On the Existence of Limit Admissible Equilibria in Discontinuous Games∗ Guilherme Carmona† University of Surrey September 8, 2017

Abstract We consider the existence of limit admissible equilibria, i.e. Nash equilibria in which each player assigns zero probability to the interior of the set of his weakly dominated strategies, in (possibly) discontinuous games. We show that standard sufficient conditions for the existence of Nash equilibrium, such as better-reply security, fail to imply the existence of limit admissible equilibria. We then modify better-reply security to obtain a new condition, admissible security, and show that admissible security is sufficient for the existence of limit admissible equilibria. This result implies the existence of limit admissible equilibria in a Bertrand competition setting with convex costs analogous to that of Maskin (1986).

∗

I wish to thank seminar participants at the SAET conference (Faro 2017) for very helpful

comments. Any remaining errors are, of course, my own. † Address: University of Surrey, School of Economics, Guildford, GU2 7XH, UK; email: [email protected].

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1

Introduction

The motivation of the refinement literature (see van Damme (1991) for a survey) is that the requirements of Nash equilibrium need to be strengthen to obtain a more appealing solution concept in normal-form games. Along this lines, and following Kohlberg and Mertens (1986), it would be desirable to have a solution concept whose existence is generally guaranteed and in undominated strategies. For games with finite action spaces, Selten’s (1975) perfect equilibrium is a way of obtaining the above properties. However, in games with infinite action spaces, several difficulties arise. First, even when players’ payoff functions are continuous, it is possible that all Nash (and, therefore, all perfect) equilibria are in weakly dominated strategies (see Simon and Stinchcombe (1995, Example 2.1)). For this reason, one can only hope to obtain general existence results for limit admissible equilibria, i.e. of Nash equilibrium in which each player assigns zero probability to the interior of the set of his weakly dominated strategies. Furthermore, when some player’s payoff function is discontinuous, existence of perfect equilibria requires conditions considerably stronger than the standard ones guaranteeing existence of Nash equilibrium (such as Reny’s (1999) better-reply security). Moreover, it is not always the case that perfect equilibria are limit admissible (see Carbonell-Nicolau (2011b)). Other solution concepts have been proposed such as Andersson, Argenton, and Weibull’s (2014) robust equilibrium to strategic uncertainty and Bich’s (2016) prudent equilibrium, but again their existence and limit admissibility have only been established under assumptions stronger than the standard ones. In this paper we directly address the existence of limit admissible equilibria. We first show that a condition different from better-reply security is needed. Indeed, we present two examples of better-reply secure games that fail to have limit admissible equilibria: One of these examples fails to have limit admissible strategies and the other, while having limit admissible strategies, fails to have one that is also a Nash equilibrium.

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We then introduce the notion of admissible security and use it to establish the existence of limit admissible equilibrium in mixed strategies.1 In contrast with the complicated task of establishing the existence of perfect equilibrium (and, thus, of establishing indirectly and under some additional assumptions the existence of limit admissible equilibria), we show that the task of establishing directly that a limit admissible equilibrium exists is a relatively simple one. Indeed, besides having to assume that limit admissible strategies exist, all it takes is that better-reply security be “appropriately relativized” to the set of limit admissible strategies. In this sense, admissible security is the natural modification of better-reply security to address the existence of limit admissible equilibria. The paper is organized as follows. Section 2 presents our notation and general definitions. Some motivating examples, including those showing that better-reply security is not sufficient for the existence of limit admissible equilibrium are in Section 3. Our new condition, admissible security, is introduced in Section 4 and used to establish the existence of limit admissible equilibrium. Several remarks in Section 4 complement this result, by showing that: (1) More general existence results can be obtained by relativizing Reny’s (2016) correspondence security and finite deviation property; (2) an extension of our result for pure strategies is possible in own-strategy quasi-concave games but requires the set of limit admissible pure strategies to be convex; (3) the relativization of Reny’s (1999) payoff security and of Bagh and Jofre’s (2006) weak reciprocal upper semicontinuity together provide sufficient conditions for admissible security; (4) better-reply secure games are admissible secure provided that each non-limit admissible strategy be weakly dominated by a limit admissible strategy (a weaker version of this property suffices). In Section 4 we also relate our existence result with those that follow from Simon and Stinchcombe (1995) for continuous games and from Carbonell-Nicolau (2011b) for discontinuous games. Our results are applied in Section 5 to a Bertrand duopoly game with strictly convex costs, modelled along the lines of Maskin (1986). Unlike 1

Admissible security also yields the existence of pure-strategy limit admissible equilibria when

the set of limit admissible pure strategies is convex for each player.

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other methods such as that of Carbonell-Nicolau (2011b, Theorem 4), applying our results requires a characterization of the set of weakly dominated strategies. Such characterization is of interest on its own as it delineates some of the additional structure that limit admissible equilibria have as compared to Nash equilibria. Moreover, once such characterization is obtained, applying our results is a relatively simple task as the Bertrand duopoly game of Section 5 illustrates.

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Notation and Definitions

Consider a normal-form game G = (Xi , ui )i∈N consisting of a finite set of players N = {1, . . . , n}, and, for each player i ∈ N , a compact metric pure strategy set Xi ∏ and a bounded and measurable real-valued payoff function ui on X = ni=1 Xi . For all i ∈ N , Xi is endowed with its Borel σ-algebra. The set of mixed strategies for player i is the set of (Borel) probability measures Mi on Xi , endowed with the narrow topology. For each xi ∈ Xi , we write δxi for the Dirac measure at xi . Let M = ∏n i=1 Mi . Given a profile σ ∈ M of mixed strategies, we write τσ for the corresponding ∫ product measure on X. Define, for all i ∈ N and σ ∈ M , ui (σ) = X ui (x)dτσ (x). Throughout the paper, the product of any number of sets is endowed with the product topology. Given a player i ∈ N , the symbol −i denotes “all players but i”. ∏ Also, X−i = j̸=i Xj . A pure strategy xi ∈ Xi is weakly dominated for i ∈ N if there exists a mixed strategy µi ∈ Mi such that ui (xi , x−i ) ≤ ui (µi , x−i ) for all x−i ∈ X−i and ui (xi , x′−i ) < ui (µi , x′−i ) for some x′−i ∈ X−i . Let Oi be the interior of the set of weakly dominated strategies for i and Ci = Xi \ Oi be its complement. We say that σ ∈ M is a Nash equilibrium of G if ui (σ) ≥ ui (σi′ , σ−i ) for each i ∈ N and σi′ ∈ Mi ; the set of Nash equilibria of G is denoted by E(G). A mixed strategy profile σ ∈ M is limit admissible if σi (Oi ) = 0 for each i ∈ N . Let L(G) denote the set of Nash equilibria of G that are limit admissible, its elements being referred to as limit admissible equilibria. The existence of Nash equilibria is guaranteed under better-reply security, a notion 4

introduced in Reny (1999) and which we now recall; we shall see that better-reply security is not sufficient for the existence of limit admissible equilibria. A game G = (Xi , ui )i∈N is better-reply secure in mixed strategies if, whenever (σ ∗ , u∗ ) ∈ cl(graph(u)) and σ ∗ is not a Nash equilibrium of G, there exist i ∈ N , µi ∈ Mi , an open neighborhood U of σ ∗ and a real number αi > u∗i such that ui (µi , σ−i ) ≥ αi for each σ ∈ U .

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Motivating examples

We illustrate with two simple examples that best-reply secure games may fail to have a limit admissible equilibrium. Example 1. Let G = (Xi , ui )i∈N be a 2-player game on the unit square (i.e. N = {1, 2} and X1 = X2 = [0, 1]) with the     1    u1 (x) = 1      0

following payoff functions: For each x ∈ X,

and

  1 if x2 = 0,

u2 (x) =

if x2 ≤ x1 < 1, if x1 = 1 and x2 = 0, otherwise,

 0 otherwise.

