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Games 2011, 2(3), 235-256; doi:10.3390/g2030235

Article
The Existence of Perfect Equilibrium in Discontinuous Games
Oriol Carbonell-Nicolau
Department of Economics, Rutgers University, 75 Hamilton Street, New Brunswick, NJ 08901, USA; E-Mail: carbonell@econ.rutgers.edu; Tel.: +1-732-932-7363; Fax: +1-732-932-7416
Received: 18 February 2011; in revised form: 27 April 2011 / Accepted: 27 June 2011 /
Published: 15 July 2011

Abstract

: We prove the existence of a trembling-hand perfect equilibrium within a class of compact, metric, and possibly discontinuous games. Our conditions for existence are easily verified in a variety of economic games.
Keywords:
trembling-hand perfect equilibrium; discontinuous game; infinite normal-form game; payoff security

1. Introduction

A Nash equilibrium is trembling-hand perfect if it is robust to the players’ choice of unintended strategies through slight trembles. That is, in a world where agents make slight mistakes, trembling-hand perfection requires that there exist at least one perturbed model of low-probability errors with an equilibrium that is close to the original equilibrium, which is then thought of as an approximate description of “slightly constrained” rational behavior, or what could be observed if the players were to interact within the perturbed game. In this regard, a Nash equilibrium that is not trembling-hand perfect cannot be a good prediction of equilibrium behavior under any “conceivable” theory of (improbable, but not impossible) imperfect choice.

Ever since it was coined by Selten [1], trembling-hand perfection has been a popular solution concept. However, the fact that Selten's treatment is valid only for finite games poses a problem, since many strategic settings are most naturally modeled as games with a continuum of actions (e.g., models of price and spatial competition (Bertrand [2], Hotelling [3]), auctions (Milgrom and Weber [4]), and patent races (Fudenberg et al. [5])).

There have been attempts to use the notion of trembling-hand perfection in infinite economic games to rule out undesirable equilibria (examples include provision of public goods (Bagnoli and Lipman [6]), credit markets with adverse selection (Broecker [7]), budget-constrained sequential auctions (Pitchik and Schotter [8]), and principal-agent problems (Allen [9])).1 However, absent a theory of trembling-hand perfection for infinite games (and given that a well-accepted formulation for finite games has long been available), there has been a general tendency to study limits of sequences of trembling-hand perfect equilibria in discretized, successively larger versions of the original (infinite) game at hand (e.g., Bagnoli and Lipman [6], Broecker [7]).2 While this is a legitimate approach to trembling-hand perfection in infinite games, Simon and Stinchcombe [10] have shown that similar limit-of-finite approaches have limitations as general solution concepts even in continuous games. Moreover, since there are alternative formulations of trembling-hand perfection for infinite games, confining attention to a limit-of-finite approach, without any comparison with other concepts, seems unsatisfactory.

For continuous games, Simon and Sinchcombe [10] offer several notions of trembling-hand perfection and compare their properties. However, infinite economic games often exhibit discontinuities in their payoffs, and a treatment for this kind of games is not available. For instance, most of the above references feature discontinuous games. In the presence of discontinuities, existence of trembling-hand perfect equilibria is not guaranteed by standard arguments. By adapting arguments from Carbonell-Nicolau [11], this paper addresses the issue of existence for an infinite-game extension of Selten's [1] original notion of trembling-hand perfection. This extension corresponds to Simon and Stinchcombe's [10] strong approach when the universe of games is restricted by continuity of the players’ payoffs. Building on the existence results obtained here, a companion paper, Carbonell-Nicolau [12], compares the properties of various notions of trembling-hand perfection within families of discontinuous games, and states the analogue of the standard characterization of trembling-hand perfection for finite games (e.g., van Damme [13], p. 28), in terms of the strong approach and other formulations. This characterization is restated in Section 2.

We first illustrate that the existence of trembling-hand perfect equilibria depends crucially on the existence of Nash equilibria in Selten perturbations. Selten perturbations are perturbed games in which the players choose any strategy in their action space with positive probability. The strategy spaces in Selten perturbations of infinite, discontinuous games exhibit peculiarities that prevent a straightforward application of the results available in the literature on the existence of Nash equilibria. In fact, in Section 2 we show that Selten perturbations need not inherit Reny's [14] better-reply security from the original game. Even the available strengthenings of better-reply security—payoff security or uniform payoff security, along with upper semicontinuity of the sum of payoffs—do not generally give better-reply security (or some of its generalizations) in Selten perturbations. Thus, one must either rely on an appropriate generalization of the main existence theorem of Reny [14] or impose a suitable strengthening of better-reply security. We seek conditions on the payoffs of a game that prove useful in applications and imply better-reply security—and hence the existence of Nash equilibria—in Selten perturbations. Ideally, to avoid dealing with expected payoffs (defined on mixed strategies) and the weak convergence of measures, one would like to have conditions that can be verified using the payoffs of the original game, rather than its mixed extension.

Carbonell-Nicolau [11] introduces a condition—termed Condition (A)—that is used to prove the existence of a pure-strategy trembling-hand perfect equilibrium. This condition is used here to establish the existence of a mixed-strategy trembling-hand perfect equilibrium. While the current paper adapts arguments from [11], the results obtained here are not implied by those of [11]. We shall provide a detailed comparison with the results in [11] in Section 2.

Roughly speaking, Condition (A) is satisfied when there exists, for each player i, a measurable map f : XiXi, where Xi represents player i's action space, with the following two properties: (1) for each pure strategy xi of player i, there is an alternative pure strategy f(xi) such that given any pure action profile yi of the other players, the action f(xi) almost guarantees the payoff player i receives at (xi, yi), even if the other players slightly deviate from yi; and (2) given any pure action profile yi of the other players, there is a subset of generic elements of Xi (which may depend upon yi) such that given any generic pure strategy xi of player i, the action profile (xi, zi), where zi is a slight deviation from yi, almost guarantees the payoff player i receives at (f(xi), zi).

We show that this condition gives payoff security of certain Selten perturbations (Lemma 2). We then combine this finding with known results to establish the existence of a trembling-hand perfect equilibrium in discontinuous games (Theorem 2). In addition, we derive (as in Carbonell-Nicolau [11]) corollaries of these results in terms of two independent conditions—generic entire payoff security and generic local equi-upper semicontinuity—that imply the existence of a map f with the above properties. In applications, verifying the two independent conditions can prove easier than checking Condition (A), for Condition (A) typically requires constructing a measurable map and verifying two conditions that depend on one another (via the said measurable map)3 The alternative hypothesis does not explicitly require the measurability of the map f, and proves easy to verify in applications.

The hypotheses of the main existence theorems are satisfied in many economic games and are often rather simple to verify. This is exemplified in Section 3.

2. Perturbed Games and Perfect Equilibria

A metric game is a collection G = ( X i , u i ) i = 1 N, where N is a finite number of players, each Xi is a nonempty metric space, and each ui : X → ℝ is bounded and Borel measurable with domain X : = x i = 1 N X i. If in addition each Xi is compact, G is called a compact metric game.

In the sequel, by Xi we mean the set ×jiXj, and, given, i, xiXi, and

x i = ( x 1 , , x i 1 , x i + 1 , , x N ) x i
we slightly abuse notation and represent the point (x1, …, xN) as (xi, xi).

