Existence of some pure-strategy Nash Equilibrium

Question

I have a question about Theorem 1.2 in Game Theory by Fudenberg and Tirole.

Theorem 1.2 in FT(Debreu, Glicksberg, Fan) Consider a strategic-form game whose strategy spaces $S_{i}$ are nonempty compact convex subsets of an Euclidean space. If the payoff functions $u_{i}$ are continuous in $s$ and quasi-concave in $s_{i}$, there exists a pure-strategy Nash Equilibrium.

Theorem 1.1 in FT (Nash): Every finite strategic-form game has a mixed strategy equilibrium.

My question here is that, how do we know that the Nash Equilibrium in theorem 1.2 is a pure-strategy one?

In the proof, they also use Kakutani's fixed point theorem, but I think it can only give us the existence of some Nash Equilibrium, and we know nothing about its characterizations. Additionally, Nash's existence theorem should be a special case of theorem 1.2 since the set of mixed strategies over finite actions is a simplex, which is compact. However, in Nash's theorem, we only know that a mixed-strategy equilibrium will exist.

My understanding of Debreu's theorem is that, the "pure strategy" in this theorem is with respect to the strategy spaces, not to action spaces. To be more specific, take the Nash Existence Theorem example above, the mixed strategy in the theorem is actually a "pure strategy" in the space of mixed strategies, so both theorems don't contradict with each other.

Answer 1

I assume you know how the proof goes using Kakutani to prove existence for finite games.

As standard, let $X_{-i} = \times_{j \not = i} X_{j}$.

Define the best response correspondence $BR_i: S_{-i} \rightrightarrows S_i$ by $$ BR_i(s_{-i}) = \arg \max_{s_i \in S_i} u_i(s_i,s_{-i}) $$

Apart from the usual existence and upper hemicontinuity, quasiconcavity of $u_i$ implies the correspondence is convex valued.

As usual, stack them up to define the correspondence $BR: S \rightrightarrows S$.

Apply Kakutani, the desired fixed point is an element of $s^* \in S$, and is thus a pure strategy.

Compare this to the proof for finite games, where we define the correspondence on the space $\times_{i} \Delta (S_i)$!