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Evolutionary Stability of Pure-Strategy Equilibria in Finite Games
GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.
21, 253]265 Ž1997.
GA970533
Evolutionary Stability of Pure-Strategy Equilibria in Finite Games E. Somanathan* Emory Uni¨ ersity, Department of Economics, Atlanta, Georgia 30322 Received January 18, 1996
Sufficient conditions for pure-strategy Nash equilibria of finite games to be ŽLyapunov. stable under a large class of evolutionary dynamics, the regular monotonic selection dynamics, are discussed. In particular, it is shown that in almost all finite extensive-form games, all the pure-strategy equilibria are stable. In such games, all mixed-strategy equilibria close to pure-strategy equilibria are also stable. Journal of Economic Literature Classification Numbers: C70, C72. Q 1997 Academic Press
1. INTRODUCTION One way of approaching the equilibrium selection problem in games that has attracted much attention recently is to model evolution. A common approach is to suppose that a game is played by large populations, one for each player. The shares of the populations playing different strategies varies over time in a payoff-related manner. The vector of shares for each population is modeled as a dynamical system with the growth rates of the shares being related to payoffs, and the limiting behavior of the system is examined. For one such class of dynamics, the regular monotonic selection dynamics, it has been shown that a necessary and sufficient condition for pure-strategy equilibria to be asymptotically stable is that they be strict ŽSamuelson and Zhang, 1992.. This is a strong restriction for extensive-form games, since no equilibrium in which there are information sets not reached in equilibrium can be strict. * I am grateful to Eric Maskin and to an anonymous referee and editor for very helpful comments and suggestions. An earlier version of this paper was part of my Ph.D. thesis at Harvard University. 253 0899-8256r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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E. SOMANATHAN
But examples have also been given to show that the replicator dynamics Žwhich fall into the above class. need not remove weakly dominated strategies Žvan Damme, 1987; Samuelson and Zhang, 1992; Sethi, 1996.. This paper shows that such situations, far from being rare, are in fact typical. A theorem giving sufficient conditions for stability of pure-strategy Nash equilibria of finite games in regular monotonic selection dynamics is presented. Here, stability Ždefined below. refers to Lyapunov stability, which is weaker than asymptotic stability.1 The theorem implies that all the pure-strategy Nash equilibria of almost all finite extensive-form games are stable. Examples of games to which the theorem applies include the well-known Chain-store Game, and the Ultimatum Game Žwhen the possible divisions are restricted to finitely many values.. An extension to some kinds of mixed-strategy equilibria is discussed.
2. STABILITY 2.1. The Model Consider a finite two-person game in normal form with strategies a1 , . . . ,a n1 and b1 , . . . , bn 2 available to players 1 and 2, respectively. Extensions to any finite number of players are straightforward, and the two-player case is used for notational convenience only. The game is played by two large populations of players of type 1 and 2. Only pure strategies are played. Let the shares of the populations playing strategies a i and bj at time t be denoted by si Ž t . and u j Ž t ., respectively. The payoff to strategy a i for a type-1 player, p 1Ž i, u., is a continuous function of the state of population 2, uŽ t . ' Ž u1Ž t ., . . . , u n 2Ž t ...2 The payoff to a type-2 player using strategy j is the analogous expression p 2 Ž s, j ., assumed to be continuous in s. Evolutionary pressure exerted by payoff differentials on the popula1 Stability has received less attention than asymptotic stability because the latter is always robust to small perturbations of the dynamics, unlike the former. ŽSee, for example, the discussion in Weibull, 1995, pp. 214]215.. However, recent work has shown that it is not generally true that the addition of payoff-unrelated drift to the model will lead to a stable equilibrium being ‘‘destabilized.’’ See Gale, Binmore, and Samuelson Ž1995. on the Ultimatum Žone-stage divide-the-dollar. Game, Sethi and Somanathan Ž1996. on a common-pool-resource exploitation game, and Binmore and Samuelson Ž1995. for general results, further examples, and a discussion of empirical tests. 2 A special case of this occurs in the random-matching model in which pairs of players 2 encounter each other at random, so that p 1Ž i, u. s Ý njs1 Bi j u j , where Bi j denotes the payoff to strategy a i for player 1 when 2 uses bj . Examples in which payoffs are possibly nonlinear functions of u, the ‘‘playing the field’’ model, are in Maynard Smith Ž1982.. An economic example is in Sethi and Somanathan Ž1996..
EVOLUTIONARY STABILITY IN GAMES
255
tion compositions or state Ž s, u. is modeled by supposing that the shares of strategies with higher payoffs grow faster. This is made precise below using terminology that follows Samuelson and Zhang Ž1992.. Let f : S n1 = S n 2 ª R n1 and g: S n1 = S n 2 ª R n 2 , where S k is the Ž k y 1.-dimensional simplex. The system
˙si s f i Ž s, u . ,
i s 1, . . . , n1 ,
Ž 1.
u ˙ j s g j Ž s, u . ,
j s 1, . . . , n 2 ,
Ž 2.
is a selection dynamic if it satisfies for all Ž s, u. g S n1 = S n 2 : f and g are Lipschitz continuous; i.e., 'M ) 0 s.t. ;Ž s, u., Ž s9, u9. g S n1 = S n 2 , max f Ž s, u . y f Ž s9, u9 . , g Ž s, u . y g Ž s9, u9 .
4 F M Ž s, u . y Ž s9, u9 .
,
Ž 3. where < ? < denotes the Euclidean norm, n1
n2
Ý fi s 0 s Ý g j , is1
Ž 4.
js1
; Ž s, u . g S n1 = S n 2 ,
si s 0 « f i Ž s, u . G 0,
Ž 5.
; Ž s, u . g S = S ,
u j s 0 « g j Ž s, u . G 0.
Ž 6.
n1
n2
The Lipschitz condition ensures that for any initial state, the selection dynamic defines a unique path originating from the state. The other conditions ensure that the shares always add up to one and remain nonnegative. The following regularity condition is also used: A selection dynamic Ž f, g . satisfies regularity if for all i, Ž s*, u*. g S n1 = S n 2 with sUi s 0: lim
Ž s, u .ª Ž s*, u* .
f i Ž s, u . si
'
f i Ž s*, u* . 0
exists and is finite, and a symmetric condition holds for g. Regularity allows the proportional growth rates of the shares to be defined and continuous on the whole of S n1 = S n 2 , not just its interior. Finally, the key idea that evolution results in strategies with higher payoffs increasing their share at the expense of strategies with lower payoffs is captured as follows: A regular selection dynamic Ž f, g . is
256
E. SOMANATHAN
monotonic if for any Ž s, u. g S n1 = S n 2 and i, i9, j, j9,
p 1 Ž i , u . ) Ž s . p 1 Ž i9, u . « p 2 Ž s, j . ) Ž s . p 2 Ž s, j9 . «
f i Ž s, u . si g j Ž s, u . uj
) Ž s. ) Ž s.
f i9 Ž s, u . si9 g j9 Ž s, u . u j9
,
.
