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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.

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

Ever since it was coined by Selten [

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 [^{1}^{2}

For continuous games, Simon and Sinchcombe [

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 [

Carbonell-Nicolau [

Roughly speaking, Condition (A) is satisfied when there exists, for each player _{i}_{i}_{i}_{i}_{i}_{−i} of the other players, the action _{i}_{i}_{−i}), even if the other players slightly deviate from _{−i}; and (2) given any pure action profile _{−i} of the other players, there is a subset of generic elements of _{i}_{−i}) such that given any generic pure strategy _{i}_{i}_{−i}), where _{−i} is a slight deviation from _{−i}, almost guarantees the payoff player _{i}_{−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 [^{3}

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.

A _{i}_{i}_{i}

In the sequel, by _{−}_{i}_{j}_{≠}_{i}_{j}_{i}_{i}_{1}, …, _{N}_{i}_{−}_{i}

The _{i}_{i}_{i}

Henceforth, the set ×_{j}_{≠}_{i}M_{j}_{−i}_{i}_{i}_{1}, …,_{N}_{i}_{−}_{i}

Given _{i}_{i}_{xi}_{i}_{i}_{i}_{xi}_{i}_{i}_{i}_{i}_{i}_{i}_{xi}_{i}_{i}_{i}_{i}_{i}_{i}^{2}_{1},…, _{N}_{1},…, _{N}

A number of definitions of trembling-hand perfection for infinite normal-form games have been proposed (

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

A Borel probability measure _{i}_{i}_{i}_{i}

For each _{i}_{i}_{1},…, _{N}^{N}_{1},…, _{N}^{N}_{1} = … = _{N}

When referring to these objects, we simply write
_{1} = … = _{N}

A strategy profile _{1},…, _{N}_{i}_{i}_{i}_{−}_{i}_{i}_{i}

Given a game

A strategy profile ^{n}^{n}^{n}^{N}^{n}^{n}^{n}

In words,

Note that, in Definition 2, we do not require that ^{4}

For _{i}_{i}

Consider the following distance function between members of _{i}

(Simon and Stinchcombe [^{n}^{n}

The following result is taken from Carbonell-Nicolau [

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

^{n}

The following example illustrates that the set of

Consider the two-player game _{1},_{2}), where _{1} and _{2} are defined by _{1}(_{1}, _{2}):= _{1}(1 – 2_{2}) and

It is easily seen that the strategy profile (0,1) is a Nash equilibrium of _{1} > 0. Therefore, player 2's best response to any tremble of player 1 in any Selten perturbation of

The _{G̅}_{G̅} and Γ_{G̅} are denoted by Γ_{G̅} and Γ̅_{G̅} respectively.

Given {

The following definition is taken from Reny [

The game _{G}_{i}_{i}_{x−}_{i}_{−}_{i}_{i}_{i}_{x}_{−}_{i}_{i}

The following proposition is analogous to Proposition 1 in Carbonell-Nicolau [^{5}

Suppose that _{δμ}

_{n−1}_{μ}^{n}^{n}^{n}

To see that all ^{n}^{n}_{δnμn}^{n}^{n}

Because ^{n}_{i}_{1}_{n}^{N}_{G̅} so if _{i}_{i}_{i}_{ϱ −i}_{−i}

We therefore have, in view of (^{n}^{n}_{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 ^{6}_{δμ}^{7}_{δμ}_{δμ}

If _{δμ}

In general, verifying the existence of
_{δμ}_{δμ}

Unfortunately, _{δμ}_{δμ}

The following definition is taken from Reny [

The game _{i}_{i}_{i}_{i}_{x−i}_{i}_{x−i}_{−}_{i}

It is well-known (Reny [

(Sion and Wolfe [_{1}, _{2}), where
_{2} : = −_{1} (

This game is zero-sum (and so

Now consider the following strengthening of payoff security (

Given _{i}_{i}_{i}_{i}_{i}_{i}_{−}_{i}_{−i}_{y}_{−}_{i}_{−}_{i}_{i}_{i}_{y}_{−}_{i}_{i}_{i}_{−i}

The game

Uniform payoff security of _{δμ}^{8}

Let (^{n}^{n}^{n}^{n}

Consider the two-player game _{1}, _{2}), where
_{2}(_{1}, _{2}) : = _{1}(_{2}, _{1}) (

It is easy to verify that
_{δμ}_{i}_{i}_{i}_{−i}_{j≠i}M_{j}_{j}_{−i} for which _{i}_{i}_{−i}_{i}_{(1-δ)αn+δμ2} of (1 − ^{n}_{2} and every _{1} ∈ [0, _{2} ∈ _{(1-δ)αn+δμ2} ⋂ _{2}(_{2}).

