Question

Ashley is negotiating an employment contract with a prospective employer, the La Jolla YMCA. The contract specifies two things: (1) Ashley's job description, which is surfing instructor or tennis instructor, and (2) Ashley's salary \(t\). Ashley is better at teaching people to surf than teaching people to play tennis, and she enjoys surfing more. If Ashley works as a surfing in structor for the YMCA, then Ashley's payoff is \(t-10,000\) and the YMCA's payoff is \(60,000-t\). If Ashley works as a tennis instructor, then her payoff is \(t-20,000\) and the YMCA's payoff is \(65,000-t\). If Ashley and the YMCA fail to reach an agreement, then the YMCA gets a payoff of 10,000 and Ashley obtains 20,000 . In other words, the disagreement outcome of this negotiation problem is given by \(d_{\mathrm{A}}=20,000\) and \(d_{\mathrm{Y}}=10,000\). The value 20,000 is due to Ashley's outside opportunity, which is to work as a writer for the San Diego Reader newspaper. The value of 10,000 for the YMCA is due to it having to hire someone who is less skilled than Ashley. Solve this bargaining problem using the standard bargaining solution, under the assumption that Ashley and the YMCA have equal bargaining weights of \(1 / 2\). What job description and salary do they agree to? Explain.

Short answer

Ashley agrees to be a surfing instructor with a salary of $37,666.67.

Step-by-step solution

Express Payoffs in Terms of Salary t

First, note Ashley's and the YMCA's payoffs based on Ashley's job role and salary. As a surfing instructor, Ashley's payoff is \( t - 10,000 \), and the YMCA's payoff is \( 60,000 - t \). As a tennis instructor, Ashley's payoff is \( t - 20,000 \), and the YMCA's payoff is \( 65,000 - t \). The disagreement outcomes are \( d_A = 20,000 \) for Ashley, and \( d_Y = 10,000 \) for the YMCA.

Calculate the Nash Bargaining Solution

The Nash bargaining solution maximizes the product of the gains of each party over their disagreement payoffs. We set up the product: \( (t - 10,000 - 20,000) \times (60,000 - t - 10,000) \) for surfing and \( (t - 20,000 - 20,000) \times (65,000 - t - 10,000) \) for tennis.

Solve for Surfing Instructor

For the surfing instructor, solve the equation: \[ (t - 30,000) \times (50,000 - t) \]. Differentiate \((t - 30,000)\) and \((50,000 - t)\) with respect to \(t\), set the derivative to zero, and solve for \(t\).\ Next, set the derivative equal to zero and solve: \(50000 - 2t + 800000 - 1600000 - t = 0 \) resulting in \(3t = 1,130,000 \). Thus, \(t = 37,666.67 \).

Solve for Tennis Instructor

For the tennis instructor, solve \((t - 40,000) \times (55,000 - t) \). Differentiate and set the derivative to zero: \(55000 - 2t + 650000 - 1600000 + 2t = 0\), resulting in \(3t = 1,150,000 \). Solving for \(t\), we get \(t = 38,333.33\).

Compare the Outcomes

For the surfing instructor, the salary \( t \) is \( 37,666.67 \), and both parties are better off than the disagreement outcome. The payoffs are higher for both the YMCA and Ashley in comparison to the disagreement result. The YMCA's payoff is \( 22,333.33 \) and Ashley's is \( 7,666.67 \). For tennis, the figures are less favorable compared to surf, particularly for Ashley.

Conclusion - Preferred Contract

Between the two job descriptions, the surfing instructor role gives better payoffs for both Ashley and the YMCA. The agreed salary is \( t = 37,666.67 \), achieving optimal gains compared to the disagreement outcome.

Key concepts

Nash Bargaining Solution

The Nash Bargaining Solution is a key concept in game theory used to solve negotiation problems. It offers a way to find an optimal agreement for both parties involved in a negotiation. This solution maximizes the product of each party's gain over their disagreement point. In plain terms, it helps the negotiators find a settlement that is the best outcome for both, depending on their respective alternatives if no agreement is reached.

In the given problem, the Nash Bargaining Solution involves calculating the salary agreement from both parties' perspectives based on different job roles. The aim is to maximize the combined payoff for Ashley, the potential employee, and the La Jolla YMCA, her prospective employer. By using mathematical formulas, like the ones shown in the step-by-step solution, one can determine which salary ensures a balanced and mutually beneficial outcome.

Game Theory

Game theory is the mathematical study of interaction strategies between rational decision-makers. Often used in economics and social sciences, it helps understand competitive and cooperative situations where the outcome for each participant depends on the actions of others.

In this context, game theory provides the framework to analyze Ashley's bargaining scenario with the YMCA. By considering both parties' options, such as potential payoffs and fallback positions, game theory offers a structured way to scrutinize and predict the negotiation's range of outcomes. This insight guides each party toward decisions that maximize their benefits based on known variables.

Negotiation Problem

A negotiation problem involves two or more parties trying to reach an agreement on matters of mutual interest. Each party has their own preferences, and a successful negotiation ideally meets the needs of all involved to some extent.

In Ashley's scenario, the negotiation problem involves choosing between two job roles, each offering different payoffs for her and the YMCA. The disagreement outcomes, which represent what each party would receive if no agreement is reached, play a crucial role here. Ashley and the YMCA both aim to surpass these disagreement payoffs, prompting them to engage in informed bargaining to improve their offers until a fair agreement is reached.

Payoff Calculation

Payoff calculation is a pivotal process in evaluating the outcomes of different decisions in a negotiation. It involves determining the value received by each party for different options. This calculation helps each party compare the benefits of potential agreements against their disagreement outcomes.

In the given exercise, calculating payoffs requires consideration of Ashley's job type and the corresponding salary. For instance, as a surfing instructor, the payoff formulas for Ashley and the YMCA are established as functions of the salary, with adjustments made for disagreement values. By solving these payoff equations for both job descriptions, one can ascertain which role offers the best salary balance, ultimately guiding both Ashley and the YMCA toward the most favorable settlement. This evaluation aligns both parties on the agreed salary and positions, providing a justification for the final negotiation decision.