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SRIHARI GOVINDAN AND ROBERT WILSON. Abstract. If a connected component of perfect equilibria of a two-player game contains a stable set as de¯ned by ...
MAXIMAL STABLE SETS OF TWO-PLAYER GAMES SRIHARI GOVINDAN AND ROBERT WILSON

Abstract.

If a connected component of perfect equilibria of a two-player game contains a stable set as de¯ned by Mertens, then the component is itself stable. Thus the stable sets maximal under inclusion are connected components of perfect equilibria.

1. Introduction The concept of stability has proved a powerful tool for equilibrium selection in economic models (e.g., Chapter 10 of van Damme (1991)). However, the topological de¯nition of stable sets in the reformulation by Mertens (1989, 1991) makes their computation di±cult. Compounding the problem is the fact that nongeneric games can have in¯nitely many stable sets. This is an important consideration in economic applications because the relevant normal-form games derive from nontrivial extensive forms and are therefore nongeneric. For generic extensive-form games, all equilibria in a component of Nash equilibria are payo® equivalent (Kreps and Wilson, 1982) so one might be satis¯ed to consider all stable sets contained in a single component of equilibria as somehow equivalent. But in extremely nongeneric games, di®erent equilibria in the same component can have di®erent payo®s, thus forcing one to re¯ne the notion of equivalence among stable sets. One natural approach is to identify those stable sets that are maximal under inclusion. In this article we show for two-player games that the maximal stable sets are connected components of perfect equilibria. A corollary is that their number is ¯nite. The restriction to two-player games is crucial to our result. The key step is the proof in Lemma 3.2 that the graph of the perturbed equilibrium correspondence near a component of perfect equilibria is path connected, but it seems unlikely that a similar result holds for games with more than two players. The bilinearity of payo®s in two-player games simpli¯es their analysis. From a theoretical viewpoint, the equilibria of two-player games have a simple geometric structure. Milham (1974) shows that the set of Nash equilibria of these games can be decomposed into a ¯nite number of maximal connected subsets that are polytopes, and Borm et al. (1993) obtain a similar result for perfect equilibria. The equivalence between admissibility and perfection in two-player games enables a further simpli¯cation: from a component of equilibria it is easy to compute the connected components of the perfect equilibria it contains. In Section 4, we provide conditions to check computationally the stability of these components of perfect equilibria. Algorithms for computing a Nash equilibrium (Lemke and Howson, 1964), a perfect equilibrium (van den Elzen and Talman, 1991), or a `simply stable' set of perfect equilibria (Wilson, 1992), are e±cient for two-player games. The algorithms in Govindan and Wilson (1999, 2001) ¯nd all equilibria accessible via a Date : October, 1999; revised version: February, 2001.

This work was funded in part by grants from the Social Sciences and Humanities Research Council of Canada and the National Science Foundation of the United States [SBR9511209]. 1

