for finite games). We work with n-person normal form games, each player having a set of strategies represented by the closed [0, 1] interval of the real line and a continuous payoff function (we call them continuous games). H o w e v e r , the results presented in this p a p e r are still true when the sets of strategies of the players are compact real intervals [a, b] and could also be extended to other classes of infinite games. It is interesting to remark that a particular collection of continuous games are the zero-sum games on the square, which were widely * Corresponding author. 0165-4896/95/$09.50 ~ 1995 - Elsevier Science B.V. All rights reserved S S D I 0165-4896(94)00768-4
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studied in the 1950s (see, for instance, Dresher et al., 1950; Bohnenblust et al., 1950; and Karlin, 1959). The paper is organized as follows. Notation and some definitions are given in Section 2. We close the section with a characterization of Nash equilibria for continuous games in terms of the set of best responses and the carrier of the strategies. In Section 3 we first introduce a natural notion of perfect equilibrium. Then, we remark that perfect equilibria are also Nash equilibria. We also state an existence result, give a characterization for perfection, and develop an example showing that, in fact, our concept provides a strict refinement of a Nash equilibrium. In Section 4 we recall some characterizations of perfect equilibria in finite games. Then, we demonstrate that similar versions for some of the implications involved in those characterizations are also true in the framework of continuous games. The other implications are proved to be false in Section 5.
2. Preliminaries
A continuous n-person game (in normal form) ~g is a system q3 = (N, { X i ) i E N , ( K i } i E N ) ,
where N = { 1 , 2 , . . . ,n} is the non-empty set of players and, for each i E N , X i = [0, 1] is the set of pure strategies for player i and Ki" IIjE N Xj---> R, a continuous function, is the payoff function for player i. Let us denote [0, 1] n =
IIj N xj. The set of mixed strategies for any player i E N is the set Si of all probability measures on [0, 1] (when considering the Borel g-algebra). Note that it is a subset of the locally convex linear space M of all signed measures on [0, 1]. In such a space, we consider the weak topology. Taking into account that M is the dual of C([0, 1]) (the set of all continuous functions from [0, 1] to ~) (see, for instance, Parthasarathy and Raghavan, 1971), such a topology is the smallest one that verifies that, for any f ~ C([0, 1]), the functional f of the form M
~,
/z
,f(/z) =
[ fdpt, [0,11
is continuous. Si turns out to be a non-empty, convex and weakly compact (compact with respect to the weak topology) subset of M. Hence it is a metrizable set (it admits a metric compatible with the weak topology) and, then, its closed subsets can be characterized by sequences. It is interesting to remark that, from the definition of the weak topology of M, it can be proved that a sequence
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{/.~}k~IC_M of measures converges weakly to / x E M (i.e. converges when considering the weak topology on M) if and only if, for every f E C([0, 1]),
lim ffdtx~=lim ffdtz. I0,11
[0,1]
For further details about weak topologies the reader is referred to Rudin (1966, 1973) or Hildenbrand (1976). Let us call S the set [lie N St of all joint mixed strategies. Given a positive measure IX on [0, 1], not necessarily a probability, and a Borel set f~ C_[0, 1], we denote by ~(A) the measure assigned by tx to the set A. The mixed extension F of the game ~ is the system /~ ~- (N, {Si}iE N, {~4}i~N), where N and every Se are as before and, for each i E N , ~.: I]j~N Sj--) R is the mathematical expectation of the payoff function Ki; more precisely, given F = ( F , , . . . , 1 7 , ) E S, ff{i(F) = f . . . f Ki(rl, . . . , r , ) d F l ( r l ) . . .dFn(r,) ,
(1)
[0, 1] n
where the right-hand side of (1) indicates the multiple integral over [0, 1]" with respect to the product measure generated by F 1 , . . . , F , (note that, in this context, Fubini's theorem holds). Clearly, ~ is a multi-linear function and, since the payoff functions Ke are continuous, it is weakly continuous (continuous when considering the product weak topology on S); namely, if {Fk)k~>l is a sequence in S converging weakly to F ~ S, then {Y{/(F~)}~>I converges to K~(F). A Nash equilibrium for the game F is a joint strategy F = (F 1. . . . , F , ) E S satisfying that ~T{~(F) >i ~ ( F I Gi)
for all i E N and all Gi E S~.
Here, F I G i denotes the joint strategy in S defined by (F 1. . . . . Fi 1, G~, F~+1. . . . , F,). A Nash equilibrium for the game F is also called a Nash equilibrium in mixed strategies for the game ~. For existence results about Nash equilibria concerning continuous games like those studied in this paper, we refer the reader to Burger (1959) and Parthasarathy and Raghavan (1971). We say that the strategy of player i, F~ E S~, is dominated if there exists G~ E S~ such that ~ ( , ~ I F,) ~<~'~(,~ I G,)
for all/5 E S ,
~(PIFi)<~(PIGi)
forsomePES.
and
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We say that a joint strategy F = ( F 1 , . . . , Fn) E S is dominated if there exists i E N such that F~ is dominated. Joint strategies which are not dominated are called undominated. Let G e be a probability measure on [0, 1]. The carrier C(G/) of Gi is the set defined by C(G~) = { r E [0, 1]:G~([r- e , r + e ] ) > 0 f o r alle > 0 } . We say that a strategy of player i, F/, is completely mixed if C(F~) = [0, 1]. We say that a joint mixed strategy F = ( F 1 , . . . , F,) E S for the game F is completely mixed if each F i (i ~ N) is completely mixed. For any r E [0, 1] we call 6(r) the probability measure on [0, 1] concentrated on r, i.e. that probability measure having C(6(r)) = {r}. Let F -- ( F 1 , . . . , Fn) E S. The set of pure best replies of player i to F, Bi(F), is the set defined by IB~(F) = {r i E [0, 1]:
~ ( g I6(r~)) i> ~ ( g 1,~(7,)) for all Y~~ [0, 1]}.
It is easy to show that both C(G~) and ~ ( F ) are non-empty closed subsets of [0, 1] for every probability measure G~ and joint strategy F ~ S. We are now ready to prove a characterization of Nash equilibria of F similar to one holding for finite games. Theorem 1. Let F be a continuous game. A joint strategy F = (F 1. . . . Nash equilibrium o f F if and only if C(Fe) C_Bi(F ) for all i @ N.
, F,) E S is a
Proof. Let us first assume that F is a Nash equilibrium and there exist i ~ N and r i ~ [ 0 , 1] such that riEC(F~) but ri~_Bi(F). Because B~(F) is a closed subset of [0, 1], there is e > 0 such that [r i - e, r i + e] A Bi(F) = 0. Since r i E C ( F i ) , F~([r~ - e, r i + e]) > 0. Now, let Fi E Be(F ). Then Y~(F 16(~i) ) ~
~(F I ~(~,.)) = (
[9~(F 16(~,) ) -
Y{,.(F18(?e))l dF,(f,)
< O.