The mixed extension of G is the game

G ¯ = ( M i , U i ) i = 1 N
where each Mi represents the set of Borel probability measures on Xi, endowed with the weak* topology, and Ui:M → ℝ is defined by
U i ( μ ) : = X u i d μ
where M : = × i = 1 N M i.

Henceforth, the set ×jiMj is denoted as M−i, and given i, μiMi, and

μ i = ( μ 1 , , μ i 1 , μ i + 1 , , μ N ) M i
we sometimes represent the point (μ1, …,μN) as (μi, μi).

Given xiXi, let δxi be the Dirac measure on Xi with support {xi}. We sometimes write, by a slight abuse of notation, xi in place of δxi. For δ ∈ [0, 1] and ( μ i , v i ) M i 2

( 1 δ ) v i + δ μ i
denotes the member σi of Mi for which σi(B) = (1 − δ)νi(B) + δμi(B) for every Borel set B. When νi = δxi for some xiXi, we sometimes write (1 − δ)xi + δμi for (1 − δ)νi + δμi. Similarly, given (ν, μ) ∈ M2
( 1 δ ) v + δ μ
denotes the point
( ( 1 δ ) v 1 + δ μ 1 , , ( 1 δ ) v N + δ μ N )
where ν = (ν1,…, νN) and μ = (μ1,…, μN).

A number of definitions of trembling-hand perfection for infinite normal-form games have been proposed (cf. Simon and Stinchcombe [10], Al-Najjar [15]). For continuous games, the refinement specification considered here is equivalent to the strong approach in [10] and to the formulation in [15]. In this paper, we focus on the issue of existence. In passing, we also illustrate certain limitations of what appears to be a natural approach to the question of existence of trembling-hand perfect equilibria in discontinuous games. This is done more transparently if we frame our discussion in terms of just one notion of trembling-hand perfection. A companion paper, Carbonell-Nicolau [12], compares the various notions of trembling-hand perfection and studies their properties, and contains the analogue of the standard three-way characterization of trembling-hand perfection for finite games (e.g., van Damme [13], p. 28), which will be stated here after several definitions.

Before presenting the formal definition of trembling-hand perfection, we need some terminology.

A Borel probability measure μi on Xi is said to be strictly positive if μi(O) > 0 for every nonempty open set O in Xi.

For each i, let M i ^ stand for the set of all strictly positive members of Mi. Set M ^ : = x i = 1 N M i ^. For νi M i ^ and δ = (δ1,…, δN) ∈ [0, 1]N, define

M i ( δ i v i ) : = { μ i M i : μ i δ i v i }
and M ( δ v ) : = x i = 1 N M i ( δ i v i ). Given δ = (δ1,…, δN) ∈ [0, 1]N and ν = ( ν 1 , , ν N ) M ^, the game
G ¯ δ ν = ( M i ( δ i v i ) , U i | M ( δ ν ) ) i = 1 N
is called a Selten perturbation of G. We often work with perturbations G ¯ δ ν satisfying δ1 = … = δN

When referring to these objects, we simply write G ¯ δ ν with δ = δ1 = … = δN.

Definition 1

A strategy profile x = (x1,…, xN) ∈ X is a Nash equilibrium of G if for each i, ui(x) ≥ ui(yi, xi) for every yiXi.

Given a game G = ( X i , u i ) i = 1 N, a Nash equilibrium of the mixed extension is called a mixed-strategy Nash equilibrium of G. By a slight abuse of terminology, we sometimes refer to a mixed-strategy Nash equilibrium of G simply as a Nash equilibrium of G.

Definition 2

A strategy profile μM is a trembling-hand perfect (thp) equilibrium of G if there are sequences (δn), (νn), and (μn) with (0, 1)Nδn → 0, ν n M ^ and μnμ, where each μn is a Nash equilibrium of the perturbed game G ¯ δ n v n.

In words, μ is a thp equilibrium of G if it is the limit of some sequence of exact equilibria of neighboring Selten perturbations of G. Intuitively, Selten perturbations of G may be interpreted as “models of mistakes”, i.e., formal descriptions of strategic interactions where any player may “tremble” and play any one of her actions. The requirement that μ be the limit of some sequence of equilibria of perturbations of G says that there exists at least one model of (low-probability) mistakes that has at least one equilibrium close to μ, so that μ is an approximate description of what the players would do (at the said equilibrium) were they to interact in the perturbed game.

Remark 1

Note that, in Definition 2, we do not require that μ be a Nash equilibrium of G. It is well-known that, for continuous games, the fact that a strategy profile μ is the limit of some sequence of equilibria of Selten perturbations of G guarantees that μ is a Nash equilibrium of G. While we do not impose continuity of payoff functions, we shall show that our conditions also ensure that the limit point is an equilibrium.4

For μM, let Bri(μ) denote player i's set of best responses in Mi to the vector of strategies μ:

Br i ( μ ) : = { σ i M i : U i ( σ i , μ i ) sup ϱ i M i U i ( ϱ i , μ i ) }

Consider the following distance function between members of Mi:

ρ i s ( μ , v ) : = sup B | μ ( B ) v ( B ) |

Definition 3

(Simon and Stinchcombe [10]). Given > 0, a strong -perfect equilibrium of G is a vector μ M ^ such that for each i

ρ i s ( μ i , Br i ( μ ) ) <
A strategy profile in G is a strong perfect equilibrium of G if it is the weak* limit as n → 0 of strong n -perfect equilibria.

The following result is taken from Carbonell-Nicolau [12] and establishes the relationship between trembling-hand perfection and strong perfection in the presence of payoff discontinuities. The equivalence of (1)-(3) is analogous to the standard characterization of trembling-hand perfect equilibria for finite games (e.g., van Damme [13], p. 28).

Theorem 1

For a metric game, the following three conditions are equivalent.

  • μ is a trembling-hand perfect equilibrium ofG.

  • μ is a strong perfect equilibrium of G.

  • μ is the limit of a sequence (μn) in M ^ with the property that for each i and every > 0

    μ i n ( { x i X i : U i ( x i , μ i n ) sup y i X i U i ( y i , μ i n ) } ) 1
    for any sufficient large n.

The following example illustrates that the set of thp equilibria of an infinite game may well be a strict refinement of the set of Nash equilibria.

Example 1

Consider the two-player game G = ([0, 1], [0, 1], u1,u2), where u1 and u2 are defined by u1(x1, x2):= x1(1 – 2x2) and u 2 ( x 1 , x 2 ) : = x 1 x 2 2.

It is easily seen that the strategy profile (0,1) is a Nash equilibrium of G. Note however that

u 2 ( x 1 , 0 ) u 2 ( x 1 , x 2 ) , for all ( x 1 , x 2 ) [ 0 , 1 ] 2
and that the inequality is strict if x1 > 0. Therefore, player 2's best response to any tremble of player 1 in any Selten perturbation of G cannot be the action 1. Thus, the equilibrium (0, 1) is not thp.

The graph of G is the set

Γ G : = { ( x , α ) X × N : u i ( x ) = α i , for each i }
The graph of the mixed extension , Γ, is defined analogously. The closures of Γ and Γ are denoted by Γ and Γ̅ respectively.

Given {A, B} ⊆ ℝ ∋ ε, we write

A > ɛ and A > B ɛ
if a > ε, for all aA, and a > bε, for all (a, b) ∈ A x B, respectively. The definitions of Aε and ABε are analogous.

The following definition is taken from Reny [14].

Definition 4

The game G is better-reply secure if for every (x, α) ∈ Γ̅G such that x is not a Nash equilibrium of G, there exist i, yiXi, a neighborhood Ox−i of xi, and β ∈ ℝ such that ui(yi, Oxi) ≥ β > αi.

The following proposition is analogous to Proposition 1 in Carbonell-Nicolau [11]5 It suggests that the existence of Nash equilibria surviving trembling-hand perfection depends crucially on the existence of Nash equilibria in Selten perturbations of G.

Proposition 1

Suppose that G is a compact, metric game. If is better-reply secure and there exists (α, μ) ∈ (0, 1) × M ^ such that δμ has a Nash equilibrium for every δ ∈(0, α], then G possesses a trembling-hand perfect equilibrium, and all trembling-hand perfect equilibria of G are Nash.

Proof. Let (α, μ) be as in the statement of the proposition. Then, for large n, each n−1μ possesses a Nash equilibrium ϱn. Because ϱnM and M is sequentially compact, we may write (passing to a subsequence if necessary) ϱnϱ for some ϱM. Therefore, ϱ is a thp equilibrium of G.

To see that all thp equilibria of G are Nash, suppose that ϱ is a thp equilibrium of G, and let ϱn be the corresponding sequence of equilibria in Selten perturbations, i.e., each ϱn is a Nash equilibrium of δnμn, where δn → 0, μ n M ^, and ϱnϱ. We wish to show that ϱ is a (mixed-strategy) Nash equilibrium of G. To this end, we assume that ϱ is not an equilibrium and derive a contradiction.

Because ϱnϱ and each ui is bounded, we may write (passing to a subsequence if necessary)

( ϱ n , ( U 1 ( ϱ n ) , , U N ( ϱ n ) ) ) ( ϱ ( α 1 , , α N ) )
for some α : = (α1,…, αn) ∈ ℝN. Consequently, (ϱ, α) Γ̅ so if ϱ is not a Nash equilibrium of G, then, since is better-reply secure, some player i can secure a payoff strictly above αi at ϱ. That is, for some σiMi, some neighborhood Oϱ −i of ϱ −i, and some γ > 0
U i ( σ i , σ i ) α i + γ , for all σ i O ϱ i

We therefore have, in view of (1)

U i ( σ i , ϱ i n ) > U i ( ϱ n ) + β
for any sufficiently large n and some β > 0. Consequently, because δn → 0, for large enough n we have
U i ( ( 1 δ i n ) σ i + δ i n μ i n , ϱ i n ) > U i ( ϱ n )
thereby contradicting that ϱn is a Nash equilibrium in δn μn

In light of Proposition 1, it is only natural to ask whether the machinery developed within the literature on the existence of Nash equilibria in discontinuous games can be employed to show that Selten perturbations of G possess Nash equilibria. Reny ([14], Theorem 3.1) proves that a compact, metric, quasiconcave, and better-reply secure game possesses a Nash equilibrium.6 If G is a compact, metric game, then, for ( δ , μ ) [ 0 , 1 ) × M ^, δμ is a compact, metric game.7 In addition, δμ is easily seen to be quasiconcave. Consequently, a Selten perturbation δμ possesses a Nash equilibrium if it is better-reply secure. This observation, together with Proposition 1, gives the following lemma.

Lemma 1

If G is a compact, metric game and there exists ( α , μ ) ( 0 , 1 ) × M ^ such that δμ is better-reply secure for every δ ∈ [0, α], then G possesses a trembling-hand perfect equilibrium, and all trembling-hand perfect equilibria of G are Nash.

In general, verifying the existence of ( α , μ ) ( 0 , 1 ) × M ^ such that δμ is better-reply secure for every δ ∈ [0, α] is cumbersome, for it entails dealing with expected payoffs, defined on mixed strategies, and the weak* convergence of measures. Consequently, rather than imposing better-reply security directly on δμ, one would like to have conditions on the payoffs of the original game G that (1) prove useful in applications and (2) imply better-reply security in perturbations of G.

Unfortunately, δμ need not inherit better-reply security from G, and even standard strengthenings of better-reply security—payoff security or uniform payoff security (to be defined below), along with upper semicontinuity of i = 1 N u i—do not generally give the desired property in δμ.

The following definition is taken from Reny [14].

Definition 5

The game G is payoff secure if for each ε > 0, xX, and i, there exists yiXi such that ui(yi, Ox−i) > ui(x) − ε for some neighborhood Ox−i of xi.

It is well-known (Reny [14], Proposition 3.2) that payoff security of G and upper semicontinuity of i = 1 N u iensure better-reply security of G. However, payoff security of G and upper semicontinuity of i = 1 N u i need not give better-reply security of the mixed extension . The following example illustrates this point.

Example 2

(Sion and Wolfe [17]). Consider the game G = ([0,1], [0, 1], u1, u2), where

u 1 ( x 1 , x 2 ) : = { 1 0 1 if x 1 < x 2 < x 1 + 1 2 , if x 1 = x 2 or x 2 = x 1 + 1 2 , otherwise .
and u2 : = −u1 (Figure 1).

This game is zero-sum (and so i = 1 N u i is constant) and payoff secure (Carmona [18], Proposition 4). Moreover, as shown by Sion and Wolfe [17], G has no mixed-strategy Nash equilibria. Hence, by Corollary 5.2 of Reny [14], fails better-reply security.

Now consider the following strengthening of payoff security (cf. Monteiro and Page [19]).

Definition 6.

Given YiXi for each i, the game G is uniformly payoff secure over × i = 1 N Y i if for each i, ε > 0, and xiYi, there exists yiXi such that for every yiX−i, there is a neighborhood Oyi of yi such that ui(yi, Oyi) > ui(xi, y−i) − ε.

The game G is uniformly payoff secure if it is uniformly payoff secure over X.

Uniform payoff security of G yields payoff security of the mixed extension ([19], Theorem 1). By standard arguments, this means that uniform payoff security of G, together with upper semicontinuity of i = 1 N u i, implies better-reply security of . Nonetheless, these two conditions need not lead to better-reply security of δμ, as illustrated by the following example.8

Example 3

Let (αn) be a sequence in ( 1 2 , 1 ) with αn ↗ 1. Let (fn) be a sequence of functions fn : [0, 1] → ℝ. with the following properties:

f n ( x ) = { 1 if x [ 1 α n , α n ] { 0 , 1 } , 0 elsewhere .
for all n.

Consider the two-player game G = ([0, 1], [0, 1], u1, u2), where

u 1 ( x 1 , x 2 ) : = { 1 f n ( x 2 ) 1 0 if x 2 = α n , n = 1 , 2 , , if x 1 = α n , n = 1 , 2 , , if x 1 = 1 or , x 2 = 1 , elsewhere .
and u2(x1, x2) : = u1(x2, x1) (Figure 2).

