Is the replicator dynamics of asymmetric evolutionary games the same as that of symmetric ones?

Question

A common assumption in evolutionary games is their symmetry based on which a rich theory exists including the formulation of replicator dynamics. Does replicator dynamics (its differential equation, to be specific) remain the same if one studies asymmetric evolutionary games? If yes, why? If no, what will that replicator dynamics be?

Answer 1

The answer is No.

Every payoff matrix representing a pair of players interacting with each other corresponds to a replicator function. Accordingly, in the case of symmetric games, there is only one replicator function because there is only one payoff matrix which is invariant subject to player relabeling. However in the case of asymmetric games, there are two payoff matrices associated with every pair of interacting players.