= Best response
{title2=$\operatorname{BR}_i(\sigma_{-i})$}
{wiki}
A <best response> maximizes a player's <expected payoff> while the other players' strategies are held fixed. In a <finite game>, write the full <set> of <mixed strategy> <best responses> as
$$
\operatorname{BR}_i(\sigma_{-i})=
\left\{\tau_i\in\Delta(S_i):U_i(\tau_i,\sigma_{-i})=
\max_{\rho_i\in\Delta(S_i)}U_i(\rho_i,\sigma_{-i})\right\}.
$$
This is generally a <set-valued mapping>, since several choices may tie. Every mixture supported on maximizing <pure strategies> is a <best response>, and mixing in a strictly worse choice reduces <expected payoff>.
For example, against $q=3/4$ in <matching pennies>, the row player's heads and tails <expected payoffs> are $1/2$ and $-1/2$, so heads is the unique <best response>. Against $q=1/2$, all mixtures are <best responses>. In the <asymmetric coordination example>, against $q=3/4$ the row player gets $9/4$ from $A$ and $1/2$ from $B$, so $A$ is optimal. A <Nash equilibrium> requires everyone to be a <best response> simultaneously; one optimal reply alone does not establish it.
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