[0,11
But this contradicts the assumption that F is a Nash equilibrium. Conversely, take F E S with C(F,.)C_ Bi(F ) for all i E N. Then, for any i E N, Y{~(F) =
f aw{i(Fl~(ri)) dF~(ri) C(Fi)
and, since we are assuming that C(F,.)C_ Bi(F), it follows that 5~(F) = Y ~ ( F I 6 ( ~ ) ) for any ?~ ~ B~(F). Besides, for all G~ E Se, it is easy to prove that
(2)
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~.(FI G,) ~<~(F I~(F/))
229
(3)
for any 7~ E [Be(F ). Taking into account that (2) and (3) hold for all i ~ N, we conclude that F is a Nash equilibrium. []
3. Perfect Nash equilibrium Let P be the class of all positive measures tt on [0, 1] satisfying /z([0, 1]) < 1 and/z(A) > 0 for all Borel sets A_C [0, 1]. Given i E N and/Xl @P we define Si(tzi) by S,(l~i) = {Fi E S~: F,(A)/>/zg(A) for all Borel sets A C [0, 1]}.
Let us denote by P" = Ilje N P the cartesian product of n copies of P. Given a joint measure/z = (/z 1. . . . ,/%) E P " , we define the/x-perturbed game F(/z) of F by
r ( ~ ) = (N, (s;o,,)}i~N,
(~c~},~N).
A Nash equilibrium for the game P(/z) is a point F E [Ije N Sj(/zj) satisfying Y~(F) >i Y{i(F I Gi)
for all i E N and all G, E Si(]Jbi) .
It is easy to see that, for each I x E P " and each i E N , the set S~(/zi) is a non-empty, convex and weakly closed subset of Si (and, hence, weakly compact). We say that a sequence {/z~}~>l C_P" converges to zero as k tends to infinity if {p.~([0, 1])}k~>1 converges to zero as k converges to infinity (for all i E N). We say that a joint strategy F E S is a perfect equilibrium if and only if there exist sequences {/zk}~>l C P" and {Fk}k>~ C_S such that: (1) {Fk}k~>l converges to F as k tends to infinity (in the weak sense). (2) {/Z~}k~>~converges to zero as k tends to infinity. (3) For each k E N, F k is a Nash equilibrium of F(/zk). We can state the following existence result: Theorem 2. A continuous n-person game F always has a perfect equilibrium. Proof. A straightforward application of Fan and Glicksberg's fixed point theorem (see Fan, 1952) shows that every /.L-perturbed game has a Nash equilibrium. Then, from the weak compactness of S, it follows that F has a perfect Nash equilibrium. [] It is easy to prove that perfect equilibria are, in fact, Nash equilibria for F (because of the weak continuity of the Y~, functions); that is the reason why sometimes we call them perfect Nash equilibria.
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Next we prove a useful characterization of a Nash equilibrium point for /x-perturbed games. Theorem 3. F ~ S(/X) is a Nash equilibrium of a tz-perturbed game of F if and only if Fi(Bi(F) ~) =/Xi(~i(F) ~) (where, for any A C_[0, 1], A~ denotes the complement of the set A with respect to [0, 1]). Proof. Let F E S(/X). We first note that, for all i @ N and G i E Si(/xi), the function G i - / x l , defined by (Gi-/Xi)(A) = Gi(A ) -/X~(A) for all Borel sets A C_[0, 1], defines a positive measure on [0, 1]. Besides, for any Gg ~ Sg(/X~),
f ~(fl6(r~))dG~(ri)+
~(FIG~)=
Bi(F)
f ~(Fl6(r~))dai(ri) ai(F) c
= f Y~(Fl6(r~))dG~(r~)+ ai(F)
+
f Y{i(Flt~(ri))d(Gi-/xi)(ri) ai(F)c
f ~C(Fl,~(ri))
d/x~(r/)
Bi(F)c
= ~(FI~(~,))G,(8~(F)) + M(Gi -/X,)(a~(F))
+ f ~'{/(Fl~(r,) ) d/xi(ri) ,
(4)
BI(F) c
where ~ is any point in B~(F) and M <~~(FIS(Fi) ). Note that the constant M in expression (4) must be smaller than 5t{/(F I ~(F/) ) whenever (G i - ~)(B,(F) ~) > 0. Thus, taking into account (4) and the fact that F~(Bi(F))+ F~(Si(F) ~) = 1 for every F, E S~, we conclude that the following assertions are equivalent: (a) F is a Nash equilibrium of F(/X); (b) Y{i(FlFi) = maxo,~s,(~i ) Y{i(F[ Gi) for all i E N; (c) Fi(Bi(F)) = max6~s,(~i ) G~(Bi(F)) for all i ~ N; (d) F~(~/(Ff) =/X~(B~(F) ¢) for all i ~ N. (To obtain that (b) implies (c) take into account that, as we remarked above, M in expression (4) is smaller than ~(F 16(F~)) whenever (G i -/xi)(~i(F) ~) > 0.) The equivalence between (a) and (d) proves our theorem. [] Using Theorem 3 we can present an interesting characterization of perfect equilibria for continuous games. Take/X = (/Xl . . . . ,/xn) E P" and consider F E S a completely mixed joint strategy of a continuous game F. We say that F is a /x-perfect equilibrium of F if, V i ~ N , Fi(Bi(F)C)~/xi(Bi(F)C). The following theorem provides the characterization mentioned above. In its proof we use the fact that a positive measure/X on [0, 1] can be introduced through its distribution function @,: R---> R defined by
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~g(r) =
0, /z([0, r]), [./x([0, 11),
I
231
ifr<0, if0~ 1.
Theorem 4. Let F be a continuous n-person game and take F @ S as a joint strategy o f F. The following two assertions are equivalent: (i) F is a perfect Nash equilibrium of F. (ii) There exist {/xk}~>l, a sequence in P converging to zero, and {Fk}k~>l, a sequence in S converging weakly to F, such that, for all k E ~, F k is a txk-perfect equilibrium of F. Proof. Let us first prove that (i) implies (ii). Since F is perfect, there exist
{tzk}k~l, a sequence in P converging to zero, and { F k } ~ l , a sequence in S converging weakly to F, such that, for all k E N, F k is a Nash equilibrium of F(tzk). Hence, for all k E N , F k is completely mixed and, from Theorem 3, F~(Bi(Fk) c) = Id,i([~i(Fk)C). Consequently, for all k E ~, F k is a/zk-perfect equilibrium of F. Let us see now that (ii) implies (i). Take F, {F~}k>~ and {/Xk}k~>~ in the conditions of (ii). Now, for each k E N and each i E N, let ~7~ be the measure having the distribution function defined by ~,f(x) = Ff((-oo, x] fq Bi(F*) c) + (1/(2k))Ff((-oo, x] N B~(Fk))
(5)
for all x E ~. Taking into account that {/Zk}k~>~ converges to zero and that Fik (Bi( F k ) c ) <~Ixi(ll~(F k ) c) for all k E ~ and all i E N, it is clear, from (5), that {rlk}k>~l converges to zero. Now, since F ik ( ~ i ( F k) c) = r/ik ( B / ( Fk) c ) (see the definition of rtf ) for all k >/1 and all i E N, from Theorem 3 we deduce that F g is a Nash equilibrium of the ~Tk-perturbed game F(r/k) (where 7/k = (rt k, . . . , r/~)) for all k/> 1. This proves that F is a perfect Nash equilibrium of F. [] We close the section with an example which shows that a perfect Nash equilibrium provides a strict refinement of a Nash equilibrium. Example 1. Let us consider the two-person continuous game F with the payoff functions defined on [0, 1] × [0, 1] by KI(X,y)=K2(x,y)'~-xy
for
allx,
y.
