OON SUSTAINABLE EQUILIBRIA
SRIHARI GOVINDAN, RIDA LARAKI, AND LUCAS PAHL
Abstract.
Following the ideas laid out in Myerson [13], Hofbauer [8] defined an equilib-rium of a game as sustainable if it can be made the unique equilibrium of a game obtained bydeleting a subset of the strategies that are inferior replies to it, and then adding others. Hof-bauer also formalized Myerson’s conjecture about the relationship between the sustainabilityof an equilibrium and its index: for a generic class of games, an equilibrium is sustainableiff its index is +1. Von Schemde and von Stengel [22] proved this conjecture for bimatrixgames. This paper shows that the conjecture is true for all finite games. More precisely,we prove that an isolated equilibrium of a given game has index +1 if and only if it can bemade unique in a larger game obtained by adding finitely many inferior reply strategies. Introduction
Myerson [13] proposes a refinement of Nash equilibria of finite games, which he calls sustainable equilibria , based on the hypothesis that most games, even if they are one-shotaffairs, should be analyzed not as if they are played in isolation, but rather as particularinstances of many plays of such games. Myerson argues that when, say, two members ofa society play a Battle-of-Sexes game, if the game has a history in this society, it becomesa “culturally familiar” game for these two players, and the past history of plays, by othermembers of the society, should inform play in this encounter. An equilibrium is then acultural norm, an institution, in this society, and the game is typically played according tothis norm. Any Nash equilibrium of the underlying game that can emerge as a norm insome society is sustainable. From this perspective, Myerson reasons, the two pure-strategyequilibria in the Battle-of-Sexes game are sustainable while the mixed equilibrium is not.In his search for a formal definition of a sustainable equilibrium, Myerson considers, andthen dismisses, on axiomatic grounds, existing refinements that yield the same prediction inthe Battle-of-Sexes game as his heuristic argument does: for example, persistent equilibria
Date : May 28, 2020.An extended version of the abstract will be published in the Proceedings of the 21st ACM conference onEconomics and Computation (EC 20), July 13-17, 2020, Virtual Event, Hungary. ACM, New York, NY,USA, 2 pages. https://doi.org/10.1145/3391403.3399514.We would like to thank Paulo Barelli, Josef Hofbauer, Andy McLennan, Klaus Ritzberger, and SylvainSorin for their comments. a r X i v : . [ ec on . T H ] M a y S. GOVINDAN, R. LARAKI, AND L. PAHL fail invariance; and evolutionary stability fails existence. Myerson concludes his paper witha conjecture that the index of an equilibrium is a determinant of its sustainability. Hofbauer [8] distills the ideas in Myerson’s paper to provide a definition of sustainableequilibria. Hofbauer posits that a minimum requirement of sustainability should be that ifan equilibrium of a game is sustainable, it should remain sustainable in the game obtained byrestricting players’ strategies to the set of best replies to the equilibrium. If one also acceptsthat an equilibrium that is unique is sustainable, then one is lead to the following definition.Say that a game-equilibrium pair is equivalent to another such pair if the restrictions of thetwo games to the set of best replies to their respective equilibria are the same game (moduloa relabelling of the players and their strategies) and the two equilibria coincide (under thesame identification). An equilibrium of a game is sustainable if it has an equivalent pairwhere the equilibrium is unique. Call an equilibrium regular if locally the equilibrium is adifferentiable function of the game payoffs (see section 2.1). Following Myerson, Hofbauerconjectured that a regular equilibrium is sustainable iff its index is +1. Von Stengel and vonSchemde [22] proved this conjecture for bimatrix games. In this paper, we show that theconjecture holds for all N -person games.To illustrate the ideas involved, consider the battle of the sexes game G below. It hasthree Nash equilibria: two are strict ( t, l ) and ( b, r ) (with index +1), and one is mixed (withindex − G = l rt (3 ,
2) (0 , b (0 ,
0) (2 , x for player 1, and y for player 2, see the game ˆ G below), b and r are now strictly dominated. Removing them yields a game where x and y are strictly dominated, making the strict equilibrium ( t, l ) of G the unique equilibrium of ˆ G .ˆ G = l r yt (3 ,
2) (0 ,
0) (0 , b (0 ,
0) (2 ,
3) ( − , x (1 ,
0) (4 , −
2) ( − , − Interestingly, Myerson speculates that one could perhaps develop a theory of index of equilibria basedon fixed-point theory, seemingly unaware, as Hofbauer [8] observes, of an extant theory in the literature (seeG¨ul, Pearce and Stacchetti [6] and Ritzberger [17]). For regular games, this is equivalent to restricting players’ strategies to the support of the equilibrium—see Section 4 for a discussion of this point.
N SUSTAINABLE EQUILIBRIA 3 in a larger game obtained by adding finitely many inferior reply strategies. Our paperextends this property to mixed equilibria: we prove that any regular equilibrium (and moregenerally any isolated equilibrium) of a given game has index +1 if and only if it can bemade the unique equilibrium in a larger game obtained by adding finitely many inferior replystrategies.What are the key properties of the index of equilibria that drive this equivalence? Toanswer this question, let us see a sketch of our proof. In one direction, suppose that anequilibrium σ of a game G is sustainable, and that ( G, σ ) is equivalent to a pair ( ¯
G, σ ) where σ is the unique equilibrium of ¯ G . Let G ∗ be the game obtained from G by deleting strategiesthat are inferior replies to σ . It follows from a property of the index that the index of σ in G can be computed as the index of σ in G ∗ . The game G ∗ is also the game obtained from ¯ G by deleting inferior replies there. Therefore, the index of σ in G ∗ can also be computed asthe index of σ in ¯ G . As σ is the unique equilibrium of ¯ G , its index is +1, which then givesus the result.Going the other way, if we have a +1 index equilibrium σ of a game G , and since the sumof indices over all equilibria is +1, then the sum of the indices of the other equilibria is zero.Now, we can take a map whose fixed points are the Nash equilibria of G and alter it outsidea neighbourhood of σ so that the new map has no fixed points other than σ . By a carefuladdition of strategies and specification of payoffs for these strategies, we obtain a game ¯ G where any equilibrium must translate into a fixed point of the modified map of G , making σ the unique equilibrium in ¯ G .A word about our methodology is in order. In a bimatrix game, a player’s payoff functionis linear in his opponent’s strategy and the index can easily be computed using the Shapleyformula [19]. Von Schemde and von Stengel [22] were able to exploit those features anduse tools from the theory of polytopes (von Schemde [23]) to prove the conjecture. In thegeneral case, their technique is inapplicable. What we do, instead, is start with a constructioninvolving a fixed-point map and then convert it into a game-theoretically meaningful one.In this respect, our approach is similar in spirit to, but different in details from, that inGovindan and Wilson [5].Equilibria with index +1 are also distinguished from their counterparts with index − To make a strict equilibrium s = ( s , ..., s N ) of a game G unique in a larger game ˆ G , it is enough to addone strategy x n per player n as in the Battle-of-Sexes, where for each player n , x n strictly dominates all itspure strategies t n (cid:54) = s n ; once the strategies t n (cid:54) = s n are eliminated, x n becomes strictly dominated by s n . The possibility of such a construction follows from a deep result in algebraic topology, the Hopf ExtensionTheorem (Corollary 8.1.18, [20]).
S. GOVINDAN, R. LARAKI, AND L. PAHL theory, that equilibria with index − Even computational dynamics likethose generated by homotopy algorithms (Lemke-Howson, linear-tracing procedure, etc.)converge to a +1 index equilibrium (Herings-Peeters [7]). While these results might be seenas eliminating − G shows. G = l m rt (10 ,
10) (0 ,
0) (0 , m (0 ,
0) (10 ,
10) (0 , b (0 ,
0) (0 ,
0) (10 , x , y , and z as in ˆ G below. The fact is that all natural dynamics increase the potential of G ; since the completely-mixed equilibrium minimizes that potential, it is unstable for allnatural dynamics—cf. Hofbauer [8] for more about his second conjecture, stating that anyregular game has at least one +1 index equilibrium which is asymptotically stable w.r.t.some natural dynamics. ˆ G = l m r x y zt (10 ,
10) (0 ,
0) (0 ,
0) (0 ,
11) (10 ,
5) (0 , − m (0 ,
0) (10 ,
10) (0 ,
0) (0 , −
10) (0 ,
11) (10 , b (0 ,
0) (0 ,
0) (10 ,
10) (10 ,
5) (0 , −
10) (0 , In fact, this observation leads McLennan to articulate his index +1 principle , which selects +1 equilibria. Note however that in G , the three pure equilibria are strict, have index +1 and can be shown to beasymptotically stable for all natural dynamics. We refer the interested reader to von Schemde [23], p. 89-91, to understand how the strategies x , y and z are geometrically constructed. It follows from Demichelis and Ritzberger [3] that Hofbauer’s second conjecture is false if we include allgames, because a necessary condition for a component of Nash equilibria to be asymptotically stable is thatits index agree with its Euler characteristic. Yet, there are examples (e.g. Ritzberger [18], p. 325) where allcomponents of equilibria are convex (and so have Euler characteristic +1), but no component has index +1.
N SUSTAINABLE EQUILIBRIA 5 both in the definition and in the result. Finally, in the Appendix we review a construct fromthe theory of triangulations that we need in the proof.2.
