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all-pay auction allocation rule analysis of auctions analyzed arg max assign the object assume assumption auction mechanism auction theory B(sj b(Vj bidding function Burguet and Sakovics buyer decides buyer with highest buyer with valuation buyers and sellers buyers will enter common value competition consider correlated defined demand function discriminatory auction dominant strategy double auction efficient entry costs entry fee equal expected payment expected price expected value game theory global bidders higher highest rival bid highest valuation Horacio Sobarzo informational rents initial conditions Jaime Maskin and Riley México multiple units obtain optimal auction order condition order statistic pj(sj;sj pj(vj winning price and second price formation private values reserve price risk aversion sealed bid auction second price auction seller's revenues Sempere sequential auctions signal symmetric symmetric equilibrium trading mechanism uniform price auction valuation Vj vector Vickrey virtual valuation willingness to pay winner winning the auction winning the object xj(Vj