It is clear that (8(0), 8(0)) is a Nash equilibrium of F. Suppose that (8(0), 8(0)) is perfect. Then, there exists a sequence {(F k, Gk)}k~l, converging weakly to (8(0), 8(0)), of Nash equilibria of a corresponding sequence of /xg-perturbed games. Since KI(1, y) > Kl(X , y) for all (x, y) E [0, 1] × [0, 1] (x ~ 1) except for y = 0 (where the equality sign holds), it follows that ~1(6(1), G k) > Y£I(F, G k) for all k/> 1 (F ~ 6 ( 1 ) ) . This means that 8(1) is the unique best reply against G k for all k/> 1. Therefore, because of Theorem 3, F k coincides with / k on [0, 1).
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Hence, taking into account that F k is a probability measure, it must have an atom in 1 (assigning probability Fk([0, 1]) --/zk([0, 1]) to that point). Thus, it is clear that, as k converges to zero, F k converges weakly to 8(1) (not to 3(0) as we assumed above). This contradiction proves that (~(0), ~(0)) is not perfect.
4. Some relationships valid for continuous games In this section we recall some relations between a perfect equilibrium and a Nash equilibrium holding for finite games, and then we prove similar statements for continuous games. We recall that a finite n-person game (in normal form) is a system J =
(N, {~i}ieN, (Pi)icN),
where N is as before and, for each i E N, ~i is a non-empty finite set and Pi is a real function defined on q~ = I-[/~N ~ . The mixed extension J of J is the system J = (N,
{~i},cN,{/~,}gEN),
where, for all i C N, ~i is the set of all probability measures defined on ~., a n d / 3 , defined on I-IjEN ~ , is the mathematical expectation of Pr For more details about this class of games, the reader is referred to van Damme (1991). The following result, which we state without proof, as well as the definitions of a perfect Nash equilibrium and undominated strategies for finite games, can also be found in van D a m m e (1991).
Theorem 5. Let J = (N, {~i}ieN, (Pi}iEN) be a finite n-person game and let be a mixed joint strategy. The following two assertions are equivalent: (i) ~b is a perfect Nash equilibrium of J; (ii) ~b is a limit point of a sequence {th(k)}k~ 1 in II/~ N ~ of completely mixed strategies with the property that qb is a best reply against every element ~b(k) in that sequence. Moreover, both (i) and (ii) imply (iii) below: (iii) ~b /s undominated. If n = 2, the three assertions are equivalent. The following theorem shows that, in the framework of continuous games, the implications ( i i ) ~ ( i ) and ( i i ) ~ ( i i i ) are also valid for statements which are natural extensions of those introduced for the finite case.
Theorem 6. Let F be a continuous n-person game and {Fk} k>>,l a sequence in S of completely mixed joint strategies converging weakly to F E S. Assume that, for each k ~ 1 and each i E N, C(F~) C_Bi(Fk). Then
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233
(a) F is a perfect Nash equilibrium o f F; (b) F is an undominated Nash equilibrium o f F. Proof. (a) For each k/> 1 and each i E N, let 7?/~ be the measure having the distribution function defined as in (5). It is clear that F ik E Si(~ik ) for all k/> 1 and all i E N. Let us show that r/~([0, 1]) tends to zero as k tends to infinity. Since C(F/) C Bi(F k) we have that 1 >i 1 - F ki ( B i ( F )k c ) = Fk(Bi(Fk)) >! F/k(C(F~)).
(6)
Now, due to the fact that F~ converges weakly to F,., we have that F~(C(F~)) converges to F,(C(F,.)) = 1. Because of (6) we obtain that F ik( B i ( F )k c ) converges to zero as k tends to infinity. This result shows that the first term in the right-hand side of (5) converges to zero when k grows to infinity. The convergence of the second term to zero is obvious. In the same way as in Theorem 4, it can be proved that, for all k E ~ , F k is a Nash equilibrium of F(r/k). Hence, F is a perfect Nash equilibrium of F. (b) If F were dominated, there would exist i E N and G i ~ S i such that ~ ( P [ F~) ~< ~ ( P [ Gi)
for all P E S ,
Y { / ( F [ F / ) < ~ ( F [ Gi)
forsome F E S .
and
These relationships imply, in particular, that Y~(8(r) [F~) ~< ~(/~(r) [ G~)
for all 8(r) = (/~(rl) . . . . .
~ ( 8 ( ~ [F~) < ~,.(8(~) [ Gi)
for some 8(r-') = (~(r"l) .
/~(rn) ) ~ S ,
(7)
and . . . .
~(F'n)) ~ S .
From the last inequality and the weak continuity of the function ~/(- [F~) we conclude that there is a neighborhood U of ~ in [0, 1]" such that, for all r ~ U, ~ ( ~ ( r ) I F~) < ~.(/~(r) I a i ) . Then, since the strategies F ~ are completely mixed and taking (7) into account, we conclude that ~.(Fk I F~) < ~ ( F k I a~) for all k/> 1. But these inequalities contradict the assumption that F~ is the best reply of player i against F k for all k/> 1 and all i E N. [] We say that F is a strongly perfect equilibrium of F if it verifies the condition of Theorem 6, i.e. if there exists {Fk}k>>_X a sequence of completely mixed joint strategies converging weakly to F such that, for all k t> 1 and all i E N, C(F~)C
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Bi(Fk). In view of Theorem 6, every strongly perfect equilibrium is perfect and every strongly perfect equilibrium is undominated. However, none of the other implications is valid for continuous games. In the next section we provide some counter-examples to prove it.
5.
Counter-examples
In this section we present two examples showing interesting features for continuous games. Example 2 exhibits the fact that an undominated Nash equilibrium does not necessarily have to be perfect (even for two-person games). Example 3 shows a two-person continuous game not having undominated Nash equilibria.
