a. Suppose players announce simultaneously whether they want to ex- change their houses. If both players agree to an exchange, the exchange takes place. Otherwise no exchange takes place. Find a Bayesian Nash equilibrium of this game in pure strategies in which each player i ac- cepts an exchange if and only if the value v, does not exceed some threshold 0₁. b. How would your answer to (a) change if player j's valuation of player i's house were v? c. Try to explain why any Bayesian Nash equilibrium of the game de- scribed in (a) must involve threshold strategies of the type postulated in (a)
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- 2. Consider the following two player game in normal form: Player 2 R 0, 5 2, 3 2, 3 Player 1 M 2, 3 0, 5 3, 2 B 5,0 3, 2 2, 3 (a) Show that for Player 1, strategy T is strictly dominated by a mixed strategy in which actions M and B are played with positive probability. (b) Find a mixed strategy Nash equilibrium of this game. 35212.7 Trading Places: Two players, 1 and 2, each own a house. Each player i values his own house at v;. The value of player i's house to the other player, i.e., to player ji, is v₁. Each player i knows the value v; of his own house to himself, but not the value of the other player's house. The values v; are drawn independently from the interval [0, 1] with uniform distribution. a. Suppose players announce simultaneously whether they want to ex- change their houses. If both players agree to an exchange, the exchange takes place. Otherwise no exchange takes place. Find a Bayesian Nash equilibrium of this game in pure strategies in which each player i ac- cepts an exchange if and only if the value v; does not exceed some threshold 0;. b. How would your answer to (a) change if player j's valuation of player i's house were {v;? c. Try to explain why any Bayesian Nash equilibrium of the game de- scribed in (a) must involve threshold strategies of the type postulated in (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.
- 2. Kier, in The scenario, wants to determine how each of the 3 companies will decide on possible new investments. He was able to determine the new investment pay off for each of the three choices as well as the probability of the two types of market. If a company will launch product 1, it will gain 50,000 if the market is successful and lose 50,000 if the market is a failure. If a company will launch product 2, it will gain 25,000 if the market is successful and lose 25,000 if the market will fail. If a company decides not to launch any of the product, it will not be affected whether the market will succeed or fail. There is a 56% probability that the market will succeed and 44% probability that the market will fail. What will be the companies decision based on EMV? What is the decision of each company based on expected utility value?[Adverse Selection] Each of the two players receives an envelope, in which there is anamount of money that is equally distributed from $0, $1, $2, ..., $100. The amounts in twoenvelopes are independent. After receiving the envelope, each individual can check exactlyhow much money is put in his/her own envelope. Then each player has the option to exchangehis/her envelope for the other individual's prize. The decisions are made simultaneously. Ifboth individuals agree to exchange, then the envelopes are exchanged; otherwise, if at leastone player chooses not to exchange, each individual keeps his/her own envelope and receivesits attached sum of money.a. Model this game as a static Bayesian game (write the normal formrepresentation) and find the Bayesian Nash equilibrium.b. Consider a new game where the probability distribution of money in eachenvelope is changed. The amount is equal to $100 with probability 90%, and is equalto each number in $0, $1, $2, ... ,$99 with probability 0.1%.…You and a coworker are assigned a team project on which your likelihood or a promotion will be decidedon. It is now the night before the project is due and neither has yet to start it. You both want toreceive a promotion next year, but you both also want to go to your company’s holiday party that night.Each of you wants to maximize his or her own happiness (likelihood of a promotion and mingling withyour colleagues “on the company’s dime”). If you both work, you deliver an outstanding presentation.If you both go to the party, your presentation is mediocre. If one parties and the other works, yourpresentation is above average. Partying increases happiness by 25 units. Working on the project addszero units to happiness. Happiness is also affected by your chance of a promotion, which is depends on howgood your project is. An outstanding presentation gives 40 units of happiness to each of you; an aboveaverage presentation gives 30 units of happiness; a mediocre presentation gives 10 units…
- Consider the game shown in Figure 3. Let A denote the probability that player 1 plays a, B the probability that player 1 plays b, C the probability that player I plays e, and D the probability that player I plays d. For player 2 X denotes the probability that player 2 plays x, Ý that he/she plays y, and Z that he she plays z. Figure 3 Player 2 O 3,7 4,6 5,4 b5,1 2,3 1,2 C 2,3 | 1,4 | 3,3 d 4,2 1,3 6,1 Player 1 In a NE what is: C, the probability that player 1 plays e a. Z, the probability that player 2 plays z b. D, the probability that player 1 plays d с. d. X, the probability that player 2 plays x e. A, the probability that player 1 plays a1. 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, 7, 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, Y, of bidder 2's valuation (and similarly, bidder 2 assumes bidder 1 will bid the same constant propor- tional value y of…Consider the game shown in Figure 3. Let A denote the probability that player I plays a, B the probability that player 1 plays b, C the probability that player I plays e, and D the probability that player I plays d. For player 2 X denotes the probability that player 2 plays x, Ý that he/she plays y, and Z that he she plays z. Figure 3 Player 2 O 3,7 4,6 5,4 b 5,1 2,3 1,2 c 2,3 1,4 3,3 d 4,2 1,3 6,1 Player 1 In a NE what is: C, the probability that player 1 plays e a. Z, the probability that player 2 plays z b. D, the probability that player 1 plays d c. d. X, the probability that player 2 plays x e. A, the probability that player 1 plays aSuppose 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?