Equilibrium refinements in games with many players

This paper introduces three notions of perfect equilibrium for games with many players, respectively, in behavioral, mixed and pure strategies. The equivalence between behavioral strategy perfect equilibrium and mixed strategy perfect equilibrium is established. More importantly, it is shown that after the resolution of strategic uncertainty, a mixed strategy perfect equilibrium leads to a pure strategy perfect equilibrium almost surely. Various properties related to limit admissibility are also considered.

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1 Introduction

Selten (1975) introduces (trembling hand) perfect equilibrium to restrict the set of Nash equilibria in finite games (i.e., games with finite players and finite actions). This refinement precludes weakly dominated actions by requiring some notion of neighborhood robustness to small perturbations of the original game. Based on the idea, Simon and Stinchcombe (1995) formulate perfect equilibrium in finite-player games with infinitely many actions, and showed its existence and several properties. In the context of games with a continuum of players (hereafter large games), Rath (1994, 1998) provides a notion of perfect equilibrium in large games with finite actions and consequently established its existence.

Sun and Zeng (2020) consider the perfect equilibrium in large games with infinitely many actions. Unlike Rath (1994, 1998), Sun and Zeng (2020) introduce a new notion of perfect equilibrium to capture the essential idea of perfection by working with perturbations of societal summaries rather than societal summaries themselves.111Whereas the notion of perfect equilibrium proposed by Rath (1994, 1998) may not be (limit) admissible (see Section 5 in Rath (1998)), the notion of perfect equilibrium in Sun and Zeng (2020) forces almost all the players to choose (limit) admissible actions via incorporating perturbations of societal summaries. To obtain the existence of pure strategy perfect equilibria, Sun and Zeng (2020) turn to the nowhere equivalence condition introduced in He et al. (2017). They show that a large game always has a pure strategy perfect equilibrium whenever the underlying player space satisfies the nowhere equivalence condition. Furthermore, they also establish the limit admissibility of perfect equilibria.

In the paper, we work with large games with saturated player spaces, where each player’s payoff function continuously depends on her own action and on the societal summary induced by other players’ actions. We present three new notions of perfect equilibrium for games with many players. A main departure from the earlier notions of perfect equilibrium in behavioral and pure strategies is that we also allow perturbations on the agent space, in addition to perturbations of individual actions as in Rath (1994, 1998) and perturbations of societal summaries as Sun and Zeng (2020). We also introduce mixed strategy perfect equilibrium for the first time.

We prove the equivalence between behavioral/randomized strategy perfect equilibrium and mixed strategy perfect equilibrium in that the first can be consolidated and lifted up into the second, and that the second can be personalized to induce the first; see Theorem 1. More importantly, we show in Theorem 2 the property of ex post perfection for mixed strategy perfect equilibria: after the resolution of strategic uncertainty, a mixed strategy perfect equilibrium leads to a pure strategy perfect equilibrium almost surely. Moreover, we also find the ex post property of limit admissibility—A mixed strategy profile is limit admissible if and only if it is ex post limit admissible.

The rest of the paper is organized as follows. In Section 2, we present the formal definitions of large games and Nash equilibria. In Section 3, we state behavioral strategy perfect equilibria and related results. In Section 4, we introduce mixed strategy perfect equilibria in the framework of Fubini extension and present the main results. The limit admissibility is discussed in Section 5. Some technological proofs are collected in Section 6.

2 Large games and Nash equilibria

In this section, we state the formal definition of large games and Nash equilibrium. It is conventional that a large (atomless) game has three basic elements: an atomless probability space modeling the space of players,222Throughout this paper, we assume that the probability space is always a complete and countably additive. a compact metric space representing the common action set for each player, and a set of payoff functions defined on the set . A large game is a measurable function from to , which assigns payoff functions for all the players. In a large game, a pure strategy profile is a measurable function from to . We explain more details in the following.

Let denote the Borel -algebra of and let denote the set of all Borel probability measures on . Notice that the space (with the Prokhorov metric) is also a weakly compact and convex metric space. A behavioral strategy profile (resp. pure strategy profile) is a measurable function from to (resp. ). Given a behavioral strategy profile , we model the societal summary as its Gelfand integral , which is an element in and represents the average action distribution of all the players. Also notice that, when is a pure strategy profile, its Gelfand integral equals , which is the action distribution induced by .

