Dominated strategy

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

A pure strategy is very weakly dominated by another strategy when its payoff is no greater against every opposing pure strategy profile, with no strict advantage required anywhere. The definition permits payoff-equivalent strategies and even domination by the original strategy itself. A weakly dominated strategy instead has a strict inequality for at least one opposing strategy profile.

In a binary game, suppose ${\sigma }_i$ very weakly dominates $s_i$ and assigns positive probability to a different pure action. At every opposing strategy profile where $s_i$ pays one, all actions in the support of a mixed strategy must also pay one. Where $s_i$ pays zero, their payoffs are automatically no smaller. Therefore every distinct supported action very weakly dominates $s_i$. If the mixed domination is weak with a strict improvement somewhere, choose a supported action paying one at a profile where $s_i$ pays zero; that action also supplies a strict improvement. The requirement of nontrivial support is indispensable for the very-weak version.

In a finite zero-sum game, removing a strictly dominated strategy preserves the value of a zero-sum game: replace any probability on it by the dominating pure strategy. This increases every payoff for the maximizing player, or decreases every payoff for the minimizing player. Repeat on the reduced game; new domination relations may appear after a deletion.

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.