Question

Consider the following two-player game:

	L	C	R
T	2,2	0,2	0,1
M	2,0	1,1	0,2
B	1,0	2,0	0,0

(a) Find all pure strategy Nash equilibria of this game.

(b) Consider the following procedure of iterated elimination of weakly dominated actions : all weakly dominated actions of each player are eliminated at each stage. What are the action profiles that survive this procedure in the above game?

I have no problem with solving (a) but (b) is so difficult. This is a question From Osborne's An introduction to game theory. Excercise 391.1. Even though there's a solution available at Chegg, I can't follow the explanation.

Answer 1

b) Since it there are weakly dominated strategies, the equilibrium depends on the order of elimination of strategies. Let us consider T and M for player 1. We see that if Player 2 plays L, Player 1 has the same payoff in T and M. If player 2 plays C, Player 1 has better payout with M and if Player 2 plays R, player A has same payoff in T and M. So M weakly dominates T. So we can eliminate T. The resulting table will be

	L	C	R
M	2,0	1,1	0,1
B	1,0	2,0	0,0

Now for Player 2, let us consider L and C.

When player 1 plays M, Player 2 has better payoff with C and when player 1 plays B, player 2 has same payoff with L and C.

So C weakly dominates L.

Thus we can eliminate L.

The resulting table is

	C	R
M	1,1	0,1
B	2,0	0,0

Again let us move to player 1. When player 2 plays C, Player 1 has a better payoff with B. When player 2 plays R, player 1 has same payoff with M and B. So B weakly dominates over M.

So the resultant table is

	C	R
B	2,0	0,0

Here we see that player 2 has the same payoff in both cases. However, since our assumption is all individuals are rational, player 2 will choose C so that Player 1 has a higher payoff.

So the effective equilibrium will be when Player 2 chooses C and player 1 chooses B.

However, this can change if our order of iterated elimination of weakly dominated strategy changes. So this is one of many solutions

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