Question

Can there be more than one equilibria in a simultaneous game? Explain and/or

give na example of why or why not?

Answer 1

Often in simultaneous games, players have more than one Nash equilibrium. In these cases, it can be much harder to narrow down all players' options to just one clear strategy.

Suppose, for instance, Albert and Betty are talking on the phone but suddenly get disconnected. If Betty tries to call Albert back, then Albert should stay off the line rather than try to call Betty. However, if Albert decides to wait for Betty to call — but Betty also decides to wait, thinking that Albert will call — the call will never be completed. So the two Nash equilibrium outcomes for this game are Betty calls/Albert waits or Betty waits/Albert calls, as shown in the following payoff matrix:

	Betty
Call	Wait
Albert	Call	0, 0	10, 10
Wait	10, 10	0, 0

Notice that with multiple Nash equilibria neither player has a dominant strategy. This illustrates the limits of the Nash concept: once multiple equilibria have been identified, the concept of Nash equilibrium cannot be used to determine which equilibrium will actually occur — or, in fact, if any will occur.

In the real world, communication, even with an opponent, can often be the solution to this problem. However, in situations where communication is not possible, the determination of what outcome (Nash or otherwise) will occur is entirely dependent on the overall conditions of the game.

For instance, in the game above, if Albert initiated the first call, it might be logical for him to initiate the second. However, maybe Betty knows Albert's phone number, but he does not know hers, in which case she has to be the one to call. Without clear communication here, the outcome is in question.