1. (3pts) Consider a modified version of solved problem 19.1 in the textbook. As before, suppose everyone is risk neutral and let ₂ be the proportion of used cars that are low-quality (lemons). Assume buyers value lemons at $2,000 and peaches at $10,000. The reservation price of owners is $1,000 for lemons and $6,000 for peaches. For what values of 6, do all used cars sell? 1/4 (a) Any (b) Only ₁ = 2/3 (c) Any 0 1/2 (d) Any 0 ≤2/3
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- 1. A dealer decides to sell a rare book by means of an English auction with a reservation price of 54. There are two bidders. The dealer believes that there are only three possible values, 90, 54, and 45, that each bidder’s willingness to pay might take. Each bidder has a probability of 1/3 of having each of these willingnesses to pay, and the probabilities for each of the two bidders are independent of the other’s valuation. Assuming that the two bidders bid rationally and do not collude, the dealer’s expected revenue is approximately ______. 2. A seller knows that there are two bidders for the object he is selling. He believes that with probability 1/2, one has a buyer value of 5 and the other has a buyer value of 10 and with probability 1/2, one has a buyer value of 8 and the other has a buyer value of 15. He knows that bidders will want to buy the object so long as they can get it for their buyer value or less. He sells it in an English auction with a reserve price which he must…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…3. Find the saddle point, if it exists, for the following game. (b) Solve the following game by using the principle of dominance and find the probabilities of strategies for each player and the value of the game. Player B Player A II III IV V 3 4 4 II 2 4 III 4 4 IV 4 4 20 2420 8760
- 1. Individual Problems 18-1 You hold an oral, or English, auction among three bidders. You estimate that each bidder has a value of either $88 or $110 for the item, and you attach probabilities to each value of 50%. The winning bidder must pay a price equal to the second highest bid. The following table lists the eight possible combinations for bidder values. Each combination is equally likely to occur. On the following table, indicate the price paid by the winning bidder. Combination Number Bidder 1 Value Bidder 2 Value Bidder 3 Value Probability Price ($) ($) ($) 1 $88 $88 $88 0.125 2 $88 $88 $110 0.125 3 $88 $110 $88 0.125 4 $88 $110 $110 0.125 5 $110 $88 $88 0.125 6 $110 $88 $110 0.125 7 $110 $110 $88 0.125 8 $110 $110 $110 0.125 The expected price paid is . Suppose that bidders 1 and 2 collude and would be willing to bid up to a maximum of their values, but the two bidders…7. Suppose the only game in town involves flipping a fair coin (so Heads and Tails are equally likely), with a $x bet. If Heads comes up, the payoff is $0.9x; if Tails comes up, you lose the $x. You have $10,000, and must win at least $5,000 by tomorrow morning to pay off a debt to a mean dude. a. Compute the likelihood of winning at least $5000 by making a single bet of $10,000. b. Compute the likelihood of winning at least $1000 by playing the game 10,0000 times and betting a dollar each time. What is the likelihood of not losing money? Message learned?Suppose that the buyers do not know the quality of any particular bicycle for sale, but the sellers do knowthe quality of the bike they sell. The price at which a bike is traded is determined by demand and supply.Each buyer wants at most one bicycle.(ii) Assuming that each buyer purchases a bike only if its expected quality is higher than the price,and each seller is willing to sell their bike only if the price exceeds their valuation, what is theequilibrium outcome in this market?
- 2 Consider the two investments listed below with possible outcomes and probabilities: INVESTMENT (in $1000) SAFE RISKY INVESTMENT AMOUNTⓇ 40+ 40+ GOOD SCENARIO OUTCOME 45+ 80+ AVERAGE+ SCENARIO PROB OUTCOME 0.40* 0.40€ 42+ 45+ BAD+ SCENARIO PROB OUTCOME PROB 0.20 35+ 0.20 10+ 0.40€ 0.40+ b) a) Suppose I have utility function U(*) = (x)2. What is the expected utility from each investment? Which investment will I choose, if any? Show and explain your work and provide the intuition. c) What is the value of the risk premium for the SAFE investment? Show and explain your work and provide the intuition. d) What is the value of the risk premium for the RISKY investment? Show and explain your work and provide the intuition.< +4 In a gambling game, Player A and Player B both have a $5 and a $10 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B's bill. If the bills match, Player B wins Player A's bill. (a) Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. Player B Player A $5 $10 $5 (b) Is there a pure strategy? Why or why not? ---Select--- . Since the maximum of the row minimums is --Select--- (c) Determine the optimal strategies and the value of this game. probability Player A selects $5 probability Player A selects $10 probability Player B selects $5 probability Player B selects $10 = = $10 = = Does the game favor one player over the other? Yes O No and the minimum of the column maximums is (d) Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the…. If you examine the decision tree in Figure 9.12 (orany other decision trees from PrecisionTree), you willsee two numbers (in blue font) to the right of each endnode. The bottom number is the combined monetaryvalue from following the corresponding path throughthe tree. The top number is the probability that thispath will be followed, given that the best strategy isused. With this in mind, explain (1) how the positiveprobabilities following the end nodes are calculated,(2) why some of the probabilities following the endnodes are 0, and (3) why the sum of the probabilitiesfollowing the end nodes is necessarily 1.
- 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.rock paper scissors гock 0. -3 1 рарer 1. -1 scissors -1 3 0. (a) Show that xT= ( ) and yT= (3) together are not a Nash equilibrium 3 3 313 for this modified game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.you and a friend decide to run a three mile race. If you agree to run together, you keep up with himfor the first mile, but you overexert yourself and run the last two miles at slower paces on your own. Tomake up for lost time, your friend runs the last two miles at a faster pace. Your mile times are 6:30, 7:00,and 7:30. Your friend’s times are 6:30, 6:00, and 6:00. If you both agree to run on your own, you run aconstant pace of 7:05 while your friend runs at a constant pace of 6:05. If you want to run together butyour friend wants to run solo, he runs his constant pace of 6:05. You, on the other hand, want to showhim that you can run faster, but you end up overexerting yourself after the first mile. You run times of6:20, 7:05, and 7:30. If he wants to run together but you do not, you both run at your pace of 7:05. Thissituation can be turned into an economic game, with the payoffs the overall race times. You each wantto run the fastest time you possibly can.(a) Who are the players in…