It is easy to verify that i = 1 N u i is upper semicontinuous and G is uniformly payoff secure. However, δμ fails payoff security whenever μ M ^ and δ ∈ (0, 1). To see this, fix μ = ( μ 1 , μ 2 ) M ^ and δ ∈ (0, 1). We need to show that there exist ε > 0, i, and νM(δμ) such that for all σiMi(δμi) there is a point σ−i ∈ Xj≠iMj(δμj) arbitrarily close to ν i for which Uii, σ−i) ≤ Ui(ν) − ε. Thus, it suffices to establish the following for ε > 0 sufficiently small: there is an n such that for each neighborhood O(1-δ)αn+δμ2 of (1 − δ)αn + δμ2 and every y1 ∈ [0, 1]

U 1 ( ( 1 δ ) y 1 + δ μ 1 , v 2 ) U 1 ( ( 1 δ ) α n + δ μ 1 , ( 1 δ ) α n + δ μ 2 ) ɛ
for some ν2O(1-δ)αn+δμ2M2(δμ2).

Choose ε > 0 with the property that for any large enough n

δ ( 1 δ ) ( μ 1 ( { 1 } ) + m = n + 1 μ 1 ( { α m } ) ) δ ( 1 δ ) ( μ 2 ( { 0 , 1 } ) + μ 2 ( [ 1 α n , α n ] ) ) ɛ
Take any neighborhood O(1-δ)αn+δμ2 of (1 − δ)αn + δμ2 and any y1 ∈ [0, 1]. Clearly, we may pick some y2 ∈ (αn, αn+1) sufficiently close to αn to ensure that (1 − δ)y2 + δμ2O(1-δ)αn+δμ2 By linearity of U1 we have
U 1 ( ( 1 δ ) y 1 + δ μ 1 , ( 1 δ ) y 2 + δ μ 2 ) = ( 1 δ ) 2 U 1 ( y 1 , y 2 ) + ( 1 δ ) δ U 1 ( y 1 , y 2 ) + δ ( 1 δ ) U 1 ( μ 1 , y 2 ) + δ 2 U 1 ( μ )

Therefore, because U1(y1, y2) ≤ 1 ≥ U1(y1, μ2) and U 1 ( μ 1 , y 2 ) μ 1 ( { 1 } ) + m = n + 1 μ 1 ( { α m } )

U 1 ( ( 1 δ ) y 1 + δ μ 1 , ( 1 δ ) y 2 + δ μ 2 ) ( 1 δ ) 2 + ( 1 δ ) δ + δ ( 1 δ ) ( μ 1 ( { 1 } ) + m = n + 1 μ 1 ( { α m } ) ) + δ 2 U 1 ( μ )

On the other hand, we have

U 1 ( ( 1 δ ) α n + δ μ 1 , ( 1 δ ) α n + δ μ 2 ) ɛ = ( 1 δ ) 2 U 1 ( α n , α n ) + ( 1 δ ) δ U 1 ( α n , μ 2 ) + δ ( 1 δ ) U 1 ( μ 1 , α n ) + δ 2 U 1 ( μ ) ɛ = ( 1 δ ) 2 + ( 1 δ ) δ ( μ 2 ( { 0 , 1 } ) + μ 2 ( [ 1 α n , α n ] ) ) + δ ( 1 δ ) + δ 2 U 1 ( μ ) ɛ
and since the right-hand side of this equation is, in light of (3), greater than or equal to the right-hand side of (4), the desired inequality (2), follows. We conclude that δμ is not payoff secure.

The perturbation δμ also fails better-reply security. To see this, choose μ = ( μ 1 , μ 2 ) M ^ and δ ∈ (0, 1), and observe that

( ( ( 1 δ ) α n + δ μ 1 , ( 1 δ ) α n + δ μ 2 ) , ( γ 1 n , γ 2 n ) )
where
γ 1 n = U 1 ( ( 1 δ ) α n + δ μ 1 , ( 1 δ ) α n + δ μ 2 )
and
γ 2 n = U 2 ( ( 1 δ ) α n + δ μ 1 , ( 1 δ ) α n + δ μ 2 )
belongs to Γ̅G̅δμ. Moreover, the strategy profile
( ( 1 δ ) α n + δ μ 1 , ( 1 δ ) α n + δ μ 2 )
is not a Nash equilibrium in δμ, for
U 1 ( ( 1 δ ) 1 + δ μ 1 , ( 1 δ ) α n + δ μ 2 ) > U 1 ( ( 1 δ ) α n + δ μ 1 , ( 1 δ ) α n + δ μ 2 )
Reasoning as in the previous paragraph one can show that for large enough n there is no νiMi(δμi) for which U i ( v i , O ( 1 δ ) α n + δ μ i ) > γ i n for some neighborhood O(1-δ)αn+δμi of (1 − δ)αn + δμi. It follows that δμ is not better-reply secure.9

In light of Example 3, any condition on the payoff functions of G leading to the hypothesis of Lemma 1 (when combined with upper semicontinuity of i = 1 N u i) must be stronger than uniform payoff security.10

The following condition appears in Carbonell-Nicolau [11].

Condition (A)

There exists ( μ 1 , , μ N ) M ^ such that for each i and every ε > 0 there is a Borel measurable map f : XiXi such that the following is satisfied:

  • For each xiXi and every y−iXi, there is a neighborhood Oyi of y−i such that

    u i ( f ( x i ) , O y i ) > u i ( x i , y i ) ɛ

  • For each y−iX−i, there is a subset Yi of Xi with μi(Yi) = 1 such that for every xiYi, there is a neighborhood Vyi of y−i such that ui(f(xi), zi) < ui(xi, zi) + ε for all ziVyi.11

It is clear that (A) strengthens the concept of uniform payoff security.

Remark 2

The following implications are immediate:

continuity ⇒(A)
uniform payoff security
payoff security

We can establish payoff security of a Selten perturbation of G from Condition (A).

Lemma 2

Suppose that a compact, metric game G satisfies Condition (A). Then there exists μ M ^ such that δμ is payoff secure for every δ ∈ [0, 1).

This result plays a central role in the proof of the main results of this paper.12 The proof of Lemma 2 can be found in Section 4.

Lemma 2 can be combined with known results to prove an existence theorem. In fact, under the hypothesis of Lemma 2, we obtain payoff security of δμ for any δ ∈ [0, 1). If in addition i = 1 N u i is upper semicontinuous, since upper semicontinuity of i = 1 N u i gives upper semicontinuity of ΣiUi (Reny [14], Proposition 5.1), it follows that δμ is better-reply secure for any δ ∈ [0, 1) (Reny [14], Proposition 3.2). Applying Lemma 1 gives a thp equilibrium in G.

The following statement summarizes this finding.

Theorem 2

Suppose that a compact, metric game G satisfies Condition (A). Suppose further that i = 1 N u i is upper semicontinuous. Then G has a trembling-hand perfect equilibrium, and all trembling-hand perfect equilibria of G are Nash.