It is easy to see that the set of Nash equilibria of G is {(σ1 , δ0 ) : σ1 ∈ M1 }. We now show that G is better-reply secure in mixed strategies. Let (σ ∗ , u∗ ) ∈ cl(graph(u)) be such that σ ∗ is not a Nash equilibrium of G. Then σ2∗ ̸= δ0 , i.e. ∗ ∗ σ2∗ ({0}) < 1. Letting {σk }∞ k=1 ⊆ M be such that (σk , u(σk )) → (σ , u ) and using

Portmanteau theorem (e.g. Billingsley (1999, Theorem 2.1, p. 16)), we obtain that u∗2 = lim u2 (σk ) = lim σk,2 ({0}) ≤ σ2∗ ({0}) < 1. k

k

Since u2 (σ1 , δ0 ) = 1 for each σ1 ∈ M1 , it follows that the condition in the definition of better-reply security is satisfied with i = 2, µ2 = δ0 , U = M and α2 = 1. 5

Despite being better-reply secure, G has no limit admissible equilibrium since C1 = ∅. Indeed, each x1 < 1 is weakly dominated for i = 1 by x′1 ∈ (x1 , 1) and x1 = 1 is weakly dominated for i = 1 by x′1 = 1/2. The previous example shows that better-reply security is not sufficient for the non-emptiness of Ci for each i ∈ N . The following example shows that better-reply secure games with Ci ̸= ∅ for each i ∈ N may fail to have limit admissible equilibria. Example 2. Let G = (Xi , ui )i∈N be as in Example 1 except that now u1 (1/2, 1/2) = 2 and u1 (x1 , 0) = 0 for each 1/4 < x1 < 3/4. It is clear that C2 = {0} and that C1 = {1/2}; for the latter, each x1 < 1 different than 1/2 is weakly dominated for i = 1 by x′1 ∈ (max{x1 , 3/4}, 1) and x1 = 1 is weakly dominated for i = 1 by x′1 = 3/4. It is also easy to see that the set of Nash equilibria of G is {(σ1 , δ0 ) : σ1 ((1/4, 3/4)) = 0}. We now show that G is better-reply secure in mixed strategies. Let (σ ∗ , u∗ ) ∈ cl(graph(u)) be such that σ ∗ is not a Nash equilibrium of G. Arguing as in Example 1, we may assume that σ2∗ = δ0 and u∗2 = 1. Hence, σ1∗ ((1/4, 3/4)) > 0. Letting ∗ ∗ {σk }∞ k=1 ⊆ M be such that (σk , u(σk )) → (σ , u ), we have that limk σk,2 ({0}) =

limk u2 (σk ) = u∗2 = 1. Thus, by the Portmanteau theorem (e.g. Billingsley (1999, ∫ Theorem 2.1, p. 16)) and by X1 u1 (x1 , x2 )dσk,1 (x1 ) ≤ 2 for each x2 > 0 and k ∈ N, we obtain that ( ) u∗1 = lim u1 (σk ) ≤ lim σk,2 ({0})σk,1 (X1 \ (1/4, 3/4)) + 2σk,2 ((0, 1]) k

k

= lim σk,1 (X1 \ (1/4, 3/4)) ≤ σ1∗ (X1 \ (1/4, 3/4)) < 1. k

∫ Let α1 ∈ (u∗1 , 1) and O2 be an open neighborhood of δ0 such that X2 1[0,3/4) dσ2 > ∫ 1 dδ0 −(1−α1 ) for each σ2 ∈ O2 .2 Let i = 1, µ1 = δ3/4 and U = M1 ×O2 ; since, X2 [0,3/4) ∫ ∫ for each σ ∈ U , u1 (µ1 , σ2 ) = σ2 ([0, 3/4]) ≥ X2 1[0,3/4) dσ2 > X2 1[0,3/4) dδ0 − (1 − α1 ) = α1 > u∗1 , it follows that G is better-reply secure. 2

The function 1[0,3/4) is defined by 1[0,3/4) (x2 ) = 1 if x2 ∈ [0, 3/4) and 1[0,3/4) (x2 ) = 0 if x2 ̸∈

[0, 3/4); it is lower semi-continuous. Thus, the existence of O2 follows by Aliprantis and Border (2006, Theorem 15.5, p. 511).

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However, many discontinuous games have limit admissible equilibria. An easy example is provided by a symmetric Bertrand duopoly with constant marginal costs. Example 3. Suppose that each firm has a constant marginal cost 0 < c < 1 and there is one commodity whose demand is d(x) = 1 − x, where x is the lowest price in the market. Each of the two firms sets a price in the unit interval and gets half of the demand when these prices are equal. Thus, this situation can be described by a game G = (Xi , ui )i∈N with N = {1, 2}, X1 = X2 = [0, 1] and,     ((x1 − c)(1 − x1 ), 0)    u(x) = (0, (x2 − c)(1 − x2 ))      ((x1 − c)(1 − x1 )/2, (x1 − c)(1 − x1 )/2)

for each x ∈ X, if x1 < x2 , if x1 > x2 , if x1 = x2 .

It is easy to check that Ci = [c, (1 + c)/2] for each i ∈ N and that (δc , δc ) is a Nash equilibrium of G and, thus, a limit admissible equilibrium of G. As we shown in the next section, the existence of a limit admissible equilibrium of G also follows from our existence result.

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Existence of limit admissible equilibria

We obtain the existence of limit admissible equilibria by relativizing the notion of better-reply security as follows. For each i ∈ N , let Mi0 = {µi ∈ Mi : µi (Oi ) = 0} ∏ and let M 0 = i∈N Mi0 . We say that a game G = (Xi , ui )i∈N is admissibly secure if (a) Ci ̸= ∅ for each i ∈ N and (b) If (σ ∗ , u∗ ) ∈ cl({(σ, u(σ)) : σ ∈ M 0 }) and σ ∗ is not a Nash equilibrium of G, then there exist i ∈ N , µi ∈ Mi0 , an open neighborhood U of σ ∗ and a real number αi > u∗i such that ui (µi , σ−i ) ≥ αi for each σ ∈ U ∩ M 0 . Analogously to Reny’s (1999) description of better-reply security (the differences are emphasized), admissible security requires that for each non-Nash equilibrium limit admissible strategy σ ∗ and every payoff vector limit u∗ resulting from limit admissible 7

strategies approaching σ ∗ , some player i has a limit admissible strategy yielding a payoff strictly above u∗i even if the others deviate slightly from σ ∗ to another limit admissible strategy. Admissible security is a relativization of better-reply security in mixed strategies and, in this way, is neither stronger nor weaker than better-reply security in mixed strategies.3 A further relativization, namely changing (b) by requiring it to hold only for non-equilibrium strategies of the game obtained by restricting players to choose limit admissible strategies, is not possible when the goal is to obtain a condition that is sufficient for the existence of limit admissible equilibria. This is shown by Example 2 which has no limit admissible equilibrium and where this alternative condition holds trivially since Mi0 is a singleton for each player i. Note further that Example 1 satisfies (b) (vacuously) but not (a) and that Example 2 satisfies (a) but not (b). Thus, both conditions in the definition of admissible security are needed for this condition to imply the existence of limit admissible equilibria. Our main result states that admissible security is indeed sufficient for the existence of limit admissible equilibria. Theorem 1. If G = (Xi , ui )i∈N is admissibly secure, then L(G) ̸= ∅. Proof. Let G be admissibly secure and consider G0 = (Mi0 , ui |M 0 )i∈N , where ui |M 0 is the restriction of ui to M 0 for each i ∈ N . Note that cl({(σ, u(σ)) : σ ∈ M 0 }) = cl(graph(u|M 0 )) and that U ∩ M 0 is open in M 0 whenever U is open in M . Furthermore, if σ ∗ ∈ M 0 is not a Nash equilibrium of G0 , then σ ∗ is not a Nash equilibrium of G. Hence, G0 is better-reply secure. It then follows by Reny (1999, Theorem 3.1) (G0 is clearly a compact and own-strategy quasi-concave game) that G0 has a Nash equilibrium σ ∗ ∈ M 0 . Suppose, in order to reach a contradiction, that L(G) = ∅. Then σ ∗ is not a Nash equilibrium of G. Thus, as (σ ∗ , u(σ ∗ )) ∈ cl({(σ, u(σ)) : σ ∈ M 0 }) and G is admissible 3

A sufficient condition for admissible security which is stronger than better-reply security in

mixed strategies is the following: If (σ ∗ , u∗ ) ∈ cl(graph(u)) and σ ∗ is not a Nash equilibrium of G, then there exist i ∈ N , µi ∈ Mi0 , an open neighborhood U of σ ∗ and a real number αi > u∗i such that ui (µi , σ−i ) ≥ αi for each σ ∈ U .

