symmetry of equilibria with heterogeneous players I have a question about game theory terminology. I am working on a model in which players are heterogeneous in two dimensions, and there are four types of players. For example one type of players receive some reward and have some cost of playing, while the other type differs in this reward and cost.
I am interested in the equilibria of the game in which the strategies of users who face the same reward and costs (i.e. same type) are the same.
I think symmetric equilibrium in this case is misleading since not all players are expected to have the same strategies, but it is something more like a separating equilibrium across types. However, there is perfect information so I'm unsure whether separating/pooling equilibria concepts can also be applied here.
Any thoughts/references would be helpful!
the equilibria of the game in which the strategies of users who face the same reward and costs (i.e. same type) are the same.

This sounds like an ex ante (or ex interim or ex post) symmetric equilibrium, depending on the timing of the realization of the types. If both dimensions of the types are realized at the beginning of the game, then I'd go with the term &quot;symmetric equilibrium&quot;, which is typically understood to mean the ex ante sort.
I think you may be confusing between &quot;strategy&quot; and &quot;action&quot;, which may be why you rejected the term &quot;symmetric equilibrium&quot;. In the context of Bayesian games, a strategy is a function of types while an action is a particular value of that function. For example, in a first-price sealed bid auction with uniformly distributed private values (types), the symmetric Bayesian Nash equilibrium involves each player bidding according to the same strategy function $s(v_i)=\frac{n-1}{n}v_i$, where $n$ is the number of players and $v_i$ is player $i$'s value of the auctioned object, even though the actual bids will differ based on each player's realized type $v_i$. Nevertheless, players with the same value will place the same bid.