Imagine a seller (auctioneer) wanting to sell an item. There are two potential buyers, bidder 1 and bidder 2. Let v₁ and v₂ denote the valuations of the bidders. If bidder i wins the painting and has to pay x for it, then bidder i's payoff is vi-x. The bidders observe their own valuations before the auction. However, they do not observe each other's valuation, but know that the other valuation can be between 0 to 90. Consider now a second-price sealed bid auction. In this auction, players simultaneously and independently submit their bids b₁ and b₂. The painting is awarded to the highest bidder at a price equal to the second-highest bid. Show that the (weakly) dominant strategy for the players is to bid their own valuation (i.e. b=v), and this is the profile which will constitute the Bayesian Nash equilibrium of this game. (
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- - First-Price and Second-Price Auctions - Consider an auction of a single indi- visible object with 5 bidders, 1, 2, 3, 4 and 5, whose personal valuations (willingness to pay) for the object are v₁ = 10, V₂ = 8, 03 7, 04 5 and 5 3. The bidders simultaneously submit their bids and the winner is the one with the highest bid. Suppose that, when the highest bidder is not unique, the bidder with the smallest number (highest valuation) wins. (a) Suppose that this is a second-price auction, where the winner pays the highest bid among those from her opponents, determine whether each of the following bidding profiles is a Nash equilibrium. Explain. (i) (b₁,b2, b3, b4, b5) = (10, 8, 7, 5, 3) (ii) (b1,b2, b3, b4, b5) = (8,8, 0, 0, 0) (iii) (b₁,b2, b3, b4, b5) = (10, 0, 0, 0, 10) (b) Now suppose that this is a first-price auction, where the winner pays their own bid, determine whether each of the bidding profiles above is a Nash equilibrium. Explain. = =6. The owner of an antique piece of furniture is looking to sell their good to a known buyer. The seller has a reservation value r whereas the buyer has valuation v > r. Suppose the buyer incurs a one-off transportation cost of t from travelling to the seller to purchase the good. Assume that r, vand t are known to both parties. The game proceeds in two stages: First, the buyer decides whether or not to travel to the seller's location in order to purchase. Second, if they travel then the seller makes a take-it-or-leave-it offer (ultimatum) of price p to the buyer, which the buyer then can either accept or reject. (a) Represent this game in extensive form (b) Find the seller's optimal offer in Stage 2 of this game given that the buyer has already travelled. Is it accepted or rejected? (c) Find a Subgame Perfect Equilibrium of this game. (d) Are there any Nash equilibria which are not subgame perfect? Give an example if one exists. (e) Suppose the seller could offer free delivery at a…2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.
- 4 A recently discovered painting by Picasso is on auction at Sotheby's. There are two main bidders Amy and Ben {1,2}. Bidding starts at £10M but the value of the painting is certainly not more than £20M. Each bidder's valuation v; is independently and uni- formly distributed on the interval [10M, 20M], and this is common knowledge among the players: A bidder knows their own valuation but not of their opponent. Consider an auction where an object is allocated to the highest bidder but the price paid by the bidder is determined randomly. With probability 3/4, the bidder pays their own bid, and with probability 1/4 the bidder pays the losing bid. The person bidding lowest pays nothing. If the bids are equal, each bidder gets the object with probability one-half, and in this case, pays their bid. Suppose that bidder 1 assumes that bidder 2 will bid a constant fraction, Y, of bidder 2's valuation (and similarly, bidder 2 assumes bidder 1 will bid the same constant propor- tional value y of…4 A recently discovered painting by Picasso is on auction at Sotheby's. There are two main bidders Amy and Ben {1,2}. Bidding starts at £10M but the value of the painting is certainly not more than £20M. Each bidder's valuation v; is independently and uni- formly distributed on the interval [10M, 20M], and this is common knowledge among the players: A bidder knows their own valuation but not of their opponent. Consider an auction where an object is allocated to the highest bidder but the price paid by the bidder is determined randomly. With probability 3/4, the bidder pays their own bid, and with probability 1/4 the bidder pays the losing bid. The person bidding lowest pays nothing. If the bids are equal, each bidder gets the object with probability one-half, and in this case, pays their bid. Suppose that bidder 1 assumes that bidder 2 will bid a constant fraction, 7, of bidder 2's valuation (and similarly, bidder 2 assumes bidder 1 will bid the same constant propor- tional value y of…4. An auctioneer holds a second-price auction for two bidders, Ann (A) and Bonnie (B), who have independent private values of the good 0, and 0g If a bidder wins, her payoff is her value 0 minus the price she pays, and if she loses, her payoff is 0. The values are independently and identically distributed, but otherwise you don't need to know the specific distributions to solve the problem. Ann and Bonnie's respective strategies are to bid some value b0), that is, bid given their privately-known value (type). e. Suppose the good had one true value for both bidders equal to the average of 0, and e, (signals that are still i.i.d.); hence, the good's true value has a common component. Suppose Ann knows Bonnie is going to bid her own evaluation 0, no matter what, but like normal, Ann doesn't know 0g. Explain why bidding 0, is now a strictly dominated strategy for Ann.
