Pure Strategy¶

Definition¶

A pure strategy of player $i$ is a deterministic plan of action: it specifies, without any randomization, which action the player takes in every situation in which he or she may have to move. The set of all pure strategies of player $i$ is denoted $S_i$ .

A pure-strategy profile is a list $s=(s_1,\ldots,s_n)$ with $s_i\in S_i$ for every player $i\in N=\{1,\ldots,n\}$ ${1, \dots, n}$ ; the set of all profiles is the Cartesian product $S=S_1\times\cdots\times S_n$ $S_{n}$ . For player $i$ ,

s_{-i}=(s_1,\ldots,s_{i-1},s_{i+1},\ldots,s_n) \in S_{-i}=\prod_{j\ne i}S_j

collects the strategies of all other players, and $s=(s_i,s_{-i})$ $(s_{i}, s_{- i})$ . Writing $(s_i',s_{-i})$ means that player $i$ switches to $s_i'$ while everyone else keeps their strategy.

Interpretation¶

A strategy is a complete description of how a player will play, detailed enough that someone else could carry it out on the player's behalf. In a static game of complete information nobody observes anything before choosing, so a pure strategy is just one action.

Grid of the four Prisoner's Dilemma profiles in S1 × S2, with the column in which player 2 plays M shaded and an arrow from the profile (F,M) up to (M,M).

The figure lays out $S=S_1\times S_2$ for the Prisoner's Dilemma of the Examples: each cell is one profile, $2\cdot2=4$ in all. In $s=(F,M)$ we have $s_1=F$ and $s_{-1}=M$ $M$ . A switch by player 1 alone to $s_1'=M$ moves within the shaded column, because $s_{-1}$ is held fixed, and lands on $(s_1',s_{-1})=(M,M)$ $(M, M)$ ; this is the comparison behind dominance, best responses and Nash equilibrium.

The notation $(s_i',s_{-i})$ is used throughout game theory: strict dominance, best responses and Nash equilibrium all compare $v_i(s_i,s_{-i})$ with $v_i(s_i',s_{-i})$ while holding $s_{-i}$ fixed.

"Pure" contrasts with mixed: a mixed strategy (Week 2) is a probability distribution over pure strategies, and a pure strategy is the special case that puts probability $1$ on one element of $S_i$ .

Key Results¶

With finite strategy sets, the number of pure-strategy profiles is $|S|=|S_1|\cdot|S_2|\cdots|S_n|$ $∣ S_{2} ∣ \dots ∣ S_{n} ∣$ , and $|S_{-i}|=\prod_{j\ne i}|S_j|$ $\prod_{j \neq = i} ∣ S_{j} ∣$ .
Strategy sets need not be finite: a firm in a Cournot duopoly chooses $q_i\in[0,\infty)$ $[0, \infty)$ , an interval.
In a static game, strategies and actions coincide.
In a dynamic game (Week 3) a pure strategy is a complete contingent plan with one action at each of the player's information sets, so the number of pure strategies is the product of the numbers of actions over those information sets; see strategy in an extensive-form game.

Examples¶

Prisoner's Dilemma. $S_1=S_2=\{M,F\}$ (Mum, Fink), so $S=\{(M,M),(M,F),(F,M),(F,F)\}$ has $4$ profiles. In the profile $(F,M)$ , $s_1=F$ and $s_{-1}=s_2=M$ $M$ .

Three players. If $|S_1|=2$ $2$ , $|S_2|=3$ and $|S_3|=2$ $2$ , there are $2\cdot3\cdot2=12$ profiles, and player 1 faces $|S_{-1}|=3\cdot2=6$ possible combinations of the others' strategies.

Contingent plan. Player 1 chooses $L$ or $R$ ; player 2 observes this and then chooses $l$ or $r$ . Player 2 has two actions at each of two decision points, hence $2\cdot2=4$ pure strategies: $ll$ , $lr$ , $rl$ , $rr$ , where $lr$ means " $l$ after $L$ , $r$ after $R$ ". Two actions per node do not mean two strategies.

Prerequisites¶

Sources¶

Game Theory · Week 1

Aliases: deterministic strategy Keywords: strategy set S_i, strategy profile, pure-strategy profile, s_{-i}, complete contingent plan, Prisoner's Dilemma Symbols: element of, infinity, product

Last updated: 22 October 2026 19:04