An object of known value V is sold through a first price sealed bid all-pay auction....

 

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An object of known value V is sold through a first price sealed bid all-pay auction. This means that each player makes a bid, and the object is given to the bidder who makes the highest bid; the originality of the game is that each player pays his bid. Let us suppose that there are two players with a revenue M, with V < M. Each player i, i = 1, 2 proposes a bid bi, which is an integer between 0 and M. M and V are known by each player. If b1 >b2, player 1 gets the object, has the payoff M-bl + V, whereas player 2 has the payoff M-b2. If b1 = b2, both players get the object with probability 0.5. Suppose V = 3 and M=5. Each player then can bid 0,1,2,3,4, and 5. a) Fill in the payoff of each player in the strategic form of the game below. Player 1 0 1235 2 4 0 Player 2 1 2 3 4 5 b) Draw the extensive form of the game assuming that player 2 observes player 1's bid before playing. c) Find the Subgame Perfect Nash Equilibria (SPNE). (Use the extensive form of the game.) d) Compare the SPNE with the elimination of dominant strategies. An object of known value V is sold through a first price sealed bid all-pay auction. This means that each player makes a bid, and the object is given to the bidder who makes the highest bid; the originality of the game is that each player pays his bid. Let us suppose that there are two players with a revenue M, with V < M. Each player i, i = 1, 2 proposes a bid bi, which is an integer between 0 and M. M and V are known by each player. If b1 >b2, player 1 gets the object, has the payoff M-bl + V, whereas player 2 has the payoff M-b2. If b1 = b2, both players get the object with probability 0.5. Suppose V = 3 and M=5. Each player then can bid 0,1,2,3,4, and 5. a) Fill in the payoff of each player in the strategic form of the game below. Player 1 0 1235 2 4 0 Player 2 1 2 3 4 5 b) Draw the extensive form of the game assuming that player 2 observes player 1's bid before playing. c) Find the Subgame Perfect Nash Equilibria (SPNE). (Use the extensive form of the game.) d) Compare the SPNE with the elimination of dominant strategies.

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