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2761Winner's-Curse Adjustment From a Symmetric Signal BandA common-value asset has expected resale value equal to the average of all bidders' signals. Your private signal is s, and you know that winning means your signal was the highest among a symmetric signal band around the truth. Why should your bid be below s even if E[value|signal=s]=s before conditioning on winning?脑筋急转弯简单derivation未尝试面试订阅2762A Discrete First-Price Mixed EquilibriumTwo bidders both value an object at 2.5. Each may bid only 0, 1, or 2 in a first-price auction; ties are broken uniformly at random. Find a symmetric mixed-strategy equilibrium.脑筋急转弯困难derivation未尝试面试订阅2763Posted Price Versus Competitive AuctionA seller can either (i) post a take-it-or-leave-it price to a single buyer with value Uniform (0,1), or (ii) run a no-reserve second-price auction between two buyers with i.i.d. values Uniform (0,1). Compute the seller's optimal expected revenue in each mechanism and compare them.脑筋急转弯中等derivation未尝试面试订阅2764Why Risk Aversion Breaks Revenue EquivalenceIn one or two paragraphs, explain why a first-price auction can yield higher expected revenue than a second-price auction when bidders are risk-averse, even though the two formats are revenue-equivalent under risk neutrality.脑筋急转弯中等derivation未尝试面试订阅2765Why Private Values Avoid the Winner's CurseExplain why the winner's curse is a feature of common-value auctions but not of independent private-value auctions. Your answer should make clear what information winning reveals in each environment.脑筋急转弯简单derivation未尝试面试订阅