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Chapter 19: Problem 2

Suppose that you are attempting to buy a house, and you are bargaining with the current owner over the sale price. The house is of value $$\$ 200,000$$ to you and $$\$ 100,000$$ to the current owner; so, if the price is between $$\$ 100,000$$ and $$\$ 200,000$$, then you would both be better off with the sale. Assume that bargaining takes place with alternating offers and that each stage of bargaining (an offer and a response) takes a full day to complete. If agreement is not reached after ten days of bargaining, then the opportunity for the sale disappears (you will have no house and the current owner has to keep the house forever). Suppose that you and the current owner discount the future according to the discount factor \(\delta\) per day. The real estate agent has allowed you to decide whether you will make the first offer. (a) Suppose that \(\delta\) is small; in particular \(\delta<1 / 2\). Should you make the first offer or let the current owner make the first offer? Why? (b) Suppose that \(\delta\) is close to 1 ; in particular \(\delta>\sqrt[9]{1 / 2}\) (which means that \(\delta^{9}>1 / 2\) ). Should you make the first offer or let the current owner make the first offer? Why?

Short Answer

Expert verified
(a) You should make the first offer as \\( \\\delta < 1/2 \\\). (b) Make the first offer as \\( \\\delta > \\sqrt[9]{1/2} \\\).

Step by step solution

01

Understanding the Scenario

In this bargaining scenario, the house is valued at $200,000 to you and $100,000 to the owner. An agreement between $100,000 and $200,000 benefits both parties. The goal is to negotiate within 10 days using alternating offers.
02

Considering \\( \\\delta < \\frac{1}{2} \\\)

When the discount factor \( \delta \) is small, such as \( \delta < 1/2 \), future offers are significantly less valued. Therefore, making the first offer is advantageous because immediate offers are more valuable than future ones.
03

Considering \\( \\\delta > \\sqrt[9]{1/2} \\\)

When \( \delta \) is close to 1, like \( \delta > \sqrt[9]{1/2} \), future profits are almost as valuable as immediate ones. Making the first offer allows you to dictate the negotiation baseline, thus it is beneficial to make the first offer to gain a strategic advantage.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Game Theory in House Bargaining
In this scenario, the concept of game theory offers a framework to understand strategic decision-making in negotiations. Game theory analyzes the choices of different players in a situation where their outcomes depend on the actions of others. When buying a house, both you and the seller are "players" in this "game."
The key aspect of game theory here is the alternating offers mechanism. This structured exchanging of proposals is crucial because you and the current owner each take turns presenting offers. Each side has to think a few steps ahead, anticipating the other's potential responses.
Understanding how game theory applies will help you decide when to make the first move and how to craft your offer strategically. Making strategic choices based on predicted responses determines how advantageous the negotiation terms could be. Consider your priorities but always aim for mutual gain, as a successful deal must benefit both players.
Understanding the Discount Factor
The discount factor, represented by \( \delta \), is a central concept in bargaining scenarios, indicating how much the parties value future earnings compared to immediate ones.
A discount factor less than 0.5 means that any delay significantly decreases the potential value of receiving a future payment. Both you and the house’s seller place much higher value on immediate transactions when \( \delta < 0.5 \). Therefore, making the first offer can be beneficial to set the negotiation in motion early, capitalizing on the immediate value.
Conversely, when \( \delta \) is high, close to 1, it indicates that delaying the agreement does not significantly reduce its value. This nearly equal weight of future and immediate gains allows the first offer to set a strategically favorable baseline. Understanding \( \delta \) helps you tailor your negotiation timing and offer strategy for the best possible outcome.
Effective Negotiation Strategies
Negotiation strategies are essential in the bargaining process, as they guide how offers are structured and presented. Given the bargaining dynamics, here are some strategies you might consider:
  • **Make the First Offer**: By making the first offer, you set the anchor point. This is particularly useful when \( \delta < 0.5 \) or \( \delta > \sqrt[9]{1/2} \), where it gives you a strategic edge.
  • **Utilize Concession Patterns**: Alternating concessions can build momentum towards a deal. If \( \delta \) suggests stable value in future gains, be prepared to make calculated concessions slowly.
  • **Understand Opponent's Valuations**: Always consider the other party's valuation and priorities. This can help in crafting offers that appeal to mutual benefits.
  • **Time Your Offers**: Accelerate or delay offers based on how much \( \delta \) values future deals. For small \( \delta \), quick exchanges are key; for larger \( \delta \), you might elongate negotiations for a better deal.
By implementing these strategies, you can enhance the effectiveness of your negotiation process, paving the way for favorable outcomes for both parties involved.

