Dominant Strategy
What is Dominant Strategy?
A dominant strategy is a strategy that results in the highest payoff for a player regardless of the strategies chosen by other players.
In game theory, a dominant strategy is the best course of action for a player in a game, no matter what the opponents do. If a player has a dominant strategy, they will always choose it. Not all games have a dominant strategy for each player.
Dominant Strategy: a worked example
Two rival gyms decide whether to run a discount campaign. Payoffs are annual profits in thousands with the row gym listed first. Both advertise: (8, 8). Row advertises and column does not: (14, 4). Row does not and column advertises: (4, 14). Neither advertises: (10, 10). Check the row gym while holding the column gym fixed. If the column gym advertises, row earns 8 by advertising and 4 by staying quiet, so advertising wins. If the column gym stays quiet, row earns 14 by advertising and 10 by staying quiet, so advertising wins again. Advertising beats the alternative in every column, making it a dominant strategy, and the same comparison holds for the column gym. Both advertise and both land on 8, below the 10 each would have had by staying quiet.
The mistake students make with dominant strategy
The comparison gets made in the wrong direction. Students look at the cell where both gyms advertise, see 8 against 8, and start comparing one player's payoff with the rival's payoff instead of comparing that player's own payoffs against a fixed rival choice. Dominance is checked one column at a time for the row player and one row at a time for the column player. A second slip is assuming a dominant strategy delivers the best number on the board. Here it delivers 8 while the cell neither gym can reach delivers 10, and plenty of games hand a player no dominant strategy at all.
Dominant Strategy questions
How do you find a dominant strategy in a payoff matrix?
Hold the other player's choice fixed and compare only your own payoffs. For the row player, cover one column at a time and circle the larger of the two row payoffs in it. If the same row gets circled in every column, that row is a dominant strategy. Repeat across the rows for the column player, comparing the second number in each pair. A strategy that wins in some columns but not others is not dominant.
Can a game have no dominant strategy?
Many games give neither player a dominant strategy, and the AP exam tests that case on purpose. Whenever the best move depends on what the rival does, such as matching a price cut only when the rival cuts, no single strategy wins in every column. The game can still have a Nash equilibrium, found by checking best responses one at a time, and a game can hand one player a dominant strategy while the other player has none.
Is a dominant strategy the same as a Nash equilibrium?
A dominant strategy belongs to one player, while a Nash equilibrium describes a combination of choices. When both players have a dominant strategy, the cell where those two strategies meet is always a Nash equilibrium, because neither side gains by moving alone. The reverse fails: many Nash equilibria rest on strategies that are best responses only to what the rival happens to be doing, not to everything the rival could do.
Related terms
Common comparisons
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