two bidders, i = 1, 2 -the good on sale has value vi for bidder i -bidders’ valuations v1, v2 are distributed independently and according to the uniform distribution on [0, 1] -bidders simultaneously submit their bids -bids must be 0 or any positive value -the bidder who submits the highest bid wins the good and pays the price she bids, the other bidder gets nothing and pays nothing -in case of a tie, the winner is determined by a flip of a coin -if bidder i gets the good and pays price p, then her payoff is vi - p -structure of the auction is common knowledge for the bidders -it is called first-price sealed-bid auction -game G = {A1, A2; T1, T2; p1, p2; u1, u2} -player i’s action is to submit a nonnegative bid bi -action space Ai = [0, 1) -player i’s type is her valuation vi -type space Ti = [0, 1] -since valuations are independent, player i believes that vj is uniformly distributed on [0, 1], no matter what vi is (posterior and prior beliefs are the same) -player i’s payoff function is: ui(bi, bj; vi) = vi - bi if bi > bj , (vi - bi)/2 if bi = bj , 0 if bi < bj The BNE for two players would be: bi(vi) = vi/2 so each bidder submits half of his valuation What would be the symmetric linear equilibrium of the first price auction with 3 Bidders and with an arbitrary number N Bidders? I kindly ask to provide a step-by-step solution for each case of 3 Bidders and N Bidders
-two bidders, i = 1, 2
-the good on sale has value vi for bidder i
-bidders’ valuations v1, v2 are distributed independently and according to the uniform distribution on [0, 1]
-bidders simultaneously submit their bids
-bids must be 0 or any positive value
-the bidder who submits the highest bid wins the good and pays the
-in case of a tie, the winner is determined by a flip of a coin
-if bidder i gets the good and pays price p, then her payoff is vi - p
-structure of the auction is common knowledge for the bidders
-it is called first-price sealed-bid auction
-game G = {A1, A2; T1, T2; p1, p2; u1, u2}
-player i’s action is to submit a nonnegative bid bi
-action space
-player i’s type is her valuation vi
-type space Ti = [0, 1]
-since valuations are independent, player i believes that vj is uniformly distributed on [0, 1], no matter what vi is (posterior and prior beliefs are the same)
-player i’s payoff function is:
ui(bi, bj; vi) = vi - bi if bi > bj , (vi - bi)/2 if bi = bj , 0 if bi < bj
The BNE for two players would be: bi(vi) = vi/2 so each bidder submits half of his valuation
What would be the symmetric linear equilibrium of the first price auction with 3 Bidders and with an arbitrary number N Bidders?
I kindly ask to provide a step-by-step solution for each case of 3 Bidders and N Bidders
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