Rubinstein Bargaining

Introduction

In game theory, the Rubinstein bargaining model is a solution to the problem of finding an optimal agreement between two parties who have conflicting interests and asymmetric information. For example lets just say two parties Bob and Alice, engage in a series of alternating offers and counteroffers on a resource that is valuable to both, until they reach an agreement or until they reach a predetermined deadline. How will they behave or what are the necessary steps must be taken by each player? What are the possibile results in the end of the bargaining? In this chapter you will find further details and deepdives about Rubinstein Bargaining concept and solution.

Problem Definition

In the Rubinstein bargaining model, two parties usually referred to as "players," are trying to reach an agreement on the division of a pie, where the pie represents a set of resources that are valuable to both parties. The players have conflicting interests and asymmetric information, meaning they have different preferences over how the pie should be divided, and they need complete information about the other player's choices. The Rubinstein model is a two-stage game.

In the first stage, each player offers the other player how the resource should be split. In the second stage, the other player can accept the offer, reject it, or make a counteroffer. The game continues with the players making alternating offers and counteroffers until they reach an agreement or until they reach a predetermined deadline. The players are assumed to be rational and have complete information about their preferences but not about the importance of the other player. The goal of each player is to maximize their utility, which is the measure of their satisfaction or happiness with the outcome of the negotiation. The Rubinstein model seeks an equilibrium, a stable agreement that either player cannot improve upon.

Rubinstein Bargaining game has 3 definitive rules that must be followed through as each stage performed.

• The negotiation begins with an initial offer from one of the parties.
• Initial offer must receive a response in form of either accept, reject, or a counteroffer.
• Bargaing must end with either an agreement is reached or a predetermined timeout deadline is set.

Fact: The model does not specify a fixed number of stages or a fixed deadline, and the actual number of stages and the length of the negotiation will depend on the specific circumstances of the negotiation.

Parameters

The key parameters in the Rubinstein model are the time discount factor and the reservation value.

Set of Ordered Pairs: The set of ordered pairs is denoted by (s, t), where (s, t) is the discrete representation of time and (s, t) represents a slice of a pie with size 1. Hence, t >  0. Therefore, the pair says, "Player 1 receives s and Player 2 receives 1 - s at time t. The following prerequisites should be met by each player's preferences on (s, t):

More Pie or Resource: The best pie is more pie. According to math, if x > y, then (x, t) > (y, t).

Time is Money: This indicates that if x > 0 and t2 > t1, then (x, t1) > (x, t2).

Continuity: Thus, there are no sudden changes in people's tastes. In terms of mathematics, a preference relation is continuous, In other words, points very near to A will also be preferred to B if we prefer a point A along a preference curve to a point B.

Stationary: This means that the preference of (x, t) over (y, t + 1) is independent of t.

Time Discount Factor: The time discount factor represents the degree to which the parties value a settlement reached sooner rather than later. A high-time discount factor means that the parties place a high value on getting an agreement quickly, while a low-time discount factor means they are willing to wait for a more favorable settlement. The time discount factor is often expressed as a decimal between 0 and 1, with higher values indicating a greater preference for settlements reached sooner rather than later. For example, a time discount factor of 0.9 means that the parties place a high value on reaching an agreement quickly, while a time discount factor of 0.1 means that they are willing to wait for a more favorable settlement. So, if (x, t) is equivalent to (y, t + 1) then y needs to be bigger than x to continue one more period with the bargaining and being immaterial to him.

Reservation Value: The reservation value is the minimum amount of resources that each party is willing to accept in the settlement. If either party's reservation value is not met, they will not agree to the settlement and the negotiation will break down. The reservation value can be thought of as a "fallback" position for each party. If the negotiation breaks down and an agreement is not reached, each party will receive their reservation value rather than nothing. For this reason, the reservation value is often referred to as the "walkaway" value or the "outside option.

Nash Equilibrium Condition

Nash Equilibrium is a solution concept in game theory, which describes a state in which all players in a game are making the best decision they can given the decisions of the other players. The relation between Rubinstein bargaining and Nash equilibrium is that a Nash equilibrium can be reached through the process of Rubinstein bargaining. In other words, if the agents in a negotiation are rational and have complete information, they will eventually reach a Nash equilibrium through the process of making offers and counter-offers .

Benefits of Rubinstein Bargaining

The Rubinstein bargaining model has several benefits and positives:

1. It provides a framework for understanding how rational agents will behave in a negotiation and can be utilized to forecast the results of a negotiation.

2. Allows for calculating the Nash Bargaining Solution, a unique and efficient solution for both parties involved in the negotiation.

3. Helps to identify the optimal outcome for both parties involved in the negotiation, which can lead to mutually beneficial agreements.

4. Applicable to various negotiation scenarios, making it a versatile tool for understanding negotiations.

5. Can be used to identify potential sources of conflict and design negotiation strategies to mitigate or avoid them.

6. It is used to understand the role of different factors, such as the outside options and reservation values, in the negotiation process.

7. Researchers have extensively studied and validated it, providing a solid theoretical foundation for its use in real-world negotiations.

