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- Во ein g Inv est C. 25 d. 200 Do n't Diagram 1 Air bus Air bus Inv est Do n't Inv est (-25,-25) (170,0) (0,190) Do (0,0) n't Suppose that both Boeing and Airbus receive a subsidy of 30 units to invest. What is Boeing's equilibrium payoff in this game? a. 0 b. 5A 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.3. Consider the following two-player game: H L T D 2,3 0,2 4,0 1,1 (a) What are (pure- and mixed-strategy) Nash equilibria of this game? (b) Suppose the game is repeated twice, and each player's payoff is the sum of the payoffs they obtain in the two periods. What are the subgame perfect equilibria of the game? (c) Suppose the game is repeated indefinitely, and each player discounts his/her payoff with a discount factor & € (0, 1). Find a subgame perfect equilibrium in which (H, T) is played in every period on the equilibrium path. Compute the discount factor & needed for this equilibrium.
- 1Y You have to decide whether to invest $100 in a friend's enterprise, where in a year's time the money will increase to $150. You have agreed that your friend will split the profits evenly each year with you. Your alternative is to keep at the bank at an interest rate of r. Suppose this game is played repeatedly infinitely often. The rate of interest between any two successive periods is r, the same as that in the bank and the same for you and your friend. What is the cut-off interest rate which will make cooperation possible?3. Consider the following two-player game: H L T D 2,3 0,2 4,0 1,1 (a) What are (pure- and mixed-strategy) Nash equilibria of this game? (b) Suppose the game is repeated twice, and each player's payoff is the sum of the payoffs they obtain in the two periods. What are the subgame perfect equilibria of the game? (c) Suppose the game is repeated indefinitely, and each player discounts his/her payoff with a discount factor & € (0, 1). Find a subgame perfect equilibrium in which (H, T) is played in every period on the equilibrium path. Compute the discount factor & needed for this equilibrium. (d) Suppose the game is repeated indefinitely, and each player discounts his/her payoff with a discount factor 8 € (0, 1). Find a subgame perfect equilibrium in which (H,T) and (L,T) are played alternately on the equilibrium path. Compute the discount factor & needed for this equilibrium. (e) Suppose the game is repeated indefinitely. Player 1 (the row player) discounts her payoff with a discount…In the 3 x 3 matrix below, what is the Nash equilibrium? Р2 N S E U 6,4 7,3 5, 5 P1 H7,3 3,7 4,6 L 8,2 | 6,4 | 3, 7 Select one: a. U,E b. 8,2 О с. 5,5 O d. L,N
- 3. Consider the following two-player game: H L D T 2,3 0,2 4,0 1,1 (a) What are (pure- and mixed-strategy) Nash equilibria of this game? (b) Suppose the game is repeated twice, and each player's payoff is the sum of the payoffs they obtain in the two periods. What are the subgame perfect equilibria of the game? (c) Suppose the game is repeated indefinitely, and each player discounts his/her payoff with a discount factor 8 € (0, 1). Find a subgame perfect equilibrium in which (H, T) is played in every period on the equilibrium path. Compute the discount factor & needed for this equilibrium. (d) Suppose the game is repeated indefinitely, and each player discounts his/her payoff with a discount factor & € (0, 1). Find a subgame perfect equilibrium in which (H,T) and (L,T) are played alternately on the equilibrium path. Compute the discount factor & needed for this equilibrium. (e) Suppose the game is repeated indefinitely. Player 1 (the row player) discounts her payoff with a discount…4. Consider the following game. There are two players and two possible simultaneous move games A and B. Player 1 first chooses which game is to be played; both player observe which game is played and choose their strategy. The payoff matrices for Game A and B are given by : where k> >0. Game A U D L (10,5) (0,0) R (0,0) (5,10) Game B: U D (a) Write down the extensive form of this game. (b) Assume k = 2. Solve for all subgame perfect Nash equilibria of this game. L R (kk) (0,0) (0,0) (0,0) (c) Can you find a value of k such that there exists a SPNE in which player 1 chooses game B? Can you find a value of k such that all SPNE involve player 1 choosing game B?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…Use the following setting for questions 4-8. Consider the following static Prisoner's Dilemma game. C D 2, 2-3, 3 3, -3 1, 1 For the following parts, suppose this game is played for infinitely many times with discount factor for both players & € [0, 1). C D Suppose 8 = 0. What is the maximum equilibrium payoff that player 1 can get in the first period in a SPNE of this game? (a) -3 (b) 0 (c) 1 (d) 2 (e) 3 Let both players adopt the following grim trigger strategy: start with cooperation; cooperate as long as no one has ever defected before; otherwise defect. What is the condition on & for this strategy profile to be a SPNE? (a) 8 >0 (b) 8 ≥ 1/4 (c) 8 > 1/3 (d) 8 ≥ 1/2 (e) 8 ≤ 1/2Player 2 E F H A 6, 5 6, 7 9, 6 7,6 В Player 1 C 6, 7 6, 9 8, 5 9, 7 5, 8 5, 6 7,5 7,5 7,9 8, 7 11, 6 5, 6 (1) In the Unique Nash equilibrium of this game, which strategy does Player1 play? And why? (2) In the Unique Nash equilibrium of this game, which strategy does Player2 play? And why? (3) Is this game dominance solvable? And Why? (4) Does this game have at least one inadmissible Nash equilibrium? And Why?