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∗ secure, there exists i ∈ N and µi ∈ Mi0 such that ui (µi , σ−i ) > ui (σ ∗ ). But this is

a contradiction since σ ∗ is a Nash equilibrium of G0 . This contradiction shows that L(G) is nonempty and completes the proof. Extensions as well as particular cases of Theorem 1 are discussed in the following remarks. Remark 1 (Admissible correspondence security). An extension of Theorem 1 can be obtained by weakening admissible security along the lines of Reny’s (2016) correspondence security. A game G = (Xi , ui )i∈N is admissible correspondence secure if Ci ̸= ∅ for each i ∈ N and if whenever σ ∗ ∈ M 0 is not a Nash equilibrium of G, then there exists an open neighborhood U of σ ∗ and a closed correspondence ψ : U ⇒ M 0 with nonempty and convex values such that, for each σ ∈ U ∩ M 0 , there is i ∈ N for ′ whom ui (µi , σ−i ) > ui (σ) for each σ ′ ∈ U ∩ M 0 and µi ∈ ψi (σ ′ ). We then have, by

an argument analogous to that of Theorem 1, that if G = (Xi , ui )i∈N is admissibly correspondence secure, then L(G) ̸= ∅. Remark 2 (Admissible finite deviation property). Another extension of Theorem 1 can also be obtained by weakening admissible security along the lines of Reny’s (2016) finite deviation property. We say that G = (Xi , ui )i∈N satisfies the admissible finite deviation property if, for each σ ∈ M 0 which is not a Nash equilibrium of G, there exists µ1 , . . . , µK ∈ M 0 and a neighborhood U of σ such that, for each σ ′ ∈ U ∩ M 0 , ′ there exists i ∈ N and k ∈ {1, . . . , K} such that ui (µki , σ−i ) > ui (σ ′ ). Adapting the

proof of Theorem 1, one can show that L(G) ̸= ∅ for each game with the admissible finite deviation property. Remark 3 (Pure strategies). Our existence result extends to the case of pure strategies in games G = (Xi , ui )i∈N that are own-strategy quasi-concave (i.e. Xi is convex and ui (·, x−i ) is quasi-concave for each i ∈ N and x−i ∈ X−i ) and are such that Ci ∏ is convex for each i ∈ N . In this case, letting C = i∈N Ci , part (b) of the definition of admissible security is specialized to the case of pure strategies as follows: If (x∗ , u∗ ) ∈ cl({(x, u(x)) : x ∈ C}) and x∗ is not a Nash equilibrium of G, then there 9

exist i ∈ N , x¯i ∈ Ci , an open neighborhood U of x∗ and a real-number αi > u∗i such that ui (¯ xi , x−i ) ≥ αi for each x ∈ U ∩ C. The Bertrand duopoly of Example 3 illustrates. Indeed, it is a quasi-concave game in which Ci is convex for each i = 1, 2, namely Ci = [c, (1 + c)/2] for each i = 1, 2 (note that (1 + c)/2 is the monopoly quantity). Furthermore, it is admissibly secure: When x∗ ∈ C is not a Nash equilibrium (i.e. x∗ ̸= (c, c)) and x∗i = x∗j > c, pick player i, x¯i ∈ [c, x∗j ) sufficiently close to x∗i and U = Xi × (¯ xi , x∗j ); if x∗i > x∗j ≥ c, pick player j, x¯j ∈ (c, x∗i ) and U = Xj × (¯ xj , x∗i ). Thus, in accordance with the above, the Bertrand duopoly of Example 3 has a pure-strategy limit admissible equilibrium, namely x∗ = (c, c). In general, however, own-strategy quasi-concave games satisfying admissible security may fail to have a pure-strategy limit admissible equilibrium when Ci fails to be convex for some i ∈ N . To illustrate, we present an example of an own-strategy quasi-concave game with continuous payoff functions that fails to have a limit admissible equilibrium.4 The game G = (Xi , ui )i∈N is a variation on the matching pennies. Let N = {1, 2} and X1 = X2 = [0, 1], where xi is interpreted as the probability of player i playing heads, i = 1, 2. Player 1 want to match his choice with that of player 2 and, thus, let u1 (x) = −|x1 − x2 | for each x ∈ X. As for player 2’s payoff function, let, for each x ∈ X, w2 (x) = (1 − 2x1 )(2x2 − 1) as in the standard matching pennies, and u2 (x) = max{w2 (x), 0}. It is easy to see that G is own-strategy quasi-concave with continuous payoff functions and that it has a unique pure-strategy Nash equilibrium x∗ = (1/2, 1/2). 4

Games with continuous payoff functions have pure-strategy limit admissible equilibria provided

that Xi is convex and ui (·, x−i ) is concave for each i ∈ N and x−i ∈ X−i ; see Carbonell-Nicolau (2011a) for this and related results.

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Moreover, C2 = {0, 1} since any x2 ∈ [1/2, 1) is weakly dominated by x′2 = 1, any x2 ∈ (0, 1/2) is weakly dominated by x′2 = 0 and neither x2 = 0 nor x2 = 1 are weakly dominated. Therefore, G has no limit admissible equilibria in pure strategies. Remark 4 (Sufficient conditions 1). We say that a game G = (Xi , ui )i∈N is admissibly 0 payoff secure if for each i ∈ N , ε > 0 and σ ∈ Mi × M−i , there is a µi ∈ Mi0 and an

open neighborhood U of σ such that ′ ) > ui (σ) − ε ui (µi , σ−i

for each σ ′ ∈ U ∩ M 0 . Moreover, we say that G is admissibly reciprocal upper semicontinuous if for each (σ ∗ , α) ∈ cl({(σ, u(σ)) : σ ∈ M 0 }) \ {(σ, u(σ)) : σ ∈ M 0 }, there ∗ exists i ∈ N and µi ∈ Mi such that ui (µi , σ−i ) > αi . Analogously to Bagh and Jofre