Definitions and statement of the theorem
A finite game in normal form is a triple ( N , ( S n ) n ∈N , G ) where: N = { , . . . , N } is theset of players, with N (cid:62)
2; for each n ∈ N , S n is a finite set of pure strategies; and, letting S ≡ (cid:81) n ∈N S n be the set of pure strategy profiles, G : S → R N is the payoff function. By aslight abuse of notation, we will refer to a game by its payoff function G .Given a game G , for each n , let Σ n be the set of n ’s mixed strategies and let Σ ≡ (cid:81) n ∈N Σ n .Also, for each n , S − n ≡ (cid:81) m (cid:54) = n S n , and Σ − n ≡ (cid:81) m (cid:54) = n Σ n . The payoff function G extends toΣ in the usual way and we will denote this extension by G as well.Define an equivalence relation on game-equilibrium pairs as follows. For i = 1 ,
2, let(( N i , ( S in ) n ∈N i , G i ) , σ i ) be a game-equilibrium pair, i.e., σ i is an equilibrium of G i . Say that( G , σ ) ∼ ( G , σ ) if, up to a relabelling of players and strategies, the restriction of G tothe set of best replies to σ is the same game as the restriction of G to the set of best repliesto σ , and the equilibria coincide under this identification. It is easily checked that ∼ is anequivalence relation. Definition 2.1.
An equilibrium σ of a game G is sustainable if ( G, σ ) ∼ ( ¯ G, ¯ σ ) for a game¯ G where ¯ σ is the unique equilibrium.Sustainability is a property of equivalence classes and we could say that the canonicalrepresentation of a sustainable equilibrium is the game-equilibrium pair where there are noinferior replies to the equilibrium.2.1. Index and degree of equilibria.
Both the index and the degree of equilibria aremeasures of the robustness of equilibria to perturbations. They differ in the space of per-turbations they consider (perturbations of fixed-point maps vs payoffs perturbations) butultimately agree with one another. We start with the degree of equilibria. For simplicitywe give a definition of degree only for regular equilibria. This approach allows to bypassthe use of algebraic topology, but more importantly it is germane to our problem, as we areconcerned only with regular equilibria in this paper. For the general definition, see for e.g.,Govindan and Wilson [5]. Note that this definition requires the number of players in an equivalence class to be the same. The reason we are defining and reviewing both concepts, when apparently one would do, is that theyare both useful in the exposition.
S. GOVINDAN, R. LARAKI, AND L. PAHL
Fix both the player set N and the strategy space S . The space of games with strat-egy space S is then the Euclidean space Γ ≡ R N × S of all payoff functions G . Let E bethe graph of the Nash equilibrium correspondence over Γ, i.e., E = { ( G, σ ) ∈ Γ × Σ | σ is a Nash equilibrium of G } . Let proj : E →
Γ be the natural projection: proj ( G, σ ) = G .By the Kohlberg-Mertens Structure Theorem [9], there exists a homeomorphism h : E →
Γsuch that h − is differentiable almost everywhere. Say that an equilibrium σ of a game G is regular if proj ◦ h − is differentiable and has a nonsingular Jacobian at h ( G, σ ); and saythat a game is regular if each equilibrium σ of G is regular. If an equilibrium σ is regular,then it is a quasi-strict equilibrium —that is, all unused strategies are inferior replies —andlocally, the equilibrium is a smooth, even analytic, function of the game; moreover, it is alsoa regular equilibrium in the space of games obtained by deleting the unused strategies or,indeed, by adding strategies that are inferior replies to the equilibrium. The set of gamesthat are not regular is a closed subset of lower dimension—actually codimension one—in Γand thus regular games are generic.If an equilibrium σ of a game G is regular, then we can assign a degree to it that is either+1 or − proj ◦ h − at h ( G, σ ) has a positive or anegative determinant. An inspection of the formula for the Jacobian shows that the degree ofa regular equilibrium σ is the same as its degree computed in the space of games obtained bydeleting the strategies that are inferior replies to σ . Therefore, if σ is a regular equilibriumof G and if ( G, σ ) ∼ ( ¯ G, ¯ σ ) then ¯ σ is a regular equilibrium of ¯ G and it has the same degreeas σ , making degree an invariant for an equivalence class.As the Kohlberg-Mertens homeomorphism extends to the one-point compactification of E and Γ, and proj ◦ h − is homotopic to the identity on this extension, the sum of the degreesof equilibria of a regular game is +1. In fixed-point theory, the index of fixed points contains information about their robustnesswhen the map is perturbed. (See McLennan [11] for an account of index theory writtenprimarily for economists.) Since Nash equilibria are obtainable as fixed points, index theoryapplies directly to them. For simplicity, suppose f : U → Σ is a differentiable map definedon a neighborhood U of Σ in R N | S | and such that the fixed points of f are the Nash equilibriaof a game G . Let d be the displacement of f , i.e., d ( σ ) = σ − f ( σ ). Then the Nash equilibriaof G are the zeros of d . Suppose now that the Jacobian of d at a Nash equilibrium σ of G See see Ritzberger [17] and van Damme [21]. If G is nongeneric, we can define the degree of a component of equilbria as the sum of the degreesof equilibria in a neighborhood of the component for a regular game that is in a neighborhood of G ; thiscomputation is independent of the neighborhoods chosen, as long as they are sufficiently small. The sum ofthe degrees of the components of equilibria of a game is +1. N SUSTAINABLE EQUILIBRIA 7 is nonsingular. Then we can define the index of σ under f as ± d is positive or negative.One potential problem with the definition of index is the dependence of the computationon the function f , as intuitively we would think of the index as depending only on the game G . But, under some regularity assumptions of f , we can show that the index is independentof f . Specifically, consider the class of continuous maps F : Γ × Σ → Σ with the property thatthe fixed points of the restriction of F to { G } × Σ are the Nash equilibria of G . Demichelisand Germano [2] show that the index of equilibria is independent of the particular map in thisclass that is used to compute it; Govindan and Wilson [4] show that the degree is equivalentto the index computed using one of the maps in this class, the fixed-point map definedby G¨ul, Pearce and Stacchetti [6]. Thus, the index and degree of equilibria coincide—seeDemichelis and Germano [2] for an alternate, more direct, proof of this equivalence. Giventhese results, for a regular equilibrium, we can talk unambiguously of its index and use theterm degree interchangeably with it.2.2. Games in strategic form.
It is convenient for us to work with a somewhat largerclass of games than normal-form games, called strategic-form games, and in this subsectionwe will define these games—cf. Pahl [16] for an extensive treatment of these games.A game in strategic form is a triple ( N , ( P n ) n ∈N , V ) where: N is the player set; for each n , P n is a polytope of strategies; V : (cid:81) n ∈N P n → R N is a multilinear payoff function. Clearlyany normal-form game is a strategic-form game. Going the other way, given a strategic-formgame ( N , ( P n ) n ∈N , V ), we can define a normal-form game ( N , ( S n ) n ∈N , G ) where for each n , S n is the set of vertices of P n and for each s ∈ S = (cid:81) n S n , G ( s ) = V ( s ); the polytope P n can be viewed as the quotient space of Σ n obtained by identifying all mixed strategies thatare duplicates of one another (i.e., induce the same payoffs for all players for any profile ofstrategies of n ’s opponents).2.3. Statement of the theorem.
The following theorem settles the Myerson-Hofbauerconjecture in the affirmative.
Theorem 2.2.
A regular equilibrium is sustainable iff its index is +1 . As the sum of the indices of the equilibria of a regular game is +1, there is at least onewith index +1. Thus, we have the following corollary.
Corollary 2.3.
Every regular game has at least one sustainable equilibrium. A polytope is a convex hull of finitely many points.
S. GOVINDAN, R. LARAKI, AND L. PAHL Proof of Theorem 2.2
We will present the proof in a sequence of steps, each of which will be carried out in aseparate subsection.3.1.
The index of a sustainable equilibrium.
We begin with a proof of the necessityof the condition. Let σ ∗ be a regular equilibrium of a game G that is sustainable. Let( G, σ ∗ ) ∼ ( ¯ G, ¯ σ ), where ¯ σ is the unique equilibrium of ¯ G . As we saw in the previous section,the index is constant on an equivalence class. Since ¯ σ is the unique equilibrium of ¯ G , itsindex is +1, and the result follows.3.2. Preliminaries.
The rest of the section is devoted to proving the sufficiency of thecondition. In this subsection, we introduce some key ideas that we exploit in the proof.First, we gather a list of notational conventions to be used. Throughout Section 3 (but notin the Appendix) we use the (cid:96) ∞ -norm on Euclidean spaces. For any subset A of a topologicalspace X , we let ∂ X A be its topological boundary and int X ( A ) its interior. If C is a convexset in a Euclidean space, then ∂C and int( C ) refer to the boundary and the interior of C inthe affine space generated by C . Definition 3.1.
Given a payoff function G , and a vector h ∈ (cid:81) n ∈N R S n , let G ⊕ h be thegame where the payoff to player n from a profile s ∈ S is G n ( s ) + h n,s n .For a game G , recall that Nash [14] obtains its equilibria as fixed points of a map on thestrategy space. This function, which we denote by f , is defined as follows. For each n, s n and σ , let φ n,s n ( σ ) ≡ max { , G n ( s n , σ − n ) − G n ( σ ) } and φ n ≡ (cid:81) s n ∈ S n φ n,s n ; then f n,s n ( σ ) ≡ σ n,s n + φ n,s n ( σ )1 + (cid:80) t n ∈ S n φ n,t n ( σ ) . For each n , f n ( σ ) = σ n iff φ n,s n = 0 for each s n ∈ S n . If f n ( σ ) (cid:54) = σ n for some n and σ , thenletting r n ( σ ) = (cid:32) (cid:88) t n ∈ S n φ n,t n ( σ ) (cid:33) − φ n ( σ )and λ n ( σ ) = 11 + (cid:80) t n ∈ S n φ n,t n ( σ ) , we have f n ( σ ) = λ n ( σ ) σ n + (1 − λ n ( σ )) r n ( σ ) . N SUSTAINABLE EQUILIBRIA 9
Thus f n ( σ ) is an average of σ and a mixed strategy r n ( σ ); r n ( σ ) has the following properties:(1) it assigns a positive probability to a pure strategy iff it does strictly better than σ —inparticular, it assigns zero probability to some strategy in the support of σ n , as f n ( σ ) (cid:54) = σ n ;(2) it assigns the highest probabilities to the best replies to σ . Definition 3.2.
