# 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.

• Can you explain how the "two dimensions of heterogeneity" and the "two types" relate to each other? When I see "two types", I think of a single dimension of heterogeneity (e.g. worker's productivity) that has two values (e.g. high and low). But I'm confused when you throw "two dimensions of heterogeneity" into the mix. Unless of course there is only one possible value in each dimension. Feb 2, 2021 at 15:32
• Thanks for the comment! It should be four, very silly of me!
– Ali
Feb 2, 2021 at 15:54
• So, for example heterogeneous in productivity and skill (I cannot think of a better example). Each aspect has only two levels, say high and low. So a worker can be highly productive and low skilled, highly productive and high skilled, etc. My interest is in the terminology for the equilibria in which players of the same type will employ the same strategy. I worked out the pure equilibria of the game but I am at a loss on how equilibria are classified in perfect but incomplete information games.
– Ali
Feb 2, 2021 at 16:00

I think you may be confusing between "strategy" and "action", which may be why you rejected the term "symmetric equilibrium". 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.