Question

Estelle has an antique desk that she does not need, whereas Joel and his wife have a new house with no furniture. Estelle and Joel would like to arrange a trade, whereby Joel would get the desk at a price. In addition, the desk could use restoration work, which would enhance its value to Joel. Specifically, the desk is worth 0 to Estelle (its current owner), regardless of whether it is restored. An unrestored desk is worth \(\$ 100\) to Joel, whereas a restored desk is worth \(\$ 900\). Neither Joel nor Estelle has the skills to perform the restoration. Jerry, a professional actor and woodworker, can perform the restoration at a personal cost of \(\$ 500\). Jerry does not need a desk, so his value of owning the restored or unrestored desk is 0 . (a) Suppose Estelle, Jerry, and Joel can meet to negotiate a spot contract specifying transfer of the desk, restoration, and transfer of money. Model this as a three-player, joint-decision problem, and draw the appropriate extensive form. Calculate the outcome by using the standard bargaining solution, under the assumption that the players have equal bargaining weights \(\left(\pi_{\mathrm{E}}=\pi_{\text {Jerry }}=\pi_{\text {Joel }}=1 / 3\right)\). Does the desk get traded? Is the desk restored? Is this the efficient outcome? (b) Suppose spot contracting as in part (a) is not possible. Instead, the players interact in the following way. On Monday, Estelle and Jerry jointly decide whether to have Jerry restore the desk (and at what price to Estelle). If they choose to restore the desk, Jerry performs the work immediately. Then on Wednesday, regardless of what happened on Monday, Estelle and Joel jointly decide whether to trade the desk for money. Model this game by drawing the extensive form. (Hint: The extensive form only has joint-decision nodes.) Assume the parties have equal bargaining weights at all joint-decision nodes. Determine the negotiation equilibrium. Compare the outcome with that of part (a). (c) Now suppose the players interact in a different order. On Monday, Estelle and Joel jointly decide whether to trade the desk for money. Trade takes place immediately. On Wednesday, if Joel owns the desk, then he and Jerry jointly decide whether to have Jerry restore the desk (and at what price to Joel). If they choose to restore the desk, Jerry performs the work immediately. Model this game by drawing the extensive form. (Hint: Again, the extensive form only has joint-decision nodes.) Assume the parties have equal bargaining weights at all jointdecision nodes. Determine the negotiation equilibrium. Compare the outcome with that of parts (a) and (b). (d) Explain the nature of the hold-up problem in this example.

Short answer

Yes, the desk is traded and restored. It's the efficient outcome as all parties benefit equally from bargaining.

Step-by-step solution

Understanding the Problem

Estelle, Joel, and Jerry are trying to negotiate a trade and restoration deal for an antique desk. Estelle values the desk at $0, Joel values it at $100 if unrestored and $900 if restored, and Jerry can restore it for $500 but doesn't need the desk himself.

Calculating Payoffs for Part (a)

The expected outcome for the desk being restored and owned by Joel is $400 (Joel's value of restored desk, $900 minus restoration cost, $500). Since Estelle doesn't value the desk, the negotiation with equal bargaining weights ($1/3$ each) aims to share this surplus efficiently across all parties.

Outcome Determination for Part (a)

The desk is transferred to Joel and restoration work is contracted with Jerry. Each participant divides the $400 surplus equally, giving each $133.33. Therefore, Joel receives a desk valued at $900, pays $266.67 ($gives up $133.33$ from surplus), making Jerry's service profitable at $500. This is efficient as all parties benefit equally.

Modeling Part (b)

In this sequential game, Estelle and Jerry decide on restoration first. The desk's potential value increase from $100 to $900 incentivizes them to agree on restoration, sharing the costs and benefits equally. on Wednesday, Estelle and Joel negotiate the trade. Both parties equally share the surplus from transferring a restored desk.

Equilibrium for Part (b)

Restoration takes place first as it adds value. Joel and Estelle agree on the trade with both Estelle and Jerry benefiting equally from the $400 gain. Joel's final win is owning a restored desk valued at $900. Resulting surplus remains $400 split equally, similar to part (a)'s outcome.