A game ( N , ¯ S, ¯ G ) embeds ( N , S, G ) if: (1) for each n : S n ⊆ ¯ S n ; and (2)the restriction of ¯ G to S equals G .Again for notational convenience, we will talk of a game ¯ G embedding G . When ¯ G embeds G , we view the set Σ of mixed strategies of G as a subset of the set ¯Σ of mixed strategies in¯ G . Obviously, if ¯ G embeds G and σ is an equilibrium of ¯ G where for each n , the strategiesthat are not in S n are inferior replies, then ( G, σ ) ∼ ( ¯ G, σ ). Our proof technique is to showthat for each regular +1 equilibrium σ ∗ of G , we can embed G in a game ¯ G where σ ∗ is theunique equilibrium and the newly added strategies are inferior replies to σ ∗ .We say that a strategic-form game ¯ V embeds G if the associated normal-form game ¯ G ,as defined in Subsection 2.2, embeds G , or equivalently for each n , each strategy in S n is avertex of the polytope ¯ P n of n ’s strategies in ¯ V , and G ( s ) = ¯ V ( s ) for all s ∈ S . In our proofwe construct embeddings of G in strategic-form games ¯ V that have a simple structure: foreach n , the strategies in S n span a face of ¯ P n .3.3. A Simple consequence of regularity.
From now on fix a game G and let σ ∗ be aregular equilibrium with index +1. For each n , let S ∗ n be the support of σ ∗ n . Our objectivein this subsection is to record the following simple, and yet consequential, property of σ ∗ .There exists ¯ ε > σ (cid:54) = σ ∗ is an equilibrium of G , then there exist two differentplayers n , n , such that for i = 1 ,
2, there exists s n i ∈ S ∗ n i with σ n i ,s ni < σ ∗ n i ,s ni − ¯ ε . Indeedif this property is not true, there exist a sequence k → ∞ , a corresponding sequence σ k ofequilibria converging to some σ , and a player n such that: (1) σ k (cid:54) = σ ∗ for all k ; and (2)for all m (cid:54) = n and s m ∈ S ∗ m , σ km,s m (cid:62) σ ∗ m,s m − k − . Therefore, σ m = σ ∗ m for all m (cid:54) = n .But σ n (cid:54) = σ ∗ n as σ ∗ is regular and, hence, isolated. This implies that λσ + (1 − λ ) σ ∗ is anequilibrium for all λ ∈ [0 , σ ∗ is isolated. Thus, thereexists ¯ ε with the stated property. For 0 < ε (cid:54) ¯ ε , and each n , let B εn be the set of σ n ∈ Σ n such that σ n,s n (cid:62) σ ∗ n,s n − ε forall s n ∈ S ∗ n ; and let B ε be the set of σ such that σ n is not in B εn for at most one n . (N.B. (cid:81) n ∈N B εn (cid:40) B ε ). Since B ε = (cid:83) n (Σ n × (cid:81) m (cid:54) = n B εm ) is the union of finitely many closed sets,it is closed. Note that this proof only uses the fact that σ ∗ is isolated. Killing all fixed points of f other than σ ∗ . From the viewpoint of fixed-pointtheory, our problem amounts to embedding Σ as a proper face of a polytope ¯Σ, extending f (the Nash map) to a function ¯ f on it, and then modifying ¯ f such that its only fixed pointis σ ∗ . From a game-theoretic viewpoint, there is an additional problem introduced by thecaveat that ¯ f should, in a sense, be realizable as a fixed-point map of a game ¯ G that embeds G —i.e., a map whose fixed points are the equilibria of game ¯ G . In this subsection, we solvethe first problem partially, by constructing a map f that coincides with f on B ε for some0 < ε (cid:54) ¯ ε and that has no fixed points outside it. We will later use this map to constructthe embedding ¯ G .If necessary by adding a strictly dominated strategy for each player, we can assume that σ ∗ n belongs to ∂ Σ n for each n . (Recall that sustainability and index are properties of equivalenceclasses of regular equilibria, so that the addition of such strategies is harmless.) Let V ≡ B ¯ ε ; X ≡ Σ \ int Σ ( V ). The boundary of X is relative to the affine space generated by Σ, i.e., ∂X ≡ ( ∂ Σ \ V ) ∪ ∂ Σ V . We claim now that ( X, ∂X ) is homeomorphic to a ball with boundary.Indeed, the desired homeomorphism can be constructed as follows. Pick a completely-mixedstrategy-profile σ such that σ n,s n < σ ∗ n,s n − ¯ ε for all n and s n ∈ S ∗ n . (Such a choice ispossible since σ ∗ n belongs to the boundary of Σ n and if necessary, we can decrease the ¯ ε defined above.) The set X is star-convex at σ : for each σ ∈ X , λσ + (1 − λ ) σ ∈ X forall λ ∈ [0 , σ in Σ \ ∂ Σ that is contained in X \ ∂X that is homeomorphic to X using radial projections from σ .Define ˜ f : Σ → Σ as follows. First, using Urysohn’s lemma, construct a continuousfunction α : Σ → [0 ,
1] that is zero on V and positive everywhere else. Then, letting τ n bethe barycenter of Σ n for each n , define ˜ f ( σ ) = (1 − α ( σ )) f ( σ ) + α ( σ ) τ . The function ˜ f equals f on V and therefore σ ∗ is the unique fixed point of ˜ f in V . The set Σ \ V is mappedby ˜ f to Σ \ ∂ Σ. Hence all the other fixed points of ˜ f belong to X \ ∂X .Let ˜ d be the displacement of ˜ f : ˜ d ( σ ) ≡ σ − ˜ f ( σ ). For each n , let A n be the hyperplanein R S n through the origin and with normal (1 , . . . , A = (cid:81) n A n . The map ˜ d mapsΣ into A . As the index of σ ∗ is +1, the sum of the indices of the other components of fixedpoints of ˜ f , which are contained in X \ ∂X , is zero. Therefore, ˜ d : ( X, ∂X ) → ( A, A − d from X to A − ∂X coincides with ˜ d . Extend ˆ d to thewhole of Σ by letting it be ˜ d outside X , i.e., on V \ ∂ Σ V .For each σ ∈ Σ, there exists λ ∈ (0 ,
1] such that σ − λ ˆ d ( σ ) ∈ Σ: indeed, this is obviousfor σ ∈ X \ ∂X , since σ belongs to the interior of Σ; for σ ∈ ∂X ∪ V , we can take λ = 1 as N SUSTAINABLE EQUILIBRIA 11 ˆ d = ˜ d . Now for each σ , let λ ( σ ) be the largest λ ∈ [0 ,
1] such that σ − λ ( σ ) ˆ d ( σ ) ∈ Σ. Notethat the map σ (cid:55)→ λ ( σ ) is continuous. Define f : Σ → Σ by: f ( σ ) = σ − λ ( σ ) ˆ d ( σ ). Themap f is continuous, coincides with f on V , and has σ ∗ as its unique fixed point.3.5. Example.
We now introduce a running example, where we can carry out our construc-tion numerically, and which we hope will aid in the understanding of the proof. The examplediffers from the text in one somewhat irrelevant respect: we focus on symmetric strategies,as it reduces the dimension of the problem and allows us to perform a two-dimensionalgraphical analysis as well.The game we study is a two-player coordination game given below.
L RL (1 ,
1) (0 , R (0 ,
0) (1 , x ∈ [0 , x is the probability of playing L . There are two pure strategy equilibria: x = 1 and x = 0,both of which have index +1; and there is a mixed equilibrium, x = 1 /
2, which has index −
1. The restriction of the Nash map f to symmetric strategies allows us to represent it asa function from [0 ,
1] to itself. By computation, we obtain: f ( x ) ≡ (cid:40) x − x + x if x ∈ [0 , / x − x +3 x − − x +3 x − if x ∈ (1 / , f , with its three fixed points corresponding to the three equilibria of thegame, is illustrated below in Figure 1. Let V ≡ [2 / , f as in the previous section, whose only fixed point is x = 1 (see the graph of f in greenin Figure 1). f ( x ) ≡ (cid:40) if x ∈ [0 , / x − x +3 x − − x +3 x − if x ∈ (2 / , From Figure 1, one can observe that it is impossible to construct a map which coincides with f in theneighbourhood of x = 1 / / Figure 1.
Graphs of f (black) and f (green)3.6. A parametrized family of perturbed games.
