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Consider a sealed-bid all-pay auction in which every buyer submits a non-negative bid, the highest bidder receives the good, and every buyer pays the seller the amount of his bid regardless of whether he wins

Economics

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Consider a sealed-bid all-pay auction in which every buyer submits a non-negative bid, the highest bidder receives the good, and every buyer pays the seller the amount of his bid regardless of whether he wins. For simplicity assume [x , x ] = [0, 1]. Use the expression for bidder equilibrium utility from the Revenue Equivalence Theorem 1 to derive a (symmetric) equilibrium bidding function for the all pay auction.  Please show all work in derivation of bidding function. Justify your answer carefully. Is the seller better or worse off using an all pay auction than a second price auction under your stated assumptions? What about bidders?