A Rubinstein bargaining model refers to a class of bargaining games that feature alternating offers through an infinite time horizon. The original proof is due to Ariel Rubinstein

Ariel Rubinstein

Ariel Rubinstein is an Israeli economist who works in game theory. He was educated at the Hebrew University of Jerusalem, 1972–1979, in both mathematics and economics...

 in a 1982 paper. For a long time, the solution to this type of game was a mystery; thus, Rubinstein's solution is one of the most influential findings in game theory

Game theory

Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

.

Requirements

A standard Rubinstein bargaining model has the following elements:

• Two players
• Complete information
• Unlimited offers—the game keeps going until one player accepts an offer
• Alternating offers—the first player makes an offer, then the second player makes a counteroffer if she rejects, then the first player makes another counteroffer if he rejects, and so forth
• Delays are costly

Solution

Although there are many different methods to solving the game, one of the simpler ones involves extending the ultimatum game

Ultimatum game

The ultimatum game is a game often played in economic experiments in which two players interact to decide how to divide a sum of money that is given to them. The first player proposes how to divide the sum between the two players, and the second player can either accept or reject this proposal. ...

 to increasingly many counteroffers. The subgame perfect equilibrium

Subgame perfect equilibrium

In game theory, a subgame perfect equilibrium is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game...

 offer for any number of periods begins to form a geometric series. Thus, we can extend the solution to infinitely many periods by solving for a geometric series. If d is a common discount factor, then the first player earns 1/(1+d) and the second player earns d/(1+d) in equilibrium.

Desirability

Rubinstein bargaining has become pervasive in the literature because it has many desirable qualities:

• It has all the aforementioned requirements, which are thought to accurately simulate real-world bargaining.
• There's a unique solution.
• That solution is pretty clean, which wasn't necessarily expected given the game is infinite.
• There's no delay in the transaction.
• As both players become infinitely patient or can make counteroffers increasingly quickly (i.e. as d approaches 1), then both sides get half of the pie.