(2006, Proposition 1), we obtain that if G is admissibly payoff secure and admissibly reciprocal upper semi-continuous, then G is admissibly secure. Remark 5 (Continuous games). Games with continuous payoff functions, considered in Simon and Stinchcombe (1995), are easily seen to be admissible secure. Indeed, admissible reciprocal upper semi-continuity is trivially satisfied as cl({(σ, u(σ)) : σ ∈ M 0 }) \ {(σ, u(σ)) : σ ∈ M 0 } is empty. Furthermore, for admissible payoff security, it suffices to show that, against any σ ∈ M , each player i has a best-reply in Mi0 . To see that this property holds, let i ∈ N and σ ∈ M be given. Letting j ̸= i ∑∞ 1 1 ¯j + and {xk,j }∞ ¯j = k=1 be dense in Xj , define σ k=1 2k δxk,j ∈ Mj and σk,j = k σ ( ) 1 − k1 σj for each k ∈ N. Fix k ∈ N and, using the continuity of ui , let xk,i ∈ Xi be such that ui (xk,i , σk,−i ) = maxσi ∈Mi ui (σi , σk,−i ). If xk,i ̸∈ Ci , then letting µi ∈ Mi weakly dominate xk,i , we have that ui (xk,i , σk,−i ) < ui (µi , σk,−i ) as τσk,−i gives strictly positive probability to each open subset of X−i ; but this is a contradiction to ui (xk,i , σk,−i ) = maxσi ∈Mi ui (σi , σk,−i ). Thus, xk,i ∈ Ci for each k ∈ N. Since Ci is compact, we may assume that {xk,i }∞ k=1 converges and let xi = limk xk,i . Then xi ∈ Ci and the continuity of ui , together with σk,−i → σ−i , implies that ui (xi , σ−i ) = maxσi ∈Mi ui (σi , σ−i ). Remark 6 (Related results). The existence of a limit admissible equilibrium can be obtained from some results in Carbonell-Nicolau (2011b), namely from Theorem 4 11

and also by combining Theorem 7 and Remark 6 in that paper. Under the assumptions of Theorem 7 in Carbonell-Nicolau (2011b), both admissible payoff security and admissible reciprocal upper semicontinuity hold, hence, the existence of a limit admissible equilibrium follows by Remark 4 and Theorem 1 above (this argument makes Remark 6 in Carbonell-Nicolau (2011b) redundant). Under the assumptions of Theorem 4 in Carbonell-Nicolau (2011b), better-reply security in mixed strategies holds and so does admissible reciprocal upper semicontinuity. However, it is unclear whether or not admissible payoff security holds. This is the case even under the assumptions of Carbonell-Nicolau’s (2011b) Theorem 4 which imply payoff security, i.e., it is the case that for each i ∈ N , ε > 0 and σ ∈ M , there is µi ∈ Mi and an ′ open neighborhood U of σ such that ui (µi , σ−i ) > ui (σ) − ε for each σ ′ ∈ U ; the

difficulty is to conclude that µi can be taken to be an element of Mi0 as required by admissible payoff security. On the other end, our Theorem 1 does not require any of the conditions imposed in Carbonell-Nicolau’s (2011b) results. Remark 7 (Sufficient conditions 2). Carbonell-Nicolau’s (2011b) Theorem 4 described in the previous remark can be seen as providing conditions that, when added to better-reply security in mixed strategies, imply the existence of Nash equilibria with additional structure. Here we ask: What additional conditions need to be added to better-reply security in mixed strategies for a game to have a limit admissible equilibrium? One such condition is: For each i ∈ N and σi ∈ Mi \ Mi0 , there exists σi0 ∈ Mi0 such that ui (σi0 , x−i ) ≥ ui (σi , x−i ) for each x−i ∈ X−i ; this is a weakening of the requirement that each strategy outside Mi0 be weakly dominated by a strategy in Mi0 . It is easy to see that any game satisfying this condition and better-reply security in mixed strategies is admissibly secure. Remark 8 (Bertrand duopoly). The conditions in the previous remark apply to the Bertrand duopoly of Example 3. It is well known that better-reply security in mixed strategies holds for this example. Moreover, for each i ∈ N and σi ∈ Mi \ Mi0 , there exists σi0 ∈ Mi0 such that ui (σi0 , x−i ) ≥ ui (σi , x−i ) for each x−i ∈ X−i . To see this, recall that Ci = [c, (1+c)/2], define f : Xi → Ci by setting, for each xi ∈ Xi , f (xi ) = c 12

when xi < c, f (xi ) = (1 + c)/2 when xi > (1 + c)/2 and f (xi ) = xi otherwise and set σi0 = σi ◦ f −1 ∈ Mi0 . For each x−i ∈ X−i , we have that ui (xi , x−i ) ≤ 0 = ui (c, x−i ) for each xi < c and ui (xi , x−i ) ≤ ui ((1 + c)/2, x−i ) for each xi > (1 + c)/2. Hence, ( ) ( ∫ ∫ ui (σi0 , x−i ) − ui (σi , x−i ) = [0,c) ui (c, x−i ) − ui (xi , x−i ) dσi (xi ) + ((1+c)/2,1] ui ((1 + ) c)/2, x−i ) − ui (xi , x−i ) dσi (xi ) ≥ 0. Thus, Theorem 1 applies. Remark 9 (An extension). In Theorem 1, Ci is the closure of the set of undominated strategies for each i ∈ N . More generally, the conclusion of Theorem 1 holds for any nonempty, closed subset of Xi as follows. Let G = (Xi , ui )i∈N be a game and, for each ˜ i = {σi ∈ Mi : σi (C˜i ) = 1} i ∈ N , C˜i be a nonempty, closed subset of Xi . Define M ˜ =∏ ˜ ˜ for each i ∈ N and M i∈N Mi . Then G has a Nash equilibrium in M provided that the following condition analogous to admissible security holds: If (σ ∗ , u∗ ) ∈ ˜ }) and σ ∗ is not a Nash equilibrium of G, then there exist cl({(σ, u(σ)) : σ ∈ M ˜ i , an open neighborhood U of σ ∗ and a real number αi > u∗ such that i ∈ N , µi ∈ M i ˜. ui (µi , σ−i ) ≥ αi for each σ ∈ U ∩ M

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An applications: Bertrand duopoly with strictly convex costs

We consider a Bertrand duopoly with strictly convex costs as an economic application of our main result, Theorem 1. We depart from the standard formalization of Bertrand competition considered in Andersson, Argenton, and Weibull (2014) and Bich (2016) because the assumption that the firm posting the lowest price serves the entire demand leads to difficulties in the presence of strictly convex costs. This is so because firms may prefer to tie to reduce the quantity produced. For this reason, it is more appropriate to allow firms to choose the quantity they want to supply. We allow for this by allowing each firm to choose a price and the maximum production level it is willing to produce, i.e. an endogenous capacity. Our formalization is thus analogous to that of Dasgupta and Maskin (1986, Section 2.2) where each firm has a capacity but which is exogenously given. Our formalization is closer to that of 13

Maskin (1986) where firms choose both prices and quantities, and firms produce to order, i.e., produce only after the entire price profile has been observed. As in Dasgupta and Maskin (1986), there is a market for a single commodity with a continuum of consumers represented by the unit interval [0, 1]. Consumers are identical, and the representative consumer’s demand for the commodity is a C 2 function d : R+ → R+ such that there exists p¯ > 0 satisfying d(p) > 0, d′ (p) < 0 and d′′ (p) ≤ 0 for all p ≤ p¯ and d(p) = 0 for all p ≥ p¯. There is a C 2 cost function c : R+ → R+ satisfying c(0) = 0, p¯ > c′ (0) and, for each q > 0, c′ (q) > 0 and c′′ (q) > 0. There are two firms. Each firm i = 1, 2 chooses a price pi and a capacity si , the latter being the maximum amount the firm is willing to produce. Let P = [0, p¯] and S = [0, d(0)]. If one firm offers a price p lower than the price p′ offered by the other firm, then it serves the entire market up to its capacity s. A fraction

(d(p)−s)+ d(p)

=

max{d(p)−s,0} d(p)

of consumers is not served and each of these consumers demands d(p′ ) from the firm offering the highest price. When both firms set the same price, then the demand at the common price is split by each firm up to its capacity. In the latter case, several allocations are possible and we will focus on allocations that maximize the total profits in the market. Formally, the quantity produced by firms is described by the correspondence Φ : (P × S)2 ⇒ R2+ defined by setting, for each x ∈ (P × S)2 ,  (d(p1 )−s1 )+   if p1 < p2 ,  (min{d(p1 ), s1 }, min{ d(p1 ) d(p2 ), s2 })   {   ϕ ∈ R2 : ϕ + ϕ ≤ d(p ), ϕ ≤ s and 1 2 1 i i + Φ(x) =   [d(p1 ) − ϕ1 − ϕ2 ][si − ϕi ] = 0 for each i = 1, 2} if p1 = p2 ,      (min{ (d(p2 )−s2 )+ d(p ), s }, min{d(p ), s }) if p1 > p2 . 1 1 2 2 d(p2 ) When prices are different, these quantities are the same as in both Dasgupta and Maskin (1986) and Maskin (1986) (with the proportional rationing rule in the latter). However, we rule out the possibility that a firm produces less than its capacity when there is unfulfilled demand (i.e. ϕ1 +ϕ2 < d(p1 ) and ϕi < si for some i is not possible); in contrast this is allowed in both Dasgupta and Maskin (1986) and Maskin (1986). 14