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S. GOVINDAN AND R. WILSON

homotopy along any line through the true game.1 And of course one can compute all equilibria by brute-force methods. Thus in the two-player case there are practical techniques for computing maximal stable sets. The paper is organized as follows. Section 2 sets up the preliminaries. Section 3 provides the main result of the paper. Section 4 indicates some of the computational aspects. Section 5 presents examples that illustrate the main ideas. 2. Preliminaries For any ¯nite set M , let §(M ) be the set of probability distributions over M , and let §± (M ) be the relative interior of §(M ). Fix a ¯nite two-player game ¡ , and let E be the set of Nash equilibria of ¡ . Use S1 and S2 to denote players 1 and 2's pure strategy sets, and by S1 = m and S2 = n their cardinalities. Let § = §(S ), S = S1 S2, and § = §1 §2. Use = 2 1 2 2 to represent the set of supports of mixed strategies. For each support T = (T1 ;T2 ) , de¯ne E (T ) as the closure of E (§± (T1 ) §± (T2 )). Denote by the collection of supports T for which E (T ) is nonempty; thus the set of equilibria can be represented as E = 2E E (T ). For each support T , the set E (T ) of equilibria with that support (or smaller) is a polytope with the property that the support and the best-reply correspondence are constant over the relative interior of each face of E (T ). The connected components of Nash equilibria admit an easy characterization in terms of these polytopes E (T ). Indeed, consider the equivalence relation given by the transitive closure of the relation T T 0 i® E (T ) E (T 0 ) is nonempty; then a component of equilibria is the union of all those E (T )s for which the T s belong to a corresponding equivalence class. We turn now to perfect equilibria. An equilibrium is perfect i® it is a best reply to a completely mixed strategy pro¯le, or equivalently, it uses no dominated strategies. Let PE be the set of perfect equilibria of ¡ and let be the corresponding supports; i.e., (T1 ;T2 ) i® it belongs to and there exists a completely mixed strategy pro¯le against which each pure strategy in T1 or T2 is a best reply. Obviously, PE = 2PE E (T ). The connected components of PE can be constructed from equivalence classes analogous to those of E . This characterization of perfect equilibria shows how to detect components of PE in a component of E . Let ¤ be the set of T 's in such that there exists a completely mixed strategy against which every strategy in T is a best reply. Computing ¤ amounts to solving systems of linear equations and inequalities. The information about ¤ , coupled with that about , helps determine the set of perfect equilibria. We conclude this Section with a de¯nition of stable sets adapted from Mertens (1989). For 0 6 " 6 1, j

£

S

j

j

j

i

i

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£

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2 S

\

£

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[T

2 E

»

PE µ E

2 PE

\

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de¯ne

P" = f ´ 2 Rm+n j 0 6 ´ij

P

6 ";



j 2Si

ij

6 1;

i = 1; 2; j 2 Si g :

Let P"± be the topological interior of P" , and let @P" = P" nP"± be its boundary. For each ´ 2 P1 de¯ne ´i = (´ij )j 2Si and ´ i = j 2Si ´ij for i = 1; 2. One interprets P" as a set of strategy perturbations, using the following construction. Given any ´ 2 P1 , de¯ne the perturbed game ¡ (´ ) as the game in which the players have the same strategy sets as in ¡ but where the payo®s to a strategy pro¯le ¾ are the payo®s in 1 Actually, in the notation of Section 4 below, a generic line in the space P~ , or perturbed if necessary so that it is generic " in Rm+n .

MAXIMAL STABLE SETS OF TWO-PLAYER GAMES

3

¡ from the strategy pro¯le ¿ = ((1 ¡ ´ )¾ + ´ ) =1 2 . We call ¿ a perturbed equilibrium of ¡ (´ ) i® ¾ is an equilibrium of ¡ (´ ). Denote by N the graph of the perturbed equilibrium correspondence in P1 §; i.e., i

i

i i

;

£

N = (´; ¿ ) P1 § ¿ is a perturbed equilibrium of ¡ (´) ; and let p : N P1 be the natural projection onto the space of perturbations. For each X N and " > 0, de¯ne (X" ; @X" ) = X p¡1 (P" ; @P" ), and X"± = X" @X" . In the following, H refers to simplicial homology theory with coe±cients in a group G. f

2

£

j

g

!

µ

\

n

De¯nition 2.1. A closed semi-algebraic subset X of N is called a germ if there exists ± (0; 1] such that 2

(®) Connexity: X"± is connected and dense in X" for each " 6 ±, and (¯ ) Essentiality: p¤ : H¤ (X± ; @X± ) H¤ (P± ; @P± ) is nonzero. !

