Consider a firm in which a boss and a worker each simultaneously get to choose their actions. The boss can choose between centralizing (C) or delegating (D). The worker can put in low effort (L) or high effort (H). The payoffs are 2 to the boss and 2 to the worker if the choices are C and L. If the boss chooses D and the worker L the payoffs are (1,6) to the boss and worker, respectively. If the actions chosen are C and H the payoff are (6, 1). Finally, if the actions are D and H, the payoffs are 4 apiece. a. If the game is played once, what is the Nash equilibrium. Explain your answer. Could this represent an outcome in a real firm? b. What if the worker and boss meet twice, in which in the first period both make their choice of action simultaneously. Following this, the outcome (and payoffs) is revealed and the game proceeds to the next and final period. In the second period, again the parties simultaneously choose their actions, the payoffs are revealed and the game ends. What outcome do we see in both periods in the subgame perfect equilibria and why? c. Now assume that instead of meeting once or twice, the two players play the simultaneous-choice stage game an infinite number of times. In each period, the two players simultaneously make their choices, the outcome in that period is revealed to all and the players proceed to the next period. Each party discounts future periods by a discount factor δ, where 0 ≤ δ ≤ 1.
Question
Consider a firm in which a boss and a worker each simultaneously get to choose their actions. The boss can choose between centralizing (C) or delegating (D). The worker can put in low effort (L) or high effort (H). The payoffs are 2 to the boss and 2 to the worker if the choices are C and L. If the boss chooses D and the worker L the payoffs are (1,6) to the boss and worker, respectively. If the actions chosen are C and H the payoff are (6, 1). Finally, if the actions are D and H, the payoffs are 4 apiece. a. If the game is played once, what is the Nash equilibrium. Explain your answer. Could this represent an outcome in a real firm? b. What if the worker and boss meet twice, in which in the first period both make their choice of action simultaneously. Following this, the outcome (and payoffs) is revealed and the game proceeds to the next and final period. In the second period, again the parties simultaneously choose their actions, the payoffs are revealed and the game ends. What outcome do we see in both periods in the subgame perfect equilibria and why? c. Now assume that instead of meeting once or twice, the two players play the simultaneous-choice stage game an infinite number of times. In each period, the two players simultaneously make their choices, the outcome in that period is revealed to all and the players proceed to the next period. Each party discounts future periods by a discount factor δ, where 0 ≤ δ ≤ 1.
Solution
a. The Nash equilibrium of this game is where neither the boss nor the worker can improve their payoff by unilaterally changing their strategy, given the other's strategy. In this case, the Nash equilibrium is (D, H), where the boss delegates and the worker puts in high effort. This results in a payoff of 4 for both the boss and the worker. This could represent an outcome in a real firm where the boss delegates tasks to the worker, and the worker responds by putting in high effort, resulting in mutual benefit.
b. In a repeated game where the boss and worker meet twice, the subgame perfect equilibrium in both periods would still be (D, H). This is because in each period, both the boss and the worker have the incentive to choose D and H respectively, given the other's choice. This results in the highest payoff for both parties in each period.
c. In an infinitely repeated game, the strategies and outcomes can be more complex. If both the boss and the worker are patient (i.e., they value future payoffs highly, indicated by a high δ), they might be able to sustain cooperation, resulting in the (D, H) outcome in every period. This is because the threat of punishment in future periods can deter deviation from the cooperative outcome. However, if either the boss or the worker is impatient (i.e., they value future payoffs less, indicated by a low δ), they might have the incentive to deviate from the cooperative outcome, leading to different outcomes in different periods. The exact strategies and outcomes would depend on the specific values of δ for the boss and the worker.
Similar Questions
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