A farmer can either invest in irrigation for their farm or not. There's a 30% chance that they will receive adequate rain such that the irrigation is not necessary, and a 70% chance that the irrigation will be useful if acquired. Assume that the farmer will receive $5,000 if their crops are adequately watered, and $1,000 otherwise. Irrigation costs $3,000. For simplicity, assume this is a one-off game and that nature is not perverse. a. Draw the extensive form of this game, with the farmer acting first and nature second. b. What is the expected value of each strategy for the farmer? c. What is the expected value of perfect information on the weather?
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7. A farmer can either invest in irrigation for their farm or not. There's a 30% chance that they will receive adequate rain such that the irrigation is not necessary, and a 70% chance that the irrigation will be useful if acquired. Assume that the farmer will receive $5,000 if their crops are adequately watered, and $1,000 otherwise. Irrigation costs $3,000. For simplicity, assume this is a one-off game and that nature is not perverse.
a. Draw the extensive form of this game, with the farmer acting first and nature second.
b. What is the expected value of each strategy for the farmer?
c. What is the expected value of perfect information on the weather?
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- Suppose it is a well-known fact that among ten-year old Ford F-150s, half the trucks are good and half of them are lemons. Suppose that it is also known to all parties that a good truck is worth $8,000 to current owners and $10,000 to potential buyers. A bad truck, on the other hand, is only worth $1,000 to current owners and $2,000 to potential buyers. Throughout, assume that buyers are risk-neutral. 4) Suppose that after much haggling, the current owner is willing to let her truck go for $6,000. What is the most likely implication? a) The truck is a lemon. b) The buyer is an excellent negotiator. c) It's a mutually beneficial transaction.12.3 Armed Conflict: Consider the following strategic situation: Two rival armies plan to seize a disputed territory. Each army's general can choose either to attack (A) or to not attack (N). In addition, each army is either strong (S) or weak (W) with equal probability, and the realizations for each army are independent. Furthermore the type of each army is known only to that army's general. An army can capture the territory if either (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is weak. If both attack and are of equal strength then neither captures the territory. As for payoffs, the territory is worth m if captured and each army has a cost of fighting equal to s if it is strong and w if it is weak, where s <w. If an army attacks but its rival does not, no costs are borne by either side. Identify all 12.7 Exercises • 267 the pure-strategy Bayesian Nash equilibria of this game for the following two cases, and briefly describe…Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.
- Consider the following game played by four individuals, players 1, 2, 3, and 4. Each individual has $10,000. Each player can donate between $0 and $10,000 to build a public park that costs $20,000. If they collect enough money, they construct the park, which is worth $9,000 to each of them. However, if they collect less than $20,000, they cannot build a park. Furthermore, regardless of whether the park is built or not, individuals lose any donations that they make. a) Describe the Nash equilibria for a simultaneous game. What makes them equilibria? Hint: There are many equilibria, so you may want to use a mathematical expression! b) Suppose that players 1, 2, and 3, each donate $4,000 for the park. How much will player 4 donate and why. What are the resulting payoffs for the players? c) Suppose instead that player 1 donated first, player 2 second, player 3 third, and player 4 last. Furthermore, players could only donate in intervals of 1,000 (0, $1,000, $2,000, etc.). How much will…5) Suppose that the letter grade earned on a test for each student in a class depends upon how well he/she does relative to other students in the class. This exhibit shows a prisoner’s dilemma setting for two representative students in the class, George and Gina. What is the expected final Nash Equilibrium? a) Study 4 hours for both Gina and George. b) Study 2 hours for both Gina and George. c) George studies for 2 hours, and Gina studies for 4 hours. d) George studies for 4 hours, and Gina studies for 2 hours.We have a group of three friends: Kramer, Jerry and Elaine. Kramer has a $10 banknote that he will auction off, and Jerry and Elaine will be bidding for it. Jerry and Elaine have to submit their bids to Kramer privately, both at the same time. We assume that both Jerry and Elaine only have $2 that day, and the available strategies to each one of them are to bid either$0, $1 or $2. Whoever places the highest bid, wins the $10 banknote. In case of a tie (that is, if Jerry and Elaine submit the same bid), each one of them gets $5. Regardless of who wins the auction, each bidder has to pay to Kramer whatever he or she bid. Does this game have a Nash Equilibrium? (If not, why not? If yes, what is the Nash Equilibrium?)