The set serves as the set of societal summaries. Each player’s payoff continuously depends on her own action and the societal summary, i.e., a continuous function on the product space of and . The set of all possible payoff functions is denoted by —the set of all continuous functions on . Endowed with its sup-norm topology and the resulting Borel -algebra, would be conceived as a measurable space .

We are now ready to present the definitions of large games and pure strategy Nash equilibria:

Definition 1.

A large game is a measurable function from to . For convenience, each is usually rewritten as .

A pure strategy Nash equilibrium of is a pure strategy profile such that for -almost all ,

To guarantee the existence of pure strategy Nash equilibria, we introduce saturated probability spaces.

Definition 2.

A probability space is said to be (essentially) countably generated if its -algebra can be generated by a countable number of subsets together with the null sets; otherwise, it is not countably generated.

A probability space is saturated if it is nowhere countably generated in the sense that for any subset with , the restricted probability space is not countably generated, where and is the probability measure rescaled from the restriction of to .

Keisler and Sun (2009) show that a saturated player space is a necessary and sufficient condition to guarantee the existence of pure strategy Nash equilibria—every large game from to has a pure strategy Nash equilibrium if and only if is a saturated probability space. Throughout this paper, we assume that the player space is always saturated.

Although each large game with a saturated player space has a pure strategy Nash equilibrium, there exist some large games that have “bad” Nash equilibria.

Example 1.

We consider a large number of smartphone sellers in a market. We identify the agent space with a saturated probability . Each seller has three options: selling high-quality smartphones (action ), selling regular smartphones (action ), and selling low-quality smartphones (action ).

For each seller , if he sells regular smartphones, his payoff is normalized to 0; If he chooses to sell low-quality smartphones, then his payoff is , where is the proportion of sellers who sell regular smartphones. That is, the payoff of a seller who sells low-quality smartphones decreases if more sellers sell regular smartphones, and the payoff for selling low-quality smartphones is less than the payoff for selling the regular smartphones. If a seller chooses to sell high-quality smartphones, then his payoff is assumed to be . This payoff function is reasonable since the payoff for selling high-quality smartphones will be higher if more sellers sell regular ones, and a seller has incentive to sell high-quality smartphones (compared with regular ones) only if more than of the sellers sell regular ones.

In summary, the payoff functions in this game are as follows:

for each player .

It is easy to check that the strategy profile is a Nash equilibrium: in this case, is zero and hence is one of the best choices for each player. However, this Nash equilibrium is a “bad” equilibrium since each seller sells low-quality smartphones, and there is neither regular smartphones nor high-quality smartphones in this market. ∎

3 Behavioral strategy perfect equilibria

To address the problems mentioned in the previous section, we consider the concept of perfect equilibrium, which is a refinement of notion of Nash equilibrium.

A behavioral strategy profile is fully supported if for -almost all , the probability measure assigns strictly positive probability to every nonempty open subset of . This notion is used to capture the idea that every player may “tremble” and play any one of her actions.

Simon and Stinchcombe (1995) introduce the strong metric and weak metric to measure the distance between behavioral strategies for finite-player game with infinitely many actions. Given two probability measures and on , and are defined as follows:

where is the -neighborhood of the Borel measurable set .

We follow the definitions of -perfect equilibrium and the perfect equilibrium in Sun and Zeng (2020), which involve societal perturbations. Given a societal summary , the perturbed societal summary, denoted by , is a full-support probability measure on . Let denote the set of player ’s best responses:

Since is weakly compact and is continuous, hence is nonempty.

Definition 3.

A behavioral strategy profile with full support is said to be a strong -perfect equilibrium if there exists a set of players with and a perturbed societal summary with at least -weight on such that for -almost all ,

where is the abbreviation of the societal summary .

A weak -perfect equilibrium is defined similarly by replacing the strong metric with the weak metric .

Then we can define the behavioral strategy strong/weak perfect equilibrium. We use to denote the set of positive integers. The limits taken in are respect to the usual weak convergence.

Definition 4.

A behavioral (resp. pure) strategy profile is said to be a behavioral strategy strong perfect equilibrium if there exists a sequence of behavioral strategy profiles and a sequence of positive constants such that

  1. each is a strong -perfect equilibrium with as goes to infinity,

  2. for -almost all , there exists a subsequence (each is associated with ) such that and ,

  3. .

A behavioral/pure strategy weak perfect equilibrium is defined similarly by replacing “each is a strong -perfect equilibrium” with “each is a weak -perfect equilibrium” in Condition (1).

Notice that the definitions above are less demanding than those in Sun and Zeng (2020) that require or . It is easy to see that a pure strategy perfect equilibrium is a pure strategy Nash equilibrium. Moreover, we will see that this modification will not affect the main results in Sun and Zeng (2020), including the existence and limit admissibility of pure strategy perfect equilibria.

In games with a finite number of players, the perfect equilibrium is defined as a pointwise limit of a sequence of -perfect equilibria; see Selten (1975) and Simon and Stinchcombe (1995). However, in large games, the pointwise convergence may break down; see Rath (1994). So we have to weaken it: we require a perfect equilibrium to be a limit point of strategies for almost all players as in Condition (2). The requirement that converges to can avoid the case where the limit of -best responses is not a best response in the limit; see Rath (1994) for more details.

Khan et al. (1997) provide an example illustrating that a pure strategy Nash equilibrium may not exist if the player space is not saturated. Clearly, a pure strategy perfect equilibrium in that example does not exist either since a perfect equilibrium is always a Nash equilibrium. Sun and Zeng (2020) systematically study the existence issue of the pure strategy perfect equilibria in large games. They prove that the nowhere equivalence condition, introduced in He et al. (2017), is sufficient and necessary to guarantee the existence. Given a large game with a saturated player space , we let denote the -algebra of that is induced by . Since the action space is compact, the space of payoff functions is polish, and hence is countably generated. Therefore, is nowhere equivalent to and a pure strategy perfect equilibrium exists. Such a result is summarized as follows:

Proposition 1.

Let be an atomless saturated probability space. Then every large game has a pure strategy strong/weak perfect equilibrium.

We revisit the example in Section 2. Although proposition above insures the existence of pure strategy perfect equilibria, it is not easy to identify a pure strategy perfect equilibrium. Nevertheless, we can identify a behavioral/randomized strategy perfect equilibrium in the following.

For each , we consider a strategy profile . Let the perturbed societal summary be , where

is the uniform distribution. Thus, it can be easily verified that

is an -perfect equilibrium. As goes to zero, converges to . Therefore, we obtain a behavioral/randomized strategy perfect equilibrium .

To get a pure strategy perfect equilibrium by purifying a behavioral/randomized strategy perfect equilibrium , we have to introduce the notion of mixed strategy perfect equilibria.

4 Mixed strategy perfect equilibria

4.1 Fubini extension

In this section, we introduce the mixed strategy perfect equilibria of large games. The mixed strategy profile in large games is first introduced in Khan et al. (2015).

As a mixed strategy profile requires the randomization to be independent across agents, it leads to a process with a continuum of independent random variables in the setting of a continuum of agents. In order to resolve the measurability issues

333See Sun (2006) for more details. See also Khan et al. (2015) for a discussion of the issues in modeling mixed-strategy Nash equilibrium in a large game. of these processes and to guarantee the existence of these processes with a variety of distributions, we adopt the framework of a rich Fubini extension as in Sun (2006).

Let be a probability space that captures all the uncertainty associated with the individual randomization for the agents in a mixed allocation or a mixed strategy profile. Throughout the rest of this section, we will assume that the agent space together with the sample space allows a rich Fubini extension.444The usual Lebesgue unit interval, as an agent space, can be extended to allow a rich Fubini extension; see Sun and Zhang (2009), and more generally Podczeck (2010). Also note that a rich Fubini extension is called a rich product probability space in Sun (2006). Recall that a Fubini extension is a probability space that extends the usual product space of the agent space and a sample space , and retains the Fubini property. Such a Fubini extension is rich if there is a -measurable process from to such that the random variables are independent and have the uniform distribution on . A process from a Fubini extension to a Polish space is said to be essentially pairwise independent if for -almost all , the random variables and are independent for -almost all .555Given that is an atomless (complete) probability space, a single point (and thus up to countably many points) has a measure zero, and thus, essential pairwise independence is more general than the usual pairwise and mutual independence.