Example 2. Let us consider the two-person continuous game F with the payoff functions defined on [0, 1] × [0, 1] by
(y-x)x, ify>x, Kl(x'Y)= - ( y - x ) Z ( 1 - x ) , i f y ~ x , K2(x, y) = 0 for all (x, y) E [0, 1] x [0, 1]. The assertion we are going to prove is that the joint strategy (6(1), 6(0)) is an undominated Nash equilibrium which is not a perfect one. It is easy to see that (6(1),6(0)) is an undominated Nash equilibrium. In order to prove that (6(1), 6(0)) is not perfect, let us consider a completely mixed strategy G E S 2 for player 2. Then, both G([0, 1/2]) and G((1/2, 1]) are greater than zero. Taking into account that Kl(X, y) is an increasing function of y for all x E [0, 1], we conclude that, if x E [0, 1/2], ~{1(6(X), G) ~
KI(X, 0)G([0, 1/2]) + K,(x, 1/2)G((1/2, 1]) = - x2(1 - x)G([0, 1/2]) + x(1/2 - x)G((a/2, 1]).
Thus, by taking x E [0, 1] small enough, we can guarantee that
ff{'l(6(X), G) > O.
(8)
Now, suppose that (6(1), 6(0)) is a perfect Nash equilibrium. Then there exists a sequence {(Fk, Gk)}k~l, converging weakly to (6(1), 6(0)), of Nash equilibria of a corresponding sequence of/zk-perturbed games. Since {Gk}k~l converges weakly to 6(0), there exists k 0 such that Gk([0, 1/50])/> 14/15 for all k/> k 0. This implies that G~((1/50,1])<~1/15 for all k>~ko. Then, for all k>~ko and all x >t 1/50, taking into account that Kl(X, y) is an increasing function of y (for all x),
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~1(8(x), G k) <~K,(x, 1)Gk((1/50,
1]) +
235
Kl(X , 1/50)Gk([o, 1/50]) ~<
~
(9)
Because of (8), (9) and Theorem 3 we conclude that, in F(/zk), any best reply of player 1 against G k (in particular F~), has to behave like t, k on [1/2, 1] for all k >/k 0. But this implies that {Fg}k.1 cannot converge to the strategy 8(1). This shows that (6(1), 8(0)) is not perfect. It is well known that (the mixed extension of) every finite n-person game has undominated Nash equilibria. The following example shows that the situation in the framework of continuous games is substantially different. In fact, it exhibits a (mixed extended) continuous game without undominated Nash equilibria. Example 3. Let us consider the two-person continuous game F with the payoff functions defined on [0, 1] × [0, 1] by: "xy -- X 2 ,
K, (x, y) = ( 1 / 2 - y ) ( x - 1), 1/2 ( y - 1 / 2 ) ,
ifx ~< 1 / 2 , ifx/> 1/2 and y ~< 1 / 2 , i f x ~ 1/2 and y/> 1 / 2 ,
2x)y, (4x 2 - 2x)y + 1/2 - y , K2(x, Y) = 1/2-y, "(4X 2 --
( - 4 y 2 + 2y)(x - 1 / 2 ) ,
if0~
First we show that (6(0), 8(0)) is the only Nash equilibrium of this game. Let us assume that (F, G) is a Nash equilibrium of F. Since K2(x ,
1/2) > K2(x, y)
for all y E (1/2, 1], and all x ~ [0, 1],
we have that
Y(2(F,a(1/Z))> Ygz(F, 6(y))
f o r a l l y E ( 1 / 2 , 1].
This implies that C(G) C_ [0, 1/2]. Now, if G = 8(0), player 1 has only two pure best replies against G: 8(0) and 8(1). It is easy to verify that (8(0), 6(0)) is a Nash equilibrium and that neither (8(1), 8(0)) nor any other joint strategy (F, 8(0)) ~ S, where F = (1 - Cl)8(0) + cl8(1 ) with 0 < c 1 < 1, can be a Nash equilibrium. On the other hand, if C(G)C_[0, 1/2] but G~8(O) we can show, like in Example 2, that there exists a pure strategy x(G) ~ (0, 1/2) which is a best reply of player 1 against G and such that 5g~(8(x(G)), G) > 0. Besides, it is easy to see that Y(~(8(x), G) <~0 for all x ~ [1/2, 1]. Hence, if F is to be a best reply against G for player 1, C(F) C_ [0, 1/2]. However, F cannot be 8(0) because Kl(0, y) = 0 for
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/Strongly \ Perfect
Undominated
A
Perfect
Ex2 ~~'* Fig. I. Overviewof the relations between concepts. all y ~ [0, 1], which implies that ~t~1(8(0), G) = 0 < $'t~x(8(x(G)), G). This condition naturally excludes (8(0), G) from being a Nash equilibrium. Besides, since 9'/'~(8(1/2), G ) < 0, (8(1/2), G) cannot be a Nash equilibrium either. Finally, it is easy to verify that the best reply of player 2 against F with C(F) C_[0, 1/2] (F being neither 8(0) nor 8(1/2)), is the pure strategy G = 8(0). But, in this case, G was chosen to be different from 8(0). From this contradiction we conclude that the point (8(0), 8(0)) is the only Nash equilibrium of the game. But now it is easy to see that, for player 1, 8(1) dominates 8(0). Hence, the only Nash equilibrium of this game (8(0), 8(0)) is not undominated. Combining Example 2 and Theorem 6 we have that an undominated Nash equilibrium does not have to be strongly perfect. Example 3 shows a two-person continuous game not having undominated Nash equilibria. Since it must have a perfect Nash equilibrium (according to Theorem 2), from this example we can conclude that perfectness does not imply undomination (for a Nash equilibrium). Moreover, in view of Theorem 6, this example also shows that a perfect Nash equilibrium does not have to be strongly perfect. Finally, observe that Examples 2 and 3 together show that the notions of a perfect Nash equilibrium and an undominated Nash equilibrium are, in fact, independent (even in the two-person case). All these relations are summarized in Fig. 1.
Acknowledgements We thank the University of Santiago de Compostela and Xunta de Galicia for financial support through projects 60902.25064(5060), XUGA20102B91 and
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237
XUGA20701B91. We are also grateful to two anonymous referees for many helpful comments and suggestions.
References H.F. Bohnenblust, S. Karlin and L.S. Shapley, Games with continuous convex pay-off, in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games, Annals of Mathematics Studies 24 (Princeton University Press, Princeton, 1950) pp. 181-192. E. Burger, Theory of Games (Prentice-Hall, Berlin, 1959). M. Dresher, S. Karlin and L.S. Shapley, Polynomial games, in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games, Annals of Mathematics Studies 24 (Princeton University Press, Princeton, 1950) pp. 161-180. K. Fan, Fixed points and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. USA 38 (1952) 121-126. W. Hildenbrand, Core and Equilibria of a Large Economies (Princeton University Press, Princeton, 1976). S. Karlin, Mathematical Methods and Theory in Games, Programming and Economy (AddisonWesley, New York, 1959). T. Parthasarathy and T.E.S. Raghavan, Some Topics in Two-person Games (Elsevier, New York, 1971). W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966). W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973). R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, Int. J. Game Theory 4 (1975) 25-55. E.E.C. van Damme, Stability and Perfection of Nash Equilibria (Springer-Verlag, Berlin, 1991).
