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2748Why Third-Price Auctions Are Not TruthfulShow by explicit counterexample that truthful bidding is not a dominant strategy in a third-price auction with three bidders, where the highest bidder wins but pays the third-highest bid.脑筋急转弯简单derivation未尝试面试订阅2749English Auction Dropout LogicFour bidders have independent private values 40, 55, 70, and 90. In an English ascending auction with no reserve, who wins, what price is paid, and why is the outcome equivalent to a second-price auction in this private-value setting?脑筋急转弯简单derivation未尝试面试订阅2750Dutch Auction as a First-Price ClockIn a Dutch descending-price auction with four bidders whose i.i.d. values are Uniform (0,1), a bidder has value 0.8. Assuming risk neutrality and the private-value benchmark, at what clock price should that bidder stop the clock in symmetric equilibrium? Briefly explain why.脑筋急转弯简单derivation未尝试面试订阅2751Entry Decision Before a Second-Price AuctionA bidder with private value v \in [0,1] considers entering a second-price auction against one rival who is known to enter. Values are i.i.d. Uniform (0,1), and entering costs a fixed fee c=0.08. If bidding is truthful after entry, for which values v should the bidder enter?脑筋急转弯中等derivation未尝试面试订阅2752Winner's Curse After Seeing High Against One Low SignalAn asset has common value V\in\ 0,100\ with prior P(V=100)=1/2. Each bidder receives a signal that equals the true state with probability 0.8, independently across bidders conditional on V. You observe a high signal and learn that your rival observed a low signal. What is E[V\mid your signal high, rival low ]? If you bid 80 in a first-price auction and win for sure in this information set, what is your expected profit?脑筋急转弯中等derivation未尝试面试订阅2753Winner's Curse Gets Stronger Against Two Low SignalsUse the same binary common-value setup as above, except now there are three bidders total. You observe a high signal and learn that the other two bidders both observed low signals. Compute E[V\mid your signal high, both rivals low ].脑筋急转弯中等derivation未尝试面试订阅2754Value of Tie-Breaking PriorityTwo bidders both plan to submit the same bid b in a first-price auction. If bidder A wins ties with probability q while bidder B wins ties with probability 1-q, what is the incremental expected value of A's tie-breaking priority at valuation v, relative to a fair 50-50 tie break?脑筋急转弯困难derivation未尝试面试订阅2755Expected Spend in an All-Pay ContestIn the two-player all-pay auction above, suppose the prize value is V=120. Compute (a) each player's expected bid, (b) the expected highest bid, and (c) the expected total spend.脑筋急转弯中等derivation未尝试面试订阅2756Bidder Utility Under Revenue EquivalenceA bidder has value 0.9 in a two-bidder private-value environment with values i.i.d. Uniform (0,1). Compute that bidder's expected utility in (a) the first-price equilibrium and (b) the truthful second-price auction. Verify that the utilities agree.脑筋急转弯中等derivation未尝试面试订阅2757Reserve Price as a Participation FilterIn a second-price auction with reserve r, a bidder with value v faces a strategic question only when v is near r. What simple cutoff rule determines whether the bidder should participate at all?脑筋急转弯中等derivation未尝试面试订阅2758First-Price Bidding With a ReserveTwo bidders have i.i.d. values Uniform (0,1), and the seller imposes a reserve price r\in(0,1). Derive the symmetric equilibrium bid function for values v\ge r in the first-price auction.脑筋急转弯困难derivation未尝试面试订阅2759Applying the First-Price Reserve FormulaIn the two-bidder first-price auction with reserve from the previous question, take r=0.5. What is the equilibrium bid of a bidder with value 0.8, and what is that bidder's expected utility?脑筋急转弯中等derivation未尝试面试订阅2760General Procurement Bid Function With Four SuppliersA procurement auction has four risk-neutral suppliers with i.i.d. costs Uniform (0,1). The lowest bid wins and is paid its own bid. Derive the symmetric equilibrium bid function, and compute the bid of a supplier with cost 0.3.脑筋急转弯中等derivation未尝试面试订阅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2766Matching PenniesIn the zero-sum game of matching pennies, Row and Column each choose H or T. Row receives +1 if the choices match and -1 otherwise. Find the mixed-strategy equilibrium and the value of the game.脑筋急转弯简单derivation未尝试面试订阅2767Weighted Matching PenniesConsider the zero-sum matrix game \[ \begin pmatrix 2 & -1 \\ -1 & 2 \end pmatrix . \] Find the optimal mixed strategies and the value of the game.脑筋急转弯中等derivation未尝试面试订阅