Economic and Game Theory
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Consider a first price, sealed bid auction. There are two bidders, 1 and 2. Bidder i values the good at v(i). If she pays p for the good then the payoff is v(i)-p. The bidder valutations are independently, uniformly distributed over the interval [0,1]. Bids are constrained to be non-negative. The bidder that offers the highest bid gets the good for that price. In the event of a tie, a coin flip decides which bid wins. The bids are risk neutral. All this is common knowledge.
Set this auction up as a Bayesian game. Prove that the following is a Bayesian equilibrium: Both bidders bid half their valuation, b(i)(v(i)) = v(i)/2, where b(i)(v(i)) is the bid of bidder i with valuation v(i).
I'm having trouble figuring out a utility function for either player, or just seeing why neither player would bid more than half their valuation. Can anyone give me some clues as to how they would go about proving that equilibrium? I'm completely stumped. I'm not asking for a solution, just for some guidance.
Thanks. [Manage messages]