We then let ϕ : (P ×S)2 ⇒ R2+ be measurable and such that, for each x ∈ (P ×S)2 , ϕ(x) solves max ϕ∈Φ(x)

2 ∑ (

) pi ϕi − c(ϕi ) .

i=1

Players’ payoffs are then ui (x) = pi ϕi (x) − c(ϕi (x))

(1)

for each i ∈ {1, 2} and x ∈ (P × S)2 . Our results, however, extend to a broad class of Bertrand games obtained by changing the function ϕ; see Remark 10 below. The existence of the function ϕ follows by combing the measurable maximum theorem (e.g. Aliprantis and Border (2006, Theorem 18.19, p. 605)) with the following lemma. Lemma 1. The graph Φ is closed. Proof. To this end, let xk = (pk1 , sk1 , pk2 , sk2 ) → (p1 , s1 , p2 , s2 ) = x and ϕk → ϕ with ϕk ∈ Φ(xk ) for each k ∈ N. It is clear that ϕ ∈ Φ(x) when p1 ̸= p2 and also when both p1 = p2 and pk1 = pk2 for infinitely many k ∈ N hold. Thus, assume that p1 = p2 and that pk1 < pk2 for all sufficiently large k ∈ N (the case pk2 < pk1 for all sufficiently large k ∈ N is analogous). We have that ϕki ≤ ski for each k and i, and hence ϕi ≤ si for each i. If s1 ≥ d(p1 ), then ϕ1 = d(p1 ) and ϕ2 = 0 = lim k

(d(pk1 ) − sk1 )+ d(pk2 ).5 k d(p1 )

Therefore, ϕ1 + ϕ2 = d(p1 ) and ϕ ∈ Φ(x). Finally, if s1 < d(p1 ), then ϕ1 = s1 and ϕ2 = min{d(p1 ) − s1 , s2 } (note that s1 < d(p1 ) implies that p1 < p¯ and hence limk

(d(pk1 )−sk1 )+ d(pk2 ) d(pk1 )

=

d(p1 )−s1 d(p2 ) d(p1 )

= d(p1 ) − s1 since p1 = p2 ). In either case,

ϕ ∈ Φ(x). This shows that Φ has a closed graph. While in the above we allowed firms to choose any capacity in S, some choices are easily seen to be redundant. Indeed, note that for each i = 1, 2, x = (pi , si , pj , sj ) and ϕ ∈ Φ(x), we have that ϕi ≤ d(pi ). Furthermore, if si > d(pi ) and x′ = 5

Note that when p1 = p¯, limk

k + (d(pk 1 )−s1 ) d(pk2 ) d(pk 1)

= 0 since limk (d(pk1 ) − sk1 )+ = 0 and 0 ≤

for each k.

15

d(pk 2) d(pk 1)

≤1

(pi , d(pi ), pj , sj ), then ϕi (x) = ϕi (x′ ).6 Hence, we may assume that si ≤ d(pi ) for each i = 1, 2 and pi ∈ P . In particular, si = 0 whenever pi = p¯. We then let Xi = {(pi , si ) ∈ P × S : si ∈ [0, d(pi )]} for each i ∈ N . Let GB = (X1 , X2 , u1 , u2 ) be the game just defined. The following is our main result of this section. Theorem 2. The game GB is admissibly secure and, thus, L(GB ) ̸= ∅. Proceeding to the proof of Theorem 2, we start by characterizing the set Ci for each i ∈ {1, 2}. Define, for each p ∈ P , ϕ∗ (p) to be the unique solution to max 0≤ϕ≤d(p)

(

) pϕ − c(ϕ) .

Note that the uniqueness of the solution follows because c is strictly convex. Define m : P → R+ by setting m(p) = pϕ∗ (p) − c(ϕ∗ (p)) for each p ∈ P . It follows by the maximum theorem (e.g. Aliprantis and Border (2006, Theorem 17.31, p. 570)) that m and ϕ∗ are continuous. Further properties of m and ϕ∗ are described in the following lemma. Lemma 2. The following holds: 1. There exists a unique pˆ ∈ P such that c′ (d(ˆ p)) = pˆ. Moreover, c′ (0) < pˆ < p¯ and p − c′ (d(p)) > 0 for each p > pˆ. 2. For each p ∈ P ,

    0    ϕ∗ (p) = (c′ )−1 (p)      d(p)

if p ≤ c′ (0), if c′ (0) < p < pˆ, if p ≥ pˆ.

3. The function m has a unique maximizer p∗ ∈ (ˆ p, p¯). Furthermore, m is strictly increasing in [c′ (0), p∗ ] and strictly decreasing in [p∗ , p¯]. This is clear when pi ̸= pj ; when pi = pj , Φ(x) = Φ(x′ ) = {ϕ ∈ R2+ : ϕ1 + ϕ2 = d(p1 ) and ϕj ≤ ) ∑2 ( sj }, a convex set. As ϕ 7→ i=1 pi ϕi − c(ϕi ) is strictly concave, ϕ(x) = ϕ(x′ ). 6

16

Proof. We start with the proof of part 1. Define, for each p ∈ P , f (p) = p − c′ (d(p)). We have that f ′ (p) = 1 − c′′ (d(p))d′ (p) ≥ 1 for each p ∈ P ; hence f is strictly increasing. Moreover, f is continuous, f (0) = −c′ (d(0)) < 0 and f (¯ p) = p¯ − c′ (0) > 0. Thus, there exists pˆ ∈ (0, p¯) such that f (ˆ p) = 0. Since f is strictly increasing, pˆ is unique and p − c′ (d(p)) > 0 for each p > pˆ. Finally, pˆ < p¯ implies that d(ˆ p) > 0 and, therefore, pˆ = c′ (d(ˆ p)) > c′ (0). We next turn to the proof of part 2 of the lemma. Let, for each p ∈ P and ϕ ∈ R+ , f (ϕ; p) = pϕ − c(ϕ). If p ≤ c′ (0), then f ′ (ϕ; p) = p − c′ (ϕ) < p − c′ (0) ≤ 0 for each ϕ > 0; hence, ϕ∗ (p) = 0. If p ≥ pˆ, then f ′ (ϕ; p) = p − c′ (ϕ) > p − c′ (d(p)) ≥ 0 for each ϕ < d(p), where the last inequality follows by part 1; hence, ϕ∗ (p) = d(p). Finally, when p ∈ (c′ (0), pˆ), the first order condition gives p = c′ (ϕ∗ (p)). We now establish part 3 of the lemma. Note first that m(p) = 0 for each p ≤ c′ (0) by part 2. Next, note that m is strictly increasing in [c′ (0), pˆ]. Indeed, for each p ∈ (c′ (0), pˆ), m is differentiable at p and, using part 2, m′ (p) = ϕ∗ (p) +

p − c′ (ϕ∗ (p)) = ϕ∗ (p) > 0. c′′ (ϕ∗ (p))