De¯nition 2.2. A subset X0 of § is called a semi-algebraic G-stable set if there exists a germ X such that X0 = ¾ (0; ¾) X . A subset of § is called G-stable if it is the Hausdor® limit of a sequence of semi-algebraic G-stable sets. f

j

2

g

It follows trivially from this de¯nition that each G-stable set is a connected subset of PE . A germ X that justi¯es calling X0 a semi-algebraic G-stable set of equilibria for the game ¡ is sometimes (as often su±ces when X0 is an entire component of PE ) simply a closed semi-algebraic neighbourhood of X0 in the graph N of the perturbed equilibrium correspondence above the game ¡ located at ´ = 0. Essentiality of the projection map from this neighbourhood implies that the projection is onto, but not conversely. Thus, the de¯nition ensures that each perturbed game has an equilibrium near X0 , but it also requires the stronger property that the local projection map is topologically essential. G-stable sets are de¯ned as Hausdor® limits to eliminate dependence on the semi-algebraicity property imposed on the germ. Mertens phrased his original de¯nition using the graph of the equilibrium correspondence, call it N , and not the graph of the perturbed equilibrium correspondence. However, the two de¯nitions are equivalent. To see this, observe that there is a homeomorphism between N and N that commutes with the projection to P1 from those spaces. 3. Maximal Stable Sets Call a G-stable set maximal if it is not properly contained in another G-stable set. The main result of this article characterizes the maximal G-stable sets. Theorem 3.1. is itself

If a connected component of perfect equilibria contains a

G-stable.

In particular, the maximal

G-stable set, then the component

G-stable sets are connected components of perfect equilibria.

The proof of Theorem 3.1 is broken up into two parts. Lemma 3.2 shows that the graph of the perturbed equilibrium correspondence near a component of perfect equilibria is path connected. The second part shows that given a germ X for a G-stable set, one can add all the points in N close to the component containing this stable set to obtain a germ for the component itself, thus verifying that the component is G-stable. To formalize these ideas, consider a polytope E (T ) for T 2 PE . As remarked in Section 2, for each face F of E (T ), all the points in the relative interior of F have the same support, say RF = (RF1 ; RF2 ), and the same set of pure best replies, say S F = (S1F ; S2F ). Obviously, RF T S . Let § be the set of completely i

µ

i µ

F i

R

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S. GOVINDAN AND R. WILSON

R

mixed strategies against which all strategies in De¯ne the correspondence 'F : §R ! P1 by

' (¾ ) = ´ P 1

0 ¿ ´ 6 ¾;

are best replies, which is nonempty because

2

j

ij

= ¾ij if

F

µ

PE .

j = S or j is not a best reply to ¾ : ' is a nonempty- and convex-valued correspondence. Let Y be the closure of the graph of ' in P1 §. Obviously Y is a closed semi-algebraic subset of N . Let Y = Y ( ) and de¯ne Y = Y , where the union is over all faces of E (T ). For each ", let (Y ; @Y ) = Y p¡1 (P ; @P ). Observe that E (T ) Y0 = Y p¡1 (0). f

F

´

F

2

F

F i

g

F

F

T

"

" > 0, Y ± = Y n@Y

Lemma 3.2. For each

"

"

\

"

[F

E T

"

F

µ

"

is path connected and dense in

"

£

F

\

Y. "

To prove the path connectedness of Y ± , it is su±cient to show that for every face F of E (T ), any point (´ 0 ; ¾0 ) in Y \ Y ± can be connected to each (´1 ; ¾ 1 ) in Y \ Y ± . Given two such points, note that all strategies in R are best replies against ¾0 because (´0 ; ¾ 0 ) 2 Y , and also against ¾ 1 because (´1 ; ¾1 ) 2 Y and R µ T ). Choose a point ¿ in the relative interior of F . Pick two completely mixed strategy pro¯les ¿ 0 and ¿ 1 on the line segments between ¾0 and ¿ , and ¾ 1 and ¿ , respectively, with the property that ¿ < " if j 2= R , for each k = 0; 1 and i = 1; 2. This construction is possible since both ¾0 and ¾1 are completely mixed strategies and the support of ¿ is R . Observe that for each k, ¿ is a perturbed equilibrium of the game ® 2 P ± with ® = ¿ if j 2= R and an arbitrary positive number strictly smaller than min(¿ ; ") otherwise. The linear path connecting (´ ; ¾ ) and (® ; ¿ ) belongs to Y ± since if ¾ > ´ then j is a best reply against both ¾ and ¿ . The linear path between (®0 ; ¿ 0 ) and (®1 ; ¿ 1 ) also belongs to Y ± for the same reason. Thus we have shown that Y ± is path connected. To show that Y ± is dense in Y , it is su±cient to show that Y \ Y ± is dense in Y \ Y . Let (´; ¾) 2 Y \ Y . Since Y is the closure of the graph of ' , we can choose ¾ 0 2 § such that if ¾ > ´ then j belongs to S and is a best reply against ¾0 . If necessary by replacing ¾ 0 with a point on the line segment between it and a point in the relative interior of F , we can additionally assume that ¾ 0 < " for j 2= R . Let ´0 2 P ± be such that ¾0 > ´0 , with ´0 = ¾ 0 if j 2= R . Then (´0 ; ¾ 0 ) 2 Y ± \ Y and the line segment between (´0 ; ¾0 ) and (´; ¾) proves that Y ± \ Y is dense in Y \ Y . Proof.

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We are now ready to prove the Theorem. Let C be a component of PE that contains a G-stable set X0 . We can assume without loss of generality that X0 is semi-algebraic, i.e., there exists a germ X N that satis¯es the conditions of De¯nition 2.1 and such that X0 = ¾ (0; ¾) X . The Theorem is proved if we can show that for every T , if E (T ) X0 is nonempty then E (T ) X0 is contained in a semi-algebraic G-stable set. Indeed, the Theorem then follows from this along with the observation that if E (T ) and E (T 0 ) belong to a component C ¤ then there exists a ¯nite sequence T1 ; : : : ; T with T1 = T and T = T 0 and such that E (T ) E (T +1 ) is nonempty for i = 1; : : : ; k 1. Suppose X0 intersects the interior of a face, say F , of E (T ). Let X~ = X Y , where Y is as de¯ned in the beginning of the Section. Obviously X~ is a semi-algebraic set and X0 E (T ) ¾ (0; ¾) X~ . We will show that ¾ (0; ¾ ) X~ is a semi-algebraic G-stable set, i.e., that X~ is a germ. Remark ¯rst that X~ satis¯es the essentiality condition since we are merely replacing X by a larger set. To prove that X~ satis¯es the connexity requirement for stability, it is enough to show that for each su±ciently small " > 0, X 0 Y 0 is nonempty. (Indeed, since X 0 is connected and dense in X by de¯nition, and Y 0 is connected Proof of Theorem 3.1.

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2 PE

j

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MAXIMAL STABLE SETS OF TWO-PLAYER GAMES

5

and dense in Y , by Lemma 3.2, X~ 0 = X 0 Y 0 would then be connected and dense in X~ .) To prove this last point, choose ¾0 X0 F . There exists a sequence of points (´ ; ¾ ) (0; ¾0 ) with (´ ; ¾ ) X1± for all n. If necessary by passing to a subsequence, we can assume that for each n and i = 1; 2, ¾ > ´ implies j S . Hence, the sequence is also contained in Y . For every " now, there exists n such that ´ P ± . Thus X 0 Y 0 is nonempty, as asserted. "

