Best Response¶

Definition¶

In a normal-form game $\langle N,\{S_i\},\{v_i\}\rangle$, fix the other players' strategies $s_{-i}\in S_{-i}$$S_{-i}$. A strategy $s_i\in S_i$ is a best response of player $i$ to $s_{-i}$ if, for every $s_i'\in S_i$$S_i$,

More generally, if player $i$ holds a belief $\pi_i$, a probability distribution over $S_{-i}$ (finite here), then $s_i$ is a best response to $\pi_i$ if it maximizes the expected payoff $\displaystyle \sum_{s_{-i}\in S_{-i}}\pi_i(s_{-i})\,v_i(s_i,s_{-i})$ over $S_i$.

Interpretation¶

A best response is what a rational agent does once a belief about the others has been formed: the opponents' play is treated as given and the player solves an ordinary maximization problem. With a probabilistic belief this is expected payoff maximization.

The figure uses the matrix of Example 1 below. If player 2's belief puts probability $\pi_2(T)$ on $T$, her expected payoffs are $3-2\pi_2(T)$ from $L$, $1+\pi_2(T)$ from $M$ and $0$ from $R$. The best response is the strategy on the highlighted upper envelope: $L$ while $\pi_2(T)<\tfrac{2}{3}$ and $M$ above $\tfrac{2}{3}$, where the two lines cross, with both optimal at $\tfrac{2}{3}$; $R$ is never a best response. At the belief $0.4$ of Example 2 the comparison is $2.2$ against $1.4$.

A best response is defined relative to $s_{-i}$; it may change when $s_{-i}$ changes. If the belief is wrong, the chosen strategy may not be optimal against what is actually played. Nash equilibrium is the situation in which every player best responds to the strategies the others actually use.

Best responses need not be unique: with ties, all maximizers are best responses, collected in the best-response correspondence $BR_i(s_{-i})$.

Key Results¶

• If $S_i$ is finite, a best response exists for every $s_{-i}$. If $S_i$ is compact and $v_i(\cdot,s_{-i})$ is continuous, existence follows from the extreme value theorem on compact sets; otherwise a best response may fail to exist (undercutting in a Bertrand duopoly).
• A strictly dominated strategy is never a best response, to any $s_{-i}$ or any belief.
• A strictly dominant strategy is the unique best response to every $s_{-i}$.
• If $S_i$ is an interval and $v_i(\cdot,s_{-i})$ is differentiable and concave, every interior point satisfying the first-order condition $\partial v_i/\partial s_i=0$ is a best response. On a closed bounded interval, continuity guarantees an optimum; if no interior point is stationary, compare the endpoints. On an open or unbounded interval, a best response may fail to exist: the payoff $v_i(s_i)=s_i$ on $(0,1)$ has no maximizer.

Examples¶

1. Matrix game. Payoffs $(v_1,v_2)$:

Player 1 compares down each column: the best response to $L$ is $T$ ($2>$$2>1$$1$), to $M$ is $T$ ($3>$$3>1$$1$), to $R$ is $B$ ($4>$$4>0$$0$). Player 2 compares along each row: the best response to $T$ is $M$ ($2>$$2>1>0$$0$), to $B$ is $L$ ($3>$$3>1>0$$0$).

2. Belief. If player 2 believes player 1 plays $T$ with probability $0.4$ and $B$ with probability $0.6$, the expected payoffs are $L$: $0.4\cdot1+0.6\cdot3=2.2$$2.2$, $M$: $0.4\cdot2+0.6\cdot1=1.4$$1.4$, $R$: $0$. The best response to this belief is $L$.

3. Cournot. Consider nonnegative quantities $q_1,q_2\ge0$ in a Cournot duopoly, using the linear price formula $100-q_1-q_2$ and unit cost $10$. Firm 1's profit is $(90-q_1-q_2)q_1$$q_2)q_1$. Its second derivative in $q_1$ is $-2<0$$0$, so the constrained maximizer is $BR_1(q_2)=\max\{0,(90-q_2)/2\}$$q_2)/2\}$. The interior first-order condition applies when $q_2<90$$90$; at $q_2\ge90$ the best response is $0$. Against $q_2=20$ it is $q_1=35$$35$.

Prerequisites¶

• Normal-Form Game
• Rational Agent

Related Concepts¶

• Best-Response Correspondence
• Nash Equilibrium
• Cournot Duopoly
• Expected Payoff Maximization

Sources¶

• Game Theory · Week 1

Aliases: best reply, BR Keywords: best response to s_{-i}, best response to a belief, belief about opponents, reaction to the rival's choice, never a best response Symbols: element of, partial, pi, sum

Last updated: 27 October 2026 11:52