Understanding Dominant Strategy
Choosing well in a game normally requires anticipating what others will do. Domination identifies the special situations where that reasoning is unnecessary, because one option is better whatever the others choose.
Russell and Norvig give the definitions precisely. A strategy s strongly dominates s′ if the outcome of s is better for that player than the outcome of s′ for every choice of strategies by the other players. Strategy s weakly dominates s′ if it is better on at least one strategy profile and no worse on any other. A dominant strategy is one dominating all others available.
The normative consequence is stated equally directly: it is irrational to play a dominated strategy, and irrational not to play a dominant strategy when one exists. Since the comparison holds across every contingency, no belief about the opponents can justify the dominated option.
When every player possesses a dominant strategy, each plays it without needing to reason about anyone else, and the resulting combination is a dominant strategy equilibrium. Such an equilibrium is always a Nash equilibrium, since nobody can gain by deviating. The reverse does not hold: many games have Nash equilibria without any dominant strategies, which is exactly why the more general concept is needed.
Example of Dominant Strategy
In the prisoner’s dilemma, Alice reasons through both cases. If Bob testifies, she gets 5 years for testifying and 10 for refusing, so testifying is better. If Bob refuses, she goes free by testifying and gets 1 year by refusing, so testifying is better again.
Testifying is better in every case, so it strongly dominates refusing and is Alice’s dominant strategy. The payoffs are symmetric, so the identical argument applies to Bob.
Both therefore testify and both receive 5 years, despite both refusing being available and giving each only 1 year. Individually irresistible reasoning produces a jointly worse outcome, which is what makes the situation a dilemma rather than merely a puzzle.
Frequently Asked Questions
What is the difference between strong and weak domination?
Strong domination requires the strategy to be strictly better in every case. Weak domination requires it to be at least as good everywhere and strictly better somewhere. Strongly dominated strategies can be eliminated safely; eliminating weakly dominated ones can remove equilibria.
Does every game have a dominant strategy?
No, and most do not. Matching pennies, for example, has none: the best choice depends entirely on the opponent’s choice. Where no dominant strategy exists, Nash equilibrium is the appropriate solution concept.
Is a dominant strategy equilibrium always good for the players?
Not at all. It is individually rational but can be collectively poor, as the prisoner’s dilemma shows: the equilibrium outcome is worse for both than an alternative they could have reached together but neither can unilaterally sustain.
The Bottom Line
A dominant strategy is best regardless of what anyone else does, so it can be chosen without predicting them. Such strategies are rare, and when they exist they can still deliver an outcome that is worse for everyone than one they jointly forgo.