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4. Computational Aspects This Section is devoted to some aspects of computing stable sets. From here on, we consider only Z-stable sets, i.e., stable sets de¯ned using simplicial homology theory with integer coe±cients. Let A1 and A2 be the m £ n and n £ m payo® matrices of the two players in ¡ . Given a vector (g1 ;g2 ) 2 R + , consider the perturbed game ¡ © g for which the payo® matrix for player i has A + g as its (j;k)-th entry. The set of perturbed games ¡ (´ ) de¯ned in Section 2 can be viewed as a subset of R + in the following sense. De¯ne f : P1 ! R + by f (´) = g with g1 = A1 ¢ ´2 and g2 = A2 ¢ ´1. Then f (´) has the same set of Nash equilibria as ¡ (´=1 + ´). Hereafter, let P~ = f (P ) for each 0 6 " 6 1 be the payo® perturbations corresponding to P . P~ is easily seen to be a polytope. Denote by @ P~ the relative boundary of P~ . By Theorem 2 in Section 2 of Mertens (1989), stable sets can be de¯ned equivalently by using (P~ ;@ P~ ) instead of (P ;@P ) as the space of perturbations. Let N µ R + £ § be the graph of the Nash equilibrium correspondence over R + , interpreted as the space of perturbed games | ¡ being viewed here as the origin of R + | and let proj be the natural projection to the space of games. Govindan and Wilson (1997) de¯ne the degree of a component C of equilibria of ¡ in terms of the local degree of the projection map and show that it agrees with the index of C . Their de¯nition can be summarized as follows. Choose an open neighbourhood U of C in N with the property that its closure U is disjoint from the other components of equilibria of ¡ . Choose an open ball V centered at the origin in Rm+n with the property that for each perturbation g in V , each equilibrium of the perturbed game ¡ © g lies either in U or in the complement of U . Fix now a generic perturbation g in V . The degree of C , denoted deg(C ), is the sum of the degrees of the equilibria of ¡ © g that belong to U . This computation turns out to be independent of the choice of g due to the fact that the projection map is a proper map over V . Call a component essential if its degree is nonzero. Then it follows immediately from Mertens (1986) and from the fact that perturbed games can be viewed as a subset of Rm+n that an essential component contains a stable set. Thus the essentiality of a component is su±cient to guarantee the existence of a stable set within it. In practical computations, it is easy to check whether a component of equilibria is essential. For a generic perturbation such as g above, the degree or index of each of its (isolated) equilibria is given by the sign of the determinant of a Jacobian matrix (GÄul, Pearce, and Stacchetti, 1993). Therefore, there is an operational procedure for determining the degree of a component, and each component with a nonzero degree must contain a stable set. There are related problems that the technique described above does not resolve. First, a component of Nash equilibria might contain multiple components of perfect equilibria, thus leaving a question as to which among these are stable. Second, even an inessential component might contain stable sets. Section 5 contains m

i jk

i j

m

m

n

j

"

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j

j

n

j

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m

n

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m

n

n

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S. GOVINDAN AND R. WILSON

examples illustrating these features. Can one use a ¯ner notion than the degree of a component to say something about stable sets in a component? The technique outlined below addresses this question. Let PE1 ; : : : ; PEK be the components of perfect equilibria of ¡ . For each k = 1; : : : ; K , choose an open neighbourhood Uk of PEk in § such that the closure U k of Uk is expressible as a ¯nite union of polytopes, and U k \ U l is empty for all l 6 = k. Choose " > 0 such that for each g 2 P~"± all its equilibria are contained in [k Uk . Theorem 4.1.

k, there exists an integer dk such that:

P deg(C ) = dk

For each

g 2 P~"± , , where the sum is over all the components C of equilibria of the game perturbed by g that are contained in Uk . 2. dk is nonzero only if PEk is stable. 3. When P~" is full dimensional: dk is nonzero if and only if PEk is stable. Proof. §n([k Uk ) is a ¯nite union of polytopes. Therefore, the set of games in Rm+n that have an equilibrium in § ( k Uk ) is expressible as a ¯nite union of convex polyhedra W1 ; : : : ; WL . For each l, Wl P~" ( @ P~" . Therefore, using a separation argument, we can ¯nd a full-dimensional polytope W with boundary @W such that (i)P~"± W @W ; (ii) for each l = 1; : : : L, Wl W ( @W . It follows from (ii) that for each g W @W all the equilibria of g are contained in k Uk . Thus, for each k, proj : (W @W Uk ) W @W is a proper map, which readily implies the ¯rst statement. Let S k be the closure of proj¡1 (W @W ) ((W @W ) Uk )and let @S k = S k proj¡1 (@W ). Observe that (W; @W ) is essential i® dk is nonzero. We claim now that if proj : (S k ; @S k ) (W; @W ) proj : (S k ; @S k ) is essential then PEk is stable. To see this, suppose the projection map from (S k ; @S k ) is essential. Let (X k ; @X k ) = proj¡1 (P~" ; @ P~" ). Then, by Mertens (1986), proj : (X k ; @X k ) (P~" ; @ P~" ) is also essential. Using the technique of Theorem 1 of Mertens (1989), we have that X0k = ¾ (0; ¾) X k contains a stable set. Obviously, X0k PEk , and by Theorem 3.1 PEk is itself stable. We can now prove statements (2) and (3) of the Theorem as follows. If dk is nonzero, then the projection map from (S k ; @S k ) is essential and, by the argument above, PEk is stable. If P~" is full dimensional, then we can take P~" to be the set W . Thus, if PEk is not stable then the projection map from (S k ; @S k ) is inessential, i.e., dk is zero. 1.

