1. assume all conditions that needed for the pure strategy nash equilibrium existence. prove that the…

Answer:

The best-response correspondence (B(a)) has a closed graph.

Explanation:

Step1: Define closed graph

A correspondence (B: A \rightrightarrows A) has a closed graph if for all sequences ({a_n}{n=1}^\infty \subset A), ({b_n}{n=1}^\infty \subset A) with (a_n \to a), (b_n \to b), and (b_n \in B(a_n)) for all (n), we have (b \in B(a)).

Step2: Set up payoff function

Let (u: A \times A \to \mathbb{R}) be the payoff function, where (A) is the non-empty compact convex strategy set (from pure strategy Nash equilibrium existence conditions). For (a \in A), (B(a) = {b \in A \mid u(b,a) = \max_{b' \in A} u(b',a)}).

Step3: Use continuity of payoff

By Nash equilibrium existence conditions, (u) is continuous in both arguments. For (b_n \in B(a_n)), (u(b_n,a_n) \geq u(b',a_n)) for all (b' \in A), all (n).

Step4: Take limits of inequalities

As (n \to \infty), (a_n \to a), (b_n \to b). Continuity of (u) gives (\lim_{n \to \infty} u(b_n,a_n) = u(b,a)) and (\lim_{n \to \infty} u(b',a_n) = u(b',a)) for all (b' \in A). Taking limits of (u(b_n,a_n) \geq u(b',a_n)) gives (u(b,a) \geq u(b',a)) for all (b' \in A).

Step5: Conclude closed graph

This implies (b \in B(a)), so (B(a)) has a closed graph.