Definition 5.

A probability space is said to be a Fubini extension of the usual product probability space if for any real-valued -integrable function on , the following statements hold:

  1. The function is -integrable on for -almost all , and is -integrable on for -almost all ;

  2. The integrals and are integrable on and , respectively. In addition, .

A Fubini extension is said to be rich if there is a -measurable process from to the interval such that is essentially pairwise independent, and induces the uniform distribution on for -almost all . Notice that the marginal probability measure of on and are and , respectively. Thus, we denote the Fubini extension by .

We connect the saturation property of a probability space to the existence of a rich Fubini extension. The following result is from (Sun, 2006, Proposition 4.2) and (Podczeck, 2010, Theorem 1).

Fact 1.

The probability space is saturated if and only if there is a rich Fubini extension based on it.

The rich Fubini extension plays an important role in large games as one can construct processes on it with essentially pairwise independent random variables that have any given variety of distributions on a general polish space. The following result is taken from (Sun, 2006, Proposition 5.3).

Fact 2.

Let be a rich Fubini extension, let be a Polish space, and let be a measurable mapping from to . Then there exists an -measurable process such that the process is essentially pairwise independent and is the induced distribution by for -almost all .

This proposition reveals the fact that unlike the Lebesgue unit interval, saturated probability spaces are hospitable to independence and measurability, moreover, this proposition guarantees that every behavioral strategy profile can be lifted to a measurable process defined in a Fubini extension space.

Now, we are able to define a mixed strategy profile of a large game. From now on, we use the Fubini extension as the framework to ensure that almost any two players play independent mixed strategies in a non-cooperative setting. Let be a saturated probability space and let be a rich Fubini extension of the product space .

Definition 6.

A mixed strategy profile of a large game is an measurable function , where the process is assumed to be essentially pairwise independent. A mixed strategy Nash equilibrium of is a mixed strategy profile , such that for -almost all ,

for any random variable from to .

The above definition of mixed strategy Nash equilibrium for a large game was firstly proposed in Khan et al. (2015), and they also demonstrated that there is a one to one correspondence between the behavioral strategy Nash equilibrium and the mixed strategy Nash equilibrium for large games.

The next result ia taken from Corollary 2.9 of Sun (2006), which is a version of the ELLN in the framework of Fubini extension. It will be necessary when we turn to inquire the relationship between a behavioral strategy perfect equilibrium and a mixed strategy perfect equilibrium.

Fact 3.

Assume that is a Fubini extension. If F is an essentially pairwise independent and -measurable process, then the sample distribution is the same as the distribution for -almost all .

4.2 Mixed strategy perfect equilibria

We are now ready to define the notion of mixed strategy perfect equilibria as we have developed the necessary background to do so. Given a mixed strategy profile , let denote player ’s best responses, i.e.,

where is a perturbed societal summary666To make sure that is well defined, the perturbation should be measurable as a function of . Throughout the rest of this paper, the perturbation is always assumed to be measurable. of such that has at least -weight on .

It is a straightforward observation that is nonempty due to the continuity of and the compactness of . We can now turn to the definitions of mixed strategy (strong/weak) -perfect equilibria and mixed strategy (strong/weak) perfect equilibria.

Definition 7.

A mixed strategy strong -perfect equilibrium is a mixed strategy profile , such that for almost all , is fully supported, and there exists a set of players , such that and for -almost all ,

A mixed strategy weak -perfect equilibrium is defined similarly by replacing the strong metric with the weak metric .

The above notion of mixed -perfect equilibria is modified from the notion of behavioral -perfect equilibria. For a particular player

, the induced probability distribution

is the induced behavioral strategy for player . Thus, a mixed perfect equilibrium should be a limit of a sequence of mixed -perfect equilibria:

Definition 8.

A mixed strategy profile is said to be a mixed strategy strong perfect equilibrium if there exists a sequence of mixed strategy profiles and a sequence of positive constants such that

  1. each is a strong -perfect equilibrium with as goes to infinity,

  2. for -almost all , there exists a subsequence (each is associated with ) such that and ,

  3. .

