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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2768Penalty Kick as a Zero-Sum GameA striker chooses Left or Right, and a goalkeeper chooses Left or Right. The striker's success probabilities are \[ \begin pmatrix 0.6 & 0.9 \\ 0.8 & 0.7 \end pmatrix , \] where rows are the striker's choices and columns are the keeper's choices. Treat this as a zero-sum game with Row = striker. Find the equilibrium mixing probabilities and the value.脑筋急转弯中等derivation未尝试面试订阅2769General 2x2 Zero-Sum FormulaFor the zero-sum matrix game \[ \begin pmatrix a & b \\ c & d \end pmatrix , \] assume there is no pure saddle point and a-b-c+d 0. Derive the mixed strategy of Row, the mixed strategy of Column, and the value of the game.脑筋急转弯困难derivation未尝试面试订阅2770Spotting a Saddle PointConsider the zero-sum matrix \[ \begin pmatrix 4 & 1 & 3 \\ 2 & 2 & 2 \\ 5 & 0 & 4 \end pmatrix . \] Does the game have a saddle point in pure strategies? If so, identify it and give the value.脑筋急转弯简单derivation未尝试面试订阅2771Searching Boxes With Unequal StakesA searcher chooses one of three boxes to inspect; a hider chooses one box to hide in. If the searcher picks the correct box, the searcher earns the box's value; otherwise the payoff is 0. The three box values are 6, 3, and 2. Find the searcher's optimal mixed strategy and the value of the game.脑筋急转弯中等derivation未尝试面试订阅2772Patrolling Two TargetsA defender can patrol either Target 1 or Target 2. An attacker chooses which target to strike. If the defender patrols the attacked target, damage is prevented and the defender's payoff is 0. If not, the defender loses the target's damage: 4 for Target 1 and 1 for Target 2. Find the defender's optimal mixed strategy and the value of the game.脑筋急转弯中等derivation未尝试面试订阅2773Continuous Defense SplitA defender allocates a fraction x of one unit of defense budget to Target 1 and the remaining 1-x to Target 2. If the attacker strikes Target 1, the defender's payoff is -9(1-x); if the attacker strikes Target 2, the defender's payoff is -6x. Find the defender's optimal allocation and the value.脑筋急转弯中等derivation未尝试面试订阅2774Weighted Rock-Paper-ScissorsConsider the zero-sum matrix \[ \begin pmatrix 0 & -1 & 2 \\ 1 & 0 & -1 \\ -2 & 1 & 0 \end pmatrix . \] Find the optimal mixed strategy and the value of the game.脑筋急转弯中等derivation未尝试面试订阅2775Adding a Constant to Every EntrySuppose A is a zero-sum payoff matrix with value v. If we add the same constant c to every entry of A, what happens to the optimal mixed strategies and to the value? Explain briefly.脑筋急转弯简单derivation未尝试面试订阅2776Scaling the Whole Payoff MatrixSuppose A is a zero-sum payoff matrix with value v, and >0. What happens to the optimal mixed strategies and the game value if the payoff matrix becomes A?脑筋急转弯简单derivation未尝试面试订阅