Best-Response Correspondence¶

Definition¶

The best-response correspondence of player $i$ assigns to every profile of the other players' strategies $s_{-i}\in S_{-i}$ the set of player $i$ 's best responses:

BR_i(s_{-i}) =\{s_i\in S_i\colon v_i(s_i,s_{-i})\ge v_i(s_i',s_{-i})\quad \text{for every } s_i'\in S_i\} =\arg\max_{s_i\in S_i}v_i(s_i,s_{-i}).

\begin{aligned} &BR_i(s_{-i})\\ &=\{s_i\in S_i\colon v_i(s_i,s_{-i})\ge v_i(s_i',s_{-i})\\ &\qquad\text{for every } s_i'\in S_i\}\\ &=\arg\max_{s_i\in S_i}v_i(s_i,s_{-i}). \end{aligned}

A correspondence is a rule that assigns to each point a set, which may contain several elements or be the empty set. If $BR_i(s_{-i})$ contains exactly one element for every $s_{-i}$ , the correspondence is an ordinary function, called a best-response function (or reaction function).

Interpretation¶

The correspondence summarizes all of player $i$ 's optimal reactions in one object. Several strategies can be optimal against the same $s_{-i}$ when payoffs tie, which is why a set is used rather than a single value.

Graph of the Bertrand best-response correspondence of firm 1 in the (p2, p1) plane: a shaded region for p2 below c, a vertical ray at p2 = c, nothing from c to p^m, and a flat line at p^m.

The figure draws Example 3 below: above each rival price $p_2$ it shows the whole set $BR_1(p_2)$ . For $p_2<c$ the set is the open region above the diagonal, at $p_2=c$ it is the vertical ray $[c,\infty)$ , and for $p_2>p^m$ it is the single point $p^m$ , where the correspondence behaves like a function. For $c<p_2\le p^m$ nothing is drawn above the axis, because the set of best responses is empty.

Move the rival price $p_2$ and read off $BR_1(p_2)$ above it: a region for $p_2<c$ $c$ , a ray at $p_2=c$ $c$ , nothing between $c$ and $p^m$ , and the single point $p^m$ beyond.

A frequent mistake is to report only one maximizer when several tie; a Nash equilibrium can then be missed. A second is to assume $BR_i(s_{-i})$ is always nonempty: in games with infinite strategy sets the supremum of the payoff may not be attained.

Key Results¶

A profile $s^*$ is a Nash equilibrium if and only if $s_i^*\in BR_i(s_{-i}^*)$ for every $i\in N$ $N$ .
For two players with interval strategy sets, the Nash equilibria are the intersection points of the graphs of $BR_1$ and $BR_2$ drawn in the same $(s_1,s_2)$ plane.
If $S_i$ is finite, $BR_i(s_{-i})$ is nonempty for every $s_{-i}$ .

Examples¶

1. Ties. Suppose player 1 has strategies $U$ , $M$ , $D$ and, against player 2's strategy $L$ , receives $1$ , $3$ , $3$ . Then $BR_1(L)=\{M,D\}$ ${M, D}$ , a two-element set.

2. Cournot reaction functions. In a Cournot duopoly with price $100-q_1-q_2$ and unit cost $10$ ,

BR_1(q_2)=\max\Big\{\frac{90-q_2}{2},\,0\Big\},\qquad BR_2(q_1)=\max\Big\{\frac{90-q_1}{2},\,0\Big\}.

Both are single-valued. Their intersection solves $q_1=(90-q_2)/2$ and $q_2=(90-q_1)/2$ $q_{1}) /2$ , so $q_1=q_2=30$ $30$ .

3. Bertrand: an empty set. In a Bertrand duopoly with identical products, unit cost $c$ , demand $D(p)=\max\{(a-p)/b,0\}$ with $a>c$ and $b>0$ $0$ , and monopoly price $p^m=(a+c)/2$ $c) /2$ :

$BR_1(p_2)=(p_2,\infty)$ if $p_2<c$ (any price that sells nothing);
$BR_1(p_2)=[c,\infty)$ if $p_2=c$ (profit is $0$ at best);
$BR_1(p_2)=\varnothing$ if $c<p_2\le p^m$ $p^{m}$ : firm 1 wants to price just below $p_2$ , but no largest price below $p_2$ exists;
$BR_1(p_2)=\{p^m\}$ if $p_2>p^m$ $p^{m}$ .

Prerequisites¶

Sources¶

Game Theory · Week 1

Aliases: Best Response Correspondence, best-reply correspondence, BR correspondence, reaction curve, best-response curve Keywords: BR_i(s_{-i}), set-valued map, best-response function, reaction function, intersection of best-response curves Symbols: element of, infinity

Last updated: 27 October 2026 11:52