As I mentioned before, not all games have a strictly dominant strategy. However, there’s another way we can use the concept of strictly dominated strategies to find equilibria.
Let’s say we have another two-player game. The players have choices A, B, C, and D. Player 1’s payoff is negative one if players play B,D, zero if they play B,C, or A,D, and one if they play A,C. Player 2’s payoffs are five if they play A,C, and two if they play A,D, and… actually there’s no point in me saying what his payoffs are in the other two cases, because we already have enough information to find an equilibrium.
But how can that be? It’s simple. Player 2 knows player 1’s payoffs, and he knows that player 1 is rational. So he knows that player 1 will not play B, because player 1 is always better off playing A. B is strictly dominated by A, so player 2 can ignore the possibility that B will be played. So we can delete B as an option, and then we see that player 2 must choose C, as A,C gives him a larger payoff than does A,D.
This might be an dominant strategy equilibrium depending on player 2’s payoffs, but it doesn’t have to be.
Here’s a more complicated example. There’s no dominant strategy equilibrium here, but let’s see if we can find an equilibrium by deleting the strictly dominated strategies.
We can see that player 1 will never pick C because he’s always better off choosing B, so let’s delete that. Now, given that C has been deleted, F can be deleted also. A,F and B,F both give player 2 a payoff of zero, and he could get a payoff of one by playing G, so he won’t play F. With F deleted, player 1 won’t ever play A, since he can always be better off playing B. That leaves player 1 playing B for sure, and player 2 can clearly see that he get his highest payoff in B,D, so he can eliminate E and G.
So there you have it, an equilibrium.
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