Mathematical Social Sciences 29 (1995) 225-237
Perfection of Nash equilibria in continuous games L. M 6 n d e z - N a y a a'*, I. G a r c i a - J u r a d o b, J . C . C e s c o c "Departamento de Econometria y Metodos Cuantitativos, Universidad de Santiago de Compostela, 15771 Santiago de Compostela, Spain bDepartarnento de Estadistica e Investigacion Operativa, Universidad de Santiago de Compostela, 15771 Santiago de Compostela, Spain CInstituto de Matematica Aplicada San Luis, Universidad Nacional de San Luis, E]ercito de Los Andes 950, 5700 San Luis, Argentina
Received October 1993 Revised April 1994
Abstract
In this paper we introduce a generalization of Selten's perfect equilibrium for continuous n-person games in normal form. We also study some properties of our new concept as well as its relationship with other equilibria. Some facts that we show contrast significantly with the finite theory. Keywords: Nash equilibrium; Perfect equilibrium; Normal form game; Infinite games
1. Introduction
In this p a p e r we define a notion of perfectness in a class of normal form games with uncountable sets of strategies (generalizing the concept introduced in Selten, 1975, for finite games). We work with n-person normal form games, each player having a set of strategies represented by the closed [0, 1] interval of the real line and a continuous payoff function (we call them continuous games). H o w e v e r , the results presented in this p a p e r are still true when the sets of strategies of the players are compact real intervals [a, b] and could also be extended to other classes of infinite games. It is interesting to remark that a particular collection of continuous games are the zero-sum games on the square, which were widely * Corresponding author. 0165-4896/95/$09.50 ~ 1995 - Elsevier Science B.V. All rights reserved S S D I 0165-4896(94)00768-4
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studied in the 1950s (see, for instance, Dresher et al., 1950; Bohnenblust et al., 1950; and Karlin, 1959). The paper is organized as follows. Notation and some definitions are given in Section 2. We close the section with a characterization of Nash equilibria for continuous games in terms of the set of best responses and the carrier of the strategies. In Section 3 we first introduce a natural notion of perfect equilibrium. Then, we remark that perfect equilibria are also Nash equilibria. We also state an existence result, give a characterization for perfection, and develop an example showing that, in fact, our concept provides a strict refinement of a Nash equilibrium. In Section 4 we recall some characterizations of perfect equilibria in finite games. Then, we demonstrate that similar versions for some of the implications involved in those characterizations are also true in the framework of continuous games. The other implications are proved to be false in Section 5.
2. Preliminaries
A continuous n-person game (in normal form) ~g is a system q3 = (N, { X i ) i E N , ( K i } i E N ) ,
where N = { 1 , 2 , . . . ,n} is the non-empty set of players and, for each i E N , X i = [0, 1] is the set of pure strategies for player i and Ki" IIjE N Xj---> R, a continuous function, is the payoff function for player i. Let us denote [0, 1] n =
IIj N xj. The set of mixed strategies for any player i E N is the set Si of all probability measures on [0, 1] (when considering the Borel g-algebra). Note that it is a subset of the locally convex linear space M of all signed measures on [0, 1]. In such a space, we consider the weak topology. Taking into account that M is the dual of C([0, 1]) (the set of all continuous functions from [0, 1] to ~) (see, for instance, Parthasarathy and Raghavan, 1971), such a topology is the smallest one that verifies that, for any f ~ C([0, 1]), the functional f of the form M
~,
/z
,f(/z) =
[ fdpt, [0,11
is continuous. Si turns out to be a non-empty, convex and weakly compact (compact with respect to the weak topology) subset of M. Hence it is a metrizable set (it admits a metric compatible with the weak topology) and, then, its closed subsets can be characterized by sequences. It is interesting to remark that, from the definition of the weak topology of M, it can be proved that a sequence
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227
{/.~}k~IC_M of measures converges weakly to / x E M (i.e. converges when considering the weak topology on M) if and only if, for every f E C([0, 1]),
lim ffdtx~=lim ffdtz. I0,11
[0,1]
For further details about weak topologies the reader is referred to Rudin (1966, 1973) or Hildenbrand (1976). Let us call S the set [lie N St of all joint mixed strategies. Given a positive measure IX on [0, 1], not necessarily a probability, and a Borel set f~ C_[0, 1], we denote by ~(A) the measure assigned by tx to the set A. The mixed extension F of the game ~ is the system /~ ~- (N, {Si}iE N, {~4}i~N), where N and every Se are as before and, for each i E N , ~.: I]j~N Sj--) R is the mathematical expectation of the payoff function Ki; more precisely, given F = ( F , , . . . , 1 7 , ) E S, ff{i(F) = f . . . f Ki(rl, . . . , r , ) d F l ( r l ) . . .dFn(r,) ,
(1)
[0, 1] n
where the right-hand side of (1) indicates the multiple integral over [0, 1]" with respect to the product measure generated by F 1 , . . . , F , (note that, in this context, Fubini's theorem holds). Clearly, ~ is a multi-linear function and, since the payoff functions Ke are continuous, it is weakly continuous (continuous when considering the product weak topology on S); namely, if {Fk)k~>l is a sequence in S converging weakly to F ~ S, then {Y{/(F~)}~>I converges to K~(F). A Nash equilibrium for the game F is a joint strategy F = (F 1. . . . , F , ) E S satisfying that ~T{~(F) >i ~ ( F I Gi)
for all i E N and all Gi E S~.
Here, F I G i denotes the joint strategy in S defined by (F 1. . . . . Fi 1, G~, F~+1. . . . , F,). A Nash equilibrium for the game F is also called a Nash equilibrium in mixed strategies for the game ~. For existence results about Nash equilibria concerning continuous games like those studied in this paper, we refer the reader to Burger (1959) and Parthasarathy and Raghavan (1971). We say that the strategy of player i, F~ E S~, is dominated if there exists G~ E S~ such that ~ ( , ~ I F,) ~<~'~(,~ I G,)
for all/5 E S ,
~(PIFi)<~(PIGi)
forsomePES.
and
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We say that a joint strategy F = ( F 1 , . . . , Fn) E S is dominated if there exists i E N such that F~ is dominated. Joint strategies which are not dominated are called undominated. Let G e be a probability measure on [0, 1]. The carrier C(G/) of Gi is the set defined by C(G~) = { r E [0, 1]:G~([r- e , r + e ] ) > 0 f o r alle > 0 } . We say that a strategy of player i, F/, is completely mixed if C(F~) = [0, 1]. We say that a joint mixed strategy F = ( F 1 , . . . , F,) E S for the game F is completely mixed if each F i (i ~ N) is completely mixed. For any r E [0, 1] we call 6(r) the probability measure on [0, 1] concentrated on r, i.e. that probability measure having C(6(r)) = {r}. Let F -- ( F 1 , . . . , Fn) E S. The set of pure best replies of player i to F, Bi(F), is the set defined by IB~(F) = {r i E [0, 1]:
~ ( g I6(r~)) i> ~ ( g 1,~(7,)) for all Y~~ [0, 1]}.
