2. Pat is selling his house in a second price sealed bid auction. That is, a written bid is submitted, the highest bid wins the item but the price paid is the second-highest bid. There are two bidders, Alex and Bert, who have valuations of £1.1lm and £1.2m respectively. a) Show that there is an equilibrium where each bidder bids his or her valuation. b) What price is the house is sold for?
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- 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…Say that you are bidding in a sealed-bid auction and that you really want the item being auctioned. Winning it would be worth $500 to you. Say you expect the next-highest bidder to bid $300.a. In a standard “highest-bid” auction, what bid would a rational person make? The rational choice is to bid $500 since that is what the item is worth to you. The rational choice is to bid a little bit more than $300 because that is the expected next-highest bid. The rational choice is to bid just under $500 so that you have a higher chance of winning the auction and would still have a net benefit. The rational choice is to bid over $500 to guarantee that you win the item. b. In a Vickrey auction, what bid would he make? The rational choice is to bid slightly more than $500. The rational choice is to bid $500. The rational choice is to bid slightly less than $500. The rational choice is to bid slightly more than $300.4. Suppose that there is a negotiation between two players over a painting. Person 1, the seller, has no interest in the painting. On the other hand the painting is worth $100 to the buyer. If the painting is sold at a price in between 0 and 100, both are better off. Player 1 proposes a price p to player 2. Then, after observing player 1's offer, player 2 decides whether to accept it or to reject it. If the offer is rejected both get zero and the game ends. Find the unique SPNE of this game.
- 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…Explain why a player in a sealed-bid, second-price auction would never submit a bid that exceeds his or her true value of the object being sold. (Hint: What if all players submitted bids greater than their valuations of the object?)
- 9. There are four agents who are trying to decide whether they will buy a publicly shared TV, of which the total cost is $200. The valuations of the TV are $30 for agent 1, S35 for agent 2, $70 for agent 3, and $70 for agent 4. Let c; be how much agent i will pay if the TV is purchased. Suppose c = 20, c2 = C3 = C4 = 60. According to Groves and Clarke mechanism, which one could not be the pivotal agent? A. agent 1 B. agent 2 C. agent 3 D. agent 41. Equilibria in an auction Suppose a single item will be sold to one of two bidders. Each bidder's value is drawn uniformly at random from the interval [0, 1]. Each bidder will simultaneously name a price and the item will be sold to the higher bidder for the price that bidder named. If this price is p and the winning bidder has a value of v then the utility of the winning bidder is v-p and the utility of the losing bidder is 0. This situation is more complex than the normal form games we studied in class for two reasons. First, the set of strategies is infinite. Second, rather than known utilities there is uncertainty over them, so we need a Bayesian model and in general the equilibrium strategy will depend on (i.e. be a function of) v. This assignment walks you through the derivation of an equilibrium in this setting. (a) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. If you bid p, what is your expected utility? (b) Suppose your value is…What is the Nash Equilibrium of the following game? |0, 0 Up Down 3, 2 |2, 1 3, 1 2, 6 4, 2 5,5 10, 0 a. (Down, D) b. (Down, C) c. (Up, A) d. (Up, B)
- 3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the first roll, you will either have $90 or $110 after the first roll. (a) Suppose you wish to maximize your profit on the first roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected profit on the first roll if your bet is b with 0 ≤ b ≤ 100? (c) Suppose you wish to maximize your expected profit on the first roll. How much should you bet? (d) Suppose you wish to maximize your expected profit betting on the nth roll. How much of your current wealth do you bet? (e) Let X₂, be your…1. Consider the following N-player game. Each agent has a choice of strategy A or B. The state variable is x, the proportion of agents choosing strategy A. For every agent, the utility of choosing strategy A is U(A) = 10 +2x and the utility of choosing strategy B is U(B) = 10+3x. What is the Nash equilibrium to this game? Explain.10 Use the expected value information to illustrate how having more bidders in an oral auction will likely result in a higher winning bid.