Lemma 2 is similar to Lemma 1 in Carbonell-Nicolau [11]. Lemma 1 in [11] states that if a compact, metric game satisfies Condition (A), then there exists μ M ^ such that the game G(δ,μ) is payoff secure for every δ ∈ [0, 1), where G(δ,μ) is defined as

G ( δ , μ ) = ( X i , u i ( δ , μ ) ) i = 1 N
and u i ( δ , μ ) : X is given by
u i ( δ , μ ) ( x ) : = u i ( ( 1 δ ) x 1 + δ μ 1 , , ( 1 δ ) x N + δ μ N )

The statement of this lemma differs from that of Lemma 2 in that G(δ,μ) and δμ are distinct objects. In fact, the latter can be shown to be homeomorphic to the mixed extension of the former. Consequently, since payoff security of a game does not generally imply payoff security of its mixed extension, Lemma 2 is not implied by Lemma 1 in [11]. On the other hand, it should be noted that Theorem 2 is not implied by Theorem 3 or Theorem 4 in [11]. In fact, both the hypothesis and the conclusion are weaker for Theorem 2.13

The remainder of this section derives a corollary of Theorem 2 in terms of two independent conditions introduced in [11]—generic entire payoff security and generic local equi-upper semicontinuity—that imply Condition (A). While stronger, these conditions prove useful in applications: they apply in a variety of economic games and do not explicitly require the measurability of the map f in Condition (A). Both Theorem 2 and its corollary (Corollary 1, in terms of generic entire payoff security and generic local equi-upper semicontinuity) are illustrated in Section 3.14

Let Ai be the set of all accumulation points of Xi (i.e., the set Ai of points xiXi such that (V ∖ {xi}) ⋂ Ai ≠ ∅ for each neighborhood V of xi). Since Xi is compact and metric, it can be written as a disjoint union AiKi, where Ai is closed and dense in itself (i.e., with no isolated points) and Ki is countable.

Let M ~ i be the set of measures μi in Mi such that μi({xi}) = 0 and μi (Ne(xi)) > 0 for each xiAi and every ∈ ≥ 0, and μi({xi}) > 0 for every xiKi. Define M ~ : = × i = 1 N M ~ i.

Clearly, M ~ i is a subset of M ˆ i. Moreover, M ~ i is nonempty In fact, it is not difficult to show that M ~ i is dense in Mi for each i.

Definition 7

Given YiXi for each i, we say that G is entirely payoff secure over × i = 1 N Y i if for each i, ∈ > 0, and xiYi, and for every neighborhood O of xi, there exist yiO and a neighborhood Oxi of xi such that for every yixi, there is a neighborhood Oyi of yi for which ui(yi, Oy_i) > ui(Oxi, yi) − ∈.

We say that G is entirely payoff secure if it is entirely payoff secure over X.

Definition 8

Given YiXi for each i, we say that the game G is generically entirely payoff secure over × i = 1 N Y i if there is, for each i, a set ZiYi with YiZi countable such that G is uniformly payoff secure over × i = 1 N Y i ( Y i \ Z i ) and entirely payoff secure over × i = 1 N Z i

A game G is generically entirely payoff secure if it is entirely payoff secure over × i = 1 N K i and generically entirely payoff secure over X i = 1 N A i (recall that Xi = AiKi, where Ai is closed and dense in itself and Ki is countable).

Remark 3

The following implications are immediate:

continuity ⇒entire payoff security
generic entire payoff security
uniform payoff security
payoff security

Definition 9

The game G is locally equi-upper semicontinuous if for each i, xiXi, and xiXi, and for each ε > 0, there exists a neighborhood Oxi of xi such that for every yiOxi there exists a neighborhood Ox−i of xi such that ui(yi, yi) < ui(xi, yi) + ε for all y−iOx−i.

Definition 10

The game G is generically locally equi-upper semicontinuous if there exists ( μ 1 , , μ N ) M ~ such that for each i and xi ∈ ×i, there exists YiXi with μi(Yi) = 1 such that for each xiYi and ε > 0, there exists a neighborhood Oxi of xi such that for every yiOxi there is a neighborhood Ox−i of x−i such that ui(yi, y−i) < ui(xi, y−i) + for all y−i ∈ Ox−i.

It turns out that generic entire payoff security and generic local equi-upper semicontinuity imply Condition (A).

Lemma 3

(Carbonell-Nicolau [11], Lemma 4). Suppose that G is generically entirely payoff secure and generically locally equi-upper semicontinuous. Then G satisfies Condition (A).

Lemma 3, combined with Theorem 2, gives the following result.

Corollary 1 (to Theorem 2)

Suppose that G is compact, metric, generically entirely payoff secure, and generically locally equi-upper semicontinuous. Suppose further that i = 1 N u i is upper semicontinuous. Then G has a trembling-hand perfect equilibrium, and all trembling-hand perfect equilibria of G are Nash.

3. Applications

The hypotheses of our main results are often satisfied in applications. This is illustrated by the following economic games.

Example 4 (Bertrand competition with discontinuous demand)

Consider a two-player Bertrand game G = ([0, 4], [0, 4], u1, u2), where

u i ( p i , p i ) : = { π ( p i ) 1 2 π ( p i ) 0 if p i < p i , if p i = p i , if p i > p i .
and
π ( p ) : = { p ( 8 p ) if 0 p 2 , p ( 4 p ) if 2 p 4 .
The map π(p) represents the operating profits that a monopolist charging a price p would earn (Figure 3). The two (identical) firms produce at zero costs, and the associated demand function is
D ( p ) = { 8 p if 0 p 2 , 4 p if 2 p 4.

Similar duopoly games can be found in Baye and Morgan [22]. See also [22] for a discussion on economic phenomena that explain demand discontinuities.

It is readily seen that i = 1 N u i is upper semicontinuous. Moreover, G is entirely payoff secure. To see this, fix i, > 0, pi ∈ [0, 4], and a neighborhood O of pi. We wish to show that there exist aiO and a neighborhood Opi around pi such that for every pi ∈ [0, 4], there is a neighborhood Op−i of pi for which

u i ( a i , O p i ) > u i ( O pi , p i ) ɛ
This is clearly true if pi = 0, for ui ≥ 0. Assume pi > 0, and choose aiO with 0 < ai < pi sufficiently close to pi to ensure that π(ai) > π(Opi) − ε for some neighborhood Opi of pi satisfying {ai} ⋂ Opi = ∅. Now fix p−i ∈ [0, 4], and pick a neighborhood Op−i of p−i satisfying the following:
  • If pi > ai, then Op−i ⋂ {ai} = ∅.

  • If piai, then Op−iOpi = ∅.

It is straightforward to verify that (5) holds.

Finally, G is generically locally equi-upper semicontinuous. In fact, take i, xi ∈ [0, 1], xi ∈ [0, 1] ∖ {2, xi}, and ε > 0. We only consider the case when xi < xi and xi < 2, for the other cases can be dealt with similarly If xi < xi and xi < 2, we have ui(yi, yi) = yi(8 − yi) for all (yi, yi) ∈ VxiVxi and for some neighborhoods Vxi and Vx−i of xi and xi respectively, so it is clear that there exists a neighborhood Oxi of xi such that for every yiOxi there is a neighborhood Oxi of x−i such that ui(yi, yi) < ui(xi, yi) + ε for all y−iOx−i.

Because G is generically locally equi-upper semicontinuous and entirely payoff secure, Corollary 1 can be invoked to establish the existence of a thp equilibrium.

Example 5 (all-pay auction)

There are N bidders competing for an object with a known value equal to 1. The highest bidder wins and every bidder pays his bid. Ties are broken via an equal probability rule. Given a profile of bids (b1, …, bN) ∈ [0, 1]N, the winning bid is maxi∈ {1,…,N} bi.

This situation can be modeled as an N-person normal-form game G = ( X i , u i ) i = 1 N, where Xi = [0, 1] and

u i ( b 1 , , b N ) : = { 1 # w ( b 1 , , b N ) b i if b i = max j { 1 , , N } b j , b i if b i < max j { 1 , , N } b j .
where W(b1,…, bN) : = {i : bi = maxj∈ {1,…, N} bj}.

