The Existence of Perfect Equilibrium in Discontinuous Games

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.

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 : X_i → X_i, where X_i represents player i's action space, with the following two properties: (1) for each pure strategy x_i of player i, there is an alternative pure strategy f(x_i) such that given any pure action profile y_−i of the other players, the action f(x_i) almost guarantees the payoff player i receives at (x_i, y_−i), even if the other players slightly deviate from y_−i; and (2) given any pure action profile y_−i of the other players, there is a subset of generic elements of X_i (which may depend upon y_−i) such that given any generic pure strategy x_i of player i, the action profile (x_i, z_−i), where z_−i is a slight deviation from y_−i, almost guarantees the payoff player i receives at (f(x_i), z_−i).

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 $G={\left(X_i,u_i\right)}_{i=1}^N$, where N is a finite number of players, each X_i is a nonempty metric space, and each u_i : X → ℝ is bounded and Borel measurable with domain $X:=x_{i=1}^N{X}_i$. If in addition each X_i is compact, G is called a compact metric game.

In the sequel, by X_−i we mean the set ×_j≠iX_j, and, given, i, x_i ∈ X_i, and

$$x_{-i}=\left(x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_N\right)\in x_{-i}$$

we slightly abuse notation and represent the point (x₁, …, x_N) as (x_i, x_−i).

The mixed extension of G is the game

$$\overline{G}={\left(M_i,U_i\right)}_{i=1}^N$$

where each M_i represents the set of Borel probability measures on X_i, endowed with the weak* topology, and U_i:M → ℝ is defined by

$$U_i\left(\mu \right):={\int }_X{u}_i\text{d}\mu$$

where $M:={\times }_{i=1}^N{M}_i$.

Henceforth, the set ×_j≠iM_j is denoted as M_−i, and given i, μ_i ∈ M_i, and

$${\mu }_{-i}=\left({\mu }_1,\dots ,{\mu }_{i-1},{\mu }_{i+1},\dots ,{\mu }_N\right)\in M_{-i}$$

we sometimes represent the point (μ₁, …,μ_N) as (μ_i, μ_−i).

Given x_i ∈ X_i, let δ_xi be the Dirac measure on X_i with support {x_i}. We sometimes write, by a slight abuse of notation, x_i in place of δ_xi. For δ ∈ [0, 1] and $\left({\mu }_i,v_i\right)\in M_i^2$

$$\left(1-\delta \right)v_i+\delta {\mu }_i$$

denotes the member σ_i of M_i for which σ_i(B) = (1 − δ)ν_i(B) + δμ_i(B) for every Borel set B. When ν_i = δ_xi for some x_i ∈ X_i, we sometimes write (1 − δ)x_i + δμ_i for (1 − δ)ν_i + δμ_i. Similarly, given (ν, μ) ∈ M²

$$\left(1-\delta \right)v+\delta \mu$$

denotes the point

$$((1-\delta )v_1+\delta {\mu }_1{,}_{\cdots },\left(1-\delta \right)v_N+\delta {\mu }_N)$$

where ν = (ν₁,…, ν_N) and μ = (μ₁,…, μ_N).

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 μ_i on X_i is said to be strictly positive if μ_i(O) > 0 for every nonempty open set O in X_i.

For each i, let $\widehat{M_i}$ stand for the set of all strictly positive members of M_i. Set $\widehat{M}:={\text{x}}_{i=1}^N\widehat{M_i}$. For ν_i ∈ $\widehat{M_i}$ and δ = (δ₁,…, δ_N) ∈ [0, 1]^N, define

$$M_i\left({\delta }_i{v}_i\right):=\left\{{\mu }_i\in M_i:{\mu }_i\ge {\delta }_i{v}_i\right\}$$

and $M\left(\delta v\right):={\text{x}}_{i=1}^N{M}_i({\delta }_i{v}_i)$. Given δ = (δ₁,…, δ_N) ∈ [0, 1]^N and $\nu =($, the game

$${\overline{G}}_{\delta \nu }={\left(M_i\left({\delta }_i{v}_i\right),U_i|M\left(\delta \nu \right)\right)}_{i=1}^N$$

is called a Selten perturbation of G. We often work with perturbations ${\overline{G}}_{\delta \nu }$ satisfying δ₁ = … = δ_N

When referring to these objects, we simply write ${\overline{G}}_{\delta \nu }$ with δ = δ₁ = … = δ_N.

Definition 1

A strategy profile x = (x₁,…, x_N) ∈ X is a Nash equilibrium of G if for each i, u_i(x) ≥ u_i(y_i, x_−i) for every y_i ∈ X_i.

Given a game $G={\left(X_i,u_i\right)}_{i=1}^N$, a Nash equilibrium of the mixed extension G̅ 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 μ ∈ M is a trembling-hand perfect (thp) equilibrium of G if there are sequences (δⁿ), (νⁿ), and (μⁿ) with (0, 1)^N ∋ δⁿ → 0, ${\nu }^n$ and μⁿ → μ, where each μⁿ is a Nash equilibrium of the perturbed game ${\overline{G}}_{{\delta }^n{v}^n}$.

