Back to Bargaining Basics: A Simple 50-50 Model for Splitting a Pie

Abstract

Nash (1950) and Rubinstein (1982) give two different justifications for a 50-50 split of surplus to be the outcome of bargaining with two players. Nash's axioms extend to n players, but the search for a satisfactory n-player non-cooperative game theory model of bargaining has been fruitless. I offer a simple static model that reaches a 50-50 split (or 1/n) as the unique equilibrium. Each player chooses a level simultaneously, but greater toughness always generates a risk of breakdown. Introducing asymmetry, a player who is more risk averse gets a smaller share in equilibrium. Bargaining strength can also be parameterized to yield an asymmetric split. The model can be expanded to resemble Rubinstein (1982) by making breakdown mere delay, but with an exact 50-50 split if the player's discount rates are equal. The model only needs minimal assumptions on breakdown probability and pie division as functions of toughness and has a clear intuition: whoever has a bigger share loses more from breakdown and hence has less incentive to be tough.