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?