This says that the proportional growth rate of a strategy with a higher payoff will be greater than that of a strategy with a lower payoff. Note that the replicator dynamic,
˙si s si Ž p 1 Ž i , u . y p 1 Ž s, u . . , u ˙ j s u j Ž p 2 Ž s, j . y p 2 Ž s, u . . ,
i s 1, . . . , n1 , j s 1, . . . , n 2 ,
Ž 7. Ž 8.
1 where p 1Ž s, u. s Ý nis1 sip 1Ž i, u., derived from biological models of evolution, is a regular monotonic selection dynamic. The description of how play will evolve is now complete. To arrive at a prediction about long-run behavior, one needs a notion of stability of the dynamics. A point Ž s*, u*. is said to be stable in the dynamics Ž f, g . if, for any neighborhood U of Ž s*, u*., there exists a neighborhood V of Ž s*, u*. with V : U such that Ž sŽ0., uŽ0.. g V « Ž sŽ t ., uŽ t .. g U for all t ) 0. Stable points therefore have the property that paths starting close to them do not drift away from them. A point Ž s*, u*. is asymptotically stable if it is stable and there exists a neighborhood U of Ž s*, u*. such that every path starting in U converges to Ž s*, u*.. By definition, asymptotic stability implies stability.
2.2. Sufficient Conditions In looking for stable states of regular monotonic selection dynamics one can confine attention to Nash equilibria since all stable states are Nash equilibria ŽSamuelson and Zhang, 1992.. It is well known that strict Nash equilibria are asymptotically stable in dynamics of the type described here. In fact, Samuelson and Zhang Ž1992. have shown that a necessary condition for pure-strategy equilibria to be asymptotically stable in regular monotonic selection dynamics is that they be strict. But in extensive-form games, strictness of equilibria is the exception, not the rule. Given a pure-strategy equilibrium, the payoff to a strategy that differs from the equilibrium strategy only at information sets that are never reached in equilibrium must be equal to the payoff to the equilibrium strategy. In such games, asymptotic stability may be too restrictive as a solution concept.
EVOLUTIONARY STABILITY IN GAMES
257
The following result provides conditions under which pure-strategy equilibria that are not strict satisfy the weaker requirement of stability. These conditions are quite weak. In fact, all the pure-strategy Nash equilibria in almost all finite extensive-form games satisfy them Žsee Corollary 2.. ‘‘Almost all’’ or ‘‘generic’’ is defined as follows: Take a game and consider the class of games with the same game tree but with possibly different payoff vectors at the terminal nodes. One can identify each game in this class with the vector of payoff vectors at its terminal nodes and, thus, the class itself with a Euclidean space. A property holds for almost all games if, for each class, the subset of games for which it holds is open and dense in the corresponding Euclidean space. Before stating the theorem, some further notation is introduced: e 1i will denote the vector in S n1 that has a 1 in the ith place and zeros elsewhere and e j2 will denote the vector in S n 2 that has a 1 in the jth place and zeros elsewhere. THEOREM 1.
Suppose Ž a1 , b1 . is a Nash equilibrium such that
p 1 Ž 1, e12 . s p 1 Ž i , e12 .
; i s 2, . . . , k 1 ,
p 1 Ž 1, e12 . ) p 1 Ž i , e12 .
; i s k 1 q 1, . . . , n1 ,
Ž 10 .
p 2 Ž e11 , 1 . s p 2 Ž e11 , j .
; j s 2, . . . , k 2 ,
Ž 11 .
p 2 Ž e11 , 1 . ) p 2 Ž e11 , j .
; j s k 2 q 1, . . . , n 2 .
Ž 12 .
Ž 9.
And suppose there exists a neighborhood X : S n1 = S n 2 of Ž e11 , e12 . such that for all Ž s, u. g X satisfying si ) 0 only if i F k 1 and u j ) 0 only if j F k 2 , the following inequalities hold:
p 1 Ž 1, u . G p 1 Ž i , u .
; i F k1 ,
Ž 13 .
p 2 Ž s, 1 . G p 2 Ž s, j .
; j F k2 .
Ž 14 .
Then Ž a1 , b1 . is stable in e¨ ery regular, monotonic selection dynamic on S n1 = S n 2 . Proof. Let Ž f, g . be a regular, monotonic selection dynamic on S n1 = S . It will be convenient to define n2
k1
sc '
Ý si ,
n1
sd '
is2 k2
uc '
Ý uj , js2
Ý
isk 1q1
si ,
n2
ud '
Ý
jsk 2q1
uj ,
258
E. SOMANATHAN k1
fc ' gc '
n1
fd '
Ý fi ,
Ý
is2
isk 1q1
k2
n2
gd '
Ý gj , js2
Ý
jsk 2q1
fi ,
gj.
We will refer to strategies satisfying Ž9. and Ž11., that is, a2 through a k 1 and b 2 through bk 2 , as type-c strategies and to strategies satisfying Ž10. and Ž12., a k 1q1 through a n1 and bk 2q1 through bn 2 , as type-d strategies. Note that the proportional growth rates of sc , s d , u c , and u d are weighted averages of the proportional growth rates of the shares of corresponding type-c and type-d strategies. Let U be a neighborhood of Ž e11, e12 . in S n1 = S n 2 . We will find a neighborhood V : U of this point such that no path starting in V ever leaves V. By continuity of the payoffs and Ž9. ] Ž12., there exists a closed neighborhood N of Ž e11, e12 ., N : X, U in which sc F s1 , u c F u1 , and
p 1 Ž i , u . ) p 1 Ž i9, u .
; i F k 1 , i9 ) k 1 ,
Ž 15 .
p 2 Ž s, j . ) p 2 Ž s, j9 .
; j F k 2 , j9 ) k 2 .
Ž 16 .
We will proceed as follows: The assumptions on the payoff structure, together with monotonicity guarantee that the shares of the equilibrium strategies do not fall when type-d strategies are not present. Monotonicity and the other properties of the selection dynamic will be used to show that Ž˙ sc q ˙ s d .rŽyu ˙d . and Ž u˙c q u˙d .rŽys˙d . are bounded above in N. These relative bounds on the growth of nonequilibrium strategy shares are then used to construct V. By Ž16. and monotonicity, g1
,
gc
u1 u c
)
gd ud
within N. Using the continuity of the proportional growth rates Žwhich follows from regularity. and the fact that N is closed and bounded, we get g1
,
gc
u1 u c
)
gd ud
ql
for some l ) 0.
