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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!

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Your first suggestion is correct: you need to multiply the amount you are prepared to pay by the probability you are prepared to pay it and then sum, or in this continuous case integrate, over all possibilities.

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.

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  • $\begingroup$ But which one of those is correct? Note that expected utility implies risk-neutrality, but we are missing the main point. $\endgroup$ – fox Apr 29 '15 at 12:25
  • $\begingroup$ Your first suggestion is correct $\endgroup$ – Henry Apr 29 '15 at 15:13

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