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- 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 Million6. 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…15 caricature artists must decide whether to operate outside the aquarium (location A) or at the beach boardwalk (location B). The more artists located at one of these locations, the more profit they earn. This is because when more artists cluster together, their collective presence becomes more significant thus attracting more customers. If n artists choose location A, each artist earns n? If m artists choose location B, each artist earns 2m? 14. Find all the pure strategy Nash equilibria in this game. (Treat artists as integers, not a continuous variable)
- L 0 4 15.1 X₂ A M R 0 1 4 0 2 B L 4 0 с X3 M R 0 1 3 FIGURE 15.5 Exercise 15.1. Equilibrium Selection: Consider the extensive-form game in Figure 15.5. a. Find all the Bayesian Nash equilibria of this game. b. Which of the Bayesian Nash equilibria are also perfect Bayesian equi- libria? Why?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.Consider a modified Traveler’s Dilemma. In terms of strategy options that the players have and the dollars they earn, it is like the standard Traveler’s Dilemma, but the players do not have endless appetite for money. Up to 100 dollars, each dollar feels like a dollar. But any moneybeyond 100 is psychologically like 100 dollars. Assuming that players are maximizers of ‘psychological’ dollars instead of real dollars, describe all the Nash equilibria of this modified Traveler’s Dilemma.
- 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?Firm 1 and firm 2 compete with each other by choosing quantities. The market demand is given by 400-Q, if Q < 400 P(Q) = 9 0, otherwise = 40q1, and firm 2 has a cost function where Q = 9₁ +92. Firm 1 has a cost function C₁(9₁) C2 (92) = 50q2. Answer the following questions.2. Suppose two players are caught in a prisoner's dilemma with the payoffs given by Nice Mean Nice 2,2 -4,5 -2,-2 Mean 5,-4 If Beta is the probability that the game will be played for another round, for what value of Beta is tit-for- tat an equilibrium strategy?
- 10. Here's another game that has interested researchers, especially those of the type who work in the Max Gluskin House on campus at UofT. It's also a two-player game, but this time the payment is integral to describing the game. There is no direct Researcher involvement in the game, other than potentially as the source of a payout. For the sake of describing the game, imagine that Anson and Kanav are our two players and that they are playing the game virtually via Zoom for bitcoins, denoted B. There are always two piles of bitcoin in play: a larger one and a smaller one. Ahead of the game Anson and Kanav decide the maximum number 2n of turns, for some n E N greater than or equal to 1. • During the first turn the large pile of bitcoin has 4 Band the smaller pile of bitcoin has 1 B. Anson can either take the bitcoin or pass. If Anson (4 B) takes the bitcoin he gets the larger pile, Kanav gets the smaller pile (1 B) and the game ends. If he passes, the size of each pile is doubled. • Now…A firm plans to expand its product line and faces a dilemma whether to build a small or largefacility to produce new products. If it builds a small facility and demand is low, the NPV afterdeducting for building costs will be four hundred thousand pesos. If demand is high, the firm caneither maintain the small facility or expand it. Expansion would have an NPV of four hundredfifty pesos while maintaining the small facility would have an NPV of fifty thousand pesos. If alarge facility is built and demand is high, the estimated NPV would be eight hundred thousandpesos. If demand turns out to be low, the NPV would be a loss of ten thousand. The probabilitythat the demand is high is estimated to be sixty percent.a. Analyze using a decision tree.b. Compute for EVPI.c. Determine the range over which each alternative would be best in terms of the valuewhen demand is low.Dynamic Game: US Tax No Tax Fiji Fiji Tax No Tax Tax No Tax 10,000,10 9900,-9989 10,100,-89.990 Sub-game_1 10,000,-99,989 Sub-game_2 Sub-game_3 1. Is there a Nash Equilibrium of this game in which Fiji improves its payoffs relative to the Sub-game Perfect Nash equilibrium? If there is, explain carefully why this strategy profile is not a Sub-game Perfect Nash equilibrium.