We now make the following claim: g d Ž s, u . ud
-0
; Ž s, u . g N.
Ž 17 .
259
EVOLUTIONARY STABILITY IN GAMES
Proof of claim. Suppose there exists Ž s, u. g N for which g dru d G 0. Then g d G 0. Also, g 1ru1 , g cru c ) 0. So, g c G 0 and g 1 ) 0 Žsince u1 ) 0 in N .. Therefore, g 1 q g c q g d ) 0, a contradiction to Ž4.. This proves the claim. We note for future reference that Ž17. implies that g d Ž s, u . F 0
; Ž s, u . g N.
Ž 18 .
Now Ž17. also implies g dru d - ym within N for some m ) 0. Hence, yg d G m u d
within N.
Ž 19 .
Now consider Ž s, u. in N. Suppose f 1Ž s, u. F 0. Then, f c Ž s, u . F Ž f c y f 1 . Ž s, u . F Ž f c y f 1 . Ž s, u . y Ž f c y f 1 . Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . F Ž f c y f 1 . Ž s, u . y Ž f c y f 1 . Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . F
1
'2
M Ž s, u . y Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . for some M ) 0
s F
1
'2 1
'2
ž
M u 2d q
1r2
n2
Ý
jsk 2q1
M Ž u 2d q u 2d .
u 2j
/
1r2
s Mu d . The second inequality above is obtained as follows: f drs d - f crsc F f 1rs1 when u d s 0 Žby Ž13., Ž15., and monotonicity.. So f 1 G 0 when u d s 0 Žotherwise Ž4. would be violated.. Hence, f c F f 1 scrs1 when u d s 0. Since f 1 G 0, this means f c F f 1 , or f c y f 1 F 0 when u d s 0. This establishes the second inequality. The fourth inequality uses the Lipschitz continuity of f. So, f c Ž s, u . F Mu d
; Ž s, u . g N for which f 1 Ž s, u . F 0.
Ž 20 .
260
E. SOMANATHAN
So by Ž19. and Ž20., within N, fc F
yg d M
m
,
as long as f 1 F 0. Since f d F 0 in N Žby Ž18. and symmetry.; therefore, within N, fc q fd F
yg d M
m
,
if f 1 F 0.
Ž 21 .
And for those Ž s, u. g N at which f 1Ž s, u. ) 0, we have f c Ž s, u. q f d Ž s, u. - 0 and so, fc q fd - 0 F
yg d M
m
.
Putting this together with Ž21. we get f c Ž s, u . q f d Ž s, u . F
yg d Ž s, u . M
m
' yg d Ž s, u . B1
; Ž s, u . g N.
Ž 22 . So Ž ˙ sc q ˙ s d . and Žby a symmetric argument. Ž u ˙c q u˙d . are bounded above . Ž . in N by Žyu B and ys B , respectively. ˙d 1 ˙d 2 We will now construct V. Let V ' Ž s, u . g S n1 = S n 2 ¬ Ž 23 . ] Ž 25 . hold4 , where 0Fs d q u d F x
Ž 23 .
0Fsc F B1 Ž x y u d . q x y s d
Ž 24 .
0Fu c F B2 Ž x y s d . q x y u d ,
Ž 25 .
where x is positive and chosen small enough that V : N. Note that V is a neighborhood of Ž e11 , e12 . in S n1 = S n 2 . Next we show that any path starting in V cannot leave V. Let Ž s, u. g V. So Ž sc , s d , u c , u d . satisfies Ž23. ] Ž25.. We show that Ž23. ] Ž25. must continue to hold as time passes. First, note that Ž23. must continue to hold since u ˙d F 0 by Ž18. and s˙d F 0 by symmetry. Next, note that Ž24. can be rewritten as sc q s d q B1 u d F B1 x q x. Now since ˙ sc q ˙ s d F Žyu ˙d . B1 , i.e., ˙sc q ˙s d q B1 u˙d F 0, therefore Ž24. must also continue to hold. And by symmetry, Ž25. must continue to hold as well. Hence, any path starting in V cannot leave V. This completes the proof. B
EVOLUTIONARY STABILITY IN GAMES
261
The conditions Ž13. and Ž14. of Theorem 1 are not necessary for stability as the following example shows.3 The pure-strategy Nash equilibrium where both players play their first strategy in the bimatrix normal-form game 1, 1
1, 1
1, 2
2, 1
is stable in the replicator dynamic Žthis is easily checked., but does not satisfy Ž13. in the hypotheses of the theorem. COROLLARY 2. All the pure-strategy Nash equilibria of almost all finite extensi¨ e-form games are stable in e¨ ery regular monotonic selection dynamic. Proof. In almost all finite extensive-form games, the vector of payoff vectors has distinct elements. Consider any pure-strategy equilibrium in such a game. Strategies that involve deviations from the equilibrium path, when everyone is playing the equilibrium strategies, must yield strictly lower payoffs. These are type-d strategies in the terminology of the theorem. All other strategies that deviate from the equilibrium yield payoffs equal to the equilibrium payoffs; hence, they are type-c strategies Žthat is, they satisfy conditions Ž9. and Ž11.. In fact, the payoff to a type-c strategy against any type-c strategy in the opposing population equals the equilibrium payoff. So type-c strategies satisfy conditions Ž13. and Ž14. with equality. Now we can apply the theorem to conclude that the equilibrium is stable. B In extensive-form games that are not generic, conditions Ž13. and Ž14. from the hypotheses of Theorem 1 may not hold, and so Nash equilibria may not be stable, as the following example shows. In Fig. 1, the game shown has a nongeneric extensive form. In every neighborhood of the equilibrium ŽT, L., the payoffs to the other strategies are strictly higher than the payoffs to T and L at all points which place positive probability on D and R. So Ž13. and Ž14. from the hypotheses of the theorem do not hold. Clearly, ŽT, L. is unstable in regular monotonic selection dynamics. An argument similar to that in Theorem 1 can be made to show that in almost all finite extensive-form games, each pure-strategy equilibrium has a neighborhood in the subspace of shares of type-c strategies that consists of stable points. More precisely, 3
I am grateful to an anonymous referee for providing this simple example.
262
E. SOMANATHAN
FIG. 1. A nongeneric extensive form with an unstable equilibrium.