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 μ ∈ M, let Br_i(μ) denote player i's set of best responses in M_i to the vector of strategies μ:

$${\mathit{Br}}_i\left(\mu \right):=\left\{{\sigma }_i\in M_i:U_i\left({\sigma }_i,{\mu }_{-i}\right)\ge \underset{\varrho i\in M_i}{\text{sup}}{U}_i\left({\varrho }_i,{\mu }_{-i}\right)\right\}$$

Consider the following distance function between members of M_i:

$${\rho }_i^s\left(\mu ,v\right):=\underset{B}{\text{sup}}|\mu \left(B\right)-v\left(B\right)|$$

Definition 3

(Simon and Stinchcombe [10]). Given ∊ > 0, a strong ∊-perfect equilibrium of G is a vector ${\mu }^{∊}$ such that for each i

$${\rho }_i^s\left({\mu }_i^{∊},{\mathrm{Br}}_i\left({\mu }^{∊}\right)\right)<∊$$

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$\widehat{M}$ with the property that for each i and every ∊ > 0

  $${\mu }_i^n\left(\left\{x_i\in X_i:U_i\left(x_i,{\mu }_{-i}^n\right)\ge \underset{y_i\in X_i}{\text{sup}}{U}_i\left(y_i,{\mu }_{-i}^n\right)\right\}\right)\ge 1-∊$$
  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 $u_2\left(x_1,x_2\right):=-x_1{x}_2^2$.

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

$$u_2\left(x_1,0\right)\ge u_2\left(x_1,x_2\right),\text{for all}\left(x_1,x_2\right)\in {\left[0,1\right]}^2$$

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

$${\Gamma }_G:=\left\{\left(x,\alpha \right)\in X\times {ℝ}^N:u_i\left(x\right)={\alpha }_i,\text{for each}i\right\}$$

The graph of the mixed extension G̅, Γ_G̅, is defined analogously. The closures of Γ_G̅ and Γ_G̅ are denoted by Γ_G̅ and Γ̅_G̅ respectively.

Given {A, B} ⊆ ℝ ∋ ε, we write

$$A>\varepsilon \text{and}A>B-\varepsilon$$

if a > ε, for all a ∈ A, and a > b − ε, for all (a, b) ∈ A x B, respectively. The definitions of A ≥ ε and A ≥ B − ε are analogous.

The following definition is taken from Reny [14].

Definition 4

The game G is better-reply secure if for every (x, α) ∈ Γ̅_G such that x is not a Nash equilibrium of G, there exist i, y_i ∈ X_i, a neighborhood O_x−i of x_−i, and β ∈ ℝ such that u_i(y_i, O_x−i) ≥ β > α_i.

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 G̅ is better-reply secure and there exists (α, μ) ∈ (0, 1) × $\widehat{M}$ such that G̅_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 G̅_n⁻¹μ possesses a Nash equilibrium ϱⁿ. Because ϱⁿ ∈ M and M is sequentially compact, we may write (passing to a subsequence if necessary) ϱⁿ → ϱ for some ϱ ∈ M. 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 Ḡ_δⁿμⁿ, where δⁿ → 0, ${\mu }^n$, and ϱⁿ → ϱ. 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 ϱⁿ → ϱ and each u_i is bounded, we may write (passing to a subsequence if necessary)

$$\left({\varrho }^n,\left(U_1\left({\varrho }^n\right),\dots ,U_N\left({\varrho }^n\right)\right)\right)\to \left(\varrho \left({\alpha }_1,\dots ,{\alpha }_N\right)\right)$$

(1)

for some α : = (α₁,…, α_n) ∈ ℝ^N. Consequently, (ϱ, α) Γ̅_G̅ so if ϱ is not a Nash equilibrium of G, then, since G̅ is better-reply secure, some player i can secure a payoff strictly above α_i at ϱ. That is, for some σ_i ∈ M_i, some neighborhood O_{ϱ −i} of ϱ _−i, and some γ > 0

$$U_i\left({\sigma }_i,{\sigma }_{-i}\right)\ge {\alpha }_i+\gamma ,\text{for all}{\sigma }_{-i}\in O_{\varrho -i}$$

We therefore have, in view of (1)

$$U_i\left({\sigma }_i,{\varrho }_{-i}^n\right)>U_i\left({\varrho }^n\right)+\beta$$

for any sufficiently large n and some β > 0. Consequently, because δⁿ → 0, for large enough n we have

$$U_i\left(\left(1-{\delta }_i^n\right){\sigma }_i+{\delta }_i^n{\mu }_i^n,{\varrho }_{-i}^n\right)>U_i\left({\varrho }^n\right)$$

thereby contradicting that ϱⁿ is a Nash equilibrium in G̅_{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 $(\delta ,\mu )$, G̅_δμ is a compact, metric game.⁷ In addition, Ḡ_δμ is easily seen to be quasiconcave. Consequently, a Selten perturbation Ḡ_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 $(\alpha ,\mu )$ such that G̅_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 $(\alpha ,\mu )$ such that G̅_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 G̅_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, G̅_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 ${\sum }_{i=1}^N{u}_i$—do not generally give the desired property in G̅_δμ.

The following definition is taken from Reny [14].

Definition 5

The game G is payoff secure if for each ε > 0, x ∈ X, and i, there exists y_i ∈ X_i such that u_i(y_i, O_x−i) > u_i(x) − ε for some neighborhood O_x−i of x_−i.

It is well-known (Reny [14], Proposition 3.2) that payoff security of G and upper semicontinuity of ${\sum }_{i=1}^N{u}_i$ensure better-reply security of G. However, payoff security of G and upper semicontinuity of ${\sum }_{i=1}^N{u}_i$ need not give better-reply security of the mixed extension Ḡ. The following example illustrates this point.