- You are one of five risk-neutral bidders participating in an independent private values auction. Each bidder perceives that bidders' valuations for the item are evenly distributed between $20,000 and $50,000. For each of the following auction t determine your optimal bidding strategy if you value the item at $35,000. a. First-price, sealed-bid auction. O Bid $20,00. O Bid $50,00. O Bid $35,00. O Bid $32,000 b. Dutch auction O Let the auctioneer continue to lower the price until it reaches $20,000, and then yell "Minel". O Let the auctioneer continue to lower the price until it reaches $35.000, and then yell "Mine!" O Let the auctioneer continue to lower the price until it reaches $32,000, and then yell "Mine!" O Let the auctioneer continue to lower the price until it reaches $50,000, and then yell "Mine!". C. ond-price, sealed-bid auction O Bid $50,00. O Bid $35.000 O Bid $32,00. b. Dutch auction. O Let the auctioneer continue to lower the price until it reaches $20,000, and then yell…2. Suppose we have 2 players in a first price auction. More specifically, suppose player i values the object being auctioned at v, where v > vz > 0. These valuations are known to both players. They simultaneously choose bids b;. Whoever bids the most gets the object and pays what she bid. In case of a tie, each player has probability of receiving the good and paying her bid and probability of a payoff of 0. Suppose the strategy set (set of all possible bids) for each player is [0, 0). Show that there is no pure strategy Nash equilibrium.A cool kid is willing to rename himself for a profit. He decides to auctionoff the naming right. Two bidders show interest. Their valuations for thenaming right are independently and uniformly distributed over [0,100].There are several possible ideas to design the auction. The auction runs as follows. Both bidders are invited to the same room; an auctioneer will start the auction with an initial price 0, and increase it by $1 every minute. The bidders are not allowed to say anything during the process, but they can walk out of the room at any moment. If one bidder walks out of the room when the price increases to p (the bidder does not need to pay), the remaining bidder will be awarded the naming right for a price of p. If both walk out when the price reaches p, the naming right is not assigned andthe two bidders do not need to pay. What should the bidders do? Explain your answer.
- A cool kid is willing to rename himself for a profit. He decides to auctionoff the naming right. Two bidders show interest. Their valuations for thenaming right are independently and uniformly distributed over [0;100]:There are several possible ideas to design the auction. The auction runs as follows. Both bidders are invited to the sameroom; an auctioneer will start the auction with an initial price 0 and increase it by $1 every minute. The bidders are not allowed to say anything during the process, but they can walk out of the room at any moment. If one bidder walks out of the room when the price increases to p(the bidder does not need to pay), the remaining bidder will be awarded the naming right for a price of p. If both walk out when the price reaches p, the naming right is not assigned and the two bidders do not need to pay. What should the bidders do? Explain your answer.A cool kid is willing to rename himself for a profit. He decides to auctionoff the naming right. Two bidders show interest. Their valuations for thenaming right are independently and uniformly distributed over [0,100].There are several possible ideas to design the auction. a) The auction runs as follows. Both bidders are invited to the sameroom; an auctioneer will start the auction with an initial price 0, and increase it by $1 every minute. The bidders are not allowed to say anything during the process, but they can walk out of the room at any moment. If one bidder walks out of the room when the price increases to p (the bidder does not need to pay), the remaining bidder will be awarded the naming right for a price of p. If both walk out when the price reaches p, the naming right is not assigned and the two bidders do not need to pay. What should the bidders do? Explain your answer. (b) Both bidders are invited to submit their bids covertly (bids are non-negative real numbers).…4. An auctioneer holds a second-price auction for two bidders, Ann (A) and Bonnie (B), who have independent private values of the good 0, and e, If a bidder wins, her payoff is her value 0 minus the price she pays, and if she loses, her payoff is 0. The values are independently and identically distributed, but otherwise you don't need to know the specific distributions to solve the problem. Ann and Bonnie's respective strategies are to bid some value b (0.). that is, bid given their privately-known value (type). a. Explain what a second price auction is, who wins given some pair of bids b, and bg. and what the winner pays. b. Why is a strategy where Ann bids above her own value 0, weakly dominated by a strategy where she bids her value? c. Why is a strategy where Ann bids below her own value e, weakly dominated by a A strategy where she bids her value? d. Applying the ideas from (b) and (c) to both Ann and Bonnie, what is the Weakly Dominant Strategy Equilibrium for this game?