Ideally, we would like a game G suchthat f is the Nash map of G . This seems to be too strong a property to hold. However, f does contain enough information for us to construct a function g : Σ → (cid:81) n ∈N R S n suchthat: (1) g ( · ) is zero on B ε for some sufficiently small ε ; (2) σ is an equilibrium of G ⊕ g ( σ )iff σ = σ ∗ .Choose 0 < ε < ¯ ε and let U ≡ B ε ; note that U is a closed subset contained in the interiorof V . For each n , let Z n be ( d n ) − (0) ∩ (Σ \ int Σ ( U )), where d is the displacement of f .Let Z n be the complement of Z n in Σ \ int Σ ( U ). Define r n : Z n → ∂ Σ n as follows. For each σ ∈ Z n , let r n ( σ ) be the unique point in ∂ Σ n on the ray from σ n through f n ( σ ), i.e., it is theunique point of the form (1 − α ) σ n + αf n ( σ ) for α (cid:62) ∂ Σ n . If σ ∈ Z n , thenthere exists some t n that is in the support of σ n but not of r n ( σ ). For each n, s n , let Z + n,s n bethe closure of the set of σ ∈ Z n for which r n,s n ( σ ) (cid:62) r n,t n ( σ ) for all t n ∈ S n . If f ( σ ) = f ( σ )(the Nash map) and σ ∈ Z n , then r n ( σ ) equals r n ( σ ) as defined in subsection 3.2; therefore, σ ∈ Z + n,s n iff s n is a best reply to σ in G .We are now ready to define the function g ( σ ). In doing so, we repeatedly invoke Urysohn’slemma to construct functions that are zero on a closed set and positive outside it. First,let v n ( σ ) = max s n G n ( s n , σ − n ). Second, let β n : Σ → [0 ,
1] be a continuous function that iszero on Z n and positive everywhere else. Third, for each n, s n , let β n,s n : Σ → [0 ,
1] be acontinuous function that is one on Z + n,s n and strictly smaller than one elsewhere. Finally, let β : Σ → [0 ,
1] be a continuous function that is one on Σ \ int Σ ( V ), zero on U and strictly N SUSTAINABLE EQUILIBRIA 13 positive everywhere else. For each n, s n and σ , define: g n,s n ( σ ) = β ( σ ) β n,s n ( σ )[ v n ( σ ) − G n ( s n , σ − n ) + β n ( σ )] . If σ ∈ U , then g ( σ ) = 0 as β ( σ ) = 0; and σ is an equilibrium of G ⊕ g ( σ ) iff σ = σ ∗ .Suppose σ / ∈ U . Since σ ∗ is the only fixed point of f , there exists some n such that f n ( σ ) (cid:54) = σ n . For this n , there exists s n such that: σ ∈ Z + n,s n (take s n s.t. r n,s n ( σ ) (cid:62) r n,t n ( σ )for all t n ∈ S n ); and there is t n in the support of σ n but not in the support of r n ( σ ). Thisimplies β n,s n ( σ ) = 1, while β n,t n ( σ ) < σ / ∈ V , then β ( σ ) = 1 and so, G n ( s n , σ − n ) + g n,s n ( σ ) = v n ( σ ) + β n ( σ ) > v n ( σ ) + β n,t n β n ( σ ) (cid:62) G n ( t n , σ − n ) + g n,t n ( σ ) , showing that σ is not an equilibrium of G ⊕ g ( σ ).If σ ∈ V \ U , then as f coincides with f , s n is a best reply against σ while t n is not. Thus G n ( s n , σ − n ) = v n ( σ ) and G n ( t n , σ − n ) < v n ( σ ). Since β ( σ ) >
0, we obtain that G n ( s n , σ − n )+ g n,s n ( σ ) = v n ( σ )+ β ( σ ) β n ( σ ) > v n ( σ )+ β ( σ ) β n,t n ( σ ) β n ( σ ) > G ( t n , σ − n )+ g n,t n ( σ ) , and again σ is not an equilibrium of G ⊕ g ( σ ). Thus the function g has the desired properties.3.7. Example.
We continue with the example of subsection 3.5. We will construct thefunction g of the previous section. (Recall our convention of dropping the player subscriptfor terms like Z n , Z n .) What we are after is a function g : [0 , → R { L,R } such that x = 1is the only (symmetric) equilibrium of the game G ⊕ g ( x ). Let U ≡ [3 / , f has no fixed point outside U , so Z is empty and Z = [0 , / x ∈ [0 , / f ( x ) > x , which implies that r n ( x ) = 1 so that Z + L = [0 , / v n ( x ) ≡ (cid:40) − x if x ∈ [0 , / x if x ∈ (1 / , . Because Z is empty, we can set β ( · ) to be a constant function equal to δ >
0. The map β ( · ) will be defined as follows: β ( x ) ≡ x ∈ [0 , / − x + 9 if x ∈ (2 / , / x ∈ (3 / , . There is no need to introduce the function β ( · ) in this example. Putting these ingredientstogether, we can define g as follows: g L ( x ) = β ( x )[ v ( x ) − x + δ ] and g R ≡
0. We show that if the payoffs are now perturbed according to the bonus function g for each player, theonly remaining equilibrium is x = 1. Let x be an equilibrium of the perturbed game. If x ∈ [0 , / β ( x ) = 1, so if player 1 plays x , player 2 gets δ more than the best payoff v ( x )in the unperturbed game from playing L whereas by playing R he will not get more than v ( x ). Therefore, x = 1, which is a contradiction. On the other hand, if x ∈ [2 / , L is already the strict best-reply in the original game and since the g is nonnegative, it followsthat x = 1 is the unique equilibrium of the perturbed game.3.8. Isolating σ ∗ . Before we can use the perturbation g , we need to first embed G in agame ˜ G where σ ∗ is the only equilibrium in the face Σ of ˜ G and in fact the only equilibriumin which the strategy of even one of the players is in Σ n . (The perturbation g is then usedon the face opposite to Σ.) Hence, the embedding ˜ G will be such that if ˜ σ is an equilibriumof ˜ G and the support of ˜ σ n is in S n , then ˜ σ = σ ∗ . The game ˜ G that embeds G will berepresented in strategic form.Choose 0 < ε ∗ < ε such that σ ∗ n,s n > ε ∗ for each n and s n ∈ S ∗ n . Let U ∗ ≡ B ε ∗ and U ∗ n ≡ B ε ∗ n , for each n ∈ N ; the set U ∗ is a proper subset of U (and is a closed subsetcontained in the interior of V ). For each n , choose an arbitrary object 0 ∗ n (not in S ∗ n ). LetΘ n be the set of distributions over (cid:81) m (cid:54) = n ( S ∗ m ∪ { ∗ m } ). For each player n , his strategy set˜Σ n in ˜ G is Σ n × Θ n . A typical element ˜ σ n ∈ ˜Σ n has coordinates ( σ n , θ n ).We will now describe the payoff functions. For each θ n ∈ Θ n and m (cid:54) = n , we let θ n,m bethe marginal distribution of θ n over S ∗ m ∪ { ∗ m } and let Θ n,m be the set of all probabilitydistributions over S ∗ m ∪ { ∗ m } . For each m (cid:54) = n , let γ n,m : Θ n,m × Σ m → R be a bilinearfunction defined as follows. For all σ m , γ n,m (0 ∗ m , σ m ) = 0, while for s m ∈ S ∗ m , γ n,m ( s m , σ m ) =1 − ( σ ∗ m,s m − ε ∗ ) − σ m,s m . For each ˜ σ , player n ’s payoff in ˜ G is: ˜ G n (˜ σ ) = G n ( σ ) + γ n ( θ n , σ − n ) , where γ n ( θ n , σ − n ) ≡ (cid:88) m (cid:54) = n γ n,m ( θ n,m , σ m ) . Notice that the payoff function of each player n is affine over each strategy set ˜Σ m , m =1 , ..., N , so ˜ G is indeed a well-defined game in strategic form. For each n , let θ n = (0 ∗ m ) m (cid:54) = n .Then G is embeddable in ˜ G as the face Σ n × { θ n } is a copy of the original face Σ n , for n = 1 , ..., N . Technically, Θ n,m and γ n,m do not depend on n . N SUSTAINABLE EQUILIBRIA 15
Suppose ˜ σ is an equilibrium of ˜ G , then σ is an equilibrium of G , as the functions γ n ofeach player n do not depend on σ n . If σ = σ ∗ , then the unique θ n,m that is optimal for each n (cid:54) = m is 0 ∗ m and thus the equilibrium uses θ n for each n . On the other hand if σ (cid:54) = σ ∗ , by theproperty of subsection 3.3, there are at least two players m for whom σ m / ∈ U ∗ m . Therefore,for each n , there is at least one m (cid:54) = n such that 0 ∗ n,m is not optimal. Thus for each n , thesupport of θ n does not include θ n .To conclude, if ˜ σ = ( σ n , θ n ) n ∈N is an equilibrium of ˜ G , then σ is an equilibrium of G andeither: (1) σ = σ ∗ and θ n = θ n , for each n ∈ N ; or (2) σ (cid:54) = σ ∗ and the support of θ n doesnot contain θ n for any n ∈ N .3.9. Example.
In the context of the example of subsection 3.5: Θ = ∆( { ∗ } ∪ { L } ). Weidentify Θ with [0 ,
1] and an element θ ∈ [0 ,
1] denotes the probability of L . We define thefunction γ : let γ (0 ∗ , x ) = 0 and γ ( L, x ) = 1 − x . Then define γ ( θ, x ) = θ γ ( L, x ) + (1 − θ ) γ (0 ∗ , x ). The graph of γ ( L, x ) is depicted below in red in Figure 2. Define ε ∗ = 1 / Figure 2.
