# Definition:Best-Response Function

## Definition

Let a strategic game $G$ be modelled by:

$G = \left\langle{N, \left\langle{A_i}\right\rangle, \left\langle{\succsim_i}\right\rangle}\right\rangle$

Let $a^*$ be a Nash equilibrium of $G$:

$\forall i \in N: \forall a_i \in A_i: \left({a^*_{-i}, a^*_i}\right) \succsim_i \left({a^*_{-i}, a_i}\right)$

For any $a_{-1} \in A_{-i}$, let $B_i \left({a_{-i} }\right)$ be the set of player $i$'s best moves, defined as:

$B_i \left({a_{-i} }\right) = \left\{ {a_i \in A_i: \forall a'_i \in A_i: \left({a_{-i}, a_i}\right) \precsim_i \left({a_{-i}, a'_i}\right)}\right\}$

Then $B_{-i}$ is known as the best-response function of player $i$.