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Most popular questions from this chapter

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

[Nash's demand game \({ }^{6}\) ] Compute and describe the Nash equilibria of the following static bargaining game. Simultaneously and independently, players 1 and 2 make demands \(m_{1}\) and \(m_{2}\). These numbers are required to be between 0 and 1 . If \(m_{1}+m_{2} \leq 1\) (compatible demands, given that the surplus to be divided equals 1), then player 1 obtains the payoff \(m_{1}\) and player 2 obtains \(m_{2}\). In contrast, if \(m_{1}+m_{2}>1\) (incompatible demands), then both players get 0 . In addition to describing the set of equilibria, offer an interpretation in terms of bargaining weights as in the standard bargaining solution.

Consider a three-player bargaining game, where the players are negotiating over a surplus of one unit of utility. The game begins with player 1 proposing a three-way split of the surplus. Then player 2 must decide whether to accept the proposal or to substitute for player l's proposal his own alternative proposal. Finally, player 3 must decide whether to accept or reject the current proposal (whether it is player 1's or player 2's). If he accepts, then the players obtain the specified shares of the surplus. If player 3 rejects, then the players each get 0 . Draw the extensive form of this perfect- information game and determine the subgame perfect equilibria.

In experimental tests of the ultimatum bargaining game, subjects who propose the split rarely offer a tiny share of the surplus to the other party. Furthermore, sometimes subjects reject positive offers. These findings seem to contradict our standard analysis of the ultimatum game. Many scholars conclude that the payoffs specified in the basic model do not represent the actual preferences of the people who participate in the experiments. In reality, people care about more than their own monetary rewards. For example, people also act on feelings of spite and the ideal of fairness. Suppose that in the ultimatum game, the responder's payoff is given by \(y+a(y-z)\), where \(y\) is the responder's monetary reward, \(z\) is the offerer's monetary take, and \(a\) is a positive constant. That is, the responder cares about how much money he gets and he cares about relative monetary amounts (the difference between the money he gets and the money the other player gets). Assume that the offerer's payoff is as in the basic model. (a) Represent this game in the extensive form, writing the payoffs in terms of \(m\), the monetary offer of the proposer, and the parameter \(a\). (b) Find and report the subgame perfect equilibrium. Note how equilibrium behavior depends on \(a\). (c) What is the equilibrium monetary split as \(a\) becomes large? Explain why this is the case.

Consider a bargaining game among three political parties-the Conservative Party, the Liberal Democratic Party, and the Labor Party (each party is a player). There are \(T\) periods of bargaining, numbered \(t=1, \ldots, T\). In each period, one party is the "proposer," one is the "responder," and the third is the "bystander." The proposer makes an offer to the responder, who may then accept or reject. If the responder accepts, then the game ends and the offer is implemented. Otherwise, the game continues to the following period unless the period is \(T\), in which case the game ends. If the game ends without an offer being accepted, all parties get zero. Suppose that the Liberal Democratic Party is always the responder. In odd- numbered periods the Conservative Party is the proposer, and in evennumbered periods the Labor Party is the proposer. The total amount of value to be divided among the parties is normalized to 1 . A proposal specifies a vector \((x, y)\), where \(x\) is the amount that the proposer will receive, and \(y\) is an amount that the responder will receive. (The bystander is left out.) The players discount the future according to the common discount factor \(\delta\). (a) Suppose that \(T=1\). Prove that in the subgame perfect equilibrium, the Conservative Party offers \(x=1\) and \(y=0\), and the Liberal Democratic Party accepts any offer with \(y \geq 0\). (b) Suppose that \(T\) is odd. Determine the unique subgame perfect equilibrium and describe what offer, if any, is accepted and in which period. (c) Suppose that \(T=\infty\). Describe a subgame perfect equilibrium in which an offer is accepted in the first period. What is the first-period offer?
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