THEOREM 3. Let Ž a1 , b1 . be a pure-strategy Nash equilibrium in a generic two-person finite extensi¨ e-form game, where, in the normal form of the game, player 1 has strategies a1 , . . . , a n1, and player 2 has strategies b1 , . . . , bn 2 . Suppose a2 , . . . , a k 1, and b 2 , . . . , bk 2 differ from a1 and b1 , respecti¨ ely, only at information sets not reached in equilibrium, while all other strategies in¨ ol¨ e de¨ iations from the equilibrium path. Then there exists a neighborhood N of Ž e11, e12 . in the subspace of shares of strategies a1 , . . . , a k , b1 , . . . , bk , such 1 2 that e¨ ery point in N is stable in e¨ ery regular monotonic selection dynamic on S n1 = S n 2 . Proof. Strategies a2 , . . . , a k 1 and b 2 , . . . , bk 2 clearly satisfy Ž9. and Ž11.; that is, they are type-c strategies. Since the game is generic, a k 1q1 , . . . , a n1 and bk 2q1 , . . . , bn 2 must satisfy Ž10. and Ž12.; that is, they are type-d strategies. Hence, by the continuity of the payoff functions, there exists an open neighborhood N0 of Ž e11 , e12 . in S n1 = S n 2 such that for all Ž s, u. g N0 Ž15. and Ž16. hold. Notice that,
p 1 Ž 1, u . s p 1 Ž i , u .
; i F k 1 , if u d s 0,
Ž 26 .
p 2 Ž s, 1 . s p 2 Ž s, j .
; j F k 2 , if s d s 0.
Ž 27 .
Let Ž f, g . be a regular monotonic selection dynamic on S n1 = S n 2 . It will be shown that the set N ' Ž s, u. g N0 : s d s 0 s u d 4 consists of points that are stable in Ž f, g .. Let Ž s*, u*. g N. Let U be a neighborhood of Ž s*, u*.. We shall find a neighborhood V : U of Ž s*, u*. such that no path starting in V ever leaves V. Using the regularity and monotonicity of f and g and Ž15. and Ž16., we conclude, as in the proof of Theorem 1, that there exists a closed
EVOLUTIONARY STABILITY IN GAMES
263
neighborhood V0 : U of Ž s*, u*. and m 1 , m 2 ) 0, such that for all Ž s, u. g V0 , f d Ž s, u . - ym 1 s d F 0
Ž 28 .
g d Ž s, u . - ym 2 u d F 0.
Ž 29 .
Note that V0 can be chosen so that
Ž s, u . g V0 « Ž Ž s1 q s d , s2 , . . . , sk 1 , 0, . . . , 0 . , Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . g V0 .
Ž 30 .
Next, note that Ž26. and monotonicity imply that when u d s 0, fi si
f1
s
s1
; i F k1 .
Now, by Ž4., k1
Ý f i s yf d . is1
So when u d s 0, ; i F k 1 , fi si
s
1 Ý kis1 fi 1 Ý kis1
si
s
yf d 1 Ý kis1
si
F
M1 s d 1 Ý kis1 si
Ž 31 .
for some M1 ) 0, where the inequality follows from the continuity, hence, boundedness of f drs d on the simplex. Hence, when u d s 0, f i F M1 s d
; i F k1 .
Ž 32 .
By Lipschitz continuity of f, for any Ž s, u., there exists M2 ) 0 such that f i Ž s, u . y f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . F M2 u d .
Ž 33 .
264
E. SOMANATHAN
So there exists B ) 0 such that for any Ž s, u. g V0 , ; i F k 1 , f i Ž s, u . F f i Ž s, u . y f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . q f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . F M2 u d q f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 .
Ž by Ž 33 . . ,
s M2 u d q f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 .
Ž by Ž 28. , Ž 30. , Ž 31. . F M 2 u d q M1 s d F
yM2
m2
Ž by 32 .
g d Ž s, u . q
yM1
m1
f d Ž s, u .
F yB Ž f d Ž s, u . q g d Ž s, u . . ,
Ž by Ž 29. , Ž 28. . Ž 34 .
with a symmetric inequality for g. Let V ' Ž s, u. g S n1 = S n 2 : Ž35., Ž36. hold4 , where s d q u d F x, < si y
sUi < ,
< uj y
uUj <
Ž 35 .
F x q B Ž x y sd y u d .
; i F k 1 , j F k 2 , Ž 36 .
and x ) 0 is chosen small enough that V : V0 . V : V0 : U is a neighborhood of Ž s*, u*.. Let ŽŽ sŽ0., uŽ0.. g V. To complete the proof, it is enough to show that ŽŽ sŽ t ., uŽ t .. g V ; t ) 0. This follows directly from Ž28., Ž29., and Ž34., and the definition of V. B It may be noted that Theorem 3 implies that in finite games with generic extensive-forms, there exists a neighborhood of each pure-strategy equilibrium such that all the mixed-strategy equilibria within it are stable. To see this, consider the equilibrium Ž a1 , b1 . in the game mentioned in the statement of Theorem 3. In the light of the theorem, it is enough to show that there exists a neighborhood U of Ž e11, e12 . such that no point Ž s, u. in U with a positive weight on any type-d strategy a i can be a Nash equilibrium. For a type-d strategy a i , p 1Ž1, e12 . ) p 1Ž i, e12 . or B11 ) Bi1 , where Bi j denotes the payoff to a i for player 1 when 2 uses bj . For Ž s, u. to be an equilibrium with si ) 0, it must be that n2
Ý js1
n2
B1 j u j F
Ý Bi j u j , js1
which is impossible if U is chosen small enough.
EVOLUTIONARY STABILITY IN GAMES
265
3. CONCLUSION It was shown that for a large class of evolutionary dynamics, the regular monotonic selection dynamics, all the pure-strategy Nash equilibria of almost all finite extensive-form games are stable. These include imperfect equilibria involving the use of weakly dominated strategies. Mixed-strategy equilibria in the neighborhood of a pure-strategy equilibrium in a generic finite extensive-form game are also seen to be stable.