A mixed strategy weak perfect equilibrium is defined similarly by replacing “each is a strong -perfect equilibrium” with “each is a weak -perfect equilibrium” in Condition (1).

We are now ready to show the relationship between a behavioral strategy (strong/weak) perfect equilibrium and a mixed strategy (strong/weak) perfect equilibrium. The theorem below is one of the main theorem in this paper. It establishes a one to one correspondence between behavioral strategy (strong/weak) perfect equilibria and mixed strategy (strong/weak) perfect equilibria.

Theorem 1.

The following equivalence holds for a large game : (i) Every mixed strategy strong (resp. weak) perfect equilibrium induces a behavioral strategy strong (resp. weak) perfect equilibrium and (ii) every behavioral strategy strong (resp. weak) perfect equilibrium can be lifted to a mixed strategy strong (resp. weak) perfect equilibrium.777Although we have not formally define the words “induce” and “lift”, their meaning is clear from the proofs.

This theorem suggests that the Fubini extension is an appropriate framework to define the mixed strategy perfect equilibrium and within the Fubini framework, the mixed strategy perfect equilibrium can be naturally connected with the behavioral strategy perfect equilibrium. The proof is in the Appendix and the ELLN plays a crucial role in the proof. As a corollary, we can show the existence of mixed strategy strong perfect equilibrium as a combination of Theorem 1 and the existence of behavioral strategy perfect equilibrium.

Corollary 1.

If is a saturated probability space, then there exists a mixed strategy strong perfect equilibrium.

The proof is quite straightforward as the saturation guarantees the existence of behavioral strategy strong perfect equilibrium, then by Theorem 1, it can be lifted to a mixed strategy strong perfect equilibrium.

4.3 Mixed and pure perfect equilibria: An ex post relationship

In this subsection, we discuss the relationship between a mixed strategy perfect equilibrium and its induced pure strategy perfect equilibrium in the realized ex post game. We present a novel result about mixed strategy perfect equilibria: after the resolution of strategic uncertainty, a mixed strategy perfect equilibrium leads to a pure strategy perfect equilibrium almost surely.

A rich literature has developed on equilibrium notions involving the ex post concept. In the context of large games, Khan et al. (2015) defines the notion of mixed strategy equilibrium and systematically studies the relationship between a mixed strategy Nash equilibrium and each pure strategy Nash equilibrium in the ex post game. They prove that every mixed strategy Nash equilibrium has the ex post Nash property. In this subsection, we try to establish a much more interesting result: every mixed strategy perfect equilibrium has the ex post perfect property.

Since a mixed strategy profile is defined as an -measurable function where is also assumed to be essentially pairwise independent, hence, it is easy to see that is a pure strategy profile for any realized sample . In the below theorem, we show that this induced pure strategy profile , for almost all , is a pure strategy perfect equilibrium.

Theorem 2.

A mixed strategy strong (resp. weak) perfect equilibrium of a large game has the ex post perfection property: for -almost all , is a pure strategy strong (resp. weak) perfect equilibrium.

The proof is in the Appendix and the intuition behind the proof is simple. The ELLN together with the Fubini property guarantee that the induced pure strategy profile is the “limit” of a sequence of -perfect equilibrium. So rather than the proof, it is the interpretation of Theorem 2 that is of interest. This Theorem rigorously develops the intuition that once uncertainty is resolved, a player has no incentive to depart ex post from his strategy taken in the ex ante game when he finds himself in the realized ex post game. We have shown that a mixed strategy perfect equilibrium has an ex post purification, and therefore, implies the existence of pure strategy perfect equilibrium, so this gives an immediate proof of Proposition 1. So Theorem 2 shows that in large games, the notion of mixed strategy perfect equilibrium is redundant as one can construct a pure strategy perfect equilibrium if given a mixed strategy perfect equilibrium.