This game is generically locally equi-upper semicontinuous. To see this, fix i and b−iXi, and choose any bi ∈ [0, 1]∖ {i} and any ∈ > 0, where i : = maxj∈ {1,…, N}∖{i}bj. We only consider the case when bi < i, for the case when bi > i can be handled analogously. Take a neighborhood (biδ,bi+ δ) of bi such that (bi − δ, bi + δ) ⋂ { i} = ∅ and δ < ∈. For each ai ∈ (bi − δi, bi + δ) ⋂[0,1] and for every aiXi in a neighborhood Ob−i of bi such that

c i < max j { 1 , , N } \ { i } c j , for all ( c i , c i ) ( b i δ , b i + δ ) × O b i
we have
u i ( a i , a i ) = a i < b i + δ < b i + δ = u i ( b i , a i ) + ɛ

We now show that G is generically entirely payoff secure.15 Fix a player i, and choose ∈ > 0, bi ∈ (0, 1), and a neighborhood O of bi (we omit the case when bi ∈ {0, 1}, which is easy to handle). We wish to show that there exist aiO and a neighborhood Obi such that for all bi ∈ [0, 1]N−1, there is a neighborhood Ob−i of b−i for which

u i ( a i , O b i ) > u i ( O bi , b i ) ɛ

Choose aiO ∩ (bi, bi + ), and fix a neighborhood Obi of bi such that Obi ⊆ [0, 1], ai ∈ [0, 1] ∩ (bi, bi + ), and {ai} ∩ Obi = ∅. Pick any bi ∈ [0, 1]N−1, and let Ob−i be a neighborhood of bi with the following property: if maxj∈ {1, …, N} bj < bi, then Obi ∩ {ai}N−1 = ∅. It is easy to verify that the choices of ai, Obi, and Obi yield Equation (6).

Finally, it is routine to verify that the sum of the bidders’ payoffs is continuous. Hence, Corollary 1 gives a thp equilibrium.

Example 6 (catalog games)

Page and Monteiro [23] consider a common agency contracting game in which firms compete for the business of an agent of unknown type tT, where T is a Borel subset of a separable, complete, and metric space. The distribution of types is represented by a Borel probability measure μ defined on T. There are two firms competing simultaneously in prices and products. The set of products each firm can offer is represented by a compact metric space X, and it is assumed that X contains an element 0, which denotes “no contracting”. The universe of prices that a firm can charge is denoted by D:= [0, ], with > 0. The agent can only contract with one firm and can choose to abstain from contracting altogether. Given i ∈ {1,2} and a closed subset Xi of X, let Ki:= Xi × D be the feasible set of products and prices that a firm i can offer. Assume the existence of a fictitious firm i = 0 with feasible set K0 : = {(0, 0)}. The agent chooses to abstain from contracting by choosing to contract with firm i = 0.

Each firm i competes by offering the agent a nonempty, closed subset CiKi, a catalog, of products and prices. Thus, each firm i's action space is P(Ki), the compact, metric space of catalogs, equipped with the Hausdorff distance. The utility of a type t agent who chooses (i, x, p) ∈ {0, 1, 2} × Ci is denoted as vt(i, x, p); we have vt(i, x, p) : = 0 if i = 0 and vt(i, x, p) : = ut(i, x) − p if i ∈ {1, 2}. It is assumed that utility is measurable in type t and continuous in contract choice (i, x, p). The agent's choice set given catalog profile (C1, C2) is given by

Γ ( C 1 , C 2 ) : = { ( i , x , p ) : i { 0 , 1 , 2 } , ( x , p ) C i }
A type t agent chooses (i, x, p) ∈ Γ(C1, C2) to maximize her utility:
max ( i , x , p ) Γ ( C 1 , C 2 ) v t ( i , x , p )
Define
v ( t , C 1 , C 2 ) : = max ( i , x , p ) Γ ( C 1 , C 2 ) v t ( i , x , p )
and
ϕ ( t , C 1 , C 2 ) : = arg max ( i , x , p ) Γ ( C 1 , C 2 ) v t ( i , x , p )
16 The map v*(t, ·) represents a type t agent's indirect utility function over profiles of catalogs, while Φ(t, ·) gives the type t agent's best responses to each catalog profile. The j-th firm's profit function is given by
π j ( i , x , p ) = { p c j ( x ) if j = i , 0 otherwise .
where the cost function cj(·) is bounded and lower semicontinuous. Let
π j ( C 1 , C 2 ) : = max ( i , x , p ) Φ ( t , C 1 , C 2 ) π j ( i , x , p )
Firm j's expected payoff under catalog profile (C1, C2) is
Π j ( C 1 , C 2 ) : = T π j ( , C 1 , C 2 ) d μ
The game G = (P(Ki), Πi) is an upper semicontinuous, compact game. Moreover, an argument similar to that provided in the proof of Theorem 5 of [23] to establish uniform payoff security of G can be utilized to prove that G satisfies Condition (A). Consequently, by Theorem 2, the game possesses a thp equilibrium.

Example 7 (provision of public goods)

Bagnoli and Lipman [6] study the following contribution game. There are I finitely many agents. By a slight abuse of notation, the set of agents is denoted by I. Each agent iI is endowed with an amount of wealth wi > 0. A collective decision d ∈ {0, 1} must be made (say, d = 1 designates the decision to provide streetlight, d = 0 represents the decision not to provide it). An outcome is a social decision together with an allocation of the private good (wealth) among the agents. The set of feasible outcomes is

{ ( d , x ) { 0 , 1 } × + I : i I x i i I w i c ( d ) }
The utility of agent i if outcome (d, x) is implemented is denoted by vi(d, xi); here, each vi is assumed strictly increasing in d and continuous and strictly increasing in xi. The cost of adopting decision d is c(d), where c(0) = 0 and c(1) = c > 0.

The agents simultaneously choose a contribution to the public project, each agent i's contribution being an element of Si : = [0, wi]. Let w denote the vector of endowments. Given a profile s = (si)iI of contributions, the public project is undertaken if ΣiI sic, in which case the realized outcome is (1, ws); otherwise (i.e.iI si < c) the outcome (0, w) obtains.

Let S : = X iI Si. The associated normal-form game is G = (Si, ui)iI, where ui : S → ℝ. is defined by

u i ( s ) : = { v i ( 1 , w i s i ) if i s i c , v i ( 0 , w i ) if i s i < c .
Bagnoli and Lipman[6] uses an equilibrium concept, termed undominated perfect equilibrium, that eliminates the set of weakly dominated strategies in the original game and applies the notion of trembling-hand perfection to the resulting game. To avoid defining trembling-hand perfection in infinite games and dealing with the issue of existence, Bagnoli and Lipman work with approximating finite versions of G.

Specifically, assuming vi(0, wi) = 0 for each i (a normalization that does not affect generality) and vi(1, 0) < 0 for each i (so that we do not need to consider cases when some agents would like to contribute more than their wealth), we can define ai implicitly by vi(1, wiai) = 0. Assume ΣiI wi > c. Clearly, the elimination of the interior of the set of weakly dominated strategies in G removes all siSi such that si > ai. Consider the “subgame” g of G in which i's strategy space is restricted to [0, ai] and g is otherwise identical to G.