- We have a group of three friends: Kramer, Jerry and Elaine. Kramer has a $10 banknote that he will auction off, and Jerry and Elaine will be bidding for it. Jerry and Elaine have to submit their bids to Kramer privately, both at the same time. We assume that both Jerry and Elaine only have $2 that day, and the available strategies to each one of them are to bid either$0, $1 or $2. Whoever places the highest bid, wins the $10 banknote. In case of a tie (that is, if Jerry and Elaine submit the same bid), each one of them gets $5. Regardless of who wins the auction, each bidder has to pay to Kramer whatever he or she bid. Does Jerry have any strictly dominant strategy? Does Elaine?Imagine that two firms in two different countries want to bring a new product to market. Due to economies of scale, if both firms do this, they will both lose £50 million. But if only one firm does this, it will gain £300 million. (a) What is the best strategy for firm A, if firm B has not yet entered the market, and why? (b) Illustrate this with a game theory diagram, showing appropriate payouts. (c) What is the welfare-maximising strategy for a government, and why?Q14. Do players have perfect information in the above game? Yes, all of them have perfect information No, player 2 has imperfect information No, player 3 has imperfect information No, no player has perfect information Q15. If we want to describe the above game with a strategic form representation, what would the strategy sets for the three players be? Player 1={a, b, c} ; Player 2={x, y}; Player 3={r, s} Player 1={a, b, c} ; Player 2={xx, xy, yx, yy}; Player 3={r, s}
- H2. One day, Sam and Ryan play odds/evens to see who gets the last doughnut. On command, they each extend one or two fingers. If the sum is odd Sam wins the doughnut, if the sum is even Ryan wins the doughnut. Suppose the payoff from winning the doughnut is 1 and the payoff from losing is 0. a) Illustrate this interaction as a game in matrix form. b) Suppose that Sam thinks that Ryan will play one finger for sure? What will Sam play? Does Sam have reason to think that Ryan will play one finger for sure? c) Do either of them have a strictly dominated strategy? d) Find the pure strategy Nash equilibria of the game, if any. e) Suppose that Sam thinks that Ryan will play one finger or two fingers with even odds. Will Ryan play one finger for sure, play two fingers for sure or play each strategy with even odds? Does Sam have good reason to believe that Ryan will play one finger or two fingers with even odds?Boris and Angela are negotiating a new trade deal, which resembles a modified centipede game. When it is their turn, each of them decides whether to cooperate (C) or to decline (D). Unfortunately, Angela is not sure how much time Boris has for the negotiations before he needs to leave to talk to Emmanuel. With probability p = 4/5, Boris will stay and insist on making the final proposal (long game, L). With probability 1 − p = 1/5, Boris will have to rush off, so that the game ends after Angela’s move (short game, S). Boris has perfect knowledge of his schedule. The structure and payoffs are represented in the game tree below (Attached picture). (a) Specify the set of possible histories H of the game. Underline terminal histories.(b) Specify the set of possible strategies for Boris and Angela.(c) How many subgames does this game have? Indicate all subgames in this gameA strategy for player 1 is a value for x1 from the set X. Similarly, a strategyfor player 2 is a value for x2 from the set X. Player 1’s payoff is V1(x1, x2) =5 + x1 - 2x2 and player 2’s payoff is V2(x1, x2) = 5 + x2 - 2x1.a. Assume that X is the interval of real numbers from 1 to 4 (including 1and 4). (Note that this is much more than integers and includes such numbers as 2.648 and 1.00037). Derive all Nash equilibria.b. Now assume that the game is played infinitely often and a player’s payoff is the present value of his stream of single-period payoffs, where dis the discount factor.(i) Assume that X is composed of only two values: 2 and 3; thus, aplayer can choose 2 or 3, but no other value. Consider the followingsymmetric strategy profile: In period 1, a player chooses the value 2. In period t(≥2), a player chooses the value 2. In period a player chooses the value 2 if both players chose 2 in all previous periods; otherwise, she chooses the value 3. Derive conditions which ensure…