For each p ∈ (ˆ p, p¯), ϕ∗ (p) = d(p) by part 2. Hence, m′ (p) = d(p) + [p − c′ (d(p))]d′ (p), and m′′ (p) = d′ (p)[2 − c′′ (d(p))d′ (p)] + d′′ (p)[p − c′ (d(p))]. Hence, using part 1, it follows that limp→ˆp m′ (p) = d(ˆ p)+[ˆ p −c′ (d(ˆ p))]d′ (ˆ p) = d(ˆ p) > 0, limp→¯p m′ (p) = [¯ p − c′ (d(¯ p))]d′ (¯ p) < 0 and m′′ (p) < 0 for each p ∈ (ˆ p, p¯). Thus, there exists a unique p∗ ∈ (ˆ p, p¯) such that m′ (p∗ ) = 0, m′ (p) > 0 for each p ∈ (ˆ p, p∗ ) and m′ (p) < 0 for each p ∈ (p∗ , p¯). The conclusion follows. We use the above lemma to characterize the interior of the set of weakly dominated strategies. The first result is that a capacity choice different from ϕ∗ (pi ) when choosing a price level pi is weakly dominated for i. Lemma 3. For each i ∈ {1, 2} and (pi , si ) ∈ Xi , if si ̸= ϕ∗ (pi ), then (pi , si ) is weakly dominated for i by (pi , ϕ∗ (pi )). 17

Proof. We start with the following claim. Claim 1. If i ∈ {1, 2}, x = (p1 , s1 , p2 , s2 ) ∈ (P × S)2 and x′ = (p′1 , s′1 , p′2 , s′2 ) ∈ (P ×S)2 are such that p′1 = p1 , p′2 = p2 , s′j = sj and s′i > si , then either ϕi (x) = ϕi (x′ ) or ϕi (x) = si < ϕi (x′ ). Proof of Claim 1. We have that: ϕi (x) = min{d(pi ), si } ≤ min{d(pi ), s′i } = ϕi (x′ ) if pi < pj , } { } { (d(pj ) − sj )+ (d(pj ) − sj )+ ′ ϕi (x) = min d(pi ), si ≤ min d(pi ), si = ϕi (x′ ) if pi > pj , d(pj ) d(pj ) ϕi (x) = si < s′i = ϕi (x′ ) if pi = pj and d(p1 ) ≥ s′i + sj . In fact, in the above three cases, we have that ϕi (x) = si whenever ϕi (x) < ϕi (x′ ). We next consider the two remaining cases. If pi = pj and d(p1 ) < s1 + s2 , then Φ(x) = {ϕ ∈ R2+ : ϕ1 + ϕ2 = d(p1 ), ϕi ≤ si and ϕj ≤ sj }, Φ(x′ ) = {ϕ ∈ R2+ : ϕ1 + ϕ2 = ∑ d(p1 ), ϕi ≤ s′i and ϕj ≤ sj } and, thus, Φ(x) ⊆ Φ(x′ ). Let f (ϕ) = 2i=1 (pi ϕi − c(ϕi )) for each ϕ ∈ R2+ and note that f is strictly concave. Thus, for each y ∈ {x, x′ }, maxϕ∈Φ(y) f (ϕ) has a unique solution since Φ(y) is convex. Suppose that ϕi (x) ̸= ϕi (x′ ). We have that f (ϕ(x′ )) > f (ϕ(x)) because ϕ(x) ̸= ϕ(x′ ) and Φ(x) ⊆ Φ(x′ ). This implies that ϕi (x′ ) > ϕi (x) because otherwise ϕ(x′ ) ∈ Φ(x) and this, together with f (ϕ(x′ )) > f (ϕ(x)), would contradict ϕ(x) being a solution of maxϕ∈Φ(x) f (ϕ). Suppose, in order to reach a contradiction, that ϕi (x) < si . Let λ > 0 be sufficiently small so that λϕ(x′ )+(1−λ)ϕ(x) ∈ Φ(x), which is possible because ϕi (x) < si . Then f (λϕ(x′ ) + (1 − λ)ϕ(x)) > λf (ϕ(x′ )) + (1 − λ)f (ϕ(x)) > f (ϕ(x)), a contradiction to ϕ(x) being a solution of maxϕ∈Φ(x) f (ϕ). This contradiction shows that if ϕi (x) ̸= ϕi (x′ ), then ϕi (x) = si . Finally, if pi = pj and s1 + s2 ≤ d(p1 ) < s′i + sj , then ϕi (x) = si and ϕi (x′ ) = d(p1 ) − ϕj (x′ ) ≥ d(p1 ) − sj ≥ si = ϕi (x). This completes the proof of the claim. We now establish the lemma. Let i ∈ {1, 2}, (pi , si ) ∈ Xi and si ̸= ϕ∗ (pi ). Let j ̸= i, (pj , sj ) ∈ Xj , x = (pi , ϕ∗ (pi ), pj , sj ) and x′ = (pi , si , pj , sj ). We clearly have ui (x) = ui (x′ ) if ϕi (x) = ϕi (x′ ). Assuming ϕi (x) ̸= ϕi (x′ ), Claim 1 implies that ϕi (x) = ϕ∗ (pi ) < ϕi (x′ ) if si > ϕ∗ (pi ) and hence ui (x) > ui (x′ ) by the definition of 18

ϕ∗ (pi ); furthermore, in the case si < ϕ∗ (pi ), we have that ϕi (x) = si < ϕi (x′ ) ≤ ϕ∗ (pi ) and thus ui (x) > ui (x′ ) by the definition of ϕ∗ (pi ) and the strict concavity of ϕ 7→ pi ϕ − c(ϕ). Recall that si ≤ d(pi ) and d(¯ p) = 0; thus, si ̸= ϕ∗ (pi ) implies that pi < p¯. Letting pj = p¯, it then follows that pi < pj and, hence, ϕi (x) = min{d(pi ), ϕ∗ (pi )} = ϕ∗ (pi ) and ϕi (x′ ) = min{d(pi ), si } = si (recall also that ϕ∗ (pi ) ≤ d(pi )). Thus, ui (x) > ui (x′ ) by the definition of ϕ∗ (pi ). This completes the proof. It follows from Lemma 3 that Ci ⊆ {(p, ϕ∗ (p)) : p ∈ P } (recall that ϕ∗ is continuous). The following lemma shows that Ci ⊆ {(pi , si ) ∈ Xi : pi ≥ c′ (0)}. Lemma 4. For each i ∈ {1, 2}, if pi < c′ (0), then (pi , ϕ∗ (pi )) is weakly dominated for i by (ˆ p, ϕ∗ (ˆ p)). Proof. Let i = 1 without loss of generality and p1 < c′ (0). Since ϕ∗ (p1 ) = 0, it follows that u1 (p1 , ϕ∗ (p1 ), p2 , s2 ) = 0 for each (p2 , s2 ) ∈ X2 . We have that u1 (ˆ p, ϕ∗ (ˆ p), p2 , s2 ) ≥ 0 for each (p2 , s2 ) ∈ X2 and, when p2 > pˆ, u1 (ˆ p, ϕ∗ (ˆ p), p2 , s2 ) = m(ˆ p) > 0. The following lemma restrict Ci further by showing that Ci ⊆ {(pi , si ) ∈ Xi : pi ≤ p∗ }. Lemma 5. For each i ∈ {1, 2}, if pi > p∗ , then (pi , ϕ∗ (pi )) is weakly dominated for i by (p∗ , ϕ∗ (p∗ )). Proof. Let i = 1 without loss of generality and p1 > p∗ . Lemma 2.2 implies that ϕ∗ (p∗ ) = d(p∗ ) and ϕ∗ (p1 ) = d(p1 ). Fix (p2 , s2 ) ∈ X2 and let x∗ = (p∗ , d(p∗ ), p2 , s2 ), x′ = (p1 , d(p1 ), p2 , s2 ) and γ = u1 (x∗ ) − u1 (x′ ). We consider three cases. The first case is when p2 < p∗ . In this case, we have that ϕ1 (x∗ ) = αd(p∗ ) and ϕ1 (x′ ) = αd(p1 ), where α = (d(p2 ) − s2 )/d(p2 ) (recall that s2 ≤ d(p2 ) and, hence, (d(p2 ) − s2 )+ = d(p2 ) − s2 ). Hence, γ = αp∗ d(p∗ ) − c(αd(p∗ )) − [αp1 d(p1 ) − c(αd(p∗ ))].