Example 2

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

$$u_1\left(x_1,x_2\right):=\{\begin{array}{c}-1\\ 0\\ 1\end{array}\begin{array}{l}\text{if}{x}_1<x_2<x_1+\frac{1}{2},\hfill \\ \text{if}{x}_1=x_2\text{or}{x}_2=x_1+\frac{1}{2},\\ \text{otherwise}.\end{array}$$

and u₂ : = −u₁ (Figure 1).

This game is zero-sum (and so ${\sum }_{i=1}^N{u}_i$ 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], G̅ fails better-reply security.

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

Definition 6.

Given Y_i ⊆ X_i for each i, the game G is uniformly payoff secure over${\times }_{i=1}^N{Y}_i$ if for each i, ε > 0, and x_i ∈ Y_i, there exists y_i ∈ X_i such that for every y_−i ∈ X_−i, there is a neighborhood O_y−i of y_−i such that u_i(y_i, O_y−i) > u_i(x_i, 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 G̅ ([19], Theorem 1). By standard arguments, this means that uniform payoff security of G, together with upper semicontinuity of ${\sum }_{i=1}^N{u}_i$, implies better-reply security of G̅. Nonetheless, these two conditions need not lead to better-reply security of G̅_δμ, as illustrated by the following example.⁸

Example 3

Let (αⁿ) be a sequence in $\left(\frac{1}{2},1\right)$ with αⁿ ↗ 1. Let (fⁿ) be a sequence of functions fⁿ : [0, 1] → ℝ. with the following properties:

$$f^n\left(x\right)=\{\begin{array}{ll}1\hfill & \text{if}x\in \left[1-{\alpha }^n,{\alpha }^n\right]\cup \left\{0,1\right\},\\ 0& \text{elsewhere}.\end{array}$$

for all n.

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

$$u_1\left(x_1,x_2\right):=\{\begin{array}{l}1\hfill \\ f^n\left(x_2\right)\\ 1\hfill \\ 0\hfill \end{array}\begin{array}{l}\text{if}{x}_2={\alpha }^n,n=1,2,\dots ,\hfill \\ \text{if}{x}_1={\alpha }^n,n=1,2,\dots ,\\ \text{if}{x}_1=1\text{or},x_2=1,\\ \text{elsewhere}.\end{array}$$

and u₂(x₁, x₂) : = u₁(x₂, x₁) (Figure 2).

It is easy to verify that ${\sum }_{i=1}^N{u}_i$ is upper semicontinuous and G is uniformly payoff secure. However, G̅_δμ fails payoff security whenever $\mu$ and δ ∈ (0, 1). To see this, fix $\mu =$ and δ ∈ (0, 1). We need to show that there exist ε > 0, i, and ν ∈ M(δμ) such that for all σ_i ∈ M_i(δμ_i) there is a point σ_−i ∈ X_j≠iM_j(δμ_j) arbitrarily close to ν _−i for which U_i(σ_i, σ_−i) ≤ U_i(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]

$$U_1\left(\left(1-\delta \right)y_1+\delta {\mu }_1,v_2\right)\le U_1\left(\left(1-\delta \right){\alpha }^n+\delta {\mu }_1,\left(1-\delta \right){\alpha }^n+\delta {\mu }_2\right)-\varepsilon$$

(2)

for some ν₂ ∈ O_{(1-δ)αⁿ+δμ2} ⋂ M₂(δμ₂).

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

$$\delta \left(1-\delta \right)\left({\mu }_1\left(\left\{1\right\}\right)+\sum _{m=n+1}^{\infty }{\mu }_1\left(\left\{{\alpha }^m\right\}\right)\right)\le \delta \left(1-\delta \right)\left({\mu }_2\left(\left\{0,1\right\}\right)+{\mu }_2\left(\left[1-{\alpha }^n,{\alpha }^n\right]\right)\right)-\varepsilon$$

(3)

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 + δμ₂ ∈ O_{(1-δ)αⁿ+δμ2} By linearity of U₁ we have

$$U_1\left(\left(1-\delta \right)y_1+\delta {\mu }_1,\left(1-\delta \right)y_2+\delta {\mu }_2\right)={\left(1-\delta \right)}^2{U}_1\left(y_1,y_2\right)+\left(1-\delta \right)\delta U_1\left(y_1,y_2\right)+\delta \left(1-\delta \right)U_1\left({\mu }_1,y_2\right)+{\delta }^2{U}_1\left(\mu \right)$$

Therefore, because U₁(y₁, y₂) ≤ 1 ≥ U₁(y₁, μ₂) and $U_1\left({\mu }_1,y_2\right)\le {\mu }_1\left(\left\{1\right\}\right)+{\sum }_{m=n+1}^{\infty }{\mu }_1\left(\left\{{\alpha }^m\right\}\right)$

$$U_1\left(\left(1-\delta \right)y_1+\delta {\mu }_1,\left(1-\delta \right)y_2+\delta {\mu }_2\right)\le {\left(1-\delta \right)}^2+\left(1-\delta \right)\delta +\delta \left(1-\delta \right)\left({\mu }_1\left(\left\{1\right\}\right)+\sum _{m=n+1}^{\infty }{\mu }_1\left(\left\{{\alpha }^m\right\}\right)\right)+{\delta }^2{U}_1\left(\mu \right)$$

(4)

On the other hand, we have

$$\begin{array}{l}{U}_1\left(\left(1-\delta \right){\alpha }^n+\delta {\mu }_1,\left(1-\delta \right){\alpha }^n+\delta {\mu }_2\right)-\varepsilon \\ ={\left(1-\delta \right)}^2{U}_1\left({\alpha }^n,{\alpha }^n\right)+\left(1-\delta \right)\delta U_1\left({\alpha }^n,{\mu }_2\right)+\delta \left(1-\delta \right)U_1\left({\mu }_1,{\alpha }^n\right)+{\delta }^2{U}_1\left(\mu \right)-\varepsilon \\ ={\left(1-\delta \right)}^2+\left(1-\delta \right)\delta \left({\mu }_2\left(\left\{0,1\right\}\right)+{\mu }_2\left(\left[1-{\alpha }^n,{\alpha }^n\right]\right)\right)+\delta \left(1-\delta \right)+{\delta }^2{U}_1\left(\mu \right)-\varepsilon \end{array}$$

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 G̅_δμ is not payoff secure.

