Problem 5: Limited Punishment Consider the game in table 3. The game is repeated infinitely. Both players discount payoff streams using the discount factor & = 0.9. Players both play limited punishment trigger strategies with a punishment phase of T periods. Determine the length of the punishment period that is required to support (4, 4) as the payoff in every stage of a
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- A principal can choose to make the allocation of tasks broad (B) or narrrow (N). At the same time an agent can choose to work put in high effort (H) or low effort (L). The payoffs are 20 to the principal and 30 to the agent is the actions chosen are B and L. For all other combinations of actions the payoffs are 0 to both players. Which of the following statements are true? DA Nash equilibrium of the game is (B, H) A Nash equilibrium is (N, L) There is no Nash equilibrium in pure strategies in this game. A Nash equilibrium is (N, H) A Nash equilibrium is (B, L)a) Find the Nash equilibria in the game (in pure and mixed strategies) and the associated payoffs for the players. b) Now assume that the game is extended in the following way: in the beginning Player 1 can decide whether to opt out (this choice is denoted by O) or whether to play the simultaneous-move game in a) (this choice is denoted by G). If Player 1 opts out (plays O) then both Player 1 and Player 2 get a payoff of 4 each and the game ends. If Player 1 decides to play G, then the simultaneous-move game is played. Find the pure-strategy Nash equilibria in this extended version of the game. (Hint: note that Player 1 now has 4 strategies and write the game up in a 4x2 matrix.) c) Write the game in (b) up in extensive form (a game tree). Identify the subgames of this game.Consider the game in Table 1 and answer the following questions. (1) Write down each player's best response functions and find all the Nash equilibria. (2) For each action of each player, determine whether it is strictly, weakly, or not domi- nated by any other actions. Justify your answer. (3) Find the set of rationalizable action profiles by iteratedly eliminate strictly dominated actions. Clearly show the steps. TABLE 1. A Strategic Game L CR 2,2 1,2 3,3 2,3 1,2 0,3 0,3 1,2 2,1 2,2 M B
- if Y = 4 (a) If ⟨a,d⟩ is played in the first period and ⟨b,e⟩ is played in the second period, what is the resulting (repeated game) payoff for the row player? (b) What is the highest payoff any player can receive in any subgame perfect Nash equilibrium of the repeated game?(1) Write down each player’s best response functions and find all the Nash equilibria.(2) For each action of each player, determine whether it is strictly, weakly, or not domi-nated by any other actions. Justify your answer.(3) Find the set of rationalizable action profiles by iteratedly eliminate strictly dominatedactions. Clearly show the steps.Consider two players, P1 and P2, each can choose A or B. The payoff matrix as a result of the players' choices are listed below: P1 O (P1, P2) (A, A) O (P1, P2) = (A,A) O (P1, P2) = (A, A) A Identify the pure-strategy Nash equilibrium (P1, P2). (P1, P2) = (A, A) B There are multiple pure-strategy Nash equilibria. O There is no pure-strategy Nash equilibrium. -25 35 A 25 -35 P2 35 -45 B -35 45
- Consider two players, P1 and P2, each can choose A or B. The payoff matrix as a result of the players' choices are listed below: P1 (P1, P2) = (B,A) A (P1, P2) = (B, B) B (P1, P2) = (A, A) -25 Identify the pure-strategy Nash equilibrium (P1, P2). There are multiple pure-strategy Nash equilibria. There is no pure-strategy Nash equilibrium. (P1, P2) = (A, B) 35 A 25 -35 P2 35 -45 B -35 45There are three players who must each choose an “effort” level from 1 to 7, that is, Si = {1, 2, 3, ..., 7}. The payoff for each player i is ui(si, s−i) = 10 max{s1, s2, s3} − si. How many pure- strategy Nash equilibria are there? Select one: a.2 b.4 c.none of the other answers d.3 e.1Here is a table representing the gains for the server if he serves on his opponent's forehand or backhand and for the receiver the gains if he returns on his forehand or backhand. Each player can choose to play the forehand or the backhand of the other player : Server D R D 50,50 80, 20 R 90,10 |20, 80 Receiver D Determine the Nash equilibrium(s) (pure strategies and mixed strategies).
- Within a voluntary contribution game, the Nash equilibrium level of contribution is zero, but in experiments, it is often possible to sustain positive levels of contribution for a long period. How might we best explain this? A) Participants are altruistic, and so value the payoff which other participants receive, benefiting (indirectly) from making a contribution. B) Participants believe that if they make a contribution, then other participants will be more likely to make a contribution. C) Participants in experiments believe that they have to make contributions in order to receive any payoff from their participation. D) Participants have experience of working in situations in which cooperation can be sustained for mutual benefit and so have internalised a social norm of cooperationInfinitely Repeated Prisoner's Dilemma Cooperate Defect (а, а) (b, с) (c, b) (d, d) Соорerate Defect Here, a = 16,6 = 1,c=18, d = 7. Assume that both of the players are playing the grim trigger strategy. Also assume that the discounting factor for both player 1 and 2 is d. 1. What is the payoff/utility of player 1 if both players decide to defect all the time? Assume 8 = 0.59. Write at least 3 digits after the decimal place. 2. What is the payoff/utility of player 1 if both players decide to cooperate all the time? Assume d = 0.65. Write at least 3 digits after the decimal place. 3. Find the minimum value of 5 for which player 1 will have no incentive to defect on the first stage, assuming player 1 and 2 both are playing grim trigger strategy. Write at least 3 digits after the decimal place.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.