Question

Suppose the president of the local teachers' union bargains with the superintendent of schools over teachers' salaries. Assume the salary is a number between 0 and 1,1 being the teachers' preferred amount and 0 being the superintendent's preferred amount. (a) Model this bargaining problem by using a simple ultimatum game. The superintendent picks a number \(x\), between 0 and 1 , which we interpret as his offer. After observing this offer, the president of the union says "yes" or "no." If she says "yes," then an agreement is reached; in this case, the superintendent (and the administration that she represents) receives \(1-x\) and the president (and the union) receives \(x\). If the president says "no," then both parties receive 0 . Using the concept of backward induction, what would you predict in this game? (b) Let us enrich the model. Suppose that before the negotiation takes place, the president of the union meets with the teachers and promises to hold out for an agreement of at least a salary of \(z\). Suppose also that both the superintendent and the president of the union understand that the president will be fired as union leader if she accepts an offer \(x

Short answer

(a) The superintendent offers slightly more than 0. (b) President accepts if \( x \geq \\text{max}(y, z) \). (c) Offer is \( \\text{max}(y, z) \). (d) Promise more than \( y \) but feasible.

Step-by-step solution

Understand the Problem

In this negotiation scenario, the superintendent makes a salary offer represented by a number between 0 and 1, and the president of the union accepts or rejects this offer. Acceptance results in the president receiving the offer amount, while rejection results in no salary for both.

Part (a): Model Using Ultimatum Game

The superintendent offers a salary, and the president decides whether to accept it based on her utility. If she accepts, the utility of the superintendent is 1 minus the offer, and the president's utility is the offer itself. Using backward induction, the superintendent will offer the smallest amount that will be accepted because if the offer is rejected, both get 0.

Backward Induction Analysis

The president will accept any offer greater than 0 because her utility from accepting is greater than zero, provided the utility is positive. Therefore, the superintendent will offer an amount marginally greater than 0, just enough to ensure acceptance.

Conclusion for Part (a)

In this ultimatum game, the superintendent will offer a salary slightly more than 0, and the president will accept it, resulting in her receiving a utility greater than 0.

Part (b): Incorporate Union's Constraint

The president has promised the union to not accept less than a salary of \( z \), and she suffers a penalty if she breaks this promise. Thus, if the offer \( x < z \), her utility is \( x - y \). She accepts an offer only if her utility is non-negative. Thus, she will accept if \( x - y \geq 0 \), meaning \( x \geq y \). Moreover, since \( x < z \) also leads to punishment, \( x \) must be at least \( \text{max}(y, z) \) for her to accept without penalty.

Evaluate Offers

The effective constraint for the superintendent's offer \( x \) that the president will accept is \( x \geq \text{max}(y, z) \). Therefore, the president will only say 'yes' to offers meeting this condition.

Part (c): Determine Game Outcome

Given the constraint determined in Part (b), the superintendent will offer \( x = \text{max}(y, z) \) as it is the minimum offer that satisfies the acceptance condition. The president will accept this offer. Therefore, the final salary depends on the greater value of \( y \) and \( z \). As \( y \) increases, the minimum acceptable offer to the president increases.

Part (d): Strategic Promise

The president should make a promise to hold out for a value of \( z \) that is realistic and aligned with possible offers, considering \( y \). Likely, \( z \) should be slightly higher than \( y \) to ensure a better offer than \( y \).

Conclusion and Overall Implication

The strategic promise influences the bargaining outcome by predefining the minimum salary accepted. The process teaches the intricate negotiation dynamics and trade-offs faced by union leaders.

Key concepts

Ultimatum Game

The Ultimatum Game is a fascinating example in game theory that demonstrates how people make decisions based on fairness and self-interest. In this game, two players are involved. One player, known as the proposer, suggests how to divide a sum of money. The other player, called the responder, decides to either accept or reject this proposal. If the offer is accepted, both players receive the proposed share. If rejected, neither player gets anything.

In the context of the exercise, the superintendent acts as the proposer by offering a salary amount, which is a number between 0 (least favorable to the union) and 1 (most favorable to the union). The president of the union, as the responder, can either accept or reject the offer. An accepted offer results in the distribution of salary according to the proposal, while rejection means both parties end up with nothing.

This model exemplifies the tension between maximizing one's outcomes and maintaining equitable balances in negotiations.

Bargaining Problem

The Bargaining Problem, as posed in economics, involves two or more parties trying to reach a mutually beneficial agreement. In this specific situation from the exercise, it represents the negotiation over teachers' salaries between the union president and the superintendent.

Both parties have different objectives: the superintendent wants the salary as low as possible, while the union president aims for the highest possible remuneration for the teachers. The challenge lies in striking an agreement where both parties achieve a satisfactory level of utility - satisfaction or financial reward - from the negotiation.

To resolve this issue efficiently, strategies such as bargaining and strategic promises are utilized to move the negotiations towards a resolution that both parties can accept. This problem highlights the necessity of finding an optimal agreement where the welfare of each party is considered and balanced.

Backward Induction

Backward Induction is a method used in game theory to predict outcomes by reasoning backwards from the end of the problem to the beginning. This approach is particularly useful in decision-making scenarios involving sequential actions and can be instrumental in understanding the exercise in question.

In our example, the backward induction process starts with understanding the last move (final decision) of the union president. Knowing that the president will accept any offer that provides her with positive utility, the superintendent can deduce the minimum amount to offer. The superintendent thus offers slightly more than zero since any higher amount is unnecessary for acceptance.

This strategic foresight ensures that the superintendent can formulate an offer that the president is likely to accept on the basis of maximizing the superintendent’s outcomes while satisfying the president's basic utility requirement. Utilizing backward induction can lead to more informed and strategic decision-making in bargaining scenarios.

Utility in Negotiation

Utility in negotiation refers to the satisfaction or benefit gained by parties during negotiations. In the context of the exercise, it represents the personal advantages or losses experienced by both the superintendent and the union president during salary talks.

For the president of the union, utility depends on obtaining an acceptable salary without personal loss. When the president promises not to accept less than a salary of \( z \) and risks losing her position, the utility calculation becomes more complex. The personal cost of \( y \) utility units if dismissed adds another layer of consideration to her decision-making process.

The superintendent's offer must be high enough to balance these elements - offering at least \( \text{max}(y, z) \) ensures that the president's utility remains positive and avoids punishment, leading to an accepted offer. This process of finding equilibrium reflects the intricate dynamics of calculating utility and its influence in forming agreements.