Dominant and Dominated Strategies

Dominant strategy

A Dominant Strategy is a best reply (earns the highest payoff) in every possible scenario, no matter what other players choose to do.

Consider the following game. Think about what you would do as Player 1.

		Player 2
		V	W	X
Player 1	A	0 , 9	0 , 7	0 , 6
	B	7 , 8	5 , 2	3 , 9

The choice should be pretty easy. By playing A, you always earn 0, but by playing B you earn at least 3. We would certainly expect Player 1 to play B. Regardless of whether Player 2 chooses V, W, or X, strategy B results in a higher payoff for you than strategy A. Therefore, B is a dominant strategy.

The choice might appear less obvious in this game:

		Player 2
		V	W	X
Player 1	A	6 , 9	4 , 7	2 , 6
	B	7 , 8	5 , 2	3 , 9

Now, strategy A can earn as much as 6 and strategy B can earn as little as 3. However, strategy B earns a higher payoff for Player 1 than strategy A does for every possible decision of Player 2. Specifically:

• If Player 2 chooses V, Player 1 earns 7 from B vs 6 from A.
• If Player 2 chooses W, Player 1 earns 5 from B vs 4 from A.
• If Player 2 chooses X, Player 1 earns 3 from B vs 2 from A.

Because B earns more in each case (7 > 6, 5 > 4, 3 > 2), B is dominant.

Dominated strategy

A Dominated Strategy is a strategy that is always worse than some other strategy no matter what other players do. If you can find even one other strategy that provides a higher payoff in every scenario, the current strategy is dominated.

Let's look at either game above but from Player 2's perspective. Player 2 looks only at their own payoffs (the second numbers in each cell):

• If Player 1 plays A: Player 2 gets 9 from V, 7 from W, and 6 from X.
• If Player 1 plays B: Player 2 gets 8 from V, 2 from W, and 9 from X.

Does Player 2 have a dominant strategy? No. There is no single strategy that is always best. V is best if Player 1 plays A, but X is best if Player 1 plays B.

However, Player 2 does have a dominated strategy: W. To see why, compare W to V. No matter what Player 1 does, strategy W always earns less than strategy V (7 < 9 and 2 < 8).

Because W is always worse than V, we say W is dominated by V. Player 2 would never choose W.

Note that to be dominated, a strategy doesn't have to be the worst in every case, or worse than every other strategy. It merely has to be dominated by one other strategy. For Player 2 above, W is sometimes better than X (7 > 6) and X is sometimes better than W (9 > 2). But the fact that W is dominated by V is enough to state that W is dominated.

Also note that if one strategy is dominant, all of that player's other strategies must be dominated, since they are all worse in every case than the dominant strategy.

When no strategy is dominant or dominated

Imagine we change the payoffs so that Player 1's payoff in the top-right outcome is 5:

		Player 2
		V	W	X
Player 1	A	6 , 9	4 , 7	5 , 6
	B	7 , 8	5 , 2	3 , 9

Now Player 1 has no dominant strategy. If Player 2 plays V or W, Player 1 prefers B. But if Player 2 plays X, Player 1 prefers A. Furthermore, neither strategy is dominated because A is better in one case (X) and B is better in others (V, W).

Strictly vs. Weakly Dominant

The distinction between strict and weak depends on whether a strategy is always strictly better (>) or just at least as good (≥).

In our second example, B was strictly dominant because the payoffs for B compared to A were:

• 7 > 6 (against V)
• 5 > 4 (against W)
• 3 > 2 (against X)

Since the payoff is strictly greater in every single case, it is a strictly dominant strategy.

Now consider this variation where we change Player 1's top-right payoff to 3:

		Player 2
		V	W	X
Player 1	A	6 , 9	4 , 7	3 , 6
	B	7 , 8	5 , 2	3 , 9

Now, comparing B to A, we get:

• 7 > 6 (against V)
• 5 > 4 (against W)
• 3 = 3 (against X)

Because 3 is not strictly greater than 3, strategy B is no longer strictly dominant. However, because 7 ≥ 6, 5 ≥ 4, and 3 ≥ 3, strategy B is weakly dominant.

For a strategy to be weakly dominant, it must be at least as good as the alternative in all cases (≥) and strictly better (>) in at least one case.

Note that if a strategy is strictly dominant then it is also weakly dominant. This is because the condition for strict dominance (B > A) satisfies the condition for weak dominance (B ≥ A) in every possible case. Similarly, any strategy that is strictly dominated is also weakly dominated, as being strictly worse in every scenario automatically means the strategy is at least as bad as (and often worse than) the alternative.

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A Common Error

Students often compare their payoff to the other player's payoff (e.g., "I get 7 and they get 8, so I'm losing"). This is irrelevant to dominance. Dominance only compares your own potential payoffs against each other. You can have a dominant strategy that results in you getting a lower payoff than your opponent.