Dominant and Dominated Strategies

A Dominant Strategy is the best strategy (a best reply / earns the highest payoff) in every case, no matter what other players do.

Consider the following game and 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. Perhaps the choice becomes a little more difficult for Player 1 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, stragegy 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 we think that Player 2 will choose V, Player 1 earns more from B (a payoff of 7) than from A (a payoff of 6).
• If we think that Player 2 will choose W, Player 1 earns more from B (a payoff of 5) than from A (a payoff of 4).
• If we think that Player 2 will choose X, Player 1 earns more from B (a payoff of 3) than from A (a payoff of 2).

Thus, whatever we expect Player 2 to do, Player 1's optimal strategy (the best response) is the same. This is what we call a Dominant Strategy. We would say that Player 1 has a dominant strategy, strategy B.

Imagine we change the above game slightly, raising Player 1's payoff in the top-right outcome:

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

Now Player 1 does not have a strategy that is always best. Specifically,

• If we think that Player 2 will choose V, Player 1 earns more from B (a payoff of 7) than from A (a payoff of 6).
• If we think that Player 2 will choose W, Player 1 earns more from B (a payoff of 5) than from A (a payoff of 4).
• But if we think that Player 2 will choose X, Player 1 earns more from A (a payoff of 5) than from B (a payoff of 3).

Player 1's best response is no longer the same regardless of what Player 2 does. In some cases, Player 1 prefers to use B, but not in every case. Therefore, the strategy is no longer dominant.

Does Player 2 have a dominant strategy? We use the same methodology to examine whether Player 2 has a strategy that is always best, for every possible choice of Player 1.

• If we think that Player 1 will choose A, Player 2 earns more from V (a payoff of 9) than from W (a payoff of 7) or X (a payoff of 6).
• If we think that Player 1 will choose B, Player 2 earns more from X (a payoff of 9) than from V (a payoff of 8) or W (a payoff of 2).

Since Player 2 does not have a single strategy that is always best (sometimes Player 2 prefers V and sometimes Player 2 prefers X), Player 2 does not have a dominant strategy.

A Dominated Strategy is a strategy that is always worse than some other strategy no matter what other players do.

While Player 2 does not have a dominant strategy in the game above (there is no one strategy that is always best), Player 2 does have a strategy that doesn't seem very useful. Consider the strategy W. Note that W is always worse than V. No matter what Player 1 does (plays A or B), Player 2's strategy of W always earns less than Player 2's strategy of V (7 is worse than 9 and 2 is worse than 8). We say that strategy W is dominated by V or, more simply, that strategy W is dominated.

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 and X is sometimes better than W. But the fact that W is dominated by V is enough to state that W is dominated.

Note that when a player has a dominant strategy, all of that player's other strategies must be dominated (by that dominant strategy). However, a player can have a dominated strategy without having a dominant strategy (like Player 2 in the example above).

Strictly & weakly dominant or dominated

We usually talk about strategies not merely as dominant or dominated but as either strictly dominant/dominated or weakly dominant/dominated. To understand this distinction, consider the following mathematical facts.

• 8 > 6 (the number 8 is greater than 6, sometimes stated as 8 is strictly greater than 6)
• 8 ≥ 6 (the number 8 is greater than or equal to 6, sometimes stated as 8 is weakly greater than 6)
• 8 ≥ 8 (the number 8 is greater than or equal to 8, therefore 8 is weakly greater than 8)
• But it is not true that 8 > 8 (the number 8 is not strictly greater than 8)

If we can say that a player's strategy earns more (strictly more, i.e., >) than every other strategy for that player, then it is strictly dominant. If we can say that a player's strategy earns at least as much as (weakly more, i.e., ≥) than every other strategy for that player, then it is weakly dominant. Reconsider an example from above:

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

In demonstrating that for Player 1's strategy B was dominant, we showed that:

• If we think that Player 2 will choose V, Player 1 earns more from B than from A (7>6).
• If we think that Player 2 will choose W, Player 1 earns more from B than from A (5>4).
• If we think that Player 2 will choose X, Player 1 earns more from B than from A (3>2).

Because we are able to use a '>' sign, we can say that B is strictly dominant.

On the other hand, if we modify the example slightly (changing the upper-right payoff for Player 1 to 3):

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

Now we can no longer conclude that B is strictly dominant because, in the case where Player 2 plays X, it is not true that 3>3. However, we can still conclude:

• If we think that Player 2 will choose V, Player 1 earns (weakly) more from B than from A (7 ≥ 6).
• If we think that Player 2 will choose W, Player 1 earns (weakly) more from B than from A (5 ≥ 4).
• If we think that Player 2 will choose X, Player 1 earns (weakly) more from B than from A (3 ≥ 3).

Since B is always at least as good as A (it is dominant using '≥'), we can conclude that strategy B is weakly dominant (and strategy A is weakly dominated).

For all of the confusion that the distinction between weakly and strictly causes, the distinction matters only in very specific cases. The only time we care if we're using '>' or '≥' to compare numbers is when the numbers are equal. In the same way, the only time that there is a distinction between a strategy being weakly or strictly dominant (or weakly or strictly dominated) is when there's a tie for best response.

Summary and Caveats

A strategy is dominant if it always earns more than every other strategy for that player no matter what others do.

A strategy is dominated if it always earns less than some other stratgey for that player no matter what others do.

The two statements above (and how to verify them) are the keys to understanding dominant and dominated strategies. Then, the issue of weak or strong just comes down to a small semantic issue of how we define "more" and "less." If the statements above are true when "more" or "less" are defined using '>' or '<' then the strategy is strictly dominant/dominated. If they are true when using '≤' or '≥' then the strategy is weakly dominant/dominated.

A few caveats:

• The definition for weakly dominant or dominated is just slightly bit more annoying than presented above. Technically, for a strategy to weakly dominate another, it can't earn the same as the other strategy in every case (the two strategies can't be equivalent). There has to exist at least some instance in which it earns strictly more.
• Note that strict dominance (>) implies weak dominance (≥). If 8>6 then it is also true that 8≥6. Thus, any time a strategy is strictly dominant (or strictly dominated) it is also weakly dominant (or weakly dominated).
• In class, we learned about the iterative deletion of strictly dominated strategies, where a strictly dominated strategy can just be crossed out leaving a smaller game in which we may find newly-dominated strategies to cross out, and so on. While we can cross out strictly dominated strategies, we cannot cross out weakly dominated strategies.

A common error

Sometimes, students compare the payoffs of the two players and incorrectly draw conclusions based on which player's payoffs are higher. Nothing in the analysis above ever requires comparing the payoffs of the two players. Dominance has nothing to do with which player is "earning" more. In fact, we can determine whether a player has a dominant strategy without even knowing the other player's payoffs.