Consider the following transferable utility cooperative game: N = {1,2,3} and v({1})= 2, v({2}) = 4, v({3}) = 2 v({1, 2}) = 8, v({1,3})= 10, v({2,3}) = 8 v({1,2,3}) = 12 = (a) Two players i and j are interchangeable if for all S that contains neither i nor j,v(SU {i})= v(SU{j}). Rewrite/explain this definition with your own words. (b) Are there any players that are interchangeable? If so, which ones?
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Cooperative Games - Game Theory
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- Some collector has a painting that he no longer values. However, there are two buyers that would be happy to acquire it. Buyer 1 assigns a value of $900 to the painting, and buyer 2 of $1,000. Explain that this situation can be represented as a cooperative game with transferable utility. Obtain the set of players and write down the characteristic function (supposing that the grand coalition’s value is $1000). Find the Core and the Shapley value of the game.Two athletes of equal ability are competing for a prize of $12,000. Each is deciding whether to take a dangerous performance-enhancing drug. If one athlete takes the drug and the other does not, the one who takes the drug wins the prize. If both or neither take the drug, they tie and split the prize. Taking the drug imposes health risks that are equivalent to a loss of XX dollars. Complete the following payoff matrix describing the decisions the athletes face. Enter Player One's payoff on the left in each situation, Player Two's on the right. Player Two's Decision Take Drug Don't Take Drug Player One's Decision Take Drug , , Don't Take Drug , , True or False: The Nash equilibrium is taking the drug if X is greater than $6,000. True False Suppose there was a way to make the drug safer (that is, have lower XX). Which of the following statements are true about the effects of making the drug safer? Check all that…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…
- In the following 3-player game, the payoffs represent the number of years in jail. The equilibrium is ________ Group of answer choices Eric and Ned denies. Eric and Tim confess, but Ned denies. Eric confesses, Ned and Tim deny. Eric and Time deny, but Ned confessesEconomic agents for example consumers or firms often do things Economic agents (for example, consumers or firms) often do things that at first glance seem to be inconsistent with their self-interest. People tip at restaurants and when they are on vacation even if they have no intention to return to the same place. Firms, sometimes, install costly pollution abatement equipment voluntarily. How can these deviations from Nash predictions be explained? Economic agents for example consumers or firms often do thingsRachel, Monica, and Phoebe are roommates; each has 10 hours of free time you could spend cleaning your apartment. You all dislike cleaning, but you all like having a clean apartment: each person’s payoff is the total hours spent (by everyone) cleaning, minus a number 1/2 times the hours spent (individually) cleaning.That is, ui(s1, s2, s3) = s1 + s2 + s3 -1/2si Assume everyone chooses simultaneously how much time to spend cleaning. a. Find the Nash equilibrium. b. Find the Nash if the payoff for each player is: ui(s1, s2, s3) = s1 + s2 + s3 − 3si Is the Nash equilibrium Pareto efficient? If not, can you find an outcome in which everyone is better off than in the Nash equilibrium outcome?
- Demonstrate the concept of coordination failure by using the payoff matrix and prove that, Coordination failure among economic agents can lead to an inefficient outcome while the opposite can guide to an efficient outcome. Use any hypothetical scenario to justify your analysis.There is a city, which looks like chopped isosceles triangle, as shown below. Citizens live uniformly distributed all over the city. Two ice-cream vendors, A and B, must independently set up stores in the city. Each citizen buys from the vendor closest to their location and when equidistant from both vendors they choose by coin toss. Each vendor’s aim is to maximize the expected number of customers. A choice of location by the two vendors is a Nash equilibrium if no vendor can do better by deviating unilaterally. Does this game have a Nash equilibrium? If so, describe it. If not, explain why notConsider a game with two players A and B and two strategies X and Z. If both players play strategy X, A will earn $300 and B will earn $700. If both players play strategy Z, A will earn $1,000 and B will earn $600. If Player A plays strategy X and player B plays strategy Z, A will earn $200 and B will earn $300. If Player A plays strategy Z and player B plays strategy X, A will earn $500 and B will earn $400. Player B finds that: a) strategy Z is a dominant strategy. b) strategy X is a dominant strategy. c) he has no dominant strategy. d) strategy X is a dominated strategy. e) strategy Z is a dominated strategy.
- Two athletes of equal ability are competing for a prize of $10,000. Each is deciding whether to take a dangerous performance enhancing drug. If one athlete takes the drug, and the other does not, the one who takes the drug wins the prize. If both or neither take the drug, they tie and split the prize. Taking the drug imposes health risks that are equivalent to a loss of X dollars. a) Draw a 2×2 payoff matrix describing the decisions the athletes face. b) For what X is taking the drug the Nash equilibrium? c) Does making the drug safer (that is, lowering X) make the athletes better or worse off? Explain.Consider the following game: you and a partner on a school project are asked to evaluate the other, privately rating them either "1 (Good)" or "0 (Bad)". After all the ratings have been done, a bonus pot of $1000 is given to the person with the highest number of points. If there is a tie, the pool is split evenly. Both players only get utility from money. Mark all of the following true statements: A. The best response to your partner rating you as Good is to rate them as Good as well. Your answer B. There is no best response in this game. C. Your partner's best response to you rating them as Bad is to also rate you as C Bad. D. Your best response to any strategy of your partner is to play "Good".Suppose China and the US are deciding whether to join an international agreement to mitigate climate change. The matrix below contains payoffs that represent each country’s net benefit from their decisions. Use this information to answer Question 24. CHINA USA Join Agreement Do Not Join Agreement Join Agreement (100,100) (0,125) Do Not Join Agreement (125,0) (25,25) [24] What does each country decide to do in a Nash equilibrium? AND What is the efficient outcome? Nash: Efficient: