a) Let $G=(A,u)$ be a strategic game such that, for each $i \in N$
$A_i$ is a nonempty, convex, compact subset of $R^{m_i}$
$u_i$ is continuous
For each $a_{-i}$, $u_i(a_{-i}, . )$ is quasi-concave on $A_i$
Prove that the set of Nash equilibria (NE) is closed.
b) For the same conditions as in a), except $A_i$ need not be concave, prove that the best response function $BR_i$ for each i need not be lower hemicontinuous.
Here is my progress so far: For a), it is trivial for games with finite NE. Assume game G has infinitely many NE, and x is a limit point of the set of NEs. Since A is compact, x is in A. Since $u_i$ is continuous, I $should$ be able to show that $u_i(x)$ is the maximum outcome for player i, provided all other players play their strategies according to the NE at that point. (I hope all that makes some sense)
For b), I imagine if at the limit point x of all the NEs, player i is indifferent among all his strategies, then I have a function $BR_i$ that is not lower hemicontinuous?
I realize I sound confusing; I expressed my thinking process the best I could.
Any help would be greatly appreciated.
The assumption that the $A_i$ are nonempty and convex and that $u_i$ is quasi-concave in $i$'s own strategy are all superfluous. Trivially, if $A_i=\emptyset$ for some player $i$, then the set of strategy profiles is empty and so is the subset of Nash equilibria. The empty set is closed.
Let $n=|N|$. So assume that $A_i$ is nonempty for all $i$. Let $B_i:A\to 2^{A_i}$ be the best response correspondence of player $i$, where $A=\prod_i A_i$. By Berge's maximum theorem, $B_i$ is upper hemicontinuous and compact-valued. Therefore, the correspondence $B:A\to 2^A$ given by $B(a)=B_1(a)\times\ldots\times B_n(a)$ is upper hemicontinuous and compact valued. A Nash equilibrium is now simply a point $a\in A$ such that $a\in B(a)$. By the closed graph theorem for correspondences, $B$ has a closed graph $\Gamma\subseteq A\times A$. Let $D\subseteq A\times A$ be the diagonal. Then $a$ is a Nash equilibrium exactly if $(a,a)\in \Gamma\cap D$. The set $\Gamma\cap D$ is clearly closed and as a closed subset of a compact set compact. The projection $\pi:A\times A$ given by $\pi(x,y)=x$ is continuous, so $\pi(\Gamma\cap D)$ is compact and therefore closed. But $\pi(\Gamma\cap D)$ is exactly the set of Nash equilibria.
The best responce correspondence of the mixed extension of matching pennies is not lower hemicontinuous (and even quasi-concave in each player's own strategy).