Modeling Part (c)

Here, Estelle and Joel first decide on the trade on Monday. Joel takes ownership of the unrestored desk, valued at $100. On Wednesday, Joel and Jerry decide on restoration. Given the $400 surplus for restoration and equal bargaining weights, Joel agrees to restoration by Jerry, sharing the surplus equally ($133.33 each).

Equilibrium for Part (c)

Restoration depends on Monday's trade decision. As Joel values a restored desk at $900 and views restoration as profitable, shares renovation gains with Jerry. Outcome is similar to (a) and (b), indicating consistent gains as restoration enhances overall surplus.

Explaining the Hold-up Problem (Part d)

The hold-up problem arises because investments must be made (restoration) before benefits are shared. This can lead to tension over the distribution of these gains, especially if contract terms fail (e.g., no restoration, desk ownership not honored). Ensuring fair distribution prevents renegotiation and disputes.

Key concepts

Negotiation Strategy

In situations where multiple parties are interested in securing a mutually beneficial agreement, negotiation strategy plays a critical role. Negotiation involves the process of making decisions and reaching an agreement that satisfies the interests of all involved parties. In the problem provided, Estelle, Joel, and Jerry each hold different interests and negotiating positions regarding the antique desk.

Developing a strong negotiation strategy requires understanding the following elements:

• Identifying key interests of each party: Estelle values cash, Joel values the desk, and Jerry values compensation for his restoration work.
• Assessing each party's bargaining power: This involves understanding what each party brings to the table (the desk, money, restoration skills).
• Formulating a plan for distributing benefits: In this scenario, using equal bargaining weight can help ensure a fair and balanced distribution of any created surplus.

A well-formulated strategy helps in avoiding conflicts and creates pathways for efficient trade-offs. Successfully navigating these aspects allows for a win-win agreement where everyone's primary interests are addressed.

Bargaining Solution

A bargaining solution in game theory is an outcome that results from all involved parties reaching an agreement that is optimal for them. In this scenario, the bargaining solution involves determining how the benefits of restoring the desk and its subsequent trade should be distributed among Estelle, Joel, and Jerry.

The concept of bargaining solutions often involves:

• Utility maximization: Each party aims to achieve the highest possible benefit from the agreement.
• Fair allocation: Here, with equal bargaining weights, the benefits such as the surplus (\(\(400\) after costs) should be equally divided among them.
• Efficiency: The bargaining solution ensures that the desk is both traded to Joel and restored by Jerry, maximizing its value from \(\)100\) to \($900\).

In this scenario, a fair bargaining solution is achieved by equally splitting the generated surplus from the restoration and trade, ensuring all parties receive a portion of the gain.

Extensive Form

The extensive form in game theory is a representation of games that shows the sequence of actions and decisions made by players. It provides a clear structure and plan of action, often visualized as a decision tree. In the provided problem, the extensive form is key to understanding the sequential nature of decisions made by Estelle, Joel, and Jerry.

The extensive form emphasizes:

• The order of moves: Which party makes a decision first and the potential outcomes of those decisions.
• Decision nodes: Key points where players make crucial choices, such as Estelle and Jerry deciding on restoration or Joel and Estelle negotiating trade.
• Payoffs at the end: Outlining what each player gains or loses depending on the choices made during the game.

By modeling the problem in an extensive form, it becomes clear how the interactions evolve and what strategies lead to efficient outcomes. This helps players anticipate the actions and reactions of others, leading to better decision-making.

Hold-up Problem

The hold-up problem is a situation in which a party is hesitant to invest because they fear not being able to fully capture the benefits or because of potential renegotiations after the investment is made. In the context of the desk restoration exercise, the hold-up problem can arise at several points.

Causes of the hold-up problem include:

• Investment risks: Jerry might be reluctant to invest time and money into restoration if there is no guaranteed benefit or if future terms change unfavorably.
• Lack of enforceable contracts: Without a well-defined agreement, Estelle or Joel might attempt to renegotiate terms after Jerry has finished the restoration.

To mitigate the hold-up problem, it is crucial for parties to:

• Establish clear upfront agreements: This helps in securing Jerry's investment by agreeing on payment before restoration begins.
• Ensure equitable sharing of returns: Sharing the benefits of a restored desk as agreed helps maintain trust and cooperation among the parties.

By addressing potential hold-up problems, the parties involved prevent future disputes, ensuring all participants' interests are protected.