Answer Happy

2. (10) Veronica has decided to sell her comic book collection in order to help pay for a new car. The only people interested in bidding for the collection are her friends, Archie and Betty. In order to get a good price and yet be fair to her friends, she has decided to have a first-price, sealed-bid auction with no reserve. Archie considers the collection worth VA and Betty considers it worth VB. For Archie, Betty's valuation is an independent random variable uniformly distributed between 0 and 1. Similarly for Betty, Archie's valua- tion is an independent random variable uniformly distributed between 0 and 1. Show that this static game has the following symmetric, pure-strategy Bayesian Nash equilibrium: 64(VA) = {VA and b(VB) = {VB. (Hint: In order to prove that this is an equilibrium strategy for both players, it is suffi- cient to show that by is Archie's best response to the belief that bg is Betty's strategy. Note also that Archie's payoff function is (VA-ba) prob(bA > bb), and Betty's is similar.)