REFERENCES Binmore, K., and Samuelson, L. Ž1995.. ‘‘Evolutionary Drift and Equilibrium Selection.’’ SSRI Working Paper 9529, University of Wisconsin, Madison; Games Econ. Beha¨ ., to appear. Gale, J., Binmore, K., and Samuelson, L. Ž1995.. ‘‘Learning to be Imperfect: The Ultimatum Game,’’ Games Econ. Beha¨ . 8, 56]90. Maynard Smith, J. Ž1982.. E¨ olution and the Theory of Games. Cambridge: Cambridge Univ. Press. Samuelson, L., and Zhang, J. Ž1992.. ‘‘Evolutionary Stability in Asymmetric Games,’’ J. Econ. Theory 57, 363]391. Sethi, R. Ž1996.. ‘‘Evolutionary Stability and Social Norms.’’ J. Econ. Beha¨ . Organ. 29, 113]140. Sethi, R., and Somanathan, E. Ž1996.. ‘‘The Evolution of Social Norms in Common Property Resource Use,’’ Amer. Econ. Re¨ . 86, 766]788. van Damme, E. Ž1987.. Stability and Perfection of Nash Equilibria. Berlin: Springer-Verlag. Weibull, J. W. Ž1995.. E¨ olutionary Game Theory. Cambridge, MA: The MIT Press.
21, 253]265 Ž1997.
GA970533
Evolutionary Stability of Pure-Strategy Equilibria in Finite Games E. Somanathan* Emory Uni¨ ersity, Department of Economics, Atlanta, Georgia 30322 Received January 18, 1996
Sufficient conditions for pure-strategy Nash equilibria of finite games to be ŽLyapunov. stable under a large class of evolutionary dynamics, the regular monotonic selection dynamics, are discussed. In particular, it is shown that in almost all finite extensive-form games, all the pure-strategy equilibria are stable. In such games, all mixed-strategy equilibria close to pure-strategy equilibria are also stable. Journal of Economic Literature Classification Numbers: C70, C72. Q 1997 Academic Press
1. INTRODUCTION One way of approaching the equilibrium selection problem in games that has attracted much attention recently is to model evolution. A common approach is to suppose that a game is played by large populations, one for each player. The shares of the populations playing different strategies varies over time in a payoff-related manner. The vector of shares for each population is modeled as a dynamical system with the growth rates of the shares being related to payoffs, and the limiting behavior of the system is examined. For one such class of dynamics, the regular monotonic selection dynamics, it has been shown that a necessary and sufficient condition for pure-strategy equilibria to be asymptotically stable is that they be strict ŽSamuelson and Zhang, 1992.. This is a strong restriction for extensive-form games, since no equilibrium in which there are information sets not reached in equilibrium can be strict. * I am grateful to Eric Maskin and to an anonymous referee and editor for very helpful comments and suggestions. An earlier version of this paper was part of my Ph.D. thesis at Harvard University. 253 0899-8256r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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E. SOMANATHAN
But examples have also been given to show that the replicator dynamics Žwhich fall into the above class. need not remove weakly dominated strategies Žvan Damme, 1987; Samuelson and Zhang, 1992; Sethi, 1996.. This paper shows that such situations, far from being rare, are in fact typical. A theorem giving sufficient conditions for stability of pure-strategy Nash equilibria of finite games in regular monotonic selection dynamics is presented. Here, stability Ždefined below. refers to Lyapunov stability, which is weaker than asymptotic stability.1 The theorem implies that all the pure-strategy Nash equilibria of almost all finite extensive-form games are stable. Examples of games to which the theorem applies include the well-known Chain-store Game, and the Ultimatum Game Žwhen the possible divisions are restricted to finitely many values.. An extension to some kinds of mixed-strategy equilibria is discussed.
2. STABILITY 2.1. The Model Consider a finite two-person game in normal form with strategies a1 , . . . ,a n1 and b1 , . . . , bn 2 available to players 1 and 2, respectively. Extensions to any finite number of players are straightforward, and the two-player case is used for notational convenience only. The game is played by two large populations of players of type 1 and 2. Only pure strategies are played. Let the shares of the populations playing strategies a i and bj at time t be denoted by si Ž t . and u j Ž t ., respectively. The payoff to strategy a i for a type-1 player, p 1Ž i, u., is a continuous function of the state of population 2, uŽ t . ' Ž u1Ž t ., . . . , u n 2Ž t ...2 The payoff to a type-2 player using strategy j is the analogous expression p 2 Ž s, j ., assumed to be continuous in s. Evolutionary pressure exerted by payoff differentials on the popula1 Stability has received less attention than asymptotic stability because the latter is always robust to small perturbations of the dynamics, unlike the former. ŽSee, for example, the discussion in Weibull, 1995, pp. 214]215.. However, recent work has shown that it is not generally true that the addition of payoff-unrelated drift to the model will lead to a stable equilibrium being ‘‘destabilized.’’ See Gale, Binmore, and Samuelson Ž1995. on the Ultimatum Žone-stage divide-the-dollar. Game, Sethi and Somanathan Ž1996. on a common-pool-resource exploitation game, and Binmore and Samuelson Ž1995. for general results, further examples, and a discussion of empirical tests. 2 A special case of this occurs in the random-matching model in which pairs of players 2 encounter each other at random, so that p 1Ž i, u. s Ý njs1 Bi j u j , where Bi j denotes the payoff to strategy a i for player 1 when 2 uses bj . Examples in which payoffs are possibly nonlinear functions of u, the ‘‘playing the field’’ model, are in Maynard Smith Ž1982.. An economic example is in Sethi and Somanathan Ž1996..
EVOLUTIONARY STABILITY IN GAMES
255
tion compositions or state Ž s, u. is modeled by supposing that the shares of strategies with higher payoffs grow faster. This is made precise below using terminology that follows Samuelson and Zhang Ž1992.. Let f : S n1 = S n 2 ª R n1 and g: S n1 = S n 2 ª R n 2 , where S k is the Ž k y 1.-dimensional simplex. The system
˙si s f i Ž s, u . ,
i s 1, . . . , n1 ,
Ž 1.
u ˙ j s g j Ž s, u . ,
j s 1, . . . , n 2 ,
Ž 2.
is a selection dynamic if it satisfies for all Ž s, u. g S n1 = S n 2 : f and g are Lipschitz continuous; i.e., 'M ) 0 s.t. ;Ž s, u., Ž s9, u9. g S n1 = S n 2 , max f Ž s, u . y f Ž s9, u9 . , g Ž s, u . y g Ž s9, u9 .
4 F M Ž s, u . y Ž s9, u9 .
,
Ž 3. where < ? < denotes the Euclidean norm, n1
n2
Ý fi s 0 s Ý g j , is1
Ž 4.
js1
; Ž s, u . g S n1 = S n 2 ,
si s 0 « f i Ž s, u . G 0,
Ž 5.
; Ž s, u . g S = S ,
u j s 0 « g j Ž s, u . G 0.