The perturbation G̅_δμ also fails better-reply security. To see this, choose $\mu$ and δ ∈ (0, 1), and observe that

$$\left(\left(\left(1-\delta \right){\alpha }^n+\delta {\mu }_1,\left(1-\delta \right){\alpha }^n+\delta {\mu }_2\right),\left({\gamma }_1^n,{\gamma }_2^n\right)\right)$$

where

$${\gamma }_1^n=U_1\left(\left(1-\delta \right){\alpha }^n+\delta {\mu }_1,\left(1-\delta \right){\alpha }^n+\delta {\mu }_2\right)$$

and

$${\gamma }_2^n=U_2\left(\left(1-\delta \right){\alpha }^n+\delta {\mu }_1,\left(1-\delta \right){\alpha }^n+\delta {\mu }_2\right)$$

belongs to Γ̅_G̅δμ. Moreover, the strategy profile

$$\left(\left(1-\delta \right){\alpha }^n+\delta {\mu }_1,\left(1-\delta \right){\alpha }^n+\delta {\mu }_2\right)$$

is not a Nash equilibrium in G̅_δμ, for

$$U_1\left(\left(1-\delta \right)1+\delta {\mu }_1,\left(1-\delta \right){\alpha }^n+\delta {\mu }_2\right)>U_1\left(\left(1-\delta \right){\alpha }^n+\delta {\mu }_1,\left(1-\delta \right){\alpha }^n+\delta {\mu }_2\right)$$

Reasoning as in the previous paragraph one can show that for large enough n there is no ν_i ∈ M_i(δμ_i) for which $U_i\left(v_i,O_{\left(1-\delta \right){\alpha }^n+\delta \mu -i}\right)>{\gamma }_i^n$ for some neighborhood O_{(1-δ)αⁿ+δμ−i} of (1 − δ)αⁿ + δμ_−i. It follows that G̅_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 ${\sum }_{i=1}^N{u}_i$) must be stronger than uniform payoff security.¹⁰

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

Condition (A)

There exists $({\mu }_1,\dots ,{\mu }_N)\in \widehat{M}$ such that for each i and every ε > 0 there is a Borel measurable map f : X_i → X_i such that the following is satisfied:

• For each x_i ∈ X_i and every y_−i ∈ X_−i, there is a neighborhood O_y−i of y_−i such that

  $$u_i\left(f\left(x_i\right),O_{y-i}\right)>u_i\left(x_i,y_{-i}\right)-\varepsilon$$

• For each y_−i ∈ X_−i, there is a subset Y_i of X_i with μ_i(Y_i) = 1 such that for every x_i ∈ Y_i, there is a neighborhood V_y−i of y_−i such that u_i(f(x_i), z_−i) < u_i(x_i, z_−i) + ε for all z_−i ∈ V_y−i.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 $\mu$ such that G̅_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 G̅_δμ for any δ ∈ [0, 1). If in addition ${\sum }_{i=1}^N{u}_i$ is upper semicontinuous, since upper semicontinuity of ${\sum }_{i=1}^N{u}_i$ gives upper semicontinuity of Σ_iU_i (Reny [14], Proposition 5.1), it follows that G̅_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 ${\sum }_{i=1}^N{u}_i$ 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 $\mu$ such that the game G_(δ,μ) is payoff secure for every δ ∈ [0, 1), where G_(δ,μ) is defined as

$$G_{(\delta ,\mu )}={(X_i,u_i^{(\delta ,\mu )})}_{i=1}^N$$

and $u_i^{(\delta ,\mu )}:X\to ℝ$ is given by

$$u_i^{(\delta ,\mu )}(x):=u_i((1-\delta )x_1+\delta {\mu }_1,\dots ,(1-\delta )x_N+\delta {\mu }_N)$$

The statement of this lemma differs from that of Lemma 2 in that G_(δ,μ) and G̅_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 A_i be the set of all accumulation points of X_i (i.e., the set A_i of points x_i ∈ X_i such that (V ∖ {x_i}) ⋂ A_i ≠ ∅ for each neighborhood V of x_i). Since X_i is compact and metric, it can be written as a disjoint union A_i ⋃ K_i, where A_i is closed and dense in itself (i.e., with no isolated points) and K_i is countable.

Let ${\tilde{M}}_i$ be the set of measures μ_i in M_i such that μ_i({x_i}) = 0 and μ_i (N_e(x_i)) > 0 for each x_i ∈ A_i and every ∈ ≥ 0, and μ_i({x_i}) > 0 for every x_i ∈ K_i. Define $\tilde{M}$.

Clearly, ${\tilde{M}}_i$ is a subset of ${\hat{M}}_i$. Moreover, ${\tilde{M}}_i$ is nonempty In fact, it is not difficult to show that ${\tilde{M}}_i$ is dense in M_i for each i.