It is easy to show that both C(G~) and ~ ( F ) are non-empty closed subsets of [0, 1] for every probability measure G~ and joint strategy F ~ S. We are now ready to prove a characterization of Nash equilibria of F similar to one holding for finite games. Theorem 1. Let F be a continuous game. A joint strategy F = (F 1. . . . Nash equilibrium o f F if and only if C(Fe) C_Bi(F ) for all i @ N.
, F,) E S is a
Proof. Let us first assume that F is a Nash equilibrium and there exist i ~ N and r i ~ [ 0 , 1] such that riEC(F~) but ri~_Bi(F). Because B~(F) is a closed subset of [0, 1], there is e > 0 such that [r i - e, r i + e] A Bi(F) = 0. Since r i E C ( F i ) , F~([r~ - e, r i + e]) > 0. Now, let Fi E Be(F ). Then Y~(F 16(~i) ) ~
~(F I ~(~,.)) = (
[9~(F 16(~,) ) -
Y{,.(F18(?e))l dF,(f,)
< O.
[0,11
But this contradicts the assumption that F is a Nash equilibrium. Conversely, take F E S with C(F,.)C_ Bi(F ) for all i E N. Then, for any i E N, Y{~(F) =
f aw{i(Fl~(ri)) dF~(ri) C(Fi)
and, since we are assuming that C(F,.)C_ Bi(F), it follows that 5~(F) = Y ~ ( F I 6 ( ~ ) ) for any ?~ ~ B~(F). Besides, for all G~ E Se, it is easy to prove that
(2)
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~.(FI G,) ~<~(F I~(F/))
229
(3)
for any 7~ E [Be(F ). Taking into account that (2) and (3) hold for all i ~ N, we conclude that F is a Nash equilibrium. []
3. Perfect Nash equilibrium Let P be the class of all positive measures tt on [0, 1] satisfying /z([0, 1]) < 1 and/z(A) > 0 for all Borel sets A_C [0, 1]. Given i E N and/Xl @P we define Si(tzi) by S,(l~i) = {Fi E S~: F,(A)/>/zg(A) for all Borel sets A C [0, 1]}.
Let us denote by P" = Ilje N P the cartesian product of n copies of P. Given a joint measure/z = (/z 1. . . . ,/%) E P " , we define the/x-perturbed game F(/z) of F by
r ( ~ ) = (N, (s;o,,)}i~N,
(~c~},~N).
A Nash equilibrium for the game P(/z) is a point F E [Ije N Sj(/zj) satisfying Y~(F) >i Y{i(F I Gi)
for all i E N and all G, E Si(]Jbi) .
It is easy to see that, for each I x E P " and each i E N , the set S~(/zi) is a non-empty, convex and weakly closed subset of Si (and, hence, weakly compact). We say that a sequence {/z~}~>l C_P" converges to zero as k tends to infinity if {p.~([0, 1])}k~>1 converges to zero as k converges to infinity (for all i E N). We say that a joint strategy F E S is a perfect equilibrium if and only if there exist sequences {/zk}~>l C P" and {Fk}k>~ C_S such that: (1) {Fk}k~>l converges to F as k tends to infinity (in the weak sense). (2) {/Z~}k~>~converges to zero as k tends to infinity. (3) For each k E N, F k is a Nash equilibrium of F(/zk). We can state the following existence result: Theorem 2. A continuous n-person game F always has a perfect equilibrium. Proof. A straightforward application of Fan and Glicksberg's fixed point theorem (see Fan, 1952) shows that every /.L-perturbed game has a Nash equilibrium. Then, from the weak compactness of S, it follows that F has a perfect Nash equilibrium. [] It is easy to prove that perfect equilibria are, in fact, Nash equilibria for F (because of the weak continuity of the Y~, functions); that is the reason why sometimes we call them perfect Nash equilibria.
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Next we prove a useful characterization of a Nash equilibrium point for /x-perturbed games. Theorem 3. F ~ S(/X) is a Nash equilibrium of a tz-perturbed game of F if and only if Fi(Bi(F) ~) =/Xi(~i(F) ~) (where, for any A C_[0, 1], A~ denotes the complement of the set A with respect to [0, 1]). Proof. Let F E S(/X). We first note that, for all i @ N and G i E Si(/xi), the function G i - / x l , defined by (Gi-/Xi)(A) = Gi(A ) -/X~(A) for all Borel sets A C_[0, 1], defines a positive measure on [0, 1]. Besides, for any Gg ~ Sg(/X~),
f ~(fl6(r~))dG~(ri)+
~(FIG~)=
Bi(F)
f ~(Fl6(r~))dai(ri) ai(F) c
= f Y~(Fl6(r~))dG~(r~)+ ai(F)
+
f Y{i(Flt~(ri))d(Gi-/xi)(ri) ai(F)c
f ~C(Fl,~(ri))
d/x~(r/)
Bi(F)c
= ~(FI~(~,))G,(8~(F)) + M(Gi -/X,)(a~(F))
+ f ~'{/(Fl~(r,) ) d/xi(ri) ,
(4)
BI(F) c
where ~ is any point in B~(F) and M <~~(FIS(Fi) ). Note that the constant M in expression (4) must be smaller than 5t{/(F I ~(F/) ) whenever (G i - ~)(B,(F) ~) > 0. Thus, taking into account (4) and the fact that F~(Bi(F))+ F~(Si(F) ~) = 1 for every F, E S~, we conclude that the following assertions are equivalent: (a) F is a Nash equilibrium of F(/X); (b) Y{i(FlFi) = maxo,~s,(~i ) Y{i(F[ Gi) for all i E N; (c) Fi(Bi(F)) = max6~s,(~i ) G~(Bi(F)) for all i ~ N; (d) F~(~/(Ff) =/X~(B~(F) ¢) for all i ~ N. (To obtain that (b) implies (c) take into account that, as we remarked above, M in expression (4) is smaller than ~(F 16(F~)) whenever (G i -/xi)(~i(F) ~) > 0.) The equivalence between (a) and (d) proves our theorem. [] Using Theorem 3 we can present an interesting characterization of perfect equilibria for continuous games. Take/X = (/Xl . . . . ,/xn) E P" and consider F E S a completely mixed joint strategy of a continuous game F. We say that F is a /x-perfect equilibrium of F if, V i ~ N , Fi(Bi(F)C)~/xi(Bi(F)C). The following theorem provides the characterization mentioned above. In its proof we use the fact that a positive measure/X on [0, 1] can be introduced through its distribution function @,: R---> R defined by
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~g(r) =
0, /z([0, r]), [./x([0, 11),
I
231
ifr<0, if0~
Theorem 4. Let F be a continuous n-person game and take F @ S as a joint strategy o f F. The following two assertions are equivalent: (i) F is a perfect Nash equilibrium of F. (ii) There exist {/xk}~>l, a sequence in P converging to zero, and {Fk}k~>l, a sequence in S converging weakly to F, such that, for all k E ~, F k is a txk-perfect equilibrium of F. Proof. Let us first prove that (i) implies (ii). Since F is perfect, there exist
{tzk}k~l, a sequence in P converging to zero, and { F k } ~ l , a sequence in S converging weakly to F, such that, for all k E N, F k is a Nash equilibrium of F(tzk). Hence, for all k E N , F k is completely mixed and, from Theorem 3, F~(Bi(Fk) c) = Id,i([~i(Fk)C). Consequently, for all k E ~, F k is a/zk-perfect equilibrium of F. Let us see now that (ii) implies (i). Take F, {F~}k>~ and {/Xk}k~>~ in the conditions of (ii). Now, for each k E N and each i E N, let ~7~ be the measure having the distribution function defined by ~,f(x) = Ff((-oo, x] fq Bi(F*) c) + (1/(2k))Ff((-oo, x] N B~(Fk))
(5)
for all x E ~. Taking into account that {/Zk}k~>~ converges to zero and that Fik (Bi( F k ) c ) <~Ixi(ll~(F k ) c) for all k E ~ and all i E N, it is clear, from (5), that {rlk}k>~l converges to zero. Now, since F ik ( ~ i ( F k) c) = r/ik ( B / ( Fk) c ) (see the definition of rtf ) for all k >/1 and all i E N, from Theorem 3 we deduce that F g is a Nash equilibrium of the ~Tk-perturbed game F(r/k) (where 7/k = (rt k, . . . , r/~)) for all k/> 1. This proves that F is a perfect Nash equilibrium of F. [] We close the section with an example which shows that a perfect Nash equilibrium provides a strict refinement of a Nash equilibrium. Example 1. Let us consider the two-person continuous game F with the payoff functions defined on [0, 1] × [0, 1] by KI(X,y)=K2(x,y)'~-xy
for
allx,
y.