Ž 6.
n1
n2
The Lipschitz condition ensures that for any initial state, the selection dynamic defines a unique path originating from the state. The other conditions ensure that the shares always add up to one and remain nonnegative. The following regularity condition is also used: A selection dynamic Ž f, g . satisfies regularity if for all i, Ž s*, u*. g S n1 = S n 2 with sUi s 0: lim
Ž s, u .ª Ž s*, u* .
f i Ž s, u . si
'
f i Ž s*, u* . 0
exists and is finite, and a symmetric condition holds for g. Regularity allows the proportional growth rates of the shares to be defined and continuous on the whole of S n1 = S n 2 , not just its interior. Finally, the key idea that evolution results in strategies with higher payoffs increasing their share at the expense of strategies with lower payoffs is captured as follows: A regular selection dynamic Ž f, g . is
256
E. SOMANATHAN
monotonic if for any Ž s, u. g S n1 = S n 2 and i, i9, j, j9,
p 1 Ž i , u . ) Ž s . p 1 Ž i9, u . « p 2 Ž s, j . ) Ž s . p 2 Ž s, j9 . «
f i Ž s, u . si g j Ž s, u . uj
) Ž s. ) Ž s.
f i9 Ž s, u . si9 g j9 Ž s, u . u j9
,
.
This says that the proportional growth rate of a strategy with a higher payoff will be greater than that of a strategy with a lower payoff. Note that the replicator dynamic,
˙si s si Ž p 1 Ž i , u . y p 1 Ž s, u . . , u ˙ j s u j Ž p 2 Ž s, j . y p 2 Ž s, u . . ,
i s 1, . . . , n1 , j s 1, . . . , n 2 ,
Ž 7. Ž 8.
1 where p 1Ž s, u. s Ý nis1 sip 1Ž i, u., derived from biological models of evolution, is a regular monotonic selection dynamic. The description of how play will evolve is now complete. To arrive at a prediction about long-run behavior, one needs a notion of stability of the dynamics. A point Ž s*, u*. is said to be stable in the dynamics Ž f, g . if, for any neighborhood U of Ž s*, u*., there exists a neighborhood V of Ž s*, u*. with V : U such that Ž sŽ0., uŽ0.. g V « Ž sŽ t ., uŽ t .. g U for all t ) 0. Stable points therefore have the property that paths starting close to them do not drift away from them. A point Ž s*, u*. is asymptotically stable if it is stable and there exists a neighborhood U of Ž s*, u*. such that every path starting in U converges to Ž s*, u*.. By definition, asymptotic stability implies stability.
2.2. Sufficient Conditions In looking for stable states of regular monotonic selection dynamics one can confine attention to Nash equilibria since all stable states are Nash equilibria ŽSamuelson and Zhang, 1992.. It is well known that strict Nash equilibria are asymptotically stable in dynamics of the type described here. In fact, Samuelson and Zhang Ž1992. have shown that a necessary condition for pure-strategy equilibria to be asymptotically stable in regular monotonic selection dynamics is that they be strict. But in extensive-form games, strictness of equilibria is the exception, not the rule. Given a pure-strategy equilibrium, the payoff to a strategy that differs from the equilibrium strategy only at information sets that are never reached in equilibrium must be equal to the payoff to the equilibrium strategy. In such games, asymptotic stability may be too restrictive as a solution concept.
EVOLUTIONARY STABILITY IN GAMES
257
The following result provides conditions under which pure-strategy equilibria that are not strict satisfy the weaker requirement of stability. These conditions are quite weak. In fact, all the pure-strategy Nash equilibria in almost all finite extensive-form games satisfy them Žsee Corollary 2.. ‘‘Almost all’’ or ‘‘generic’’ is defined as follows: Take a game and consider the class of games with the same game tree but with possibly different payoff vectors at the terminal nodes. One can identify each game in this class with the vector of payoff vectors at its terminal nodes and, thus, the class itself with a Euclidean space. A property holds for almost all games if, for each class, the subset of games for which it holds is open and dense in the corresponding Euclidean space. Before stating the theorem, some further notation is introduced: e 1i will denote the vector in S n1 that has a 1 in the ith place and zeros elsewhere and e j2 will denote the vector in S n 2 that has a 1 in the jth place and zeros elsewhere. THEOREM 1.
Suppose Ž a1 , b1 . is a Nash equilibrium such that
p 1 Ž 1, e12 . s p 1 Ž i , e12 .
; i s 2, . . . , k 1 ,
p 1 Ž 1, e12 . ) p 1 Ž i , e12 .
; i s k 1 q 1, . . . , n1 ,
Ž 10 .
p 2 Ž e11 , 1 . s p 2 Ž e11 , j .
; j s 2, . . . , k 2 ,
Ž 11 .
p 2 Ž e11 , 1 . ) p 2 Ž e11 , j .
; j s k 2 q 1, . . . , n 2 .
Ž 12 .
Ž 9.
And suppose there exists a neighborhood X : S n1 = S n 2 of Ž e11 , e12 . such that for all Ž s, u. g X satisfying si ) 0 only if i F k 1 and u j ) 0 only if j F k 2 , the following inequalities hold:
p 1 Ž 1, u . G p 1 Ž i , u .
; i F k1 ,
Ž 13 .
p 2 Ž s, 1 . G p 2 Ž s, j .
; j F k2 .
Ž 14 .
Then Ž a1 , b1 . is stable in e¨ ery regular, monotonic selection dynamic on S n1 = S n 2 . Proof. Let Ž f, g . be a regular, monotonic selection dynamic on S n1 = S . It will be convenient to define n2
k1
sc '
Ý si ,
n1
sd '
is2 k2
uc '
Ý uj , js2
Ý
isk 1q1
si ,
n2
ud '
Ý
jsk 2q1
uj ,
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E. SOMANATHAN k1
fc ' gc '
n1
fd '
Ý fi ,
Ý
is2
isk 1q1
k2
n2
gd '
Ý gj , js2
Ý
jsk 2q1
fi ,
gj.
We will refer to strategies satisfying Ž9. and Ž11., that is, a2 through a k 1 and b 2 through bk 2 , as type-c strategies and to strategies satisfying Ž10. and Ž12., a k 1q1 through a n1 and bk 2q1 through bn 2 , as type-d strategies. Note that the proportional growth rates of sc , s d , u c , and u d are weighted averages of the proportional growth rates of the shares of corresponding type-c and type-d strategies. Let U be a neighborhood of Ž e11, e12 . in S n1 = S n 2 . We will find a neighborhood V : U of this point such that no path starting in V ever leaves V. By continuity of the payoffs and Ž9. ] Ž12., there exists a closed neighborhood N of Ž e11, e12 ., N : X, U in which sc F s1 , u c F u1 , and
p 1 Ž i , u . ) p 1 Ž i9, u .
