Question

A firm has two divisions, each of which has its own manager. Managers of these divisions are paid according to their effort in promoting produc- tivity in their divisions. The payment scheme is based on a comparison of the two outcomes. If both managers have expended “high effort,” each earns $150,000 a year. If both have expended “low effort,” each earns “only” $100,000 a year. But if one of the two managers shows “high effort” whereas the other shows “low effort,” the “high effort” manager is paid $150,000 plus   $50,000 bonus, but the second (“low effort”) manager gets a reduced salary (for subpar performance in comparison with her competition) of $80,000. Managers make their effort decisions indepen- dently and without knowledge of the other manager’s choice. (a) Assume that expending effort is costless to the managers and draw the payoff table for this game. Find the Nash equilibrium of the game and explain whether the game is a prisoners’ dilemma. (b) Now suppose that expending high effort is costly to the manag- ers (such as a costly signal of quality). In particular, suppose that “high effort” costs an equivalent of $60,000 a year to a manager who chooses this effort level. Draw the game table for this new version of the game and find the Nash equilibrium. Explain whether the game is a prisoners’ dilemma and how it has changed from the game in part (a). (c) If the cost of high effort is equivalent to $80,000/year, how does the game change from that described in part (b)? What is the new equi- librium? Explain whether the game is a prisoners’ dilemma and how it has changed from the games in parts (a) and (b).

Answer 1

A) no efforts cost.

Manager 2
Manager 1		High effort	Low effort
High effort	150,000; 150,000	200,000; 80,000
Low effort	80,000; 200,000	100,000; 100,000

for any strategy of the other player, it is always in the best interest of each manager to put in high efforts to draw maximum possible salary. therefore, [high effort, high effort] is the nash equilibrium strategy set. high effort is strictly dominant strategy.

the game's payoff table is not similar to that of prisoner's dilemma because unlike prisoner's dilemma, both manager are choosing the strategy which would yield better payoffs for both of them and their choice is rational. in PD game, both prisoners choose to defect (or fink) while the best payoff is awarded when both rationally choose to cooperate.

b) cost of high effort is 60,000. so net payoff for high effort would be salary less 60,000. (150000-60000 = 90000; 200000-60000 = 140000)

Manager 2
Manager 1		High effort	Low effort
High effort	90,000;90,000	140,000;80,000
Low effort	80,000;140,000	100,000;100,000

the game has not changed much except for payoffs. [high effort, high effort] remains the nash equilibrium here. it is similar to PD game now because both player are choosing the strategy of high effort while the best pair would be [low effort, low effort] as both would be better of in that case. their choice is not rational.

c) cost of high effort is 80,000. so net payoff for high effort would be salary less 80,000. (150000-80000 = 70000; 200000-80000 = 120000)

Manager 2
Manager 1		High effort	Low effort
High effort	70,000;70,000	120,000;80,000
Low effort	80,000;120,000	100,000;100,000

the game has changed in terms of payoffs.

d) now the game has two different nash equlibria than part (a) and (b) - [high effort, low effort] and [low effort, high effort]. that is to say that if one of the manager is putting in high effort, the other one would choose to put low effort.

again this game is similar to prisoner's dilemma in the sense that the managers are not choosing the strategy actions of [low effort, low effort] which yields maximum payoff for both of them. their choice is not rational.