19

Letting mα (p) = αpd(p) − c(αd(p)) for each p ∈ [p∗ , p¯], we have that p∗ is the unique maximizer of mα in [p∗ , p¯]. Indeed, for each p ∈ (ˆ p, p¯), ( ) m′α (p) = α d(p) + [p − c′ (αd(p))]d′ (p) , ( ) m′′α (p) = α d′ (p)[2 − αc′′ (αd(p))d′ (p)] + d′′ (p)[p − c′ (αd(p))] and, recall, m′ (p) = d(p) + [p − c′ (d(p))]d′ (p). Since −d′ (p∗ )c′ (αd(p∗ )) < −d′ (p∗ )c′ (d(p∗ )), we have that m′α (p∗ ) < αm′ (p∗ ) = 0. Moreover, for each p > pˆ, p > c′ (d(p)) > c′ (αd(p)) implies that m′′α (p) < 0. Thus, mα is decreasing in [p∗ , p¯] and, thus, p∗ is the unique maximizer of mα in [p∗ , p¯]. It then follows that γ > 0. The second case is when p2 = p∗ . Then ϕ1 (x′ ) = αd(p1 ) where α = (d(p2 ) − s2 )/d(p2 ) and αd(p∗ ) ≤ ϕ1 (x∗ ) ≤ d(p∗ ). Indeed, if ϕ1 (x∗ ) < d(p∗ ), then ϕ1 (x∗ ) + ϕ2 (x∗ ) = d(p∗ ). Hence, using p2 = p∗ and ϕ2 (x∗ ) ≤ s2 , ϕ1 (x∗ ) ≥ d(p2 ) − s2 = d(p2 )−s2 d(p∗ ). d(p2 )

Thus, γ ≥ mα (p∗ ) − mα (p1 ) > 0.

The final case is p2 > p∗ . In this case, γ = m(p∗ ) − u1 (p1 , d(p1 ), p2 , s2 ) ≥ m(p∗ ) − m(p1 ) > 0. This completes the proof. Lemma 6. For each i ∈ {1, 2} and pi ∈ (c′ (0), p∗ ), (pi , ϕ∗ (pi )) is not weakly dominated for i. Proof. Let i = 1 without loss of generality and p1 ∈ (c′ (0), p∗ ). Suppose that there exists µ1 ∈ M1 that weakly dominates (p1 , ϕ∗ (p1 )). Consider a sequence {p2,k }∞ k=1 such that p2,k ↓ p1 and µ1 ({p2,k }) = 0 for each k ∈ N; such sequence exists because the set of atoms of µ1 is countable. Let s2,k = d(p2,k ) for each k ∈ N. Then, for ∫ each k ∈ N, u1 (p1 , ϕ∗ (p1 ), p2,k , s2,k ) = m(p1 ), u1 (µ1 , p2,k , s2,k ) ≤ [0,p2,k ) m(p)dµ1 (p) (equality holds if µ1 ({(p, s) : s = ϕ∗ (p)}) = 1) and ∫ m(p)dµ1 (p) ≥ u1 (µ1 , p2,k , s2,k ) ≥ u1 (p1 , ϕ∗ (p1 ), p2,k , s2,k ) = m(p1 ). [0,p2,k )

Letting f = m1[0,p1 ] and, for each k ∈ N, fk = m1[0,p2,k ) , we have that fk ↓ f . Hence, by the monotone convergence theorem, ∫ ∫ m(p)dµ1 (p) = lim [0,p1 ]

k

m(p)dµ1 (p) ≥ m(p1 ). [0,p2,k )

20

By Lemma 2.3, we have that m(p) < m(p1 ) for each p < p1 . This, together with ∫ m(p)dµ1 (p) ≥ m(p1 ) then implies that µ1 ({p1 } × [0, d(p1 )]) = 1. But then µ1 [0,p1 ] does not weakly dominate (p1 , ϕ∗ (p1 )). Indeed, this is clear if µ1 = δ(p1 ,ϕ∗ (p1 )) since then u1 (p1 , ϕ∗ (p1 ), p2 , s2 ) = u1 (µ1 , p2 , s2 ) for each (p2 , s2 ) ∈ X2 ; if µ1 ̸= δ(p1 ,ϕ∗ (p1 )) , then u1 (p1 , ϕ∗ (p1 ), p2 , s2 ) ≥ u1 (µ1 , p2 , s2 ) for each (p2 , s2 ) ∈ X2 by Lemma 3. Combining Lemmas 3–6, it follows that Ci = {(pi , si ) ∈ Xi : pi ∈ [c′ (0), p∗ ], si = ϕ∗ (pi )}. We turn now to showing that GB is admissibly secure, which we will do by establishing the conditions in Remark 4. Lemma 7. The game GB is admissibly reciprocal upper semicontinuous. Proof. It follows by Aliprantis and Border (2006, Lemma 17.30, p. 569) that u1 + u2 is upper semicontinuous. Thus, the conclusion follows by Carmona (2013, Theorem 3.20, p. 39). Lemma 8. The game GB is admissibly payoff secure. Proof. We first note that it suffices to show that the following condition (∗) holds: For each i ∈ {1, 2}, ε > 0 and σ ∈ Mi × Mj0 , there exists β ∈ [c′ (0), p∗ ] such that σj ({(β, ϕ∗ (β))}) = 0 and ui (β, ϕ∗ (β), σj ) > ui (σ) − ε. Indeed, suppose that condition (∗) holds and that G fails to be admissibly payoff secure. Then there is i ∈ {1, 2}, ε > 0 and σ ∈ Mi × Mj0 such that, for each µi ∈ Mi0 and open neighborhood U of σ, ui (µi , σj′ ) ≤ ui (σ) − ε for some σ ′ ∈ U ∩ M 0 . By condition (∗), let β ∈ [c′ (0), p∗ ] such that σj ({(β, ϕ∗ (β))}) = 0 and ui (β, ϕ∗ (β), σj ) > ui (σ) − ε. As δ(β,ϕ∗ (β)) ∈ Mi0 , for each k ∈ N, pick σk ∈ M 0 such that ui (β, ϕ∗ (β), σk,j ) ≤ ui (σ) − ε and σk,j → σj . We have that τδ(β,ϕ∗ (β)) ,σj ({x ∈ X : ui is discontinuous at x}) = σj ({(β, ϕ∗ (β))}) = 0, and hence ui (β, ϕ∗ (β), σk,j ) → ui (β, ϕ∗ (β), σj ). Thus, we get ui (σ)−ε < ui (β, ϕ∗ (β), σj ) ≤ ui (σ) − ε, a contradiction. This contradiction shows that condition (∗) is sufficient for G to be admissibly secure.

21

In what follows we verify that condition (∗) hold. Let i = 1 without loss of generality, ε > 0 and σ ∈ M1 × M20 . Let A2 = {p ∈ P : σ2 ({(p, ϕ∗ (p))}) > 0} be the set of atoms of σ2 and note that A2 is countable. If u1 (σ) ≤ 0, pick β ∈ [c′ (0), p∗ ] \ A2 and note that u1 (β, ϕ∗ (β), σ2 ) ≥ 0. Thus, we may assume that u1 (σ) > 0. There exists α ∈ P such that u1 (α, ϕ∗ (α), σ2 ) ≥ u1 (σ) since σ1 is a probability measure and using Lemma 3. As ϕ∗ (α) = 0 if α ≤ c′ (0) by Lemma 2.2, it follows that α > c′ (0). Moreover, we may take α ≤ p∗ . Indeed, if α > p∗ , it follows by Lemma 5 that α can be replaced with p∗ . Let h : [0, α]2 → [0, d(0)] be defined by setting, for each p1 , p2 ∈ [0, α], { } d(p2 ) − ϕ∗ (p2 ) ∗ h(p1 , p2 ) = min d(p1 ), ϕ (p1 ) . d(p2 ) We have that (p1 , p2 ) 7→ p1 h(p1 , p2 ) − c(h(p1 , p2 )) : [0, α]2 → R and m are uniformly continuous. Hence, let δ > 0 be such that |p − α| < δ implies that |[ph(p, p2 ) − c(h(p, p2 ))] − [αh(α, p2 ) − c(h(α, p2 ))]| < ε for all p2 ∈ [0, α]

(2)

|m(p) − m(α)| < ε.

