Consider the sealed, first-price auction in which each bidder simultaneously proposes a bid and the object goes to the highest bidder at a price equal to his bid. Suppose there are two bidders and their valuations for the object are chosen independently from a uniform distribution over [0,1]. Think of the player's type as the valuation the bidder places on the item. The player's payoff is 𝑣 − 𝑏 when he wins the object by placing a stance 𝑏 and his valuation is 𝑣. His payoff is 0 if he does not win the object. to. Formulate this game as a Bayesian game. b. Let 𝑏𝑖(𝑣) be the posture performed by player 𝑖 of type 𝑣. Show that there exists a Bayesian Nash equilibrium in which 𝑏𝑖 (𝑣) = 𝑦 + 𝑧𝑣 for all 𝑖 and all 𝑣. Determine the values of 𝑦 and 𝑧. 2. Modify the sealed first-price auction from question 1 so that the loser must also pay his bid (but does not win the object). This modified auction is known as an all-pays auction. to. Show that there exists a Bayesian Nash equilibrium in which 𝑏𝑖 (𝑣) = 𝑥 + 𝑦𝑣 + 𝑧𝑣 for all 𝑖 and all 𝑣. Determine the values of 𝑥, 𝑦 and 𝑧 b. How do the players' positions in this auction compare to those in the sealed first-price auction? What is the intuition behind the difference in postures? c. Show that the first-price auction and the all-pay auction generate the same expected revenue for the seller, ex ante (that is, before bids are received).
