Bargaining with Unequal