Question

Jack, Jill, Mike and Molly are deciding whether or not to attend their high school reunion. However, not all four of them get along well and they would prefer to stay at home rather than meeting someone they do not like at the reunion. In particular, Jack prefers not to attend the reunion if Molly is attending the reunion, otherwise he prefers to attend. Molly prefers not to attend if Jack and/or Jill attends otherwise she prefers to attend. Mike prefers not to attend if Jack attends otherwise he prefers to attend. Finally, Jill prefers not to attend if Molly and/or Mike attends otherwise she prefers to attend. Assuming that all of them make their decision simultaneously find all of the pure strategy Nash equilibria of this game. You need to argue who will end up attending the reunion and who will stay at home in Nash equilibrium. Clearly explain your reasoning.

Answer 1

Here we have four players: Jack (JA), Jill (JL), Mike (MI), and Molly(ML). This order of names will be followed in strategy profiles.

Lets denote if they attend reunion by A and if not by N

Thus, every player has two strategies to play A or N.

There are 16 strategy profiles in total:

(A, A , A , A), (A, A, A, N), (A, A, N, N), (A, N, N, N), (A, N, A, N), (A, N, A, A), (A, N, N, A), (A, A, N, A)

(N, N, N, N), (N, N, N, A), (N, N, A, A), (N, A, A, A), (N, A, N, A), (N, A, N, N), (N, A, A, N), (N, N, A, N)

(A, A, A, A) can not be the equilibrium as not all prefer to get along.

As jack and molly can not be present in equilibrium, (A, N, A, A), (A, N, N, A), (A, A, N, A) can not be equilibrium.

As molly does not prefer to attend with jack and jill thus, (N, A, N, A), (N, A, A, A) can not be the equilibrium.

As mike prefers not to attend if jack attends (A, N, A, N), (A, N, A, A), (A, A, A, N) can not be the equilibrium.

As jill does not prefer mike or molly then (A, A, A, N), (A, A, N, A), (N, A, A, A), (N, A, N, A), (N, A, A, N) can not be the equilibrium.

Then what's left :  (A, A, N, N), (A, N, N, N), (N, N, N, N), (N, N, N, A), (N, N, A, N), (N, N, A, A), (N, A, N, N)

Now, We have to see them and ask if no one wants to deviate from each profile.

(A, A, N, N): No one wants to deviate as mike and molly would not attend because of jack and jill. NE

(A, N, N, N): Jill will want to deviate to A as mike and molly are not attending. Not a NE.

(N, N, N, N): Everyone wants to deviate, as no one is coming, they would like to come. Not a NE.

(N, N, N, A): Mike will deviate as jack is not coming so he would like to come. Not a NE.

(N, N, A, N): Molly would like to deviate to A as Jack and jill are not coming. Not a NE.

(N, N, A, A): No one will devaite. Thus, it is a NE

(N, A, N, N): Jack will deviate to A as Mike and molly are not coming.

​​​There are 2 Pure strategy Nash Equilibrium those are (N, N, A, A), and (A, A, N, N)