Strictly Dominated Strategy¶

Definition¶

In a normal-form game $\langle N,\{S_i\},\{v_i\}\rangle$, let $s_i,s_i'\in S_i$ be two strategies of player $i$. Strategy $s_i'$ strictly dominates $s_i$ if, for every $s_{-i}\in S_{-i}$$S_{-i}$,

Then $s_i$ is strictly dominated (by $s_i'$). A strategy is strictly dominated if at least one other strategy of the same player strictly dominates it.

Interpretation¶

A strictly dominated strategy is worse than some alternative whatever the other players do. A rational agent therefore never plays it, and this conclusion needs no knowledge of the other players' rationality or beliefs.

Two conditions must hold together: the comparison is against one fixed alternative $s_i'$, and the inequality is strict for every $s_{-i}$. A single tie is enough to break strict dominance; the strategy may then still be weakly dominated, which is a different and weaker notion (see weakly dominant strategy, Week 6).

A common error is to call $s_i$ dominated because it is never the best choice in any column. Being worse than different strategies in different columns is not strict dominance by a pure strategy (Example 2).

Key Results¶

• A strictly dominated strategy is never a best response, neither to any $s_{-i}$ nor to any probabilistic belief about $s_{-i}$, because the dominating strategy gives a strictly higher payoff in every case and hence a strictly higher expected payoff.
• Strict dominance is transitive: if $s_i''$ strictly dominates $s_i'$ and $s_i'$ strictly dominates $s_i$, then $s_i''$ strictly dominates $s_i$.
• Removing strictly dominated strategies never removes a strategy used in a Nash equilibrium; repeating the removal gives iterated elimination of strictly dominated strategies.
• A strategy that is not dominated by any pure strategy can still be strictly dominated by a mixed strategy (Week 2).

Examples¶

1. Prisoner's Dilemma. With the payoffs of the normal-form game page, for player 1:

so Mum is strictly dominated by Fink. By symmetry the same holds for player 2.

2. Not dominated by a pure strategy. Player 1's payoffs (player 2 chooses $L$ or $R$):

$D$ is not dominated by $U$ ($1>$$1>0$ against $R$) nor by $M$ ($1>$$1>0$ against $L$). Yet the mixture "$U$ and $M$ with probability $\tfrac{1}{2}$ each" gives $1.5>1$ against both columns, so $D$ is strictly dominated by a mixed strategy.

In the figure, each strategy of player 1 is a point (payoff against $L$, payoff against $R$), and the shaded region contains the points that beat $D=(1,1)$ in both columns. Neither $U=(3,0)$ nor $M=(0,3)$ lies in it, but the segment between them consists of their mixtures, and its midpoint $\tfrac{1}{2}U+\tfrac{1}{2}M=(1.5,1.5)$ does.

3. A tie. If $v_1(F,F)$ in the Prisoner's Dilemma were changed to $-5$, then $F$ and $M$ would tie against $F$, and $M$ would no longer be strictly dominated.

Prerequisites¶

• Normal-Form Game
• Rational Agent

Related Concepts¶

• Strictly Dominant Strategy
• Iterated Elimination of Strictly Dominated Strategies
• Weakly Dominant Strategy
• Mixed Strategy

Sources¶

• Game Theory · Week 1

Aliases: strict dominance, strictly dominated action Keywords: strictly dominates, dominated strategy, s_i' strictly dominates s_i, never a best response, domination by a mixed strategy, Prisoner's Dilemma Symbols: alpha, element of

Last updated: 22 October 2026 19:04