Graph of γ ( L, · ) (red)The strategic-form game is now defined by letting each player’s strategy set be the square[0 , × [0 , G ( x, θ , y, θ ) = G ( x, y ) + γ ( θ , y ); the payoffs for player 2 are defined symmetri-cally. If player 2 plays y < /
8, it follows that the (strict) best-reply of player 1 is to play θ = 1, in order to capture the positive bonus coming from γ ( L, x ); if y (cid:62) /
8, then thebonus γ ( L, y ) is nonpositive, and θ = 0 is a best-reply, which makes the payoffs of the per-turbed game equal to the original payoffs. This game has a copy of the original equilibrium x = 1: both players choose θ = 0 and x = 1. Any other equilibrium is such that θ = 1. The embedding ¯ G δ . The embedding that allows us to obtain σ ∗ as the unique equi-librium (and regular as well) will be built from ˜ G by adding a finite number of mixedstrategies as pure strategies and by defining their payoffs to eliminate all other equilibria. The set Σ \ int Σ ( U ∗ ) is compact, g ( · ) is continuous, and no σ ∈ Σ \ int Σ ( U ∗ ) is an equilibriumof G ⊕ g ( σ ), as shown in subsection 3.6. Hence, there exists η > σ ∈ Σ \ int Σ ( U ∗ )is an equilibrium of G ⊕ g for any g with (cid:107) g − g ( σ ) (cid:107) (cid:54) η . Also, since g is uniformly continuous,there exists 0 < ζ < / (cid:107) g ( σ ) − g ( σ (cid:48) ) (cid:107) (cid:54) η , if (cid:107) σ − σ (cid:48) (cid:107) (cid:54) ζ . Reduce ζ to ensurethat it is also smaller than the distance between U ∗ n and ∂ Σ n U n for each n .For each n , take a triangulation T n of Σ n × Θ n with the following properties. (See theAppendix for the details.) (1) The only vertices in Σ n × { θ n } of T n are pure strategies( s n , θ n ), s n ∈ S n ; (2) letting Θ n be the face of Θ n where θ n has zero probability, if T n ∈ T n is a simplex either with a face in Σ n × Θ n , or shares a face with such a simplex, then thediameter of T n is less than ζ ; (3) there exists a convex function (cid:37) n : Σ n × Θ n → R + suchthat: (a) (cid:37) n ( λx + (1 − λ ) y ) = λ(cid:37) n ( x ) + (1 − λ ) (cid:37) n ( y ) iff x and y belong to a simplex T n of T n ; (b) (cid:37) − n (0) = Σ n × { θ n } .Let ¯ S n be the set of vertices of the triangulation T n . Let ¯ S n ≡ ¯ S n +1 modulo N . A typicalelement of ¯ S n is a pair ( σ n , θ n ) in Σ n × Θ n that is a vertex of T n . We fix a pure strategy s n ∈ S n for each n . These pure strategies will be used below to define the perturbation ofpayoffs π n for each player n . We denote a typical element of ¯ S n by ( σ n,n +1 , θ n,n +1 ), which isa vertex in T n +1 . For i = 0 ,
1, let ¯Σ in be the set of mixtures over ¯ S in . The pure strategy setof player n in the game ¯ G δ in normal form is ¯ S n ≡ ¯ S n × ¯ S n . The set of mixed strategies isdenoted ¯Σ n . For each mixed strategy ¯ σ n , and i = 0 ,
1, we let ¯ σ in be the marginals over ¯ S in .Define ¯ S ≡ (cid:81) n ¯ S n and ¯Σ ≡ (cid:81) n ¯Σ n . Also, let ¯ S i ≡ (cid:81) n ¯ S in and ¯Σ i ≡ (cid:81) n ¯Σ in for i = 0 , δ >
0. We will now define the payoff function ¯ G δ . For each n , let T n be the collectionof simplices of T n that have nonempty intersection with Σ n × Θ n . Given a pure strategyprofile ¯ s ∈ ¯ S with ¯ s n = ( σ n , θ n , σ n,n +1 , θ n,n +1 ) for each n , the payoff ¯ G δn (¯ s ) has five distinctcomponents:¯ G δn (¯ s ) = G n ( σ ) + (cid:88) s n ∈ S n g n,s n (¯ s − n ) σ n,s n + γ n ( θ n , σ − n ) + π n (¯ s n , ¯ s n +1 ) − δ(cid:37) n (¯ s n ) . von Schemde and von Stengel have an explicit bound on the number of strategies they add, which isthree times the number of pure strategies in G . In our construction, we have no way of obtaining such abound: the construction depends on the fixed-point map f obtained in subsection 3.4, which in turn relieson an existence result, the Hopf Extension Theorem (cf. Corollary 8.1.18, [20]). N SUSTAINABLE EQUILIBRIA 17
The first and the third terms have been defined before. The function (cid:37) n in the last term isthe convex function defined above. We will specify the other two terms. g n,s n (¯ s − n ) = ξ n (¯ s − n ) g n,s n ( σ , . . . , σ n − , σ n − ,n , σ n +1 , . . . , σ N ) , where ξ n (¯ s − n ) is one if for each m (cid:54) = n , ¯ s m is a vertex of some simplex in T m ; otherwiseit is zero. The function π n is 0 if either: (1) ¯ s n +1 is a vertex of some simplex in T n +1 and¯ s n = ¯ s n +1 ; or (2) ¯ s n +1 is not a vertex of such a simplex, but ¯ s n = ( s n +1 , θ n +1 ) ( s n +1 is afixed pure strategy chosen above, while θ n +1 is the collection (0 ∗ n,m )); elsewhere it is −
1. Thedefinition of ¯ G δ clearly implies that it embeds G .We want to make a couple of remarks about the payoffs. First, the function π n incentivizesplayer n to mimic player n +1 whenever the latter is choosing a strategy close to Σ n +1 × Θ n +1 :if n + 1 randomizes over the vertices of a simplex T n +1 ∈ T n +1 , then player n ’s best repliesmust be among the vertices of the simplex. This will be a crucial property, since the choicesin Σ n − ,n play a role in the evaluation of the bonus function g n . The idea is that wheneverthe bonus function g n is active, meaning that all players m (cid:54) = n randomize over the verticesof a simplex T m in T m , then each pure best-reply for player n must choose a vertex of T n +1 .On the other hand, if player n + 1 is randomizing over S n +1 × { θ n +1 } then it follows that theunique best-reply for player n is to choose the previously fixed strategies ¯ s n = ( s n +1 , θ n +1 ).Our second remark concerns the nature of the payoffs for mixed strategies. For n andeach i = 0 ,
1, there is a linear map p in : ¯Σ n : → Σ n + i × Θ n + i that sends each pure ¯ s n to thecorresponding mixed strategy in Σ n + i × Θ n + i . For each n , the first and the third terms ofthe payoffs depend on ¯ σ only through their images under p = (cid:81) n ∈N p n ; the fourth termdepends on all the information in ¯ σ n and ¯ σ n +1 . The second term depends on ¯ σ n only through p n , but requires the entire information in ¯ σ − n , while the last term requires the informationin ¯ σ n .3.11. Wrapping up the proof.