As a direct application, we are now able to give a pure-strategy perfect equilibrium in the smartphone sells game. Based on the discuss at the end of Section 3, we have a (randomized) behavioral strategy perfect equilibrium . Thus by Theorem 1, can be lifted to a mixed strategy perfect equilibrium . Finally, by Theorem 2, we can obtain a pure-strategy perfect equilibrium for almost all . Moreover, by using the ELLN, we can conclude that in perfect equilibrium , of the players choose (sell good smartphone), and of the players choose (sell regular smartphone). This result coincides with the real smartphone markets in most countries.

As a corollary, the below result shows that a mixed-strategy weak perfect equilibrium is a mixed Nash equilibrium. This result is compatible with the fact that the perfect equilibrium is a refinement of Nash equilibria in games with finite players.

Corollary 2.

In any large game, a mixed strategy weak perfect equilibrium is a mixed strategy Nash equilibrium.

The proof is divided into two steps. In Step 1, suppose is a mixed-strategy weak perfect equilibrium, then by Theorem 2, for almost all , the induced pure strategy profile is a pure strategy weak perfect equilibrium; In Step 2, by the (Sun and Zeng, 2020, Proposition 1), is a Nash equilibrium for almost all , and hence, itself is a mixed-strategy Nash equilibrium as it has the ex post Nash property ((Khan et al., 2015, Theorem 2)).

5 Limit admissibility

In this section, we will study the limit admissibility of mixed strategy perfect equilibria. For games with finite players and finite actions, it is well known that perfect equilibria form a nonempty set of Nash equilibria that are admissible, which means that they put no mass on weakly dominated actions. Below we start with the definition of weakly dominated strategy for large games.

Definition 9.

For each , a pure strategy is said to be a weakly dominated strategy for player if there exists a behavioral strategy such that

  1. for each ,

  2. for some ,

where .

For each , let be the set of weakly dominated strategies for player . Then we can state the formal definition of admissible strategy.

Definition 10.

A strategy profile is said to be admissible if for -almost all , , where is the complement of the set .

For games with finite players and finite actions, it is easy to verify that each pure strategy perfect equilibrium is admissible; however, For finite-player games with infinitely many actions, Simon and Stinchcombe (1995) showed that there exists a game that has a unique Nash equilibrium in weakly dominated strategies; see Example 2.1 therein. Moving on to the context of large games, Sun and Zeng (2020) provide an example of a large game with infinitely many actions, in which the unique strong perfect equilibrium is not admissible. These examples suggest that the admissibility and the existence of perfect equilibrium may not be compatible in games with infinitely many actions.

To solve this incompatibility problem, Simon and Stinchcombe (1995) introduced a weaker property called the limit admissibility that is compatible with the existence: a strategy is limit admissible if it puts no mass on the interior of the set of weakly dominated strategies.

For a player who has finitely many actions, a limit admissible strategy is indeed an admissible strategy, therefore, for finite-player game with finite actions, every perfect equilibrium is limit admissible. For finite-player game with infinitely many actions, Simon and Stinchcombe (1995) showed that every perfect equilibrium is limit admissible. In the framework of the large game, Sun and Zeng (2020) proved that every pure strategy weak/strong perfect equilibrium is limit admissible.888Their result can be easily generalized to every behavioral strategy weak/strong perfect equilibrium. Below we will show that every mixed strategy perfect equilibrium is also limit admissible. We begin with the definition of limit admissibility for mixed strategy profile.

Definition 11.

A mixed strategy profile is said to be limit admissible if for almost all , , where is the topological interior of the set . A pure strategy profile is said to be limit admissible if for -almost all , , where is the complement of .

For each , each action in is called a limit admissible action of player , where is the complement of .

The following result shows that each mixed strategy weak perfect equilibrium is limit admissible. Since a strong perfect equilibrium is also a weak perfect equilibrium, the same result holds for every mixed strategy strong perfect equilibrium.

Theorem 3.

Every mixed-strategy weak perfect equilibrium is limit admissible.

We conclude this section with an interesting theorem: we can prove that a strategy profile is limit admissible if and only if it is ex post limit admissible.

Theorem 4.

A mixed-strategy profile of a large game is limit admissible if and only if it is ex post limit admissible: for -almost all , the pure strategy is limit admissible.

As an application of the Theorem 4, we can strengthen the main result of Khan et al. (2015): a mixed strategy profile of a large game is a Nash equilibrium if and only if it is ex post Nash. By using our Theorem 4, we can see that a mixed strategy profile is a limit admissible Nash equilibrium if and only if it is ex post limit admissible Nash.