It is clear that (8(0), 8(0)) is a Nash equilibrium of F. Suppose that (8(0), 8(0)) is perfect. Then, there exists a sequence {(F k, Gk)}k~l, converging weakly to (8(0), 8(0)), of Nash equilibria of a corresponding sequence of /xg-perturbed games. Since KI(1, y) > Kl(X , y) for all (x, y) E [0, 1] × [0, 1] (x ~ 1) except for y = 0 (where the equality sign holds), it follows that ~1(6(1), G k) > Y£I(F, G k) for all k/> 1 (F ~ 6 ( 1 ) ) . This means that 8(1) is the unique best reply against G k for all k/> 1. Therefore, because of Theorem 3, F k coincides with / k on [0, 1).
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Hence, taking into account that F k is a probability measure, it must have an atom in 1 (assigning probability Fk([0, 1]) --/zk([0, 1]) to that point). Thus, it is clear that, as k converges to zero, F k converges weakly to 8(1) (not to 3(0) as we assumed above). This contradiction proves that (~(0), ~(0)) is not perfect.
4. Some relationships valid for continuous games In this section we recall some relations between a perfect equilibrium and a Nash equilibrium holding for finite games, and then we prove similar statements for continuous games. We recall that a finite n-person game (in normal form) is a system J =
(N, {~i}ieN, (Pi)icN),
where N is as before and, for each i E N, ~i is a non-empty finite set and Pi is a real function defined on q~ = I-[/~N ~ . The mixed extension J of J is the system J = (N,
{~i},cN,{/~,}gEN),
where, for all i C N, ~i is the set of all probability measures defined on ~., a n d / 3 , defined on I-IjEN ~ , is the mathematical expectation of Pr For more details about this class of games, the reader is referred to van Damme (1991). The following result, which we state without proof, as well as the definitions of a perfect Nash equilibrium and undominated strategies for finite games, can also be found in van D a m m e (1991).
Theorem 5. Let J = (N, {~i}ieN, (Pi}iEN) be a finite n-person game and let be a mixed joint strategy. The following two assertions are equivalent: (i) ~b is a perfect Nash equilibrium of J; (ii) ~b is a limit point of a sequence {th(k)}k~ 1 in II/~ N ~ of completely mixed strategies with the property that qb is a best reply against every element ~b(k) in that sequence. Moreover, both (i) and (ii) imply (iii) below: (iii) ~b /s undominated. If n = 2, the three assertions are equivalent. The following theorem shows that, in the framework of continuous games, the implications ( i i ) ~ ( i ) and ( i i ) ~ ( i i i ) are also valid for statements which are natural extensions of those introduced for the finite case.
Theorem 6. Let F be a continuous n-person game and {Fk} k>>,l a sequence in S of completely mixed joint strategies converging weakly to F E S. Assume that, for each k ~ 1 and each i E N, C(F~) C_Bi(Fk). Then
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233
(a) F is a perfect Nash equilibrium o f F; (b) F is an undominated Nash equilibrium o f F. Proof. (a) For each k/> 1 and each i E N, let 7?/~ be the measure having the distribution function defined as in (5). It is clear that F ik E Si(~ik ) for all k/> 1 and all i E N. Let us show that r/~([0, 1]) tends to zero as k tends to infinity. Since C(F/) C Bi(F k) we have that 1 >i 1 - F ki ( B i ( F )k c ) = Fk(Bi(Fk)) >! F/k(C(F~)).
(6)
Now, due to the fact that F~ converges weakly to F,., we have that F~(C(F~)) converges to F,(C(F,.)) = 1. Because of (6) we obtain that F ik( B i ( F )k c ) converges to zero as k tends to infinity. This result shows that the first term in the right-hand side of (5) converges to zero when k grows to infinity. The convergence of the second term to zero is obvious. In the same way as in Theorem 4, it can be proved that, for all k E ~ , F k is a Nash equilibrium of F(r/k). Hence, F is a perfect Nash equilibrium of F. (b) If F were dominated, there would exist i E N and G i ~ S i such that ~ ( P [ F~) ~< ~ ( P [ Gi)
for all P E S ,
Y { / ( F [ F / ) < ~ ( F [ Gi)
forsome F E S .
and
These relationships imply, in particular, that Y~(8(r) [F~) ~< ~(/~(r) [ G~)
for all 8(r) = (/~(rl) . . . . .
~ ( 8 ( ~ [F~) < ~,.(8(~) [ Gi)
for some 8(r-') = (~(r"l) .
/~(rn) ) ~ S ,
(7)
and . . . .
~(F'n)) ~ S .
From the last inequality and the weak continuity of the function ~/(- [F~) we conclude that there is a neighborhood U of ~ in [0, 1]" such that, for all r ~ U, ~ ( ~ ( r ) I F~) < ~.(/~(r) I a i ) . Then, since the strategies F ~ are completely mixed and taking (7) into account, we conclude that ~.(Fk I F~) < ~ ( F k I a~) for all k/> 1. But these inequalities contradict the assumption that F~ is the best reply of player i against F k for all k/> 1 and all i E N. [] We say that F is a strongly perfect equilibrium of F if it verifies the condition of Theorem 6, i.e. if there exists {Fk}k>>_X a sequence of completely mixed joint strategies converging weakly to F such that, for all k t> 1 and all i E N, C(F~)C
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Bi(Fk). In view of Theorem 6, every strongly perfect equilibrium is perfect and every strongly perfect equilibrium is undominated. However, none of the other implications is valid for continuous games. In the next section we provide some counter-examples to prove it.