Let ¯ σ ∗ be the profile where for each player n , the marginalon ¯Σ n is ( σ ∗ , θ n ) and the marginal on ¯Σ n is ( s n +1 , θ n +1 ). For δ (cid:62)
0, ¯ σ ∗ is an equilibrium of¯ G δ , and ( G, σ ∗ ) ∼ ( ¯ G δ , ¯ σ ∗ ). We will now show that for δ sufficiently small, this is the onlyequilibrium of ¯ G δ , which completes the proof.Say that a strategy ¯ σ n is admissible for player n if the support of its marginal ¯ σ n ∈ ¯Σ n is the set of vertices of a simplex T n in T n . Observe that for any δ >
0, every best replyfor player n is admissible. Indeed, the first three components of n ’s payoff function dependon n ’s strategy only through its projection to Σ n × Θ n and the fourth is independent ofthese choices. Therefore, any two strategies for n that project under p n to the same pointin Σ n × Θ n yield the same payoffs for these four terms, leaving the fifth to decide which one is better. But the map (cid:37) n is convex, and it is linear precisely on the simplices of T n , whichthen forces each best reply to be a mixture over the vertices of a simplex of T n .We claim that if δ = 0 and the only admissible equilibrium of ¯ G δ is ¯ σ ∗ , then for sufficientlysmall δ >
0, ¯ σ ∗ is the only equilibrium of ¯ G δ . To prove this claim, suppose that we havea sequence ¯ σ δ of equilibria of ¯ G δ converging to some equilibrium ¯ σ of ¯ G , then as we sawabove ¯ σ δ must be admissible, and hence also its limit ¯ σ . As we have assumed that ¯ σ ∗ is theunique admissible equilibrium of ¯ G , ¯ σ = ¯ σ ∗ . Observe now that for each n , every pure bestreply in ¯ G to ¯ σ ∗ is of the form ( s n , θ n , s n +1 , θ n +1 ) where s n ∈ S n is a best reply to σ ∗ ; andthis property holds for best replies to ¯ σ δ , for small δ . Thus for each such δ , and for each n ,¯ σ δn is of the form ( σ δn , θ n , s n +1 , θ n +1 ), where σ δn is a best reply to σ ∗ in G . In other words, σ δ is an equilibrium of G . As σ δ converges to σ ∗ and as σ ∗ is an isolated equilibrium of G , σ δ = σ ∗ for all small δ . Thus the claim follows and it is sufficient to show that ¯ σ ∗ is the onlyadmissible equilibrium for δ = 0.To prove this last point, fix now an admissible equilibrium ¯ σ with marginals (¯ σ , ¯ σ ) ∈ ¯Σ × ¯Σ of the game ¯ G . For each n , let ( σ n , θ n ) and ( σ n,n +1 , θ n,n +1 ) be the image of ¯ σ n under p n and p n , resp. Also, let T n be the simplex of T n generated by the support of ¯ σ n for each n .Suppose first for each n , T n belongs to T n . For each n , θ n assigns probability less than ζ ,which is smaller than one, to θ n . Also, for at least two n , σ n / ∈ int Σ n ( U ∗ n ): indeed, otherwisethere is one player n all of whose opponents m are choosing in int Σ n ( U ∗ m ), making θ n theunique optimal choice, which is impossible. Thus, σ n / ∈ int Σ n ( U ∗ n ) for at least two n , i.e., σ / ∈ int Σ ( U ∗ ) and, hence, σ is not an equilibrium of G ⊕ g ( σ ). For each n , and each ¯ s − n inthe support of ¯ σ − n , ξ n (¯ s − n ) = 1 as ¯ s m is a vertex of the simplex T m , which is in T m , for each m ; because of the function π n , the optimality of ¯ σ n − implies that each ¯ s n − in the supportof ¯ σ n − is a vertex of T n . Therefore, for each ¯ s − n in the support of ¯ σ − n , (cid:107) g (¯ s − n ) − g ( σ ) (cid:107) (cid:54) η and then (cid:107) g (¯ σ − n ) − g ( σ ) (cid:107) (cid:54) η . As σ is not an equilibrium of G ⊕ g ( σ ), by the choice of η in subsection 3.10, it is not an equilibrium of G ⊕ g (¯ σ ), which contradicts the fact that ¯ σ isan equilibrium of ¯ G .Now suppose that for exactly one n , say n = 1, T n does not belong to T n . Then, θ has positive probability under θ . Therefore, because of the definition of γ n , σ n ∈ U ∗ n for n >
1, i.e., σ ∈ U ∗ . For n >
1, the fact that σ n ∈ U ∗ n and T n belongs to T n imply thatfor each ¯ s n = ( σ n , θ n ) in the support of ¯ σ n , σ n belongs to U n (as the diameter of T n is lessthan ζ , which is smaller than the distance between U ∗ and ∂ Σ U ). Thus, g ( σ ) = 0. We willnow show that g n ( σ ) = 0 for n >
1. The payoff function π n for each n (cid:54) = N forces eachstrategy ¯ s n = ( σ n,n +1 , θ n,n +1 ) in the support of ¯ σ n to be a vertex of T n +1 and hence σ n,n +1 is in U n +1 . Therefore, for n > σ n − ,n ∈ U n . Recall that g n ( · ) was constructed to be 0 N SUSTAINABLE EQUILIBRIA 19 on U . Consequently, for each n > g n (¯ s − n ) = 0 for each ¯ s − n in the support of ¯ σ − n , i.e., g n (¯ σ − n ) = 0.The fact that g ( σ ) = 0, implies that σ = σ ∗ . Optimality of θ n for n > θ n . This is a contradiction: since T n ∈ T n , its diameter issmaller than ζ (and hence one), putting it at positive distance from Σ n × { θ n } .Finally, suppose that for at least two players n , T n does not belong to T n . Then, againbecause of γ n , for each n , σ n ∈ U ∗ n . We claim that for each n , g n (¯ s − n ) = 0 for each ¯ s − n in thesupport of ¯ σ − n . Indeed, if for some m (cid:54) = n , T m has no vertex that belongs to a simplex in T m , then ξ n is zero by construction at each ¯ s − n in the support and we are done. Otherwise,if for each m (cid:54) = n , T m has a vertex in T m then, letting ¯ s m = ( σ m , θ m ) be an arbitrary vertexof T m , it follows from the fact that the diameter of each T m is less than ζ that σ m ∈ U m ,which implies σ ∈ U . Therefore, g n ( · ) is again zero on the support of ¯ σ − n .It follows from the previous paragraph that σ is an equilibrium of G , i.e., σ = σ ∗ , making θ n = θ n . Finally, optimality of ¯ σ n implies that it is ( s n,n +1 , θ n,n +1 ), as it yields zero withothers yielding −
1. Thus, ¯ σ = ¯ σ ∗ , which concludes the proof.3.12. Example.
We can triangulate the strategy set Σ × Θ ≡ [0 , × [0 ,
1] of each player asin Figure 3. The horizontal axis represents Θ = [0 ,
1] and the vertical axis Σ = [0 , Figure 3.
Triangulation of [0 , Payoffs are defined in the exact same way as in subsection 3.10 with the following mod-ifications: the function π ≡
0, since there is no need to duplicate the strategy set of each player (by our construction, g n depends only on σ − n ); the function ξ is equal to 1 at allvertices of the triangulation that lie on the face θ = 1.Paralleling the proof of subsection 3.10, we show that the only admissible equilibrium ofthe perturbed game ¯ G δ with δ = 0 is the (symmetric) equilibrium ( θ, x ), where x = 1 and θ = 0. To see this, let ( x, θ ) be an equilibrium. Suppose first x < /
8. Then θ = 1 is a strictbest-reply. The support of the equilibrium ( θ, x ) is then a subset of one of the 1-dimensionalsimplices that subdivide { } × [0 , g and the fact that it is linear ineach of these simplices, it follows that x = 1, which is a contradiction. Therefore, x (cid:62) / L is a strict best-reply if x (cid:62) /
8. Since γ ( L, x ) (cid:54) g L (cid:62) g R ≡ L is a strict-best reply in ¯ G δ and thus x = 1.Finally, given x = 1, θ = 0 is the optimal choice in Θ.4. Discussion
There are two aspects of genericity—one concerning the definition of sustainability andthe other the statement of the theorem—that we would like to highlight. First, in thedefinition of equivalence between game-equilibrium pairs, we require that the games obtainedby restricting the strategies to the best replies to the equilibria, rather than to the supportof the equilibria, be the same. Of course, if both games are regular, then we could haveused either requirement. Also, if we are dealing with two-player games, then, too, wecould use just the support due to the fact that a unique equilibrium of a bimatrix game isquasi-strict (Norde [15]). However, in N -person games, uniqueness does not guarantee quasi-strictness (Brandt and Fischer [1]). We now construct an example that exploits this featureof N -person games to show that our theorem would fail if we merely restrict strategies tothe support of equilibria. Consider the following three-player game G , where player 3 is adummy player, whose unique action is W . l rt (6 , ,
1) (0 , , b (0 , ,
1) (6 , , t, l ) and ( b, r ), which have index +1,and a mixed equilibrium σ ∗ where players 1 and 2 mix uniformly, which has index − G where each player has three choices. Player 1’s strategy set is { ( T, t ) , ( T, b ) , B } ; 2’s strategy set is { ( L, l ) , ( L, r ) , R } ; 3’s strategy set is { W, E w , E ε } .The payoffs are: Hofbauer states his definition in the body of his paper using the supports, but he adds a footnote thatwe need to include best replies if the game is not regular.
N SUSTAINABLE EQUILIBRIA 21 W : L RT l rt (6 , ,
1) (0 , , b (0 , ,
1) (6 , ,
1) (3 , , B (3 , ,
1) (0 , , E w : L RT l rt ( − , ,
4) (1 , , b (1 , ,
0) (1 , ,
0) (1 , , B (3 , ,
0) (0 , , E e : L RT l rt (1 , ,
0) (1 , , b (1 , ,
0) ( − , ,
4) (3 , , B (3 , ,
0) (0 , , B , R , E w , E e , we get the game G and so ¯ G embeds G . The unique equilibriumof this game is the mixed equilibrium σ ∗ of G . The pair ( ¯ G, σ ∗ ) is not equivalent to ( G, σ ∗ )under the definition of this paper since, for instance strategy B is a best reply to σ ∗ but itis not a strategy in G . However, if we merely ask for equivalence using supports, then thetwo pairs are equivalent and we would make a regular − The second point about genericity is that we focus on regular equilibria in our theorem inorder to align our paper with the Hofbauer-Myerson conjecture. But what really matters, asa careful reading of the proof shows, is one particular implication of regularity, namely thatregular equilibria are isolated. Thus, we could obtain a slightly stronger result: an isolatedequilibrium is sustainable iff its index is +1.When the game is not regular, in particular if it has nontrivial components of equilibria,then there is an obvious extension of the definition of sustainability to components, whichreplaces equilibria with components. However, such a refinement may fail to exist: forexample, we could have two components of equilibria in a game , one with index +2 andthe other with index −
1; it is impossible to obtain either as the unique solution of a largergame.Even when a given game has a unique component of equilibria, which has index +1,indeterminacy of equilibria may persist in any equivalent game with an expanded set ofstrategies, as we will now show. Consider the following game G : L RT (1 ,
1) (0 , This example shows that in a three-player game G , a regular equilibrium σ with index − G , but then it is not quasi-strict in ¯ G and thus ( G, σ ) and ( ¯
G, σ ) are not equivalent.This cannot happen in two-player games because if an equilibrium σ of G can be made unique in ¯ G , it mustbe quasi-strict in ¯ G from Norde [15] and so ( G, σ ) and ( ¯
G, σ ) must be equivalent. Ritzberger [18], p. 325.
