# How much is it worth to participate in a second price auction?

You have a valuation for an object (say $v_a$), which you don't know yet but you know is distributed U[0,1]. You will be competing in a second price auction against a completely identical guy as you, who has a valuation $v_b$~U[0,1].

Assume you and him will both bid truthfully in the auction. How much are you willing to pay to enter the auction, i.e. what is the expected gain for a bidder who doesn't know his valuation?

Your expected payoff is $\frac{v_a^2}{2}$, because the probability your valuation is higher is simply $F(v_a)=v_a$. Given that you win, you get your valuation minus the expected bid, which is uniform on [0, $v_a$]. I have two answers and would like to know which one is right and why the other is wrong. It maybe be that both are wrong!

1. Take $\int_0^1$ $\frac{v_a^2}{2}$ d $v_a$ = $\frac{v_a^3}{6}$ = 0.166
2. Evaluate $\frac{v_a^2}{2}$ at the expected value of $v_a$, and get 0.125.

Thanks!

But I would not be prepared to pay this amount as it reduces my expected gain to $0$ with some risk while not participating gives the same expected gain with no risk.