Dominant Strategy

Relevant Courses: Game Theory

Domination

Let $s_i$ and $s_i^{\prime }$ be two strategies for player $i$, and let $S_{-i}$ be the set of all possible strategy profiles for the other players.

Definition: Strict Domination

$s_i$ strictly dominations $s_i^{\prime }$ if $\forall s_{-i}\in S_{-i}$, $u_i(s_i,s_{-i})>u_i(s_i^{\prime },s_{-i})$. In other words, for all situations (no matter what the other players do), person $i$ is strictly better off using strategy $s_i$ over strategy $s_i^{\prime }$ (without fail).

Definition: Very Weak Domination

$s_i$ very weakly dominates $s_i^{\prime }$ if $\forall s_{-i}\in S_{-i}$, $u_i(s_i,s_{-i})\ge u_i(s_i^{\prime },s_{-i})$. In other words, for all situations (no matter what the other players do), person $i$ is better off or the same using strategy $s_i$ over strategy $s_i^{\prime }$.

Equilibria and Dominance

• If one strategy dominates all other strategies, we say it is dominant.
• A strategy profile consisting of dominant strategies for every player must be Nash equilibrium
  • An equilibrium in strictly dominant strategies must be unique.