Rubinstein Bargaining and Nash Bargaining Solution

Rubinstein bargaining model can be used to calculate the Nash bargaining solution, it is not the only way to do so, and the relationship between the two models is somewhat complex. The Nash bargaining solution is a theoretical concept that describes the outcome of a negotiation in which both agents are making their best decisions given the decisions of the other agent. It can be derived from the Nash equilibrium, a state in which all players in a game are making the best decision they can, given the findings of the other players. The Rubinstein bargaining model is a more specific negotiation model that considers the agents' time discounting and complete information. The Nash bargaining solution, however, does not make any assumptions about the agents' preferences or information and does not consider the surplus's time discounting.

General Solution

The solution of the Rubinstein bargaining model is determined by the following formulas:

1. The first formula is for the time-discounted value of the surplus, which is used to calculate the value of an agreement as time passes. The formula is: V(t) = S / (1 + d*t) Where:

• V(t) is the time-discounted value of the surplus
• S is the total surplus
• d is the discount factor, represents the agents' time preferences.
• t is the time elapsed since the beginning of the negotiation

2. The second formula is for the disagreement point, which represents the value of the best alternative for each agent if they don't reach an agreement. The formula is: D = (1- d) * S Where:

• D is the disagreement point
• S is the total surplus
• d is the discount factor, represents the agents' time preferences.

3. The third formula is for the Rubinstein bargaining solution, which describes the division of the surplus between the two agents. The formula is: x = D + (S - D) * (b1 / (b1 + b2)) Where:

• x is the share of the surplus that goes to the first agent
• D is the disagreement point
• S is the total surplus
• b1 and b2 are the bargaining power of the agents, where a high value of b indicates that one agent has more bargaining power than the other.

It's important to notice that this model is a simplified version of the bargaining process, in the real world the negotiation process can be more complicated and factors such as the agents' emotions, trust and communication can affect the outcome.

Rubinstein Bargaining Game Examples

A quick and simple example about Rubinstein Barganing game:

Game Example 1:

A landlord and the tenant bargains over the price of rent. The landlord and the tenant have a common objective of reaching a rental agreement that is mutually beneficial. Both sides have complete information about the issue at hand and their own preferences, and are rational and will choose strategies that maximize their expected payoffs.

The tenant's outside option is to rent a similar property at a different location, while the landlord's outside option is to keep the property vacant. The potential surplus in this negotiation is the difference between the landlord's profit with a tenant and without a tenant.

The payoffs for each party can be represented by the following formulas:

Tenant: uT(r) = r - r0

Landlord: uL(r) = r - r0 - c

Where r is the agreed rent, r0 is the tenant's reservation price, and c is the landlord's cost of keeping the property vacant.

The Nash Bargaining Solution can be calculated using the following formula:

r = (r0 + c)/2 + (r0 - c)/2 = (r0 + c)

This means that the tenant and the landlord will agree on a rent that is halfway between the tenant's reservation price and the landlord's cost of keeping the property vacant.

For example, if the tenant's reservation price is $1000 and the landlord's cost of keeping the property vacant is $800, the Nash Bargaining Solution is:

r = ($1000 + $800)/2 = $900

This means that the tenant and the landlord will agree on a rent of $900, and both parties will be better off than their outside options. The tenant will be paying less than their reservation price, while the landlord will be earning more than their cost of keeping the property vacant.

Game Example 2: This example is about the negotiation process between a buyer and a seller over the price of a car with using Rubinstein Bargaining Solution. The buyer and the seller have a common objective of reaching a sale agreement that is mutually beneficial. Both sides have complete information about the issue at hand and their own preferences, and are rational and will choose strategies that maximize their expected payoffs.

The buyer's outside option is to purchase a similar car from a different seller, while the seller's outside option is to keep the car unsold. The potential surplus in this negotiation is the difference between the seller's profit from selling the car and keeping it unsold.

The payoffs for each party can be represented by the following formulas:

Buyer: uB(p) = p - p0

Seller: uS(p) = p - c - p0

Where p is the agreed price, p0 is the buyer's reservation price, and c is the seller's cost of keeping the car unsold.

The Nash Bargaining Solution can be calculated using the following formula:

p = (p0 + c)/2 + (p0 - c)/2 = (p0 + c)

This means that the buyer and the seller will agree on a price that is halfway between the buyer's reservation price and the seller's cost of keeping the car unsold.

For example, if the buyer's reservation price is $15,000 and the seller's cost of keeping the car unsold is $12,000, the Nash Bargaining Solution is:

p = ($15,000 + $12,000)/2 = $13,500

This means that the buyer and the seller will agree on a price of $13,500, and both parties will be better off than their outside options. The buyer will be paying less than their reservation price, while the seller will be earning more than their cost of keeping the car unsold.

Relevency of Rubinstein Bargaining

Rubinstein's bargaining is utilized in various fields, including economics, political science, and management. In economics, the model is used to study the market negotiation process and understand the role of information in shaping market outcomes. In political science, the model is used to study the negotiation process in international relations and understand power's role in shaping outcomes. In management, the model is used to study the negotiation process in organizations and to know how different types of administration affect the outcome of negotiations. In economics, the Rubinstein bargaining theory helps to understand how prices are shaped in markets and how changes in supply and demand affect the price. In international relations, the approach is used to know how countries negotiate treaties and agreements. In practice, Rubinstein bargaining is also used to study the negotiation process in different sectors, such as labor negotiation, mergers and acquisitions, and international trade.