I'm still a student so a bit hesitant to try to make a contribution here, but as an economist quite enamored by topology, I've wondered similar things. The below is based on an old attempt at a solution to an exercise in a rather challenging book (reference at end).
Consider first a result on coincidences of certain mappings.
Theorem Let $X$ be a finite dimensional compact Hausdorff space, $S$ a compact, convex subset of Euclidean space. Let $f,g: X\to S$ be continuous maps, with $f$ a closed, surjective map with $\tilde{H}^q(f^{-1}(s))=0$ for all $q\ge 0$ and $s \in S$ (where $\tilde{H}^*$ denotes reduced Cech cohomology with coefficients in some module $G$). Then there exists a coincidence point $f(x) = g(x)$.
Proof: Define $\partial S$ as the boundary of $S$ and $\partial X = f^{-1}(\partial S)$. Consider the two long exact sequences in relative cohomology for the pair $(X, \partial X)$ and $(S, \partial S)$. By the Vietoris-Begle Theorem and the Five Lemma, $f^*$ acting on the relative groups is an isomorphism, and in particular, is non-zero, given $\tilde{H}^{n-1}(S,\partial S; G) \neq 0$.
Now, if there were no $x \in X$ for which $f(x) = g(x)$ then using the ray from $g(x)$ through $f(x)$ we can define a function $\bar{f}: X \to \partial S$ that is homotopic to $f$, and which agrees with $f$ restricted to $\partial X$. But $\bar{f}^*$ is zero on the relative groups because $\bar{f}$ takes X onto $\partial S$, hence $f^*$ is zero, contradicting the above. QED
Now, consider an upper-hemicontinuous correspondence $\phi: S\to S$. Let the graph of this correspondence play the role of $X$ in the above, the projection to the domain $f$, and the projection to the codomain $g$. Call $\phi$ "acyclic valued" if the projection to the domain obeys the conditions on $f$ above. Then coincidence points of $f$ and $g$ are precisely those fixed points $s^* \in \phi(s^*)$.
If you are interested, the above proof is set as an exercise in Repeated Games by Mertens, Sorin, and Zamir (p. 54-55), as well as a number of extensions.