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 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.

Requirements

A standard Rubinstein bargaining model has the following elements:

Solution

Consider the typical Rubinstein bargaining game in which two players decide how to divide a pie of size 1. An offer by a player takes the form x = (x1, x2) with x1 + x2 = 1 and. Assume the players discount at the geometric rate of d, which can be interpreted as cost of delay or "pie spoiling". That is, 1 step later, the pie is worth d times what it was, for some d with 0<d<1. Any x can be a Nash equilibrium outcome of this game, resulting from the following strategy profile: Player 1 always proposes x = (x1, x2) and only accepts offers x' where x1' ≥ x1. Player 2 always proposes x = (x1, x2) and only accepts offers x' where x2' ≥ x2. In the above Nash equilibrium, player 2's threat to reject any offer less than x2 is not credible. In the subgame where player 1 did offer x2' where x2 > x2' > d x2, clearly player 2's best response is to accept. To derive a sufficient condition for subgame perfect equilibrium, let x = (x1, x2) and y = (y1, y2) be two divisions of the pie with the following property: i.e. Consider the strategy profile where player 1 offers x and accepts no less than y1, and player 2 offers y and accepts no less than x2. Player 2 is now indifferent between accepting and rejecting, therefore the threat to reject lesser offers is now credible. Same applies to a subgame in which it is player 1's turn to decide whether to accept or reject. In this subgame perfect equilibrium, player 1 gets 1/(1+d) while player 2 gets d/(1+d). This subgame perfect equilibrium is essentially unique.

A Generalization

When the discount factor is different for the two players, d_1 for the first one and d_2 for the second, let us denote the value for the first player as v(d_1, d_2). Then a reasoning similar to the above gives yielding. This expression reduces to the original one for.

Desirability

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