Definition 7

Given Y_i ⊆ X_i for each i, we say that G is entirely payoff secure over${\times }_{i=1}^N{Y}_i$ if for each i, ∈ > 0, and x_i ∈ Y_i, and for every neighborhood O of x_i, there exist y_i ∈ O and a neighborhood O_xi of x_i such that for every y_−i ∈ x_−i, there is a neighborhood O_y−i of y_−i for which u_i(y_i, O_y__i) > u_i(O_xi, y_−i) − ∈.

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

Definition 8

Given Y_i ⊆ X_i for each i, we say that the game G is generically entirely payoff secure over${\times }_{i=1}^N{Y}_i$ if there is, for each i, a set Z_i ⊆Y_i with Y_i∖Z_i countable such that G is uniformly payoff secure over ${\times }_{i=1}^N{Y}_i(Y_i\backslash Z_i)$ and entirely payoff secure over ${\times }_{i=1}^N{Z}_i$

A game G is generically entirely payoff secure if it is entirely payoff secure over ${\times }_{i=1}^N{K}_i$ and generically entirely payoff secure over $X_{i=1}^N{A}_i$ (recall that X_i = A_i ⋃ K_i, where A_i is closed and dense in itself and K_i 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, x_−i ∈ X_−i, and x_i ∈ X_i, and for each ε > 0, there exists a neighborhood O_xi of x_i such that for every y_i ∈ O_xi there exists a neighborhood O_x−i of x_−i such that u_i(y_i, y_−i) < u_i(x_i, y_−i) + ε for all y_−i ∈ O_x−i.

Definition 10

The game G is generically locally equi-upper semicontinuous if there exists $({\mu }_1,\dots ,{\mu }_N)\in \tilde{M}$ such that for each i and x_−i ∈ ×_−i, there exists Y_i ⊆ X_i with μ_i(Y_i) = 1 such that for each x_i ∈ Y_i and ε > 0, there exists a neighborhood O_xi of x_i such that for every y_i ∈ O_xi there is a neighborhood O_x−i of x_−i such that u_i(y_i, y_−i) < u_i(x_i, y_−i) + ∈ for all y_−i ∈ O_x−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 ${\sum }_{i=1}^N$ 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

$$u_i(p_{i,}{p}_{-i}):=\{\begin{array}{c}\pi (p_i)\\ \frac{1}{2}\pi (p_i)\\ 0\end{array}\begin{array}{c}\text{if}{p}_i<p_{-i},\\ \text{if}{p}_i=p_{-i},\\ \text{if}{p}_i>p_{-i}.\end{array}$$

and

$$\pi (p):=\{\begin{array}{cc}p(8-p)& \text{if}0\le p\le 2,\\ p(4-p)& \text{if}2\le p\le 4.\end{array}$$

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

$$D(p)=\{\begin{array}{cc}8-p& \text{if}0\le p\le 2,\\ 4-p& \text{if}2\le p\le 4.\end{array}$$

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 ${\sum }_{i=1}^N{u}_i$ is upper semicontinuous. Moreover, G is entirely payoff secure. To see this, fix i, ∈ > 0, p_i ∈ [0, 4], and a neighborhood O of p_i. We wish to show that there exist a_i ∈ O and a neighborhood O_pi around p_i such that for every p_−i ∈ [0, 4], there is a neighborhood O_p−i of p_−i for which

$$u_i(a_i,O_{p-i})>u_i(O_{\mathrm{pi}},p_{-i})-\varepsilon$$

(5)

This is clearly true if p_i = 0, for u_i ≥ 0. Assume p_i > 0, and choose a_i ∈ O with 0 < a_i < p_i sufficiently close to p_i to ensure that π(a_i) > π(O_pi) − ε for some neighborhood O_pi of p_i satisfying {a_i} ⋂ O_pi = ∅. Now fix p_−i ∈ [0, 4], and pick a neighborhood O_p−i of p_−i satisfying the following:

• If p_−i > a_i, then O_p−i ⋂ {a_i} = ∅.

• If p_−i ≤ a_i, then O_p−i ⋂ O_pi = ∅.

It is straightforward to verify that (5) holds.

Finally, G is generically locally equi-upper semicontinuous. In fact, take i, x_−i ∈ [0, 1], x_i ∈ [0, 1] ∖ {2, x_−i}, and ε > 0. We only consider the case when x_i < x_−i and x_i < 2, for the other cases can be dealt with similarly If x_i < x_−i and x_i < 2, we have u_i(y_i, y_−i) = y_i(8 − y_i) for all (y_i, y_−i) ∈ V_xi − V_x−i and for some neighborhoods V_xi and V_x−i of x_i and x_−i respectively, so it is clear that there exists a neighborhood O_xi of x_i such that for every y_i ∈ O_xi there is a neighborhood O_x−i of x_−i such that u_i(y_i, y_−i) < u_i(x_i, y_−i) + ε for all y_−i ∈ O_x−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, …, b_N) ∈ [0, 1]^N, the winning bid is max_{i∈ {1,…,N}} b_i.

This situation can be modeled as an N-person normal-form game $G={(X_i,u_i)}_{i=1}^N$, where X_i = [0, 1] and

$$u_i(b_1,\dots ,b_N):=\{\begin{array}{cc}\frac{1}{\#w(b_1,\dots ,b_N)}-b_i& \text{if}{b}_i={\text{max}}_{j\in \left\{1,\dots ,N\right\}}{b}_j,\\ -b_i& \text{if}{b}_i<{\text{max}}_{j\in \left\{1,\dots ,N\right\}}{b}_j.\end{array}$$

where W(b₁,…, b_N) : = {i : b_i = max_{j∈ {1,…, N}} b_j}.

