Answer to Question #63682 in Microeconomics for boset

Question

Suppose that there are n bidders whose valuations vis are drawn independently and identically from the distribution F over [0, ω]. Describe and derive the symmetric , monotonic equilibrium in the first price auction. Derive an expression for the expected payment by a bidder.

Answer 1

A symmetric equilibrium is an equilibrium where all players use the same strategy (possibly mixed) in the equilibrium. In the Prisoner's Dilemma game where is the only Nash equilibrium, both players use the same strategy and the equilibrium is symmetric.
Given no externalities all mixed-strategy equilibria in the auctions must be ex post­ allocation and interim expected payment-equivalent to some monotone pure strategy equilibrium.
For a bidder i with value vi expected payment should be:
E[maxvj(j≠i)∣vj≤vi,j≠i]
Pr(vj≤vi,j≠i)