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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 $F:X\rightrightarrows Y$ is upper hemicontinuous at $x$ when every open set $U$ containing $F(x)$ also contains $F(x^{\prime })$ for all $x^{\prime }$ in some neighborhood of $x$. Empty values are permitted. For a single-valued map this is the usual neighborhood definition of continuity.

If $Y$ is compact and the graph of a set-valued mapping is closed, upper hemicontinuity follows without a sequential-countability assumption. For each $y\in Y\setminus U$, the point $(x,y)$ is outside the closed graph, so there is a product neighborhood $N_y\times V_y$ disjoint from it. Finitely many $V_y$ cover the compact set $Y\setminus U$. Intersect their $N_y$: every nearby value is then contained in $U$.

Compact control is essential for this implication. The graph of $f(x)=1/x$ for $x\ne 0$, $f(0)=0$, is closed in ${\mathbb{R}}^2$, but the corresponding singleton-valued map is not upper hemicontinuous at zero. Its nearby outputs escape every bounded neighborhood of zero.

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.