This game is generically locally equi-upper semicontinuous. To see this, fix i and b_−i ∈ X_−i, and choose any b_i ∈ [0, 1]∖ {b̅_−i} and any ∈ > 0, where b̅_−i : = max_{j∈ {1,…, N}∖{i}}b_j. We only consider the case when b_i < b̅ _−i, for the case when b_i > b̅ _−i can be handled analogously. Take a neighborhood (b_i − δ,b_i+ δ) of b_i such that (b_i − δ, b_i + δ) ⋂ {b̅ _−i} = ∅ and δ < ∈. For each a_i ∈ (b_i − δ_i, b_i + δ) ⋂[0,1] and for every a_−i ∈ X_−i in a neighborhood O_b−i of b_−i such that

$$c_i<\underset{j\in \left\{1,\dots ,N\right\}\backslash \left\{i\right\}}{max}{c}_j,\text{for}\text{all}(c_i,c_{-i})\in (b_i-\delta ,b_i+\delta )\times O_{b-i}$$

we have

$$\begin{array}{cc}{u}_i(a_i,a_{-i})\hfill & =-a_i\hfill \\ & <-b_i+\delta \hfill \\ & <-b_i+\delta \hfill \\ & =u_i(b_i,a_{-i})+\varepsilon \hfill \end{array}$$

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

$$u_i(a_i,O_{b-i})>u_i(O_{\mathrm{bi}},b_{-i})-\varepsilon$$

(6)

Choose a_i ∈ O ∩ (b_i, b_i + ∈), and fix a neighborhood O_bi of b_i such that O_bi ⊆ [0, 1], a_i ∈ [0, 1] ∩ (b_i, b_i + ∈), and {a_i} ∩ O_bi = ∅. Pick any b_−i ∈ [0, 1]^N−1, and let O_b−i be a neighborhood of b_−i with the following property: if max_{j∈ {1, …, N}} b_j < b_i, then O_b−i ∩ {a_i}^N−1 = ∅. It is easy to verify that the choices of a_i, O_bi, and O_b−_i 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 t ∈ T, 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, d̅], with d̅ > 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 X_i of X, let K_i:= X_i × 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 C_i ⊆ K_i, a catalog, of products and prices. Thus, each firm i's action space is P(K_i), 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} × C_i is denoted as v_t(i, x, p); we have v_t(i, x, p) : = 0 if i = 0 and v_t(i, x, p) : = u_t(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

$$\Gamma (C_1,C_2):=\left\{(i,x,p):i\in \left\{0,1,2\right\},(x,p)\in C_i\right\}$$

A type t agent chooses (i, x, p) ∈ Γ(C₁, C₂) to maximize her utility:

$$\underset{(i,x,p)\in \Gamma (C_1,C_2)}{\text{max}}{v}_t(i,x,p)$$

Define

$$v^{\ast }(t,C_1,C_2):=\underset{(i,x,p)\in \Gamma (C_1,C_2)}{\text{max}}{v}_t(i,x,p)$$

and

$$\varphi (t,C_1,C_2):=\underset{(i,x,p)\in \Gamma (C_1,C_2)}{\text{arg}\text{max}}{v}_t(i,x,p)$$

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

$${\pi }_j(i,x,p)=\{\begin{array}{cc}p-c_j(x)& \text{if}j=i,\\ 0& \text{otherwise}.\end{array}$$

where the cost function c_j(·) is bounded and lower semicontinuous. Let

$${\pi }_j^{\ast }(C_1,C_2):=\underset{(i,x,p)\in \Phi (t,C_1,C_2)}{\text{max}}{\pi }_j(i,x,p)$$

Firm j's expected payoff under catalog profile (C₁, C₂) is

$${\Pi }_j(C_1,C_2):={\int }_T{\pi }_j^{\ast }(\cdot ,C_1,C_2)\text{d}\mu$$

The game G = (P(K_i), Π_i) 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 i ∈ I is endowed with an amount of wealth w_i > 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

$$\left\{(d,x)\in \left\{0,1\right\}\times {ℝ}_{+}^I:{\sum }_{i\in I}{x}_i\le {\sum }_{i\in I}{w}_i-c(d)\right\}$$

The utility of agent i if outcome (d, x) is implemented is denoted by v_i(d, x_i); here, each v_i is assumed strictly increasing in d and continuous and strictly increasing in x_i. 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 S_i : = [0, w_i]. Let w denote the vector of endowments. Given a profile s = (s_i)_i∈I of contributions, the public project is undertaken if Σ_i∈I s_i ≥ c, in which case the realized outcome is (1, w – s); otherwise (i.e.,Σ_i∈I s_i < c) the outcome (0, w) obtains.

Let S : = X _i∈I S_i. The associated normal-form game is G = (S_i, u_i)_i∈I, where u_i : S → ℝ. is defined by

$$u_i(s):=\{\begin{array}{cc}{v}_i(1,w_i-s_i)& \text{if}{\sum }_i{s}_i\ge c,\\ v_i(0,w_i)& \text{if}{\sum }_i{s}_i<c.\end{array}$$

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 v_i(0, w_i) = 0 for each i (a normalization that does not affect generality) and v_i(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 v_i(1, w_i − a_i) = 0. Assume Σ_i∈I w_i > c. Clearly, the elimination of the interior of the set of weakly dominated strategies in G removes all s_i ∈ S_i such that s_i > a_i. Consider the “subgame” g of G in which i's strategy space is restricted to [0, a_i] and g is otherwise identical to G.