(3)

and

We may take δ to be such that α − δ > c′ (0). Choose β ∈ (α − δ, α) such that β ̸∈ A2 . Hence, β ∈ [c′ (0), p∗ ] \ A2 . Let γ(p2 ) = u1 (β, ϕ∗ (β), p2 , ϕ∗ (p2 )) − ui (α, ϕ∗ (α), p2 , s2 ) for each p2 ∈ [c′ (0), p∗ ]. We consider four cases. Case 1: p2 < β. We then have that |γ(p2 )| = |[βh(β, p2 )−c(h(β, p2 ))]−[αh(α, p2 )− c(h(α, p2 ))]| < ε by (2). Case 2: β < p2 < α. Then u1 (α, ϕ∗ (α), p2 , ϕ∗ (p2 )) = αh(α, p2 ) − c(h(α, p2 )) ≤ αϕ∗ (α) − c(ϕ∗ (α)) < βϕ∗ (β) − c(ϕ∗ (β)) + ε = u1 (β, ϕ∗ (β), p2 , ϕ∗ (p2 )) + ε by (3). Case 3: p2 = α. Let ϕi ≤ ϕ∗ (α) ≤ d(α) be such that u1 (α, ϕ∗ (α), α, ϕ∗ (α)) = αϕi − c(ϕi ). Then, exactly as in Case 2 above, we obtain that γ(p2 ) > −ε. 22

Case 4: p2 > α. We then have that |γ(p2 )| = |m(β) − m(α)| < ε by (3). We then follows that u1 (β, ϕ∗ (β), σ2 ) > u1 (α, ϕ∗ (α), σ2 ) − ε since σ2 ∈ M20 and (β, ϕ∗ (β)) ̸∈ A2 . As u1 (α, ϕ∗ (α), σ2 ) ≥ u1 (σ), it follows that u1 (β, ϕ∗ (β), σ2 ) > u1 (σ)− ε, establishing condition (∗). This completes the proof of the lemma. Theorem 2 now follows by combining Lemmas 7 and 8, Remark 4 and Theorem 1. Remark 10. Using a result in Carmona and Podczeck (2017), we now show that the set of limit admissible equilibria is independent of the choice of the allocation ϕ that determines players’ payoff functions in (1), in the following sense. Let ϕ˜ be a measur˜

able selection of Φ such that the conclusion of Claim 1 holds. Furthermore, let uϕi be ˜ ˜ ˜ defined by replacing ϕ with ϕ˜ in (1) for each i = 1, 2 and let GϕB = (X1 , X2 , uϕ1 , uϕ2 ). ˜

Then the closure of the set of undominated strategies of player i in GϕB is still Ci for each i = 1, 2; this holds because the specific definition of ϕ was only used to establish ˜

Claim 1. More importantly, L(GϕB ) = L(GB ). ˜ i = {(pi , si ) ∈ Xi : si = ϕ∗ (pi )} for each i = 1, 2, The see the above, let X ˜ B = (X ˜1, X ˜ 2 , u1 , u2 ) and G ˜ ϕ˜ = (X ˜1, X ˜ 2 , uϕ1˜, uϕ2˜). Note that X ˜ i ⊆ Xi and, hence, G B ˜ i ⊆ Mi for each i = 1, 2 and M ˜ ⊆ M . Carmona and Podczeck (2017, Section 5.1) M ˜

˜

˜ B ) = E(G ˜ ϕ ). For each G ∈ {GB , Gϕ }, we clearly have that have shown that E(G B B ˜ ⊆ E(G); ˜ furthermore, we have that E(G) ˜ ⊆ E(G) ∩ M ˜ due to Lemma E(G) ∩ M ˜ , we then obtain that L(Gϕ˜ ) = E(Gϕ˜ ) ∩ M 0 = E(G ˜ ϕ˜ ) ∩ M 0 = 3.7 As M 0 ⊆ M B B B ˜ B ) ∩ M 0 = E(GB ) ∩ M 0 = L(GB ). E(G 7

˜ . De˜ i ∈ {1, 2} and σ ′ ∈ Mi . We clearly have that σ ∈ M To see this, let σ ∈ E(G), i

˜ i by setting, for each (pi , si ) ∈ Xi , f (pi , si ) = (pi , ϕ∗ (pi )). Then, for each fine fi : Xi → X (pi , si , pj , xj ) ∈ X, ui (pi , si , pj , xj ) ≤ ui (f (pi , si ), pj , xj ) by Lemma 3. Hence, ui (σi′ , σj ) ≤ ∫ ∫ ˜i u (f (pi , si ), σj )dσi′ (pi , si ) = X˜ i ui (pi , si , σj )dσi′ ◦ f −1 (pi , si ) = ui (σi′ ◦ f −1 , σj ). As σi′ ◦ f −1 ∈ M Xi i ˜ we have that ui (σ ′ ◦ f −1 , σj ) ≤ ui (σ). Thus, ui (σ ′ , σj ) ≤ ui (σ) and, therefore, and σ ∈ E(G), i i σ ∈ E(G).

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References Aliprantis, C., and K. Border (2006): Infinite Dimensional Analysis. Springer, Berlin, 3rd edn. Andersson, O., C. Argenton, and J. Weibull (2014): “Robustness to Strategic Uncertainty,” Games and Economic Behavior, 85, 272–288. Bagh, A., and A. Jofre (2006): “Reciprocal Upper Semicontinuity and Better Reply Secure Games: A Comment,” Econometrica, 74, 1715–1721. Bich, P. (2016): “Strategic Uncertainty and Equilibrium Selection in Discontinuous Games,” Paris School of Economics. Billingsley, P. (1999): Convergence of Probability Measures. Wiley, New York, 2nd edn. Carbonell-Nicolau, O. (2011a): “On the Existence of Pure-Strategy Perfect Equilibrium in Discontinuous Games,” Games and Economic Behavior, 71, 23–48. (2011b): “Perfect and Limit Admissible Perfect Equilibria in Discontinuous Games,” Journal of Mathematical Economics, 47, 531–540. Carmona, G. (2013): Existence and Stability of Nash Equilibrium. World Scientific, Singapore. Carmona, G., and K. Podczeck (2017): “Invariance of the Equilibrium Set of Games with an Endogenous Sharing Rule,” University of Surrey and Universität Wien. Dasgupta, P., and E. Maskin (1986): “The Existence of Equilibrium in Discontinuous Economic Games, II: Applications,” Review of Economics Studies, 53, 27–41. Kohlberg, E., and J.-F. Mertens (1986): “On the Strategic Stability of Equilibria,” Econometrica, 54, 1003–1037.

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Maskin, E. (1986): “The Existence of Equilibrium with Price-Setting Firms,” American Economic Review, 76, 382–386. Reny, P. (1999): “On the Existence of Pure and Mixed Strategy Equilibria in Discontinuous Games,” Econometrica, 67, 1029–1056. (2016): “Nash Equilibrium in Discontinuous Games,” Economic Theory, 61, 553–569. Selten, R. (1975): “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games,” International Journal of Game Theory, 4, 25–55. Simon, L., and M. Stinchcombe (1995): “Equilibrium Refinements for Infinite Normal-Form Games,” Econometrica, 63, 1421–1443. van Damme, E. (1991): Perfection and Stability of Nash equilibrium. Springer Verlag, Berlin.

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