Bagnoli and Lipman replace each Si by finite counterparts of varying grid sizes, and consider sequences of finite games in which the grid size converges to zero. They define an undominated perfect equilibrium in G as the limit of some sequence of undominated perfect equilibria of approximating finite versions of G.

The authors’ main result is that the game form G fully implements the core of the associated economy in undominated perfect equilibrium (i.e., any undominated perfect equilibrium of G induces a core allocation and vice versa). In view of our results, one may ask the following: Can one apply the characterization exercise conducted in [6] directly on the infinite game g? Can one obtain a similar theorem on the full implementation of the core in terms of trembling-hand perfection? While answering these questions requires a thorough analysis, Theorem 2 can be used to establish the existence of a thp equilibrium in g.

It is easily seen that the restriction of ui to [0, ai] is upper semicontinuous, so the sum of payoffs for g is upper semicontinuous.

We now show that g is entirely payoff secure. Take i, ∈ > 0, si [0, ai], and a neighborhood O of si. We need to show that there exist biO and a neighborhood Osi of si such that for every s-iXj≠i[0, aj], there is a neighborhood Os−i for which

u i ( b i , O s i ) > u i ( O si , O s i ) ɛ
The cases when si ∈ {0, ai} are easy to handle, so suppose that si ∈ (0, ai). Take biO with ai > bi > si close enough to si to ensure that
v i ( 1 , w i b i ) > v i ( 1 , w i O si ) ɛ
for some sufficiently small neighborhood Osi. Given si ∈ ×ji[0, aj], fix a neighborhood Osi with the following property: if Σj sjc, then bi + Σji s~jc for all -iOsi. Now, verifying that the choices of bi, Osi, and Osi give (7) is straightforward.

Finally, we show that g is generically locally equi-upper semicontinuous. For each i, let μi be the normalized Lebesgue measure over [0, ai]. Fix i and s-i ∈ ×j≠i[0, aj]. Consider the set of all si ∈ [0, ai] such that Σj sjc, a set that has full Lebesgue measure (i.e., it has μi-measure 1), and take any si in this set, and > 0. We only consider the case when Σj sj > c (the case when Σj sj < c can be dealt with analogously). Clearly, we may choose a neighborhood Osi of si in [0, ai] such that bi + Σj≠i sj > c and vi(1, wibi) < vi(1, wisi) + for all biOsi. Further, given biOsi, we may choose a neighborhood Osi of si in ×j≠i[0, aj] such that

b i + j i b j > c < s i + j i b j , for all b i O s i
Consequently, for every biOsi, we have
u i ( b i , b i ) = v i ( 1 , w i b i ) < v i ( 1 , w i s i ) + ɛ = u i ( s i , b i ) + ɛ

In light of Theorem 2, therefore, we obtain the non-emptiness of the set of trembling-hand perfect equilibria in g.

4. Proof of Lemma 2

To begin, we state a number of intermediate results.

Given a metric space X and YX, ℙ(Y) denotes the set of Borel probability measures on Y, and ℙ*(Y) is the subset of finitely supported measures in ℙ(Y) that assign rational values to each Borel set.

Lemma 4 (Carbonell-Nicolau [11], Lemma 6)

Let X be a compact metric space. Suppose that f : X → ℝ is bounded and Borel measurable. For each μ ∈ ℙ(X) and every > 0, there exists v* ∈ ℙ* (X) ∩ Nɛ (μ) such that | ∫X fdμ− ∫X fdv*| < ɛ.

Lemma 5 (Carbonell-Nicolau [11], Lemma 7)

Suppose that G is compact, metric, and satisfies Condition (A). Then there exists ( μ 1 , , μ N ) M ^ such that for each i and every ε > 0 there is a map f : XiXi such that the following is satisfied:

  • For each xiXi and every σiMi, there is a neighborhood Oσi of σi such that

    U i ( f ( x i ) , O σ i ) > U i ( x i , σ i ) ɛ

  • For every σiMi, there is a neighborhood Vσi of σi such that U i ( μ i f , p i ) < U i ( μ i , p i ) + ɛ for all p−iVσ−i, where μ i f M i is defined by μ i f ( B ) : = μ i ( f 1 ( B f ( X i ) ) ) .

Lemma 2

Suppose that a compact, metric game G satisfies Condition (A). Then there exists μ such that δμ is payoff secure for every δ ∈ [0, 1).

Proof. Fix δ ∈ [0, 1), and let ( μ 1 , , μ N ) M ^ be the measure given by Condition (A). We fix ε > 0, σ = (σ1, …, σN) ∈ M(δμ), and i, and show that there exists νiMi(δμi) such that Ui (νi, Oσ − i) > Ui(σ) – ε for some neighborhood Oσ − i of σ −i.

Lemma 5 gives a Borel measurable map f : XiXi satisfying the following:

  • For every yiXi, there is a neighborhood Oσi of σi such that U i ( f ( y i ) , O σ i ) > U i ( y i , σ i ) ɛ 4 .

  • There is a neighborhood Vσ −i of σ −i such that U i ( μ i f , p i ) < U i ( μ i , p i ) + ɛ 2 for all p-iVσ–i, where μ i f M i is defined by

    μ i f ( B ) : = μ i ( f 1 ( B f ( X i ) ) )

Claim 1

There exists a neighborhood Oσi of σi such that

X i U i ( f ( ) , O σ i ) d σ i > X i U i ( , σ i ) d σ i ɛ 2

Proof. By (i), for every yiXi there is a neighborhood Oσi of σi such that

U i ( f ( y i ) , O σ i ) > U i ( y i , σ i ) ɛ 4
For each n ∈ ℕ, define
X i n : = v i N 1 n ( σ i ) { y i X i : U i ( f ( y i ) , v i ) ) < U i ( y i , σ i ) ɛ 4 }
Each X i n is Borel measurable. In fact, Lemma 4 gives
X i n = v i N 1 n ( σ i ) ( X i ) X i ( v i )
where X i ( v i ) : = { y i X i : U i ( f ( y i ) , V i ) ) < U i ( y i , σ i ) ɛ 4 } Now, since ui and f are Borel measurable, for each v i N 1 n ( σ i ) the set Xi (V−i) is Borel measurable. Therefore, each X i n is (by (8)) a countable union of Borel sets, and hence a Borel set itself.

Now observe that we have n X i n = and X i n X i 2 . Consequently, for any large enough n,

σ i ( X i n ) sup ( v , ϱ ) M 2 [ U i ( v ) U i ( ϱ ) ] < ɛ 4
Hence, for any sufficiently large n,
X i U i ( f ( ) , N 1 n ( σ i ) ) d σ i = X i \ X i n U i ( f ( ) , N 1 n ( σ i ) ) d σ i + X i n U i ( f ( ) , N 1 n ( σ i ) ) d σ i > X i \ X i n U i ( , σ i ) d σ i + ɛ 4 + X i n U i ( f ( ) , N 1 n ( σ i ) ) d σ i > U i ( σ i , σ i ) ɛ 2
as desired.

Because σM(δμ), there exists, for each i, ϱiMi such that σi = (1δ) ϱi + δμi. Define

p i f : = ( 1 δ ) ϱ i f + δ μ i and υ i f : = ( 1 δ ) ϱ i f + δ μ i f
where ϱ i f M i is defined by ϱ i f ( B ) : = ϱ i ( f 1 ( B f ( X i ) ) ) .

By (ii), there exists a neighborhood Oσi of σi such that

U i ( μ i , p i ) > U i ( μ i f , p i ) ɛ 2 , for all p i O σ i
This, together with the definitions of p i f and υ i f gives, for any pi in some neighborhood of σi
U i ( p i f , p i ) = ( 1 δ ) U i ( ϱ i f , p i ) + δ U i ( μ i , p i ) > ( 1 δ ) U i ( ϱ i f , p i ) + δ U i ( μ i f , p i ) ɛ 2 = U i ( υ i f , p i ) ɛ 2
In addition, the definition of υ i fand the equality σ i = ( 1 δ ) ϱ i + δ μ i entail
U i ( υ i f , p i ) = X i U i ( , p i ) d υ i f = ( 1 δ ) X i U i ( , p i ) d ϱ i f + δ X i U i ( , p i ) d μ i f = ( 1 δ ) X i U i ( f ( ) , p i ) d ϱ i + δ X i U i ( f ( ) , p i ) d μ i = X i U i ( f ( ) , p i ) d σ i
Consequently, for every p-i in some neighborhood of σi we have
U i ( p i f , p i ) > U i ( υ i f , p i ) ɛ 2 = X i U i ( f ( ) , p i ) d σ i ɛ 2 > U i ( σ i , σ i ) ɛ
Here, the first inequality follows from (9), the second inequality is given by Claim 1, and the equality is a consequence of (10). Hence, because p i f M i ( δ μ i ), δμ is payoff secure.

Games 02 00235f1 1024
Figure 1. Example 2: The payoff functions of G.

Click here to enlarge figure

Figure 1. Example 2: The payoff functions of G.
Games 02 00235f1 1024
Games 02 00235f2 1024
Figure 2. Example 3: The payoff function for player 1.

Click here to enlarge figure

Figure 2. Example 3: The payoff function for player 1.
Games 02 00235f2 1024
Games 02 00235f3 1024
Figure 3. Example 4: Operating profit as a function of price.

Click here to enlarge figure

Figure 3. Example 4: Operating profit as a function of price.
Games 02 00235f3 1024

I am indebted to Efe Ok for his insights and encouragement; Efe read previous drafts and provided detailed comments. I also thank Rich McLean and Joel Sobel for several conversations, several anonymous referees for very useful remarks, and seminar participants at Barcelona Jocs and Rutgers for their comments. Part of this research was conducted while the author was visiting Universitat Autonoma de Barcelona. The author is grateful to this institution for its hospitality.

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  • 1For instance, sometimes the Nash equilibrium concept is too weak to sustain a given result, and the notion of trembling-hand perfection constitutes a natural refinement of the set of Nash equilibria. Beyond its intuitive appeal, trembling-hand perfection is weaker than other refinements, and therefore permits more general theories.
  • 2Allen [9] and Pitchik and Schotter [8] finitize their respective games at the outset, rather than approaching an infinite game by a series of successively larger finite games. However, their models are most conveniently analyzed in terms of continua of actions.
  • 3Constructing a measurable map can sometimes be cumbersome, especially if pure strategies are, say, maps between metric spaces rather than points in Euclidean space.
  • 4In Definition 2, each μn is an exact equilibrium of the perturbed game G ̅ δ n v n. Should one insist upon requiring that these equilibria be exact? While letting each μn be an n-equilibrium (with (n, δn) → 0) would still give a (weak) refinement of Nash equilibrium, any Nash equilibrium would survive this weakening of Definition 2. In fact, given a Nash equilibrium μ of G, take ν ∈ Mˆ and a sequence (0, 1) ∋ δn → 0, and observe that each
    ( 1 δ n ) μ + δ n ν = ( ( 1 δ n ) μ 1 + δ n ν 1 , , ( 1 δ n ) ) μ N + δ n ν N )
    is an n -equilibrium of G ̅ δ n ν for some n → 0, and we have (1 − δn)μ + δnνμ.
  • 5The reader is referred to the discussion following the statement of Theorem 2 for a comparison between Proposition 1 and Proposition 1 in [11].
  • 6A game G = ( X i , u i ) i = 1 N is quasiconcave if each Xi is a convex subset of a vector space and for each i and every xiXi, ui (.,xi) is quasiconcave of Xi.
  • 7If Xi is compact and metric, the weak* topology on Mi coincides with the topology induced by the Prokhorov metric on Mi. Hence, if Xi is nonempty, compact, and metric, then Mi(δiμi) is nonempty and metric. In addition, if Xi is nonempty, compact, and metric, Mi(δμi) is a nonempty convex subset of the weakly* compact set Mi. It is easy to check that Mi(δμi) is strongly closed, and therefore (Dunford and Schwartz ([16], Theorem V.3.13, p. 422)) weakly* closed, so Mi(δμi) is weakly* compact.
  • 8Even the generalized notion of better-reply security of Barelli and Soza [20] or the conditions for existence of Baye et al. [21] need not hold for the perturbation δμ when G is uniformly payoff secure and i = 1 N u i is upper semicontinuous.
  • 9While G is quasi-symmetric in the sense of Reny [14], and so an appropriate choice of μ renders G ̅ δ μ quasi-symmetric, G ̅ δ μ also violates diagonal better-reply security (as defined in [14]).
  • 10This means that the machinery developed in the literature on the existence of Nash equilibria cannot be employed to establish the existence of a Nash equilibrium in G ̅ δ μ under the assumption that G is uniformly payoff secure and i = 1 N u i is upper semicontinuous. We ignore if uniform payoff security of G and upper semicontinuity of i = 1 N u i implies the existence of a thp equilibrium in G. If this were true, its proof would require an appropriate generalization of the main theorem ofReny [14].
  • 11The following generalization of Condition (A) leaves all of our results intact.Condition(A′).There exists ( μ 1 , , μ N ) M ^ such that for each i and every ε > 0 there is a sequence (fk) of Borel measurable maps fk : XiXi such that the following is satisfied:
    • For each k, xiXi and y−iX−i there is a neighborhood Oy−iof y−i such that ui(fk(xi), Oy−i > ui(xi, y−i) – ε.

    • For each y−iX−i there is a subset Yi of Xi with μi (Yi) = 1 such that for each xiYi and every sufficiently large k, there is a neighborhood Vy−i of y−i such that ui(fk(xi), z−i)< ui(xi, z−i) + ε for all z−iVy−i

  • 12Lemma 2 is similar to Lemma 1 in Carbonell-Nicolau [11]. We provide a comparison between these two results after the statement of Theorem 2.
  • 13The hypothesis is weaker because it does not assume concavity or quasiconcavity-like conditions, while the conclusion is weaker because trembling-hand perfect equilibria may be in mixed strategies.
  • 14The relationship between Corollary 1 and Corollaries 1 and 3 in [11] is similar to that between Theorem 2 and Theorems 3 and 4 in [11]. In particular, Corollary 1 is not implied by the results in [11].
  • 15This game fails entire payoff security.
  • 16It is shown in [23] that v* is measurable in types and continuous in catalog profiles, while the correspondence Φ is jointly measurable in types and catalog profiles and upper hemicontinuous in catalog profiles.
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