User:Benja/2008/04/Notes on Fudenberg and Levine article

Notes on:

Random stuff that I'm writing down to help me figure out the details of the paper. Also, incidentally, to learn LaTeX math syntax.

Correctness of definition of Nash equilibrium
Page 528 (page 7 of PDF) states that a Nash equilibrium can be defined as a mixed profile &sigma; such that for each $$s_i$$ with nonzero probability, there's a $$\mu_i$$ such that


 * 1) $$s_i \mbox{ maximizes } u_i(\,\cdot\,, \mu_i)$$, and
 * 2) $$\mu_i\Big[\Big\{\pi_{-i} | \pi_j(h_j) = \hat{\pi}_j(h_j|\sigma_j)\Big\}\Big] = 1 \quad \mbox{for all } h_j \in H_{-i}$$.

In the next paragraph, the authors state that (2) says that each player's beliefs $$\mu_i$$ are "a point mass on the true distribution," i.e., that there is a single $$\pi_{-i}$$ such that $$\mu_i(\{\pi_{-i}\}) = 1$$. Why is this true?

Lemma 1. For any probability distribution $$\mbox{Pr}$$ and non-empty collection of events $$\mathbf{E}$$ such that $$\forall A \in \mathbf{E}: \mbox{Pr}(A) = 1$$, it holds that $$\mbox{Pr}(\bigcap \mathbf{E}) = 1$$.

Proof. Omitted.

Corollary 1.1. (2) is equivalent to:
 * $$\mu_i\Big[\bigcap_{h_j \in H_{-i}} \Big\{\pi_{-i} | \pi_j(h_j) = \hat{\pi}_j(h_j|\sigma_j)\Big\}\Big] = \mu_i\Big[\Big\{\pi_{-i} | \forall h_j \in H_{-i}: \pi_j(h_j) = \hat{\pi}_j(h_j|\sigma_j)\Big\}\Big] = 1$$.

Lemma 2. There is exactly one deleted behavior strategy profile $$\pi_{-i}$$ such that $$\forall h_j \in H_{-i}: \pi_j(h_j) = \hat{\pi}_j(h_j|\sigma_j)$$.

Proof. $$\pi_{-i}$$ is by definition uniquely determined by the values of $$\pi_j(h_j)$$ for $$h_j \in H_{-i}$$.

The authors' assertion now follows immediately from Corollary 1.1 in combination with Lemma 2.