; i F k 1 , i9 ) k 1 ,
Ž 15 .
p 2 Ž s, j . ) p 2 Ž s, j9 .
; j F k 2 , j9 ) k 2 .
Ž 16 .
We will proceed as follows: The assumptions on the payoff structure, together with monotonicity guarantee that the shares of the equilibrium strategies do not fall when type-d strategies are not present. Monotonicity and the other properties of the selection dynamic will be used to show that Ž˙ sc q ˙ s d .rŽyu ˙d . and Ž u˙c q u˙d .rŽys˙d . are bounded above in N. These relative bounds on the growth of nonequilibrium strategy shares are then used to construct V. By Ž16. and monotonicity, g1
,
gc
u1 u c
)
gd ud
within N. Using the continuity of the proportional growth rates Žwhich follows from regularity. and the fact that N is closed and bounded, we get g1
,
gc
u1 u c
)
gd ud
ql
for some l ) 0.
We now make the following claim: g d Ž s, u . ud
-0
; Ž s, u . g N.
Ž 17 .
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EVOLUTIONARY STABILITY IN GAMES
Proof of claim. Suppose there exists Ž s, u. g N for which g dru d G 0. Then g d G 0. Also, g 1ru1 , g cru c ) 0. So, g c G 0 and g 1 ) 0 Žsince u1 ) 0 in N .. Therefore, g 1 q g c q g d ) 0, a contradiction to Ž4.. This proves the claim. We note for future reference that Ž17. implies that g d Ž s, u . F 0
; Ž s, u . g N.
Ž 18 .
Now Ž17. also implies g dru d - ym within N for some m ) 0. Hence, yg d G m u d
within N.
Ž 19 .
Now consider Ž s, u. in N. Suppose f 1Ž s, u. F 0. Then, f c Ž s, u . F Ž f c y f 1 . Ž s, u . F Ž f c y f 1 . Ž s, u . y Ž f c y f 1 . Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . F Ž f c y f 1 . Ž s, u . y Ž f c y f 1 . Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . F
1
'2
M Ž s, u . y Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . for some M ) 0
s F
1
'2 1
'2
ž
M u 2d q
1r2
n2
Ý
jsk 2q1
M Ž u 2d q u 2d .
u 2j
/
1r2
s Mu d . The second inequality above is obtained as follows: f drs d - f crsc F f 1rs1 when u d s 0 Žby Ž13., Ž15., and monotonicity.. So f 1 G 0 when u d s 0 Žotherwise Ž4. would be violated.. Hence, f c F f 1 scrs1 when u d s 0. Since f 1 G 0, this means f c F f 1 , or f c y f 1 F 0 when u d s 0. This establishes the second inequality. The fourth inequality uses the Lipschitz continuity of f. So, f c Ž s, u . F Mu d
; Ž s, u . g N for which f 1 Ž s, u . F 0.
Ž 20 .
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E. SOMANATHAN
So by Ž19. and Ž20., within N, fc F
yg d M
m
,
as long as f 1 F 0. Since f d F 0 in N Žby Ž18. and symmetry.; therefore, within N, fc q fd F
yg d M
m
,
if f 1 F 0.
Ž 21 .
And for those Ž s, u. g N at which f 1Ž s, u. ) 0, we have f c Ž s, u. q f d Ž s, u. - 0 and so, fc q fd - 0 F
yg d M
m
.
Putting this together with Ž21. we get f c Ž s, u . q f d Ž s, u . F
yg d Ž s, u . M
m
' yg d Ž s, u . B1
; Ž s, u . g N.
Ž 22 . So Ž ˙ sc q ˙ s d . and Žby a symmetric argument. Ž u ˙c q u˙d . are bounded above . Ž . in N by Žyu B and ys B , respectively. ˙d 1 ˙d 2 We will now construct V. Let V ' Ž s, u . g S n1 = S n 2 ¬ Ž 23 . ] Ž 25 . hold4 , where 0Fs d q u d F x
Ž 23 .
0Fsc F B1 Ž x y u d . q x y s d
Ž 24 .
0Fu c F B2 Ž x y s d . q x y u d ,
Ž 25 .
where x is positive and chosen small enough that V : N. Note that V is a neighborhood of Ž e11 , e12 . in S n1 = S n 2 . Next we show that any path starting in V cannot leave V. Let Ž s, u. g V. So Ž sc , s d , u c , u d . satisfies Ž23. ] Ž25.. We show that Ž23. ] Ž25. must continue to hold as time passes. First, note that Ž23. must continue to hold since u ˙d F 0 by Ž18. and s˙d F 0 by symmetry. Next, note that Ž24. can be rewritten as sc q s d q B1 u d F B1 x q x. Now since ˙ sc q ˙ s d F Žyu ˙d . B1 , i.e., ˙sc q ˙s d q B1 u˙d F 0, therefore Ž24. must also continue to hold. And by symmetry, Ž25. must continue to hold as well. Hence, any path starting in V cannot leave V. This completes the proof. B
EVOLUTIONARY STABILITY IN GAMES
261
The conditions Ž13. and Ž14. of Theorem 1 are not necessary for stability as the following example shows.3 The pure-strategy Nash equilibrium where both players play their first strategy in the bimatrix normal-form game 1, 1
1, 1
1, 2
2, 1
is stable in the replicator dynamic Žthis is easily checked., but does not satisfy Ž13. in the hypotheses of the theorem. COROLLARY 2. All the pure-strategy Nash equilibria of almost all finite extensi¨ e-form games are stable in e¨ ery regular monotonic selection dynamic. Proof. In almost all finite extensive-form games, the vector of payoff vectors has distinct elements. Consider any pure-strategy equilibrium in such a game. Strategies that involve deviations from the equilibrium path, when everyone is playing the equilibrium strategies, must yield strictly lower payoffs. These are type-d strategies in the terminology of the theorem. All other strategies that deviate from the equilibrium yield payoffs equal to the equilibrium payoffs; hence, they are type-c strategies Žthat is, they satisfy conditions Ž9. and Ž11.. In fact, the payoff to a type-c strategy against any type-c strategy in the opposing population equals the equilibrium payoff. So type-c strategies satisfy conditions Ž13. and Ž14. with equality. Now we can apply the theorem to conclude that the equilibrium is stable. B In extensive-form games that are not generic, conditions Ž13. and Ž14. from the hypotheses of Theorem 1 may not hold, and so Nash equilibria may not be stable, as the following example shows. In Fig. 1, the game shown has a nongeneric extensive form. In every neighborhood of the equilibrium ŽT, L., the payoffs to the other strategies are strictly higher than the payoffs to T and L at all points which place positive probability on D and R. So Ž13. and Ž14. from the hypotheses of the theorem do not hold. Clearly, ŽT, L. is unstable in regular monotonic selection dynamics. An argument similar to that in Theorem 1 can be made to show that in almost all finite extensive-form games, each pure-strategy equilibrium has a neighborhood in the subspace of shares of type-c strategies that consists of stable points. More precisely, 3
I am grateful to an anonymous referee for providing this simple example.
