Best response

Any strategy $s$ of player $i$ following $B{R}_i({\mathbf{s}}_{-i})={\text{argmax}}_{s\in S_i}{u}_i(s,{\mathbf{s}}_{-i})\forall {\mathbf{s}}_{-i}\in {\mathbf{S}}_{-i}$ is a best response of player i given opponents strategies.

Strategy profile $s^{\ast }$ is Nash Equilibrium if and only if $s_i^{\ast }$ is best response for each player i.

Mixed Best response

Choose the best probability set from the Mixed strategies simplex (any set of probabilities, which sum is 1). $B{R}_i({\mathbf{p}}_{-i})={\text{argmax}}_{p\in {\Delta }_i}{U}_i(p,{\mathbf{p}}_{-i})\forall {\mathbf{p}}_{-i}\in {\mathbf{\Delta }}_{-i}$