You might be interested in a way of ensuring that 2 players always have the same mixed strategy in all Nash equilibria of some game:
Assume we have a player A and a player B. Player A has some already-specified utility function; we would like player B to play the same mixed strategy as A. Introduce a new player C who gets to observe either A or B’s action (unknown with 50% probability for each), and tries to determine who took this action (getting a utility of 1 for guessing correctly and 0 otherwise). B’s utility function is 1 if C guesses incorrectly, and 0 if C guesses correctly. B will use the same mixed strategy as A in all Nash equilibria.
You might be interested in a way of ensuring that 2 players always have the same mixed strategy in all Nash equilibria of some game:
Assume we have a player A and a player B. Player A has some already-specified utility function; we would like player B to play the same mixed strategy as A. Introduce a new player C who gets to observe either A or B’s action (unknown with 50% probability for each), and tries to determine who took this action (getting a utility of 1 for guessing correctly and 0 otherwise). B’s utility function is 1 if C guesses incorrectly, and 0 if C guesses correctly. B will use the same mixed strategy as A in all Nash equilibria.
A similar method is used in the appendix A of the reflective oracles paper.