Bagnoli and Lipman replace each S_i 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 u_i to [0, a_i] 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, s_i [0, a_i], and a neighborhood O of s_i. We need to show that there exist b_i ∈ O and a neighborhood O_si of s_i such that for every s_-i ∈ X_j≠i[0, a_j], there is a neighborhood O_s−i for which

$$u_i(b_i,O_{s-i})>u_i(O_{\mathrm{si}},O_{s-i})-\varepsilon$$

(7)

The cases when s_i ∈ {0, a_i} are easy to handle, so suppose that s_i ∈ (0, a_i). Take b_i ∈ O with a_i > b_i > s_i close enough to s_i to ensure that

$$v_i(1,w_i-b_i)>v_i(1,w_i-O_{\mathrm{si}})-\varepsilon$$

for some sufficiently small neighborhood O_si. Given s_−i ∈ ×_j≠_i[0, a_j], fix a neighborhood O_s−i with the following property: if Σ_j s_j ≥ c, then b_i + Σ_j≠i s~_j ≥ c for all s̃_-i ∈ O_s−i. Now, verifying that the choices of b_i, O_si, and O_s−i give (7) is straightforward.

Finally, we show that g is generically locally equi-upper semicontinuous. For each i, let μ_i be the normalized Lebesgue measure over [0, a_i]. Fix i and s_-i ∈ ×_j≠i[0, a_j]. Consider the set of all s_i ∈ [0, a_i] such that Σ_j s_j ≠ c, a set that has full Lebesgue measure (i.e., it has μ_i-measure 1), and take any s_i in this set, and ∈ > 0. We only consider the case when Σ_j s_j > c (the case when Σ_j s_j < c can be dealt with analogously). Clearly, we may choose a neighborhood O_si of s_i in [0, a_i] such that b_i + Σ_j≠i s_j > c and v_i(1, w_i − b_i) < v_i(1, w_i − s_i) + ∈ for all b_i ∈ O_si. Further, given b_i ∈ O_si, we may choose a neighborhood O_s−i of s_−i in ×_j≠i[0, a_j] such that

$$b_i+\sum _{j\ne i}{b}_j>c<s_i+\sum _{j\ne i}{b}_j,\text{for}\text{all}{b}_{-i}\in O_{s-i}$$

Consequently, for every b_−i ∈ O_s−i, we have

$$u_i(b_i,b_{-i})=v_i(1,w_i-b_i)<v_i(1,w_i-s_i)+\varepsilon =u_i(s_i,b_{-i})+\varepsilon$$

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 Y ⊆ X, 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) ∩ N_ɛ (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 $({\mu }_1,\dots ,{\mu }_N)\in \widehat{M}$ such that for each i and every ε > 0 there is a map f : X_i → X_i such that the following is satisfied:

• For each x_i ∈ X_i and every σ_−i ∈ M_−i, there is a neighborhood O_σ−i of σ_−i such that

  $$U_i(f(x_i),O_{\sigma -i})>U_i(x_i,{\sigma }_{-i})-\varepsilon$$

• For every σ_−i ∈ M_−i, there is a neighborhood V_σ−i of σ_−i such that $U_i({\mu }_i^f,p_{-i})<U_i({\mu }_i,p_{-i})+\varepsilon$ for all p_−i ∈ V_σ−i, where ${\mu }_i^f\in M_i$ is defined by ${\mu }_i^f(B):={\mu }_i(f^{-1}(B\cap f(X_i))).$

Lemma 2

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

Proof. Fix δ ∈ [0, 1), and let $({\mu }_1,\dots ,{\mu }_N)\in \widehat{M}$ be the measure given by Condition (A). We fix ε > 0, σ = (σ₁, …, σ_N) ∈ M(δμ), and i, and show that there exists ν_i ∈ M_i(δμ_i) such that U_i (ν_i, O_{σ − i}) > U_i(σ) – ε for some neighborhood O_{σ − i} of σ _−i.

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

• For every y_i ∈ X_i, there is a neighborhood O_σ−i of σ_−i such that $U_i(f(y_i),O_{\sigma -i})>U_i(y_i,{\sigma }_{-i})-\frac{\varepsilon }{4}.$

• There is a neighborhood V_{σ −i} of σ _−i such that $U_i({\mu }_i^f,p-i)<U_i({\mu }_i,p-i)+\frac{\varepsilon }{2}$ for all p_-i ∈ V_σ–i, where ${\mu }_i^f\in M_i$ is defined by

  $${\mu }_i^f(B):={\mu }_i(f^{-1}(B\cap f(X_i)))$$

Claim 1

There exists a neighborhood O_σ−i of σ_−i such that

$${\int }_{X_i}{U}_i(f(\cdot ),O_{\sigma -i})\text{d}{\sigma }_i>{\int }_{X_i}{U}_i(\cdot ,{\sigma }_{-i})\text{d}{\sigma }_i-\frac{\varepsilon }{2}$$