Choose _{(1-δ)αn+δμ2} of (1 − ^{n}_{2} and any _{1} ∈ [0, _{2} ∈ (^{n}^{n+1}) sufficiently close to ^{n}_{2} + _{2} ∈ _{(1-δ)αn+δμ2} By linearity of _{1} we have

Therefore, because _{1}(_{1}, _{2}) ≤ 1 ≥ _{1}(_{1}, _{2}) and

On the other hand, we have
_{δμ}

The perturbation _{δμ}_{G̅δμ}_{δμ}_{i}_{i}_{i}_{(1-δ)αn}_{+δμ−}_{i}^{n}_{−i}. It follows that _{δμ}^{9}

In light of Example 3, any condition on the payoff functions of ^{10}

The following condition appears in Carbonell-Nicolau [

There exists
_{i}_{i}

For each _{i}_{i}_{−i}_{−}_{i}_{y}_{−}_{i}_{−i}

For each _{−i}_{−i}_{i}_{i}_{i}_{i}_{i}_{i}_{y}_{−}_{i}_{−i}_{i}_{i}_{−}_{i}_{i}_{i}_{−}_{i}_{−}_{i}_{y}_{−}_{i}^{11}

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

The following implications are immediate:

We can establish payoff security of a Selten perturbation of

Suppose that a compact, metric game _{δμ}

This result plays a central role in the proof of the main results of this paper.^{12}

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}_{i}U_{i}_{δμ}

The following statement summarizes this finding.

Suppose that a compact, metric game

Lemma 2 is similar to Lemma 1 in Carbonell-Nicolau [_{(δ,μ)}_{(δ,μ)}

The statement of this lemma differs from that of Lemma 2 in that _{(δ,μ)} and _{δμ}^{13}

The remainder of this section derives a corollary of Theorem 2 in terms of two independent conditions introduced in [^{14}

Let _{i}_{i}_{i}_{i}_{i}_{i}}) ⋂ _{i}_{i}_{i}_{i}_{i}_{i}_{i}

Let
_{i}_{i}_{i}_{i}_{i}_{e}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

Clearly,
_{i}

Given _{i}_{i}_{i}_{i}_{i}_{i}_{xi}_{i}_{−i} ∈ _{−i}, there is a neighborhood _{y}_{−i} of _{−i} for which _{i}_{i}, O_{y}__{i}_{i}_{xi}_{−i})

We say that

Given _{i}_{i}_{i}_{i}_{i}_{i}

A game _{i}_{i}_{i}_{i}_{i}

The following implications are immediate:

The game _{−}_{i}_{−}_{i}_{i}_{i}_{xi}_{i}_{i}_{xi}_{x−i}_{−}_{i}_{i}_{i}, y_{−}_{i}_{i}_{i}, y_{−}_{i}_{−i}_{x−i}

The game _{−}_{i}_{−}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{xi}_{i}_{i}_{xi}_{x−i}_{−i}_{i}_{i}, y_{−i}_{i}_{i}, y_{−i}_{−i} ∈ O_{x−i}

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

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

Suppose that

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

Consider a two-player Bertrand game _{1}, _{2}), where

Similar duopoly games can be found in Baye and Morgan [

It is readily seen that
_{i}_{i}_{i}_{pi}_{i}_{−i} ∈ [0, 4], there is a neighborhood _{p−i} of _{−i} for which
_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{pi}_{pi}_{i}_{i}_{pi}_{−i}_{p−i}_{−i}

If _{−i} > _{i}_{p−i}_{i}

If _{−i} ≤ _{i}_{p−i}_{pi}

Finally, _{−i} ∈ [0, 1], _{i}_{−i}}, and _{i}_{−i} and _{i}_{i}_{−i} and _{i}_{i}_{i}, y_{−}_{i}_{i}_{i}_{i}, y_{−}_{i}_{xi}_{x}_{−i} and for some neighborhoods _{xi}_{x}_{−i} of _{i}_{−i} respectively, so it is clear that there exists a neighborhood _{xi}_{i}_{i}_{xi}_{x}_{−}_{i}_{−i}_{i}_{i}, y_{−}_{i}_{i}_{i}, y_{−}_{i}_{−i}_{x−i}

Because

There are _{1}, …, _{N}^{N}_{i∈ {1,…,N}} _{i}

This situation can be modeled as an _{i}_{1},…, _{N}_{i}_{j∈ {1,…, N}} b_{j}}.

This game is generically locally equi-upper semicontinuous. To see this, fix _{−i}_{−}_{i}_{i}_{−}_{i}_{−}_{i}_{j∈ {1,}_{…, N}∖}_{{}_{i}_{}}_{j}_{i}_{−}_{i}_{i}_{−}_{i}_{i}_{i}_{i}_{i} − δ, b_{i}_{−}_{i}_{i} ∈ (_{i} − δ_{i}, b_{i}_{−i} ∈ _{−}_{i}_{b−i}_{−}_{i}

We now show that ^{15}_{i}_{i}_{i}_{i}_{bi}_{−}_{i}^{N}^{−1}, there is a neighborhood _{b−}_{i}_{−i}

Choose _{i}_{i}, b_{i}_{bi}_{i}_{bi}_{i}_{i}, b_{i}_{i}_{bi}_{−}_{i}^{N}^{−1}, and let _{b}_{−i} be a neighborhood of _{−i} with the following property: if max_{j∈ {1, …, N}} _{j} < b_{i}_{b}_{−}_{i}_{i}^{N}^{−1} = ∅. It is easy to verify that the choices of _{i}_{bi}_{b}_{i}

