This paper adds to the discussion, in a general setting, that given a Nash-Schmeidler (nonanonymous) game it is not always possible to define a Mas-Colell (anonymous) game. In the two games, the players have different strategic behaviours and the formulations of the two problems are different. Also, we offer a novel explanation for the lack of a Nash equilibrium in an infinite game. We consider this game as the limit of a sequence of approximate, finite games for which an equilibrium exists. However, the limiting pure strategy function is not measurable.

This paper is in the context of “large games.” First, we discuss the relationship of a particular Nash-Schmeidler, nonanonymous, formulation to a Mas-Colell anonymous game. We explain that, starting with the former, it is not always possible to define a Mas-Colell game. This is due to the different strategic behaviours of the players in the two formulations. The approach of Mas-Colell [

The description “anonymous” refers to the fact that a player in Mas-Colell has no individual identity. What matters is the overall density of the choices, on which a set of possible utility functions depends. “Nonanonymous” refers to the fact that in Schmeidler [

The ideas above have an affinity to the issues discussed in the area of general equilibrium in markets with a continuum of traders. In the Schmeidler formulation, a game is defined as a measurable function from an atomless set of players to the set of players’ characteristics. In the Mas-Colell alternative formulation, a game is defined anonymously, as a distribution on the players’ characteristics.

The relationship and the link between the two different formulations have been studied in the literature, under various assumptions, a number of times. Rath [

The two formulations have different implications when the set of actions is infinite. Rath et al. [

This paper adds to the discussion in a different direction. It points out that a nonanonymous game cannot always be formulated in an anonymous, distributional form.

Second, we discuss a game for which it is known that there exists no equilibrium. It is due to Schmeidler, and it is also in the comprehensive survey by Ali Khan and Sun [

In games with a continuum of players, we look at the connection between Nash (NE) and Mas-Colell’s Cournot-Nash Equilibria (CNE), given a particular definition of the former which makes also a connection with the agents. We then point out that this definition is more restricted than the usual one and this implies that one cannot always relate the two types of games.

It is assumed that

A Borel measure

A pure strategy is measurable function

We now state the following useful result. The proof is straightforward and hence omitted.

If

However, in general, the two types of large games are clearly different and not compatible. In Mas-Colell’s anonymous games, only the average distribution of opponents’ actions matters. In Schmeidler’s nonanonymous games, the strategic behaviour of players can depend on the specific choice of each of the opponents. As we see below,

We now look at two Schmeidler type examples, one without a NE and one with a NE, which cannot be cast in the Mas-Colell form.

Consider

However, Example

The following question arises: which part of the mathematical discussion above breaks down? So we cannot define a Mas-Colell game. The example above shows that, given a measurable function

It is not always possible to go from a Nash-Schmeidler game to a Mas-Colell one, as their strategic structures are different. A utility function appropriate for Nash-Schmeidler might not be defined on

We also point out that we can define the utility function such that

We now consider the question of existence of a NE in Example

Then, in an attempt to give a further explanation, we consider a sequence of approximate games. We show that a NE exists for the sequence and explain why it fails to do so for the limiting game.

We quote Schmeidler’s result without proof.

There is no measurable function which can serve as a NE in Example

The proof there is rather dense and relies on a result concerning the nonexistence of a certain Lebesgue-measurable set. An alternative, more direct, and simpler proof was produced, but it became unnecessarily long.

An intuitive interpretation of the result is as follows. Let

We now consider Example

There is a NE in finite approximations of Example

In the first type of approximation, the interval is divided to equal segments and in the second one to arbitrary segments.

Dividing

Consider that

We have

To maximize

For any arbitrary partition of

Hence, by considering finite approximations (recent contributions by Khan and coworkers on approximations and on exact equilibria in games with hyperfinite spaces and with a biosocial topology are discussed in the works of JME and JET. See, for example, [

The author declares that there is no conflict of interests regarding the publication of this paper.

The author thanks Allan Muir for invaluable discussions and suggestions and Roman Kozhan for penetrating comments. He is very grateful to M. Ali Khan for the most valuable, detailed comments. He also acknowledges with thanks the help and encouragement by Dimitrios Tsomokos. The two referees of the journal made very incisive, detailed comments which helped to shape the final version of this paper, and the author is very grateful. Responsibility stays solely with the author.