Dominated strategy

A dominated strategy can be replaced by another strategy that performs at least as well against every opponent action.

A strategy is weakly dominated if another gives at least as much payoff against every opponent strategy and strictly more against at least one. Unlike a strictly dominated strategy, it may remain a best response to opponents who never choose a strategy where the comparison is strict. Thus weak domination does not exclude equilibrium strategies.

Starting from the full finite bimatrix game, sequentially remove actions weakly dominated by current mixed strategies, with a strict improvement somewhere. At least one Nash equilibrium survives on the final action sets. Take an Nash equilibrium of the reduced game and restore actions in reverse. A restored action is no better against the fixed surviving opponent strategy than its dominating mixture of already available actions, so it introduces no profitable deviation. Any self-weight in a dominating mixture can be removed and renormalized because strict improvement forces that weight below one. Using dominance tested only against an arbitrary initially restricted opponent set does not guarantee an Nash equilibrium of the original full game.

In the zero-sum game with row payoff matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, the second row is weakly dominated by the first. Nevertheless second row and first column form a Nash equilibrium: both rows tie against that column, and both columns tie against that row. Weak dominance therefore cannot be used to discard all possible equilibrium strategies without further qualifications.