6 Appendix

6.1 Proofs of results in Section 4

Proof of Theorem 1.

We focus on the case of strong perfect equilibrium, the case of weak perfect equilibrium can be proved similarly.

Step 1.

Suppose that is a mixed strategy strong perfect equilibrium of game . Let . We want to show that is a behavioral strategy strong perfect equilibrium.

By definition, there exists a sequence of mixed strategy profiles and a sequence of positive constants such that

  1. each is a strong -perfect equilibrium with as goes to infinity,

  2. for -almost all , there exists a subsequence (each is associated with ) such that and ,

  3. .

Define , we claim that is a behavioral strategy strong -perfect equilibrium. By the ELLN in Fact 3, it is clear that for -almost all ,

which implies that

holds for -almost all .

Therefore, we have that

Hence by the definition of , we can derive that for -almost all ,

Since is fully supported by construction, hence is a strong -perfect equilibrium.

To prove that is a behavioral strategy strong perfect equilibrium, we need to verify the following two conditions:

  1. for -almost all , there exists a subsequence (each is associated with ) such that and ,

  2. .

By definition, these two conditions can be derived from (2) and (3) directly. Therefore, is a behavioral strategy strong perfect equilibrium.

Step 2.

Now suppose is a behavioral strategy strong perfect equilibrium. By definition, there exists a sequence of behavioral strategy profiles and a sequence of positive constants such that

  1. each is a strong -perfect equilibrium with as goes to infinity,

  2. for -almost all , there exists a subsequence (each is associated with ) such that and ,

  3. .

Since and are measurable functions from to , thus given that is a rich Fubini extension, by Fact 2, there exist -measurable functions and from to such that and are essentially pairwise independent functions and for -almost all :

In addition, we can assume are independent.

Therefore, similar to the proof of Step 1, we can apply the ELLN to obtain that:

which implies that is a mixed strategy strong -perfect equilibrium for each . The remaining proof is the same as the proof of Step 1 and hence is a mixed strategy strong perfect equilibrium. ∎

Proof of Theorem 2.

We focus on the case of weak perfect equilibrium, the case of strong perfect equilibrium can be proved similarly. Suppose that is a mixed strategy weak perfect equilibrium. We shall show that has the ex post perfection property, i.e., is a pure strategy weak perfect equilibrium for -almost all .

By definition, there exists a sequence of mixed strategy profiles and a sequence of positive constants such that

  1. each is a weak -perfect equilibrium with as goes to infinity,

  2. for -almost all , there exists a subsequence (each is associated with ) such that and ,

  3. .

Based on Fact 2, without loss of generality, we can assume that for -almost all , the sequence of strategies are independent.

To show that is a pure strategy weak perfect equilibrium, we will construct a sequence of behavioral strategy profiles , where is a weak -perfect equilibrium for each .

Lemma 1.

Let , then for -almost all , the strategy profile

is a weak -perfect equilibrium.

Proof of Lemma 1.

According to the proof of Theorem 1, is a weak -perfect equilibrium and hence by the definition, for almost all :

by the definition of weak measure, we have

for some , which implies:

for each , let , by the ELLN, for almost all ,

therefore,

the last inequality is due to and . Let , then . Now we can show that is a weak -perfect equilibrium.

It is easy to see that has full support by its construction. And by the ELLN,

hence for any ,

The first inequality is due to the property of a metric. The last inequality is because . Therefore, ia a weak -perfect equilibrium. ∎

Go back to the proof of Theorem 2, to prove that is a weak perfect equilibrium, we only need to verify the following two conditions:

  1. for -almost all , there exists a subsequence (each is associated with ) such that and ,

  2. .

Condition (2’) is easy to be verified by using the ELLN:

Then we verify condition (1’). Let , by condition (3), we have:

where . Let be a countable dense subset of the support of , and one can easily check that for almost all , .

For any and , let , here is an open ball centered at with radius . Since and , hence there exist such that for ,

Since from Condition (2), are independent, hence by the second Borel-Cantelli lemma, we conclude that for almost all , and for any ,