You are a bidder in an independent private values auction, and you value the object at $4,000. Each bidder perceives that valuations are uniformly distributed between $1,500 and $9,000. Determine your optimal bidding strategy in a first-price, sealed-bid auction when the total number of bidders (including you) is: a. 2 bidders. Bid: $ b. 10 bidders. Bid: $ c. 100 bidders. Bid: $
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- 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. 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).…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.
- What amount does each bidder bid in the Bayesian Nash equilibrium of a 2nd price auction? O The expected value of the second highest bidder. One half of the expected value of the second highest bidder. Their own value. One half of their own value.You are one of five risk-neutral bidders participating in an independent private values auction. Each bidder perceives that all other bidders’ valuations for the item are evenly distributed between $10,000 and $30,000. For each of the following auction types, determine your optimal bidding strategy if you value the item at $22,000. a. First-price, sealed-bid auction. b. Dutch auction. c. Second-price, sealed-bid auction. d. English auction.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.
- 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?4. Consider a first-price, sealed-bid auction in which the bidders’ valuations are independently and uniformly distributed on [0,1]. Suppose that each bidder uses a strategy of b(vi) = avi. What is the symmetric Bayesian Nash equilibrium of this game when there are n bidders?Your company is bidding for mineral rights to a tract of land for drilling oil. Based on your geological survey reports, your valuation of the mineral rights is $38 million. You believe the distribution of bids will be uniform for the mineral rights, with a high value of $45 million and a low value of $20 million. In a second-price sealed-bid auction, how much should you for bid if there are 5 bidders? a. $41.40 Million O b. $45.00 Million Oc. $38.00 Million O d. $34.40 Million
- David wants to auction a painting, and there are two potential buyers. The value for eachbuyer is either 0 or 10, each value equally likely. Suppose he offers to sell the object for $6, and the two buyers simultaneously accept or reject. If exactly one buyer accepts, the object sold to that person for $6. If both accept, the object is allocated randomly to the buyers, also for $6. If neither accepts, the object is allocated randomly to the bidders for $0. (a) Identify the type space and strategy space for each buyer. (b) Show that there is an equilibrium in which buyers with value 10 always accept. (c) Show that there is an equilibrium in which buyers with value 10 always reject.Assume you are one of the two bidders in a second-price sealed bid auction for a preserved grilled cheese sandwich which purportedly bore the portrait of the Virgin Mary: • This sandwich is worth $8000 to you. • You know that your rival bidder’s valuation is uniformly distributed between $0 and $12,000. a) What is the probability that your bid exceeds your rival’s bid given that you know your rival’s valuation is uniformly distributed between $0 and $12,000? (In other words, what is the probability that your rival’s bid is lower than your valuation?) Remember the equilibrium bidding strategies for a SPA in weakly dominant strategies ! b) What is your rival’s average bid, if you win the auction (considering that he must have bid below your bid)? c) What is your expected payoff from this auction, if you win the bidding ? (Hint: Combine b) and c), i.e. compute the payoff from winning giving the probability of winning)Exercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.