For each

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2

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As in the case of checking for essentiality of Nash components, it is easy to verify whether a component of perfect equilibria satis¯es the above criterion. Choose a neighbourhood Uk of PEk whose closure is disjoint from every other component of perfect equilibria. Choose also a perturbation g P~"± for a small " > 0. Now to compute the sum of the degrees of all components of equilibria of ¡ g that are in Uk , choose a generic game ¡ 0 close to ¡ g and compute the sum of the indices of its equilibria in Uk . If this sum is nonzero, then PEk is stable. In part 3 of Theorem 4.1, verifying that P~" is full dimensional amounts to checking whether the dimension of the values of (A1 ´2 ; A2 ´1 ) for ´ P1 is m + n. When P~" is not full dimensional, Theorem 4.1 gives only the su±cient condition in part 2. It seems possible that if the set P~" is a low-dimensional subset of Rm+n then the projection map from S k might be inessential while the restriction to the inverse image of P~" might still be essential. It would be useful to ¯nd an example of this phenomenon. 2

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¢

¢

2

MAXIMAL STABLE SETS OF TWO-PLAYER GAMES

7

Mertens (1989) provides an algorithm to compute stable sets. This algorithm seems computationally di±cult since it requires full knowledge of the perturbed equilibrium correspondence. The advantage of the techniques here is that they are easily implemented. However, in as much as they are only su±cient, and not necessary, conditions, they are not guaranteed to detect all stable components. 5. Examples The games considered in this section are all 3 3 games. Player 1's pure strategy set is T;M;B , which labels the rows of the payo® matrix, and Player 2's pure strategy set is L;C;R , which labels the columns. The ¯rst example is called \Do the Right Thing" in Wilson (1992). Its payo® matrix is: £

f

f

g

g

03; 6 3; 6 3; 61 @4; 3 2; 1 1; 4A 4; 3 1; 4 2; 1

This game has two components of equilibria: C1 = ((0;x2 ;x3 );L) x2 > 1=3; x3 > 1=3 and C2 = (T; (y1 ;y2 ;y3 ) y2 > y1 ; y3 > y1 . Consider the perturbed game in R6+ given by g = "(3; 9=4; 2; 3; 5=2; 5=2), where " is a small positive scalar. It is easily checked that this perturbed game has a unique equilibrium close to C1 . Hence, C1 has degree 1 and C2 has degree zero. Also, P~" is full-dimensional for this game so Theorem 4.1, part 3, veri¯es that C2 is not stable, and because the perturbed game has no equilibrium close to C2 , it cannot contain a stable set. Finally, all equilibria in C1 are perfect, so Theorem 3.1 implies that C1 is the only maximal stable set of this game; in fact, C1 is the only stable set of this game. [We could have obtained this result from the fact that the unique proper equilibrium ((0; 1=2; 1=2); L) of this game is contained in C1 and Mertens' (1991) theorem that each stable set contains a proper equilibrium.] These conclusions are evidently obtained much more easily than directly verifying that a closed semi-algebraic neighborhood of C1 provides a germ satisfying connexity and essentiality, as in De¯nition 2.1. The second example merely transposes the payo®s to Player 1 for the strategies M and B. Its payo® matrix is: 03; 6 3; 6 3; 61 @4; 3 1; 1 2; 4A 4; 3 2; 4 1; 1 f

