Strictly Dominant Strategy¶

Definition¶

In a normal-form game, a strategy $s_i\in S_i$ is strictly dominant for player $i$ if, for every $s_i'\in S_i\setminus\{s_i\}$ and every $s_{-i}\in S_{-i}$ $S_{- i}$ ,

v_i(s_i,s_{-i})>v_i(s_i',s_{-i}).

Equivalently, $s_i$ strictly dominates every other strategy of player $i$ , so every other strategy is a strictly dominated strategy.

A profile $s^D=(s_1^D,\ldots,s_n^D)$ in which every $s_i^D$ is strictly dominant is a strictly dominant strategy equilibrium.

Interpretation¶

A strictly dominant strategy is the best choice regardless of what the others do, so a rational player needs no belief about the opponents to choose it. A prediction based on dominant strategies requires only that each player is rational, not that rationality is common knowledge.

Prisoner's Dilemma payoff matrix with arrows showing that each player gains by switching from M to F against every strategy of the opponent; the cell (F,F) is shaded.

In the figure, the blue arrows compare player 1's payoffs within each column: switching from $M$ to $F$ raises her payoff from $-2$ to $-1$ against $M$ and from $-5$ to $-4$ against $F$ . The red arrows make the same comparison for player 2 within each row. Every arrow points to $F$ whatever the opponent does, so $F$ is strictly dominant for both players, and the shaded cell $(F,F)$ is the strictly dominant strategy equilibrium, even though $(M,M)$ would give both players more.

The concept is demanding: the same strategy must be strictly best against every opponent profile. Many games have no strictly dominant strategy for at least one player. A frequent misunderstanding is to treat "best against what the opponent actually does" as dominance; that is only a best response.

Dominant strategies do not guarantee a good outcome. In the Prisoner's Dilemma both players use their dominant strategy, and both would be better off if both deviated.

Key Results¶

A player has at most one strictly dominant strategy: if $s_i$ and $t_i\ne s_i$ were both strictly dominant, each would give a strictly higher payoff than the other.
If $s_i$ is strictly dominant, then $s_i$ is the unique best response to every $s_{-i}$ (and to every belief).
A strictly dominant strategy equilibrium, if it exists, is unique, and it is the unique Nash equilibrium of the game.
Iterated elimination of strictly dominated strategies reaches it in a single round.
Existence is not guaranteed: in Battle of the Sexes and in Matching Pennies no player has a strictly dominant strategy.
The equilibrium may fail to be Pareto efficient.

Examples¶

1. Prisoner's Dilemma. With payoffs $(M,M)=(-2,-2)$ $(- 2, - 2)$ , $(M,F)=(-5,-1)$ $(- 5, - 1)$ , $(F,M)=(-1,-5)$ $(- 1, - 5)$ , $(F,F)=(-4,-4)$ $(- 4, - 4)$ :

v_1(F,M)=-1>-2=v_1(M,M),\qquad v_1(F,F)=-4>-5=v_1(M,F).

So $F$ is strictly dominant for player 1, and by symmetry for player 2. The strictly dominant strategy equilibrium is $(F,F)$ with payoffs $(-4,-4)$ , although $(M,M)$ gives $(-2,-2)$ to both.

2. No dominant strategy.

	$L$	$M$	$R$
$T$	$2,1$	$3,2$	$0,0$
$B$	$1,3$	$1,1$	$4,0$

Player 1's best choice is $T$ against $L$ and $M$ but $B$ against $R$ ; player 2's best choice is $M$ against $T$ but $L$ against $B$ . Neither player has a strictly dominant strategy, even though $R$ is strictly dominated by $M$ .

3. Cournot. In a Cournot duopoly a firm's best quantity falls as the rival's quantity rises, so no single quantity is best against every rival quantity.

Prerequisites¶

Sources¶

Game Theory · Week 1

Aliases: strictly dominant action Keywords: strictly dominant strategy equilibrium, dominant-strategy equilibrium, strictly dominates every other strategy, Prisoner's Dilemma, Fink, unique best response Symbols: element of

Last updated: 22 October 2026 19:04