Definition:Best-Response Function

Definition
Let a strategic game $G$ be modelled by:
 * $G = \stratgame N {A_i} {\succsim_i}$

Let $a^*$ be a Nash equilibrium of $G$:
 * $\forall i \in N: \forall a_i \in A_i: \tuple {a^*_{-i}, a^*_i} \succsim_i \tuple {a^*_{-i}, a_i}$

For any $a_{-1} \in A_{-i}$, let $\map {B_i} {a_{-i} }$ be the set of player $i$'s best moves, defined as:
 * $\map {B_i} {a_{-i} } = \set {a_i \in A_i: \forall a'_i \in A_i: \tuple {a_{-i}, a_i} \precsim_i \tuple {a_{-i}, a'_i} }$

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