This game has a unique component of equilibria { ( T, yL + (1 − y ) R ), y ∈ [0 , } and twopure equilibria ( T, L ) and (
T, R ). We claim that no equilibrium of G can be made unique byadding strategies. To see this, let (cid:98) G be obtained from G by adding rows r ∈ R and columns l ∈ L with associated payoffs g ( r, l ) and suppose that an equilibrium σ of G can be madeunique in (cid:98) G . Observe that σ cannot be pure because otherwise it will be non quasi-strictin G , and so non-quasi strict in (cid:98) G , a contradiction with Norde’s result. Suppose now that σ = ( T, yL + (1 − y ) R ), y ∈ ]0 , τ = ( T, L ) be a pure equilibrium. Then τ is the uniqueequilibrium of the following perturbation G ε . L RT (1 , ε ) (0 , G ε the extension of G which are the rows r ∈ R and the columns l ∈ L with associated payoffs g ( r, l ). This defines a perturbed game (cid:98) G ε of (cid:98) G . Since (cid:98) G has aunique equilibrium σ , by upper-hemi-continuity of the equilibrium correspondence, (cid:98) G ε hasan equilibrium σ ε that converges to σ as ε goes to zero. Because τ is the unique equilibriumof G ε , necessarily there is r ∈ R (or l ∈ L ) in the support of σ ε (otherwise, σ ε wouldbe an equilibrium of G ε and so σ ε = τ , a contradiction). By continuity, r (or l ) is stilla best response to σ , showing that σ is not a quasi-strict equilibrium in (cid:98) G , establishing acontradiction (again with Norde’s result).It is noteworthy that, if we can add also players then any equilibrium of the last gamecan be made unique. For example, the pure and non quasi-strict equilibrium (
T, L ) of thegame above is the projection of the unique equilibrium (
T, L, W ) of the following 3-playergame. W : L RT (1 , ,
1) (1 , , B (1 , ,
1) (0 , , E : L RT (0 , ,
1) (1 , , B (1 , ,
0) (0 , , References [1] Brandt, F., and F. Fischer (2008): “On the Hardness and Existence of Quasi-Strict Equilibria,” SAGT2008, LNCS 4997, 291-302. We thank Joseph Hofbauer for raising this issue. This is a mild strengthening of the example in Brandt and Fischer [1], where the unique equilibrium—which also fails to be quasi-strict—was in mixed strategies.
N SUSTAINABLE EQUILIBRIA 23 [2] Demichelis, S., and F. Germano (2000): “On the Indices of Zeros of Vector Fields,”
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Eco-nomic Theory , 42, 119–156.[8] Hofbauer, J. (2000): “Some Thoughts on Sustainable/Learnable Equilibria,” Mimeo.[9] Kohlberg, E., and J.-F. Mertens (1986): “On the Strategic Stability of Equilibria,”
Econometrica , 54,1003–37.[10] Loera, J.A., J. Rambau, and F. Santos (2010):
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Advanced Fixed Point Theory for Economics . Singapore: Springer-Verlag.[12] McLennan, A. (2016):
The Index +1 principle . Mimeo.[13] Myerson, R.B. (1996): “Sustainable Equilibria in Culturally Familiar Games,” in
Understanding Strate-gic Interaction: Essays in Honor of Reinhard Selten , edited by W. Albers et al, Springer, 111–21.[14] Nash, J.F. (1951): “Noncooperative Games,”
Annals of Mathematics , 54, 286–95.[15] Norde, H. (1999): “Bimatrix Games Have a Quasi-Strict Equilibria,”
Mathematical Programming , 85,35–49.[16] Pahl, L. (2019): Index Theory for Strategic-form Games with an Application to Extensive-form Games,Mimeo.[17] Ritzberger, K. (1994): “The Theory of Normal Form Games from the Differentiable Viewpoint,”
Inter-national Journal of Game Theory , 23: 207-236.[18] Ritzberger, K. (2002):
Foundations of Non-Cooperative Game Theory . Oxford University Press.[19] Shapley L.S. (1974): “A note on the Lemke-Howson algorithm. Mathematical Programming Study 1:Pivoting and Extensions,” 175-189[20] Spanier, E.H. (1966):
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Stability and Perfection of Nash Equilibria . Springer-Verlag[22] von Schemde, A., and B. von Stengel (2008): “Strategic Characterization of the Index of an Equilib-rium,” SAGT 2008, LNCS 4997, 242–54.[23] von Schemde, A. (2005):
Index and Stability in Bimatrix Games , Lecture Notes in Economics andMathematical Systems, Springer.
Appendix
Construction of the triangulation.
Here we construct a triangulation T n of Σ n × Θ n foreach n with the properties stated in subsection 3.10. We start with some definitions. A simplex T in R d is the convex hull of affinely independent points x , x , . . . , x k ( k (cid:54) d );a face of T is the convex hull of a subset of the points x i . A triangulation T of a polytope C ⊂ R d is a finite collection of simplices T in R d such that: (1) if T ∈ T , so is every faceof T ; (2) the intersection of two simplices in T is a face of both (possibly empty); (3) theunion of the simplices in T equals C .Throughout this Appendix, we use the (cid:96) norm, unless we specify differently. Given afinite collection { x , x , . . . , x k } of points, (where k is now an arbitrary positive integer) let C be its convex hull. Suppose that the x i ’s are in general position for spheres in R d —i.e.,no d + 2 points lie in any ( d − R d .We can construct a triangulation of the convex hull C , called the Delaunay triangulation ,as follows (cf. Loera et al [10] for details). Let D be the convex hull of the set of points( x i , (cid:107) x i (cid:107) ) ∈ R d +1 , i = 0 , , . . . k . Let D be the lower envelope of D . The natural projection( x, y ) (cid:55)→ x from D to R d is C and D is the graph of a piece-wise linear and convex function (cid:37) : C → R with the property that the subsets on which (cid:37) is linear are simplices, whoseprojections then yield the simplices of a triangulation of C .There is a dual representation of the Delaunay triangulation, known as the Voronoi Di-agram , which works as follows. For each i = 0 , , . . . k , let P i be the polyhedron in R d consisting of points y in R d such that (cid:107) y − x i (cid:107) (cid:54) (cid:107) y − x j (cid:107) for all j (cid:54) = i . We then havea polyhedral complex (which is like a simplicial complex but with polyhedra rather thansimplices) where the maximal polyhedra are the P i . There is an edge between two vertices x i and x j in the Delaunay triangulation iff the polyhedra P i and P j have a nonempty inter-section. Also, the intersection of d + 1 of these polyhedra when nonempty is a single point(because of genericity), that is then the center p of a ball that contains d + 1 points of thecollection on its boundary and no other point in the ball itself—these d + 1 points span a d -dimensional simplex in the Delaunay triangulation.For our purposes, we need a triangulation with the diameter of certain simplices to besmaller than ζ , as specified in Subsection 3.10. To obtain that, we begin with an auxiliaryconstruction. Let C be a full-dimensional polytope in R d whose set of vertices, X , are ingeneral position for spheres. Let B be a proper face of C with X the set of vertices of B and X = X \ X . Let H be a hyperplane that strictly separates B from the vertices of C that are not in B . Let B be the intersection of C with the halfspace generated by H thatcontains B in its interior, i.e., B is of the form C ∩ H − where H − = { x ∈ R d | a · x (cid:54) b } and a · x < b for all x ∈ B . Let Y be the set of vertices of B that are not in X .We claim now that if the normal a and constant b of the hyperplane H are in generalposition, then C admits a Delaunay triangulation with the vertex set X ∪ Y . To see this N SUSTAINABLE EQUILIBRIA 25 claim, let E be the set of edges e of C that have a vertex x ( e ) in X and one x ( e ) in X .Each vertex y ∈ Y is of the form α ( e ) x ( e ) + (1 − α ( e )) x ( e ) for some e ∈ E . The genericityof H is equivalent to making a generic choice of α ( e ) for d of these edges e . Take now a set V of d + 2 of the vertices in X ∪ Y . If all of the vertices are in X then the intersection oftheir Voronoi polyhedra is empty, as the vertices of C are in general position for spheres.If now a subset y , . . . , y l of l of these vertices lie on edges, then for each y i , at most oneof the two endpoints of the edge can belong to the collection V if their Voronoi polyhedraare to intersect. For each of the y i ’s there is a choice of α (which is one or zero) for whichthe Voronoi polyhedra do not intersect; therefore, for generic α ’s the intersection is empty.Thus, if H is in generic position, then the vertices of X and Y can be used to carry out aDelaunay triangulation of C .Fix now C and B as above. Let δ > (cid:107) x − y (cid:107) (cid:62) δ/
