Two players play a game which involves holding out a nickel or a dime simultaneously. If the sum of the coins is more than 10 cents, Player I gets both the coins; otherwise, Player II gets both the coins Find the optimal strategy for the column player.
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- . In a gambling game, Player A and Player B both have a $1 and a $5 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. b. Is there a pure strategy? Why or why not? c. Determine the optimal strategies and the value of this game. Does the game favor one player over the other? d. Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A’s winnings? Comment on why it is important to follow an optimal game theory strategy.Two workers are on a production line. They each have two actions: exert effort, E, or shirk, S. Effort costs a worker e > 0 and shirking costs them nothing. If two workers do action E a lot of output is produced and the workers earn £3 each. If only one worker chooses action E less output is produced and they both earn £1. The workers earn nothing if they both shirk. (i) Describe this situation as a strategic form game (assuming the workers do not observe each other's effort choice when making their own decision). (ii) For what values of e does this game have strictly dominant strategies? (iii) Describe the Nash equilibria of this game for e = 0,1, 2, 3. (iv) The workers now are re-arranged into a production line. First worker 1 moves and then worker 2 moves. Worker 2 can now see worker l's effort level before they choose their effort. Draw this extensive form game. (v) Find the subgame perfect equilibria of the production-line game for c = 1/2 and c = 3/2.The next 3 questions involve the following game. There are two players, a husband and wife. They can either be selfish (S) or selfless (U) in their marriage. If they choose to be selfish, then there is a negative ʻguilt’ payoff of g. The payoff matrix is below. Figure 1: The Marriage Game Wife S U S 10-g, 10-g 15-g, 2 U 2, 15-g 12, 12 Number left (right) of comma refers to H's (W's) payoff. 23. Suppose that g = 0. What is the Nash equilibrium (or equilibria)? (A) (S, S). (B) (U, S) and (S, U). (C) (S, S) and (U, U) (D) (U, U). 24. Suppose that g= 5. What is the Nash equilibrium (or equilibria)? (A) (S, S). (B) (U, S) and (S, U). (C) (S, S) and (U, U) (D) (U, U). 25. Suppose that g = 10. What is the Nash equilibrium (or equilibria)? (A) (S, S). (B) (U, S) and (S, U). (C) (S, S) and (U, U) (D) (U, U). Husband
- 4. Consider a two player game with Fred and Barney, who tal turns removing matchsticks from a pile. They start with 33 matchsticks, and Fred goes first. On each turn, ecach player may remove either one, two, three, four, or five matchsticks. The player to remove the last matchstick wins the game. What are the optimal strategies for each player? Who will win? b. Suppose now that they can remove up to six matchsticks, how will the optimal strategies change for- each player? a.Let there be two players in a game, Player 1 and Player 2. Consider a jar containing 5 snakes. 3 of the snakes in the Kjar are venomous, while the remaining 2 are non-venomous. In the game, both the players have to put their hand in the jar one after the other and pick a snake out. Each snake, if picked out of the jar, will bite the player's hand. The event of picking a venomous snake, or equivalently, a venomous snake's bite will earn the player zero points. On the other hand, the event of picking a non-venomous snake, or equivalently, a non-venomous snake's bite will earn the player one point. Let X denote Player 1's pick and let Y denote Player 2's pick. Suppose Player 1 is the first to pick out a snake. The expected value of Player 1's pick is: E(X)= (Express your answer as a fraction or round your answer to two decimal places)Two companies are playing an entry game. In the first stage of the game the incumbent firm decides whether to invest in new robotic equipment, which will lower its marginal cost of production. In the second stage, a potential rival decides whether to enter the market. o If the incumbent firm chooses to invest and the rival firm does not enter the market, the incumbent makes 8 million in profits. o If the incumbent firm chooses to invest and the rival firm enters the market, the incumbent makes 2 million in profits, and the rival firm looses 1 million. o If the incumbent firm chooses NOT to invest and the rival firm does not enter the market, the incumbent makes 10 million in profits. o If the incumbent firm chooses NOT to invest and the rival firm enters the market, both the incumbent firm and the rival firm make 4 million each. (a) Draw a decision tree for this game. (b) Use backward induction to solve the game. What is the rival best strategy? What is the incumbent best strategy? (C)…
- Consider the following sequential game: Player 1 plays first and can choose between L and R. Player 2 plays second and can choose between U and D. • If player 1 chooses L, player 2 does not get to play and both players get £1. If player 1 plays R, player 2 gets to play. If player 2 plays U then both players get £2. If player 2 plays D then player 2 gets £3 and player 1 gets £0. What the the outcome of the subgame perfect equilibrium? O a. £3-£0 O b. £1-£1 O c. This game does not have a subgame-perfect Nash-equilibrium. O d. £0-£3 O e. £2-£2A game is played as follows: First Player 1 decides (Y or N) whether or not to play.If she chooses N, the game ends. If she chooses Y, then Player 2 decides (Y or N) whetheror not to play. If he chooses N the game ends. If he chooses Y, then they go ahead and playanother game with the payoffs shown below. A player who opts out by choosing N gets 2 andthe other player gets 0. Draw the tree of this game and then find the two subgame-perfect Nashequilibria.Consider a repeated game with the following stage game: Player 1 O O O Cooperate Not Cooperate This game is repeated T-120 times, and t each period both players must choose their actions simultaneously. The players are deciding whether or not to cooperate specifically, whether or not to follow this cooperation strategy: r≤ 3 S|S|AW|N N|H "Each player will play cooperate as long as no one has played Not Cooperate before. If someone has played Not Cooperate before, then each player will play Not Cooperate forever." rs 5 rs Player 2 In this case, cooperation will be possible if the interest rate r is Not Cooperate 5,15 10, 10 None of the above Cooperate 12, 12 15,5
- A Nash equilibrium occurs when O no player has an incentive to unilaterally change strategies. each player has an incentive to unilaterally change strategies. O both players can cooperate to increase their payoffs. O no player can earn a higher payoff from any other strategy.$11. Anne and Bruce would like to rent a movie, but they can't decide what kind of movie to get: Anne wants to rent a comedy, and Bruce wants to watch a drama. They decide to choose randomly by playing "Evens or Odds." On the count of three, each of them shows one or two fingers. If the sum is even, Anne wins and they rent the comedy; if the sum is odd, Bruce wins and they rent the drama. Each of them earns a payoff of 1 for winning and 0 for losing "Evens or Odds." (a) Draw the game table for "Evens or Odds." (b) Demonstrate that this game has no Nash equilibrium in pure strategies.2. Consider the following game. Two criminals are thinking about pulling off a bank robbery. The take from the bank would be $20,000 each, but the job requires two people (one to rob the bank and one to drive the getaway car. Each criminal could instead rob a liquor store. The take from robing a liquor store is only $1000 but can be done with one person acting alone. Write the payoff matrix of this game Player A Bank job Liqour store Player B Bank job Liquor store a. What are the Nash equilibria in this game? b. Explain why there can be multiple equilibria in this game. c. How the game will be played if mixed strategies are allowed? Discuss.