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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 asThis 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.

A nonempty compact convex set $K$ in a finite-dimensional space and an upper hemicontinuous set-valued mapping $F:K\rightrightarrows K$ with nonempty closed convex values admit $x\in F(x)$. A closed graph suffices when the whole output space is compact.

For $v\in Z$, write $M(v)={\max }_i{v}_i$. Maximizing a linear functional on the price simplex givesThus $g(v)$ is a nonempty closed convex face of the compact price simplex. Its graph of a set-valued mapping is closed: if $v_n\to v$, $p_n\to p$ and $p_n\in g(v_n)$, then, for every fixed $q\in P$, the inequalities $p_n\cdot v_n\ge q\cdot v_n$ pass to $p\cdot v\ge q\cdot v$.

Each value $z(p)$ is closed in $Z$ by the closed-graph hypothesis, hence compact; it is nonempty and convex by assumption. Therefore $F(p,v)=g(v)\times z(p)$ has nonempty closed convex values in the nonempty compact convex set $K=P\times Z$.

Its graph of a set-valued mapping is also closed. If $(p_n,v_n)\to (p,v)$ and $(q_n,w_n)\to (q,w)$ with $q_n\in g(v_n)$ and $w_n\in z(p_n)$, the two closed-graph properties give $q\in g(v)$ and $w\in z(p)$. Finally, this closed graph implies upper hemicontinuity because the whole output space $K$ is compact. If an open set containing $F(p,v)$ failed to contain the nearby values, select outputs outside it along a convergent input sequence. A subsequence of those outputs converges in $K$; closedness puts its limit in $F(p,v)$, whereas closedness of the open set's complement puts it outside. This contradiction proves upper hemicontinuity.

Consequently all the hypotheses of the Kakutani fixed-point theorem hold: the domain is nonempty, compact and convex, and the set-valued mapping maps it into itself with nonempty closed convex values and upper hemicontinuity.

Let $S:{\mathbb{R}}^p\rightrightarrows {\mathbb{R}}^q$ be a set-valued mapping and let $\bar{v}\in S(\bar{u})$. The Aubin property at $(\bar{u},\bar{v})$ means there are neighborhoods $U$ of $\bar{u}$, $V$ of $\bar{v}$, and a finite constant $\kappa \ge 0$ such thatHere ${\mathbb{B}}_q$ is the closed unit ball. Equivalently, every solution $v\in S(u)$ near $\bar{v}$ can be matched to a solution $v^{\prime }\in S(u^{\prime })$ within $\kappa \Vert u-u^{\prime }{\Vert }_2$. The localization is on the left-hand side: the matching point need not be in $V$. The property is also called the Lipschitz-like property.

For sensitivity analysis, let $u$ represent data or perturbations and $S(u)$ the set of feasible or optimal solutions. The Aubin property bounds how far a nearby solution can move when the data change. Taking $u=\bar{u}$, $v=\bar{v}$ also guarantees a nearby solution for each sufficiently small perturbation $u^{\prime }$. It is a stability estimate for a relation, and by itself does not imply uniqueness or differentiability. For a single-valued solution map, it reduces to local Lipschitz continuity.

A set-valued mapping $F$ is upper hemicontinuous at $x$ when every open set containing $F(x)$ also contains $F(x^{\prime })$ for all nearby $x^{\prime }$. With values in a fixed compact space, a closed graph of a set-valued mapping implies upper hemicontinuity: a violating sequence has a convergent subsequence of outputs whose limit lies in the limiting value.

Variational analysis studies optimization, perturbation stability and generalized differentiation through functions, sets and set-valued mappings. Convex analysis is a foundational special case; limiting normal constructions also handle nonconvex geometry.