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851Subtraction Game State 1Two players alternate removing stones from one pile of 27. A legal move is to remove any number in 1, 4, 6 . The player who takes the last stone wins. Under optimal play, does the first player win? If yes, what is an optimal first move?脑筋急转弯简单数值题未尝试免费861Locate the Losing Position 1In a subtraction game with legal moves 1, 4, 6 , positions are classified as winning or losing in the usual last-move-wins sense. What is the 8th positive losing position?脑筋急转弯简单brainteaser未尝试免费866End-Pick Coin Row 1Two players alternately take one number from either end of the row [4, 9, 2, 7, 5]. Each player wants to maximize the sum of the numbers they personally collect. Under optimal play, what should the first player take first, and what total can the first player guarantee?脑筋急转弯简单数值题未尝试免费867End-Pick Coin Row 2Two players alternately take one number from either end of the row [8, 1, 6, 3, 9, 2]. Each player wants to maximize the sum of the numbers they personally collect. Under optimal play, what should the first player take first, and what total can the first player guarantee?脑筋急转弯中等数值题未尝试免费2741Bid Ceiling With a Fixed Outside OptionA trader values a data license at v. In a sealed first-price auction against one rival, winning with bid b occurs with probability p(b). If skipping the auction preserves an outside payoff of c, what inequality must b satisfy for participating with bid b to weakly dominate skipping?脑筋急转弯中等derivation未尝试面试订阅2742When a Rebate Makes Aggressive Bidding RationalIn a first-price auction, a bidder of value v receives a fixed rebate r if she loses, and nothing if she skips. If bidding b wins with probability p(b), derive the expected payoff of bidding b and compare it with skipping.脑筋急转弯中等derivation未尝试面试订阅2743Scaled Uniform Values and Win ProbabilityThree bidders have i.i.d. private values from Uniform (20,80). In a first-price auction, find the symmetric equilibrium bid of a bidder with value 56. Also compute that bidder's probability of winning under equilibrium play.脑筋急转弯中等derivation未尝试面试订阅2744First-Price Equilibrium Under a Quadratic CDFTwo bidders have i.i.d. private values on [0,1] with CDF F(v)=v 2. In a first-price auction, derive the symmetric equilibrium bid function and compute the bid for value 0.6.脑筋急转弯中等derivation未尝试面试订阅2745Reverse Auction With Two SuppliersA buyer runs a sealed-bid procurement auction with two risk-neutral suppliers. Each supplier's cost is i.i.d. Uniform (0,1), and the lowest bid wins and is paid its own bid. Derive the symmetric equilibrium bid function s(c), and compute the expected profit of a supplier with cost 0.2.脑筋急转弯中等derivation未尝试面试订阅2746Truthful Bidding Under a Capacity PenaltyIn a second-price auction for one extra inventory slot, paying above your true value v can backfire because if you win while your realized usage is low, you incur a deterministic penalty k. If the penalty is independent of the auction price and only applies when you win, what is the effective value that should replace v in the usual Vickrey truth-telling argument?脑筋急转弯困难derivation未尝试面试订阅2747All-Pay Contest Break-Even EffortTwo equally skilled quants enter an all-pay contest for a prize worth V. A contestant who spends effort e while the rival spends e 0 wins with probability e/(e+e 0), and everyone pays their own effort regardless of outcome. Holding e 0 fixed, write the payoff of effort e and solve the equation that defines break-even effort.脑筋急转弯简单derivation未尝试面试订阅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