I am having trouble with Rubinstein's electronic email game (proposition 83.1) in the textbook(first edition).
My question: Line 7 starting from the "proof", "...player 2's expected payoff is at least $\frac{(1-p)M}{(1-p)+p\epsilon}$ How does Rubinstein arrive at this result?
To me, the payoff should have been $(1-p)M+0p\epsilon=(1-p)M$, but I have no idea. Please help me with this. I have not eaten anything for straight 6 hours reading through pages and pages, and I intend to stop and find something to eat after finishing this Proposition, but now... I feel lost.
Thanks a lot for your help.
They are doing Bayesian updating and so they are using the conditional distribution given the signal that was received. You are not. Notice that $(1-p)+p\cdot\varepsilon<1$ so you are not computing an expectation with respect to a probability measure.