2, for all x ∈ B andvertices y ∈ C \ B . Let X δ be a finite collection of points in C such that: (1) X δ containsthe vertices of C but no other point in C \ B ; (2) for x ∈ B , there is a point x δ ∈ B ∩ X δ such that (cid:107) x − x δ (cid:107) < δ/ x δ belongs to the face of B that contains x in its interior; (3)every point in int C ( B ) ∩ X δ is at least δ/ ∂ C B ; (4) the points in X δ are in generalposition for spheres, call T δ the associated Delaunay triangulation of C . (The argument thatgenerically there exists a Delaunay triangulation as required in (4) is similar to the argumentin the previous paragraph.)The triangulation T δ above achieves two properties: (i) every simplex with vertices in B has diameter at most δ ; (ii) every simplex of T δ that has a vertex outside B does not intersectint C ( B ). To prove these properties, define r : R d → B by letting r ( x ) be the point in B thatis closest to x . If r ( x ) (cid:54) = x , r ( x ) belongs to a proper face of B , and then we can write r ( x )as x − p where p is a normal for a supporting hyperplane at r ( x ) with p · r ( x ) (cid:62) p · y for all y ∈ B . If in addition r ( x ) ∈ int C ( B ), then r ( x ) is at the boundary of C and so p · r ( x ) (cid:62) p · y for all y ∈ C as well. Suppose r ( x ) (cid:54) = x and let r ( x ) = x − p . Let y be a point such that p · y (cid:54) p · r ( x ). Let z be the nearest-point projection of y onto the line from x through r ( x ).Then (cid:107) x − y (cid:107) = ( (cid:107) x − r ( x ) (cid:107) + (cid:107) r ( x ) − z (cid:107) ) + (cid:107) z − y (cid:107) (cid:62) (cid:107) r ( x ) − x (cid:107) + (cid:107) r ( x ) − y (cid:107) , with the inequality being an equality iff z = r ( x ), i.e., p · y = p · r ( x ).We are now ready to prove that T δ has the requisite properties. Let x δ be a point in X δ ∩ B and let x be a point in R d that belongs to the Voronoi polyhedron P ( x δ ) of x δ . We claimthat (cid:107) r ( x ) − x δ (cid:107) < δ/
2. If r ( x ) = x , this follows directly from Property (2) of X δ . Supposethat r ( x ) (cid:54) = x . Then r ( x ) belongs to the interior of a proper face B (cid:48) of B and as we saw in the last paragraph, r ( x ) can be written as x − p . By definition of r ( x ), p · x δ (cid:54) p · r ( x ) andthus: (cid:107) x − x δ (cid:107) (cid:62) (cid:107) r ( x ) − x (cid:107) + (cid:107) r ( x ) − x δ (cid:107) . By Property (2), there exists y δ in B (cid:48) ∩ X δ such that (cid:107) r ( x ) − y δ (cid:107) < δ/
2. Obviously p · y δ = p · r ( x ) and since x ∈ P ( x δ ), it followsthat (cid:107) x − x δ (cid:107) (cid:54) (cid:107) x − y δ (cid:107) < (cid:107) r ( x ) − x (cid:107) + δ /
4; therefore, (cid:107) r ( x ) − x δ (cid:107) < δ/
2, as claimed.Observe that we proved that (cid:107) x − x δ (cid:107) < (cid:107) r ( x ) − x (cid:107) + δ /
4, a fact we will use below.From the above paragraph, for each x δ ∈ X δ ∩ B and each x ∈ P ( x δ ), the distance between r ( x ) and x δ is less than δ/
2; We claim finally that the diameter of each simplex with verticesin B is less than δ . Indeed, letting x δ and y δ be two vertices of a simplex in B , then theirVoronoi cells intersect, so we can take x in the intersection. Since r ( x ) is of distance δ/ x δ and δ/ y δ , x δ and y δ are distant less than δ . This concludes the proof that T δ satisfies (i).We now prove that T δ satisfies (ii): for this, it is sufficient to show that the intersection of P ( x δ ) and P ( y δ ) is empty for all x δ ∈ int C ( B ) ∩ X δ and y δ in X δ \ B . Take such a pair x δ , y δ .Fix x ∈ P ( x δ ). If r ( x ) = x , then (cid:107) x − x δ (cid:107) < δ/
2, while by the definition of δ , (cid:107) x − y δ (cid:107) (cid:62) δ/ x / ∈ P ( y δ ). Suppose r ( x ) (cid:54) = x . Since x δ ∈ int C ( B ), by Property (3) of the set X δ , r ( x ) cannot belong to ∂ C B , since in this case the distance between x δ and r ( x ) is greaterthan δ/
2. Therefore, r ( x ) belongs to a face of C . Writing r ( x ) as x − p , we then have that p is a normal to a hyperplane containing one of the faces of C and thus p · y δ (cid:54) p · r ( x ). Hence, (cid:107) x − y δ (cid:107) (cid:62) (cid:107) r ( x ) − x (cid:107) + (cid:107) r ( x ) − y δ (cid:107) (cid:62) (cid:107) r ( x ) − x (cid:107) + δ / δ , whileas we saw in the previous paragraph, (cid:107) x − x δ (cid:107) < (cid:107) r ( x ) − x (cid:107) + δ /
4; thus again x / ∈ P ( y δ )and we are done.For our problem of triangulating Σ n × Θ n , we recall the properties that the triangulationshould satisfy: (1) The only vertices in Σ n × { θ n } of T n are pure strategies ( s n , θ n ), s n ∈ S n ;(2) letting Θ n be the face of Θ n where θ n has zero probability, if T n ∈ T n is a simplexeither with a face in Σ n × Θ n , or shares a face with such a simplex, then the diameter of T n is less than ζ ; (3) there exists a convex function (cid:37) n : Σ n × Θ n → R + such that: (a) (cid:37) n ( λx + (1 − λ ) y ) = λ(cid:37) n ( x ) + (1 − λ ) (cid:37) n ( y ) iff x and y belong to a simplex T n of T n ; (b) (cid:37) − n (0) = Σ n × { θ n } .Let ˆ S n ≡ S n × { θ n } ; Let T n be the set of vertices of Θ n and ˆ S n ≡ { s n } × ( T n \ { θ n } ).Let ˆ S n ≡ ˆ S n ∪ ˆ S n \ { ( s n , θ n ) } where θ n is a vertex of Θ n that is different from θ n . Note that d ≡ dim(Σ n × Θ n ) = | S n | + | T n | − | ˆ S n | . For each ˆ s n ∈ ˆ S n , let x (ˆ s n ) be the unit vectorin R ˆ S n for the coordinate ˆ s n ; for each ˆ s n ∈ ˆ S n , let x (ˆ s n ) be a point in R ˆ S n to be determinedlater, and X ≡ { x (ˆ s n ) } ˆ s n ∈ ˆ S n . N SUSTAINABLE EQUILIBRIA 27
Define an affine function F Xn : Σ n × Θ n → R ˆ S n as follows: for each ˆ s n ∈ ˆ S n , F Xn (ˆ s n ) = x (ˆ s n );for a vertex ( s n , θ n ) of Σ n × Θ n that is not in ˆ S n ∪ ˆ S n , define F Xn ( s n , θ n ) = F Xn ( s n , θ n ) + F Xn ( s n , θ n ) − F Xn ( s n , θ n ). The map F X extends to the whole of Σ n × Θ n by linear interpolation.If the collection X ∪ { x (ˆ s n ) } ˆ s n ∈ ˆ S n is affinely independent (which holds for any choice of( x (ˆ s n )) ˆ s n ∈ ˆ S n in an open and dense set of R d ), then F Xn is an affine homeomorphism with itsimage C ( X ) ≡ F Xn (Σ n × Θ n ) and the dimension of C ( X ) is | ˆ S n | .We claim now that the set X can be chosen generically for a Delaunay triangulation of C ( X ). In fact, the condition we need is the following: for each collection of distinct vertices v i i = 1 , . . . , K with K (cid:62)
2, of S n × T n that do not belong to ˆ S n , (cid:80) Ki =1 (cid:107) F Xn ( v i ) (cid:107) (cid:54) = (cid:80) Ki = n +1 (cid:107) F Xn ( v i ) (cid:107) . To prove the claim, we have to show that the intersection of d + 2 of theVoronoi polyhedra is empty. Let v , . . . v d +2 be d + 2 vertices of Σ n × Θ n . The collection ofthe v i ’s is not affinely independent. In fact, there are an even number of vertices v , . . . v K ofthe collection such that (cid:80) Ki =1 v i = (cid:80) Ki = K +1 v i ; and of these there are as many vertices in ˆ S n among the first K as there are in the second, and there is at least one vertex on each side ofthe equality that does not belong to ˆ S n . Therefore, (cid:80) Ki =1 (cid:107) F Xn ( v i ) (cid:107) = (cid:80) Ki = K +1 (cid:107) F Xn ( v i ) (cid:107) ,which contradicts our assumption given that vertices in ˆ S n have norm one. Thus C ( X ) hasa Delaunay triangulation for generic X .Take now a generic set X with the property that the norm of x ˆ s n is strictly greater thanone for ˆ s n ∈ ˆ S n . Let B = F Xn (Σ n × Θ n ) and consider a generic hyperplane H = { x ∈ R d | a · x = b } that separates strictly B from the other vertices of C ( X ) so that we cantriangulate C ( X ) using the vertices of C ( X ) plus those of the polytope B = H − ∩ C ( X ).Since F Xn is an affine homeomorphism, (cid:107) x − y (cid:107) ∞ (cid:54) M (cid:107) F Xn ( x ) − F Xn ( y ) (cid:107) for some M > x, y ∈ Σ n × Θ n . Let δ > M ζ . Using the construction describedabove, we now have a triangulation T δ of C ( X ) where each point in int C ( B ) belongs to asimplex with diameter less than δ/M , giving us properties (1) and (2) of Subsection 3.10.As for property (3), if (cid:37) n is the convex function associated to the Delaunay triangulation T δ ,the composition (cid:37) n ◦ F Xn is convex and linear precisely on each cell of the triangulation ofΣ n × Θ n induced by the inverse mapping ( F Xn ) − : C ( X ) → Σ n × Θ n . Our convex function takes value one on Σ n × { θ n } and is strictly above one elsewhere. Subtracting now 1 from (cid:37) n ◦ F Xn we have a convex function satisfying property (3). Department of Economics, University of Rochester, NY 14627, USA.
E-mail address : [email protected] CNRS, Lamsade, University of Paris Dauphine-PSL, 75016 Paris, France, and Departmentof Computer Science, University of Liverpool, Liverpool L69 3BX, UK
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