262
E. SOMANATHAN
FIG. 1. A nongeneric extensive form with an unstable equilibrium.
THEOREM 3. Let Ž a1 , b1 . be a pure-strategy Nash equilibrium in a generic two-person finite extensi¨ e-form game, where, in the normal form of the game, player 1 has strategies a1 , . . . , a n1, and player 2 has strategies b1 , . . . , bn 2 . Suppose a2 , . . . , a k 1, and b 2 , . . . , bk 2 differ from a1 and b1 , respecti¨ ely, only at information sets not reached in equilibrium, while all other strategies in¨ ol¨ e de¨ iations from the equilibrium path. Then there exists a neighborhood N of Ž e11, e12 . in the subspace of shares of strategies a1 , . . . , a k , b1 , . . . , bk , such 1 2 that e¨ ery point in N is stable in e¨ ery regular monotonic selection dynamic on S n1 = S n 2 . Proof. Strategies a2 , . . . , a k 1 and b 2 , . . . , bk 2 clearly satisfy Ž9. and Ž11.; that is, they are type-c strategies. Since the game is generic, a k 1q1 , . . . , a n1 and bk 2q1 , . . . , bn 2 must satisfy Ž10. and Ž12.; that is, they are type-d strategies. Hence, by the continuity of the payoff functions, there exists an open neighborhood N0 of Ž e11 , e12 . in S n1 = S n 2 such that for all Ž s, u. g N0 Ž15. and Ž16. hold. Notice that,
p 1 Ž 1, u . s p 1 Ž i , u .
; i F k 1 , if u d s 0,
Ž 26 .
p 2 Ž s, 1 . s p 2 Ž s, j .
; j F k 2 , if s d s 0.
Ž 27 .
Let Ž f, g . be a regular monotonic selection dynamic on S n1 = S n 2 . It will be shown that the set N ' Ž s, u. g N0 : s d s 0 s u d 4 consists of points that are stable in Ž f, g .. Let Ž s*, u*. g N. Let U be a neighborhood of Ž s*, u*.. We shall find a neighborhood V : U of Ž s*, u*. such that no path starting in V ever leaves V. Using the regularity and monotonicity of f and g and Ž15. and Ž16., we conclude, as in the proof of Theorem 1, that there exists a closed
EVOLUTIONARY STABILITY IN GAMES
263
neighborhood V0 : U of Ž s*, u*. and m 1 , m 2 ) 0, such that for all Ž s, u. g V0 , f d Ž s, u . - ym 1 s d F 0
Ž 28 .
g d Ž s, u . - ym 2 u d F 0.
Ž 29 .
Note that V0 can be chosen so that
Ž s, u . g V0 « Ž Ž s1 q s d , s2 , . . . , sk 1 , 0, . . . , 0 . , Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . g V0 .
Ž 30 .
Next, note that Ž26. and monotonicity imply that when u d s 0, fi si
f1
s
s1
; i F k1 .
Now, by Ž4., k1
Ý f i s yf d . is1
So when u d s 0, ; i F k 1 , fi si
s
1 Ý kis1 fi 1 Ý kis1
si
s
yf d 1 Ý kis1
si
F
M1 s d 1 Ý kis1 si
Ž 31 .
for some M1 ) 0, where the inequality follows from the continuity, hence, boundedness of f drs d on the simplex. Hence, when u d s 0, f i F M1 s d
; i F k1 .
Ž 32 .
By Lipschitz continuity of f, for any Ž s, u., there exists M2 ) 0 such that f i Ž s, u . y f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . . F M2 u d .
Ž 33 .
264
E. SOMANATHAN
So there exists B ) 0 such that for any Ž s, u. g V0 , ; i F k 1 , f i Ž s, u . F f i Ž s, u . y f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . q f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 . F M2 u d q f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 .
Ž by Ž 33 . . ,
s M2 u d q f i Ž s, Ž u1 q u d , u 2 , . . . , u k 2 , 0, . . . , 0 .
Ž by Ž 28. , Ž 30. , Ž 31. . F M 2 u d q M1 s d F
yM2
m2
Ž by 32 .
g d Ž s, u . q
yM1
m1
f d Ž s, u .
F yB Ž f d Ž s, u . q g d Ž s, u . . ,
Ž by Ž 29. , Ž 28. . Ž 34 .
with a symmetric inequality for g. Let V ' Ž s, u. g S n1 = S n 2 : Ž35., Ž36. hold4 , where s d q u d F x, < si y
sUi < ,
< uj y
uUj <
Ž 35 .
F x q B Ž x y sd y u d .
; i F k 1 , j F k 2 , Ž 36 .
and x ) 0 is chosen small enough that V : V0 . V : V0 : U is a neighborhood of Ž s*, u*.. Let ŽŽ sŽ0., uŽ0.. g V. To complete the proof, it is enough to show that ŽŽ sŽ t ., uŽ t .. g V ; t ) 0. This follows directly from Ž28., Ž29., and Ž34., and the definition of V. B It may be noted that Theorem 3 implies that in finite games with generic extensive-forms, there exists a neighborhood of each pure-strategy equilibrium such that all the mixed-strategy equilibria within it are stable. To see this, consider the equilibrium Ž a1 , b1 . in the game mentioned in the statement of Theorem 3. In the light of the theorem, it is enough to show that there exists a neighborhood U of Ž e11, e12 . such that no point Ž s, u. in U with a positive weight on any type-d strategy a i can be a Nash equilibrium. For a type-d strategy a i , p 1Ž1, e12 . ) p 1Ž i, e12 . or B11 ) Bi1 , where Bi j denotes the payoff to a i for player 1 when 2 uses bj . For Ž s, u. to be an equilibrium with si ) 0, it must be that n2
Ý js1
n2
B1 j u j F
Ý Bi j u j , js1
which is impossible if U is chosen small enough.
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3. CONCLUSION It was shown that for a large class of evolutionary dynamics, the regular monotonic selection dynamics, all the pure-strategy Nash equilibria of almost all finite extensive-form games are stable. These include imperfect equilibria involving the use of weakly dominated strategies. Mixed-strategy equilibria in the neighborhood of a pure-strategy equilibrium in a generic finite extensive-form game are also seen to be stable.
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