f

j

j

g

g

This game has the same components as the previous game. But when we consider the perturbation g0 obtained from g above by transposing the second and the third coordinates, we see that there is an equilibrium with index ¡1 near C1 and two with index +1 near C2 . Hence C1 has degree ¡1 and C2 has degree +2. As before, all equilibria in C1 are perfect and hence it is a maximal stable set. On the other hand, C2 contains two components of perfect equilibria: the set PE1 consisting of points where Player 2 randomizes between L and C and another PE2 where 2 randomizes between L and R. The game g0 belongs to P~" and has one equilibrium near PE1 and one near PE2 , each with index +1. Applying Theorem 4.1, part 2, we see that both PE1 and PE2 are maximal stable sets. This example shows that an essential component of Nash equilibria might contain multiple maximal stable sets. Again, the conclusion is obtained more readily than by constructing germs for the three stable components of perfect equilibria.

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The third example comes from van Damme (1989): 02; 2 2; 2 2; 21 @0; 0 3; 2 0; 3A 3; 3 0; 2 3; 0 This game has two components: (B,L) and C2 = (T; (y1 ; y2 ; y3 )) 1=3 6 y2 6 2=3 . Because (B,L) is a strict pure-strategy equilibrium, it has index +1 and is therefore a singleton stable set. Therefore C2 has degree zero. There are two components of perfect equilibria in C2 : one where player 2 mixes between C and R, and the other where 2 mixes between C and L. For each su±ciently small " > 0, the perturbation g = "(2; 1; 2; 5=3; 2; 5=3) belongs to P~1± and has a unique equilibrium with nonzero index near each of the two components of perfect equilibria in C2 . Of course these two indices have opposite signs, +1 and 1. Hence, by Theorem 4.1, part 2, both of these are maximal stable sets. This game illustrates the possibility of stable sets residing inside inessential components. Of related interest is the striking example in Hauk and Hurkens (1999) in which every payo® perturbation yields equilibria near a component whose degree is zero. That is, the projection map is onto but not essential. However, they demonstrate that the component is not hyperstable, as de¯ned by Kohlberg and Mertens (1986). The fourth example is the \Circle" game from Govindan and Wilson (1997): 01; 1 1; 1 0; 01 @1; 1 0; 0 1; 1A 0; 0 1; 1 1; 1 This game has two components of equilibria: one is homeomorphic to a circle and consists of all equilibria where both players receive payo® 1; and the other is an isolated completely mixed equilibrium with each ¾ = 1=3. The mixed equilibrium has index +1 so the component has degree zero. All equilibria are perfect because the game has a completely mixed equilibrium; i.e., S ¤ = S . But the circular component is not stable, even though it contains twelve proper equilibria. To see this, one veri¯es that the set P~ of perturbed games is full dimensional, then Theorem 4.1, part 3, implies that the circular component is not itself stable, and then Theorem 3.1 implies that it cannot contain a stable set. This conclusion would be hard to obtain by showing directly that no subset of the circular component has a germ. In particular, every perturbation in P~ yields equilibria near the circular component, yet the perturbed game 01 + "; 1 1; 1 + " 0; 0 1 @1; 1 + " 0; 0 1 + "; 1A 0; 0 1 + "; 1 1; 1 + " has no equilibrium near the circular component. f

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References Borm P, M Jansen, J Potters, and S Tijs

games. OR Spektrum 15:17-20

(1993) On the structure of the set of perfect equilibria in bimatrix

(1997) Equivalence and invariance of the index and degree of Nash equilibria. Games and Economic Behavior 21:56-61 Govindan S, and R Wilson

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S. GOVINDAN AND R. WILSON

Department of Economics, The University of Western Ontario, London, Ontario, Canada N6A 5C2 E-mail address :

[email protected]

Stanford Business School, Stanford University, Stanford, California, USA 94305-5015 E-mail address : [email protected]