Every year, management and labor renegotiate a new employment contract by sending their proposals to an arbitrator, who chooses the best proposal (effectively giving one side or the other $3 million). Each side can choose to hire, or not hire, an expensive labor lawyer (at a cost of $300,000) who is effective at preparing the proposal in the best light. If neither hires a lawyer or if both hire lawyers, each side can expect to win about half the time. If only one side hires a lawyer, it can expect to win nine tenths, or 0.9, of the time.
Use the given information to fill in the expected payoff, in dollars, for each cell in the matrix. (Hint: To find the expected payoff, multiply the probability of winning by the dollar amount of the payoff. Be sure to account for lawyer costs, which are incurred with certainty if a lawyer is hired.)
| Management (M) | |||
| No Lawyer | Lawyer | ||
| Labor (L) | No Lawyer | L:
, M:
|
L:
, M:
|
| Lawyer | L:
, M:
|
L:
, M:
|
|
The Nash equilibrium for this game is for Management to a lawyer, and for Labor to a lawyer.
In this scenario, management and labor engage in a yearly negotiation process to determine the terms of their employment contract. They both have the option to hire an expensive labor lawyer at a cost of $300,000 to prepare their proposals, and an arbitrator ultimately selects the best proposal, awarding $3 million to the winning side. The key to analyzing this situation is to calculate the expected payoffs for each possible combination of choices, taking into account both the probability of winning and the costs associated with hiring a lawyer.
Let’s break down the expected payoffs for each cell in the matrix:
If both management and labor choose not to hire a lawyer (No Lawyer, No Lawyer):
Management’s probability of winning without a lawyer is 0.5 (50%).
Labor’s probability of winning without a lawyer is also 0.5 (50%).
Both sides incur no lawyer costs in this scenario.
If management hires a lawyer, but labor does not (Lawyer, No Lawyer):
Management’s probability of winning with a lawyer is 0.9 (90%).
Labor’s probability of winning without a lawyer remains 0.5 (50%).
Management incurs a lawyer cost of $300,000.
If management does not hire a lawyer, but labor does (No Lawyer, Lawyer):
Management’s probability of winning without a lawyer is 0.5 (50%).
Labor’s probability of winning with a lawyer is 0.9 (90%).
Labor incurs a lawyer cost of $300,000.
If both management and labor hire lawyers (Lawyer, Lawyer):
Management’s probability of winning with a lawyer is 0.9 (90%).
Labor’s probability of winning with a lawyer is also 0.9 (90%).
Both sides incur lawyer costs of $300,000 each.
Now, let’s calculate the expected payoffs for each cell:
Expected Payoff for (No Lawyer, No Lawyer):
Management: $3 million * 0.5 = $1.5 million
Labor: $3 million * 0.5 = $1.5 million
Expected Payoff for (Lawyer, No Lawyer):
Management: ($3 million – $300,000) * 0.9 = $2.43 million
Labor: $3 million * 0.5 = $1.5 million
Expected Payoff for (No Lawyer, Lawyer):
Management: $3 million * 0.5 = $1.5 million
Labor: ($3 million – $300,000) * 0.9 = $2.43 million
Expected Payoff for (Lawyer, Lawyer):
Management: ($3 million – $300,000) * 0.9 = $2.43 million
Labor: ($3 million – $300,000) * 0.9 = $2.43 million
Now, let’s identify the Nash equilibrium, which is a situation where neither side has an incentive to change their strategy given the strategy of the other side.
In this case, the Nash equilibrium is for Management to hire a lawyer, and for Labor to hire a lawyer as well. This is because in the (Lawyer, Lawyer) cell, both sides have the highest expected payoff of $2.43 million, and neither side would benefit from changing their strategy.
In conclusion, the Nash equilibrium for this negotiation game is for both Management and Labor to hire lawyers. This ensures that both sides have the highest expected payoff while taking into account the costs associated with hiring lawyers.
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