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

Page and Monteiro [_{i}_{i}_{i}_{0}

Each firm _{i}_{i}_{i}_{i}_{t}_{t}_{t}_{t}_{1}_{2}) is given by
_{1}_{2}) to maximize her utility:
^{16}_{j}_{1}_{2}) is
_{i}_{i}) is an upper semicontinuous, compact game. Moreover, an argument similar to that provided in the proof of Theorem 5 of [

Bagnoli and Lipman [_{i}_{i}_{i}_{i}_{i}

The agents simultaneously choose a contribution to the public project, each agent _{i}_{i}]. Let _{i}_{i∈I} of contributions, the public project is undertaken if Σ_{i∈I} _{i}_{i∈I} _{i}

Let _{i∈I} _{i}_{i}, u_{i}_{i∈I}, where _{i}

Specifically, assuming _{i}_{i}_{i}_{i}_{i}_{i}_{i∈I} _{i}_{i}_{i}_{i}_{i}_{i}

Bagnoli and Lipman replace each _{i}

The authors’ main result is that the game form

It is easily seen that the restriction of _{i}_{i}

We now show that _{i}_{i}_{i}_{i}_{si}_{i}_{-i}_{j≠i}_{j}_{s−i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{si}_{−i} ∈ ×_{j}_{i}_{j}_{s}_{−i} with the following property: if Σ_{j} _{j}_{i}_{j≠i} _{j} ≥ _{-i} ∈ _{s}_{−i}. Now, verifying that the choices of _{i}_{si}_{s}_{−i} give (

Finally, we show that _{i}_{i}]. Fix _{-i} ∈ ×_{j≠i}_{j}_{i}_{i}] such that Σ_{j} s_{j}_{i}-measure 1), and take any _{i}_{j} s_{j}_{j} s_{j}_{si}_{i}_{i}] such that _{i}_{j≠i} s_{j}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{si}_{i}_{si}_{s}_{−i} of _{−i} in ×_{j≠i}_{j}_{−i} ∈ _{s}_{−i}, we have

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

To begin, we state a number of intermediate results.

Given a metric space _{*}(

Let ^{*} ∈ ℙ_{*} (_{ɛ}_{X}_{X}^{*}| <

Suppose that _{i}_{i}

For each _{i}_{i}_{−i} ∈ _{−i}, there is a neighborhood _{σ}_{−i} of _{−i} such that

For every _{−i} ∈ _{−i}, there is a neighborhood _{σ}_{−i} of _{−i} such that
_{−i} ∈ _{σ−i}

Suppose that a compact, metric game _{δμ}

_{1}, …, _{N}_{i}_{i}_{i}_{i}_{i}, O_{σ − i}_{i}_{σ − i}_{−i}.

Lemma 5 gives a Borel measurable map _{i}_{i}

For every y_{i}_{i}_{σ}_{−i} of _{−i} such that

There is a neighborhood _{σ −}_{i}_{−i} such that
_{-i}_{σ}_{–i}

There exists a neighborhood _{σ}_{−i} of _{−i} such that

_{i}_{i}_{σ}_{−i} of _{−i} such that
_{i}_{i}_{−i}

Now observe that we have

Because _{i}_{i} =_{i}_{i}

By (_{σ}_{−i} of _{−i} such that
_{−}_{i}_{−i}
_{-}_{i}_{−i} we have
_{δμ}

Example 2: The payoff functions of

Example 3: The payoff function for player 1.

Example 4: Operating profit as a function of price.

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.

For 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.

Allen [

Constructing a measurable map can sometimes be cumbersome, especially if pure strategies are, say, maps between metric spaces rather than points in Euclidean space.

In Definition 2, each ^{n}^{n}^{n}^{n}^{n}^{n}^{n}^{n}^{n}^{n}ν

The reader is referred to the discussion following the statement of Theorem 2 for a comparison between Proposition 1 and Proposition 1 in [

A game _{i}_{−i} ∈ _{−i}, _{i}_{−i}) is quasiconcave of _{i}

If _{i}_{i} coincides with the topology induced by the Prokhorov metric on _{i}_{i}_{i}_{i}μ_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

Even the generalized notion of better-reply security of Barelli and Soza [_{δμ}

While

This 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

The following generalization of Condition (A) leaves all of our results intact.

Condition(A′).

There exists
_{k}_{k}_{i}_{i}

For each _{i}_{i}_{−i}_{−i}_{y−i}_{−i}_{i}_{k}_{i}_{y−i}_{i}_{i}_{−i}

For each _{−i}_{−i}_{i}_{i}_{i}_{i}_{i}_{i}_{y−i}_{−i}_{i}_{k}_{i}_{−i}_{i}_{i}_{−i}_{−i}_{y−i}

Lemma 2 is similar to Lemma 1 in Carbonell-Nicolau [

The 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.

The relationship between Corollary 1 and Corollaries 1 and 3 in [

This game fails entire payoff security.

It is shown in [