5.
Counter-examples
In this section we present two examples showing interesting features for continuous games. Example 2 exhibits the fact that an undominated Nash equilibrium does not necessarily have to be perfect (even for two-person games). Example 3 shows a two-person continuous game not having undominated Nash equilibria.
Example 2. Let us consider the two-person continuous game F with the payoff functions defined on [0, 1] × [0, 1] by
(y-x)x, ify>x, Kl(x'Y)= - ( y - x ) Z ( 1 - x ) , i f y ~ x , K2(x, y) = 0 for all (x, y) E [0, 1] x [0, 1]. The assertion we are going to prove is that the joint strategy (6(1), 6(0)) is an undominated Nash equilibrium which is not a perfect one. It is easy to see that (6(1),6(0)) is an undominated Nash equilibrium. In order to prove that (6(1), 6(0)) is not perfect, let us consider a completely mixed strategy G E S 2 for player 2. Then, both G([0, 1/2]) and G((1/2, 1]) are greater than zero. Taking into account that Kl(X, y) is an increasing function of y for all x E [0, 1], we conclude that, if x E [0, 1/2], ~{1(6(X), G) ~
KI(X, 0)G([0, 1/2]) + K,(x, 1/2)G((1/2, 1]) = - x2(1 - x)G([0, 1/2]) + x(1/2 - x)G((a/2, 1]).
Thus, by taking x E [0, 1] small enough, we can guarantee that
ff{'l(6(X), G) > O.
(8)
Now, suppose that (6(1), 6(0)) is a perfect Nash equilibrium. Then there exists a sequence {(Fk, Gk)}k~l, converging weakly to (6(1), 6(0)), of Nash equilibria of a corresponding sequence of/zk-perturbed games. Since {Gk}k~l converges weakly to 6(0), there exists k 0 such that Gk([0, 1/50])/> 14/15 for all k/> k 0. This implies that G~((1/50,1])<~1/15 for all k>~ko. Then, for all k>~ko and all x >t 1/50, taking into account that Kl(X, y) is an increasing function of y (for all x),
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~1(8(x), G k) <~K,(x, 1)Gk((1/50,
1]) +
235
Kl(X , 1/50)Gk([o, 1/50]) ~<
~
(9)
Because of (8), (9) and Theorem 3 we conclude that, in F(/zk), any best reply of player 1 against G k (in particular F~), has to behave like t, k on [1/2, 1] for all k >/k 0. But this implies that {Fg}k.1 cannot converge to the strategy 8(1). This shows that (6(1), 8(0)) is not perfect. It is well known that (the mixed extension of) every finite n-person game has undominated Nash equilibria. The following example shows that the situation in the framework of continuous games is substantially different. In fact, it exhibits a (mixed extended) continuous game without undominated Nash equilibria. Example 3. Let us consider the two-person continuous game F with the payoff functions defined on [0, 1] × [0, 1] by: "xy -- X 2 ,
K, (x, y) = ( 1 / 2 - y ) ( x - 1), 1/2 ( y - 1 / 2 ) ,
ifx ~< 1 / 2 , ifx/> 1/2 and y ~< 1 / 2 , i f x ~ 1/2 and y/> 1 / 2 ,
2x)y, (4x 2 - 2x)y + 1/2 - y , K2(x, Y) = 1/2-y, "(4X 2 --
( - 4 y 2 + 2y)(x - 1 / 2 ) ,
if0~
First we show that (6(0), 8(0)) is the only Nash equilibrium of this game. Let us assume that (F, G) is a Nash equilibrium of F. Since K2(x ,
1/2) > K2(x, y)
for all y E (1/2, 1], and all x ~ [0, 1],
we have that
Y(2(F,a(1/Z))> Ygz(F, 6(y))
f o r a l l y E ( 1 / 2 , 1].
This implies that C(G) C_ [0, 1/2]. Now, if G = 8(0), player 1 has only two pure best replies against G: 8(0) and 8(1). It is easy to verify that (8(0), 6(0)) is a Nash equilibrium and that neither (8(1), 8(0)) nor any other joint strategy (F, 8(0)) ~ S, where F = (1 - Cl)8(0) + cl8(1 ) with 0 < c 1 < 1, can be a Nash equilibrium. On the other hand, if C(G)C_[0, 1/2] but G~8(O) we can show, like in Example 2, that there exists a pure strategy x(G) ~ (0, 1/2) which is a best reply of player 1 against G and such that 5g~(8(x(G)), G) > 0. Besides, it is easy to see that Y(~(8(x), G) <~0 for all x ~ [1/2, 1]. Hence, if F is to be a best reply against G for player 1, C(F) C_ [0, 1/2]. However, F cannot be 8(0) because Kl(0, y) = 0 for
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/Strongly \ Perfect
Undominated
A
Perfect
Ex2 ~~'* Fig. I. Overviewof the relations between concepts. all y ~ [0, 1], which implies that ~t~1(8(0), G) = 0 < $'t~x(8(x(G)), G). This condition naturally excludes (8(0), G) from being a Nash equilibrium. Besides, since 9'/'~(8(1/2), G ) < 0, (8(1/2), G) cannot be a Nash equilibrium either. Finally, it is easy to verify that the best reply of player 2 against F with C(F) C_[0, 1/2] (F being neither 8(0) nor 8(1/2)), is the pure strategy G = 8(0). But, in this case, G was chosen to be different from 8(0). From this contradiction we conclude that the point (8(0), 8(0)) is the only Nash equilibrium of the game. But now it is easy to see that, for player 1, 8(1) dominates 8(0). Hence, the only Nash equilibrium of this game (8(0), 8(0)) is not undominated. Combining Example 2 and Theorem 6 we have that an undominated Nash equilibrium does not have to be strongly perfect. Example 3 shows a two-person continuous game not having undominated Nash equilibria. Since it must have a perfect Nash equilibrium (according to Theorem 2), from this example we can conclude that perfectness does not imply undomination (for a Nash equilibrium). Moreover, in view of Theorem 6, this example also shows that a perfect Nash equilibrium does not have to be strongly perfect. Finally, observe that Examples 2 and 3 together show that the notions of a perfect Nash equilibrium and an undominated Nash equilibrium are, in fact, independent (even in the two-person case). All these relations are summarized in Fig. 1.
Acknowledgements We thank the University of Santiago de Compostela and Xunta de Galicia for financial support through projects 60902.25064(5060), XUGA20102B91 and
L. M~ndez-Naya et al. / Mathematical Social Sciences 29 (1995) 225-237
237
XUGA20701B91. We are also grateful to two anonymous referees for many helpful comments and suggestions.
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