Dominance
Compare actions against every opponent action before using a distributional guess.
11 minutes · advanced · Original Poker4Quant lessonWhat you will learn
Distinguish strict dominance, weak dominance and an action that merely has higher average value.
Prerequisites: Actions and payoffs
Compare cell by cell
Action A strictly dominates B for a maximizing player if A gives a strictly higher payoff against every opponent action. It weakly dominates B if it is never worse and is better against at least one opponent action. These comparisons do not need an assumed distribution of the opponent’s choices.
A higher payoff under one guessed distribution is different. An action can be attractive against a frequent opponent choice yet perform worse against another. It is then a best response to that distribution, not necessarily a dominating action.
Elimination has conditions
In a finite game where players maximize the stated expected payoffs, a strictly dominated pure action cannot receive positive probability in a Nash equilibrium. The dominating alternative would improve that player’s payoff against the equilibrium opponent mixture.
Weak dominance is more delicate. A weakly dominated action can be tied on the opponent’s actual support and can appear in an equilibrium. Eliminating weakly dominated actions without care can remove equilibria or make iterative outcomes depend on elimination order.
Worked example
The maximizing row player has A=[3,2] and B=[1,0] against columns L and R. Is A dominant?
- Against L, A exceeds B by 3−1=2.
- Against R, A exceeds B by 2−0=2.
- Every opponent action gives A a strictly larger payoff.
A strictly dominates B. For any mixture over L and R, switching from B to A gains exactly two units.
Common mistakes
- Comparing only one column.
- Calling a larger guessed average strict dominance.
- Treating weak and strict elimination as interchangeable.
Keep these ideas
- Dominance compares every possible opponent action.
- Strict means strictly better everywhere.
- Weak dominance allows ties and needs extra care in equilibrium arguments.
Further reading
- MIT OpenCourseWare — Mixed Strategy Nash Equilibrium, lecture 3 ↗
University lecture background on expected utility, mixed strategies and indifference. Poker4Quant’s toy payoffs and exercises are original; this link does not imply affiliation or endorsement. Checked 2026-10-21.
- NYU — Game Theory, Problem Set 1 (2003) ↗
University problem set on best responses, dominance and the limitations of weak-dominance elimination. Background reading; the exercises on this site use their own specified payoffs. Checked 2026-10-21.