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.

Understanding the Procedure of Barganing

Before discovering further deep concepts, a quick and simple example about Rubinstein Barganing:

Setup for the Game Example:

we have an infinite horizon game between two players call him Dave and Sally and in all odd periods, Dave will make an offer to Sally, which Sally accepts or rejects, and in all even periods. Sally makes an offer to Dave which Dave accepts or rejects, the game continues until one player accepts when that player accepts that's the division and the game ends and as long as the players keep rejecting then we keep moving to the next period, where the other players will make the offer and we'll keep doing that until one of them finally accepts, and if no one ever accepts, then the payoffs are just zero for both players

• Odd Periods, Dave makes an offer
• Even Periods, Sally makes an offer
• Infinite Horizon and discounting factors

Rubinstein bargaining alternatively picks up on is the idea that when we're in this situation at a bargaining discussion and it's never exactly clear who has that last offer, opposed to a game with an arbitrary fixed cutoff to the negotiations. This feature is called Stationary Strategies

Stationary Strategy is a strategy that doesn't change from period to period when those periods are identical so all odd periods here are identical and all odd periods Dave is making an offer to Sally which Sally accepts or rejects and all the even periods are identical as well and there we have Sally making an offer to Dave which Dave accepts or rejects.

• Solution: Rubinstein bargaining can conceivably go on forever, meaning there is no fixed ending period that can start at and work the way back. start off by taking Sally's continuation value for some odd period as a value known as VB. so what is this continuation value well it's the non discounted amount she will receive if no agreement is made in the current period