An appropriate notion of Nash equilibrium: the best response against a pool of players

Question

Consider a pool of players. Player A plays against only one players from the pool, but player A does not know who it is. Player A knows a general information about the pool. What is the notion of Nash equilibrium when the payoff is computed as the expectation against a pool of players?

To make things concrete, let's consider a coordination game. Player A know that on average players from the pool choose Left, say, with probability 0.3. Then, player A's best response is, say, play Up with probability 0.4. If a player from the pool know that A plays Up with 0.4 probability, then their best response is to play Left with 0.3 probability. So there is a Nash equilibria, but the payoff of player A is computed against the pool. Individual players can play differently, but on average they play Left with 0.3 probability. Is it a simple Nash equilibria or is it Bayes Nash equilibria? Or may be it has a different name?

Answer 1

You did not specify the game completely, but in general it is a NE. (But see option 3.)

Let me specify and explain. Suppose you have three players $A$, $B$ and $C$. Each player chooses one of two actions, for example. Now I need to be specific about what exactly the game is.

Option 1: Player $A$ always moves and nature/randomness chooses whether she playes with $B$ or $C$ and after choice by nature, $A$ observes whom she plays against. The appropriate notion here is NE (taking into account that $A$'s strategy can depend on history - nature's move). But essentially you can 'solve' the game by looking separately at $A$ playing with $B$ and with $C$.

Option 2: All players choose their action and after that nature determines whether $A$, who always plays, plays with $B$ or $C$. This is probably closer to your notion of 'playing against a pool'. Again, NE is an appropriate solution concept here, what you have to extend the game with is $A$'s expected utility (since when she chooses her action, she is essentially choosing a lottery).

Option 3: Suppose there are only two players $A$ and $B$ but nature determines whether $B$ is nice or mean. After nature determines $B$'s type, both player simultaneously choose their actions, $B$ knowing her type, $A$ uncertain about $B$'s type. Now the appropriate solution concept is BE. You think of $B$ being nice/mean as of her type. $B$'s strategy is a mat from types to actions, which is just another way to say that her action can depend on her type. Again, you need expected utility for $A$ since uncertainty is involved.

Answer 2

Yes, it would be a Bayesian Nash equilibrium. You would model this as a game where "Nature" first moves and matches players to play with each other according to a probability distribution (say, uniform). Then players play against their matched opponent. For an example of a game tree in a Bayesian equilibrium, see for example here on page 4.

On your example of a coordination game: There should still be the same equilibria as in the full information game: All go left or all go right. Because then it does not matter who you are matched with; the entire population "coordinated". I think you would need a different game (in particular some asymmetry between player payoffs) in order to get the feature you describe, namely that players are not sure how to coordinate when they play against a "heterogeneous" population.