Proof. By (i), for every y_i ∈ X_i there is a neighborhood O_σ−i of σ_−i such that

$$U_i(f(y_i),O_{\sigma -i})>U_i(y_i,{\sigma }_{-i})-\frac{\varepsilon }{4}$$

For each n ∈ ℕ, define

$$X_i^n:=\underset{v-i\in N\frac{1}{n}(\sigma -i)}{\cup }\{y_i\in X_i:U_i(f(y_i),v-i))<U_i(y_i,{\sigma }_{-i})-\frac{\varepsilon }{4}\}$$

Each $X_i^n$ is Borel measurable. In fact, Lemma 4 gives

$$X_i^n=\underset{v-i\in N\frac{1}{n}(\sigma -i)\cap {\mathbb{P}}_{\ast }(X-i)}{\cup }{X}_i(v-i)$$

(8)

where $X_i(v-i):=\{y_i\in X_i:U_i(f(y_i),V_{-i}))<U_i(y_i,{\sigma }_{-i})-\frac{\varepsilon }{4}\}\cdot$ Now, since u_i and f are Borel measurable, for each $v_{-i}\in N\frac{1}{n}({\sigma }_{-i})$ the set X_i (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 ${\cap }_n{X}_i^n=\oslash$ and $X_i^n\supseteq X_i^2\supseteq \cdots .$ Consequently, for any large enough n,

$${\sigma }_i(X_i^n)\underset{(v,\varrho )\in M^2}{\text{sup}}\left[U_i(v)-U_i(\varrho )\right]<\frac{\varepsilon }{4}$$

Hence, for any sufficiently large n,

$${\int }_{X_i}{U}_i(f(\cdot ),N_{\frac{1}{n}}({\sigma }_{-i}))\text{d}{\sigma }_i={\int }_{X_i\backslash X_i^n}{U}_i(f(\cdot ),N_{\frac{1}{n}}({\sigma }_{-i}))\text{d}{\sigma }_i+{\int }_{X_i^n}{U}_i(f(\cdot ),N_{\frac{1}{n}}({\sigma }_{-i}))\text{d}{\sigma }_i>{\int }_{X_i\backslash X_i^n}{U}_i(\cdot ,{\sigma }_{-i})\text{d}{\sigma }_i+\frac{\varepsilon }{4}+{\int }_{X_i^n}{U}_i(f(\cdot ),N_{\frac{1}{n}}({\sigma }_{-i}))\text{d}{\sigma }_i>U_i({\sigma }_i,{\sigma }_{-i})-\frac{\varepsilon }{2}$$

as desired.

Because σ ∈ M(δμ), there exists, for each i, ϱi ∈ M_i such that σ_i = (1 − δ) ϱ_i + δμ_i. Define

$$p_i^f:=(1-\delta ){\varrho }_i^f+\delta {\mu }_i\text{and}{\upsilon }_i^f:=(1-\delta ){\varrho }_i^f+\delta {\mu }_i^f$$

where ${\varrho }_i^f\in M_i$ is defined by ${\varrho }_i^f(B):=\varrho i(f^{-1}(B\cap f(X_i))).$

By (ii), there exists a neighborhood O_σ−i of σ_−i such that

$$U_i({\mu }_i,p_{-i})>U_i({\mu }_i^f,p_{-i})-\frac{\varepsilon }{2},\text{for all}{p}_{-i}\in O_{\sigma -i}$$

This, together with the definitions of $p_i^f$ and ${\upsilon }_i^f$ gives, for any p_−i in some neighborhood of σ_−i

$$\begin{array}{cc}{U}_i(p_i^f,p_{-i})& =(1-\delta )U_i({\varrho }_i^f,p_{-i})+\delta U_i({\mu }_i,p_{-i})\hfill \\ & >(1-\delta )U_i({\varrho }_i^f,p_{-i})+\delta U_i({\mu }_i^f,p_{-i})-\frac{\varepsilon }{2}\hfill \\ & =U_i({\upsilon }_i^f,p_{-i})-\frac{\varepsilon }{2}\hfill \end{array}$$

(9)

In addition, the definition of ${\upsilon }_i^f$and the equality ${\sigma }_i=(1-\delta )\varrho i+\delta {\mu }_i$ entail

$$\begin{array}{cc}{U}_i({\upsilon }_i^f,p_{-i})& ={\int }_{X_i}{U}_i(\cdot ,p_{-i})\text{d}{\upsilon }_i^f\hfill \\ & =(1-\delta ){\int }_{X_i}{U}_i(\cdot ,p_{-i})\text{d}{\varrho }_i^f+\delta {\int }_{X_i}{U}_i(\cdot ,p_{-i})\text{d}{\mu }_i^f\hfill \\ & =(1-\delta ){\int }_{X_i}{U}_i(f(\cdot ),p_{-i})\text{d}{\varrho }_i+\delta {\int }_{X_i}{U}_i(f(\cdot ),p_{-i})\text{d}{\mu }_i\hfill \\ & ={\int }_{X_i}{U}_i(f(\cdot ),p_{-i})\text{d}{\sigma }_i\hfill \end{array}$$

(10)

Consequently, for every p_-i in some neighborhood of σ_−i we have

$$\begin{array}{cc}{U}_i(p_i^f,p_{-i})& >U_i({\upsilon }_i^f,p_{-i})-\frac{\varepsilon }{2}\hfill \\ & ={\int }_{X_i}{U}_i(f(\cdot ),p_{-i})\text{d}{\sigma }_i-\frac{\varepsilon }{2}\hfill \\ & >U_i({\sigma }_i,{\sigma }_{-i})-\varepsilon \hfill \end{array}$$

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 $p_i^f\in M_i(\delta {\mu }_i)$, G̅_δμ is payoff secure.