Question

Consider a game in which two players bid for a single item. Each player i places value vi which is known to that player but not the other player. Assume v1 and v2 are independent, each distributed on [0,1] according to the probability distribution function Prob[v1 ? x] = x² for 0 ? x ? 1. Each player, after observing vi, makes a bid bi on the item. The item goes to the highest bidder (you may ignore the possibility that both players make the same bid). The winner's payoff is vi – bi, the loser's payoff is 0. Very carefully prove that a Bayesian-Nash equilibrium is for each player i to make a bid equal to (2/3)vi.

          Consider a game in which two players bid for a single item. Each player i places value vi which is known to that player but not the other player. Assume v1 and v2 are independent, each distributed on [0,1] according to the probability distribution function Prob[v1 ? x] = x² for 0 ? x ? 1. Each player, after observing vi, makes a bid bi on the item. The item goes to the highest bidder (you may ignore the possibility that both players make the same bid). The winner's payoff is vi – bi, the loser's payoff is 0. Very carefully prove that a Bayesian-Nash equilibrium is for each player i to make a bid equal to (2/3)vi.

Consider a game in which two players bid for a single item. Each player i places value vi which is known to that player but not the other player. Assume v1 and v2 are independent, each distributed on [0,1] according to the probability distribution function Prob[v1 ? x] = x² for 0 ? x ? 1. Each player, after observing vi, makes a bid bi on the item. The item goes to the highest bidder (you may ignore the possibility that both players make the same bid). The winner's payoff is vi – bi, the loser's payoff is 0. Very carefully prove that a Bayesian-Nash equilibrium is for each player i to make a bid equal to (2/3)vi.

Added by Daniel S.

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