# Allocation game optimal strategy

There are two players, Alice and Bob. There is an initial pool of 100 dollars. Alice proposes an allocation of the dollars (real numbers, not necessarily integers), and Bob can either accept or reject. If he accepts, the allocation stands. If he rejects, half of the money is destroyed and Bob proposes an allocation. If Alice rejects, half the money is again destroyed. This continues until one of them accepts. What allocation should Alice propose?

Example play through of the game: 1) Alice proposes (100 - pi, pi) 2) Bob rejects and proposes (e, 50 - e). Since half the money was destroyed, only 50 remains. 3) Alice rejects and proposes (20, 5). Since half the money was destroyed, only 25 remains. 4) Bob rejects and proposes (6.25, 6.25). Alice accepts and each walks away with 6.25.

• This is a classic game theory example and is in no way research-level. It is probably homework. It might be appropriate for math.stackexchange.com Jul 24, 2012 at 3:53
• This is Ariel Rubinstein's bargaining model: arielrubinstein.tau.ac.il/papers/11.pdf Aug 14, 2012 at 0:06

As already mentioned by Michael Greinecker, this is Ariel Rubinstein's bargaining model arielrubinstein.tau.ac.il/papers/11.pdf . More specifically, the variant with a fixed discounting factor. In your case, the discounting factor is $$\frac12$$ for both players and all timesteps. Following Rubinstein, in the single perfect Nash equilibrium of your game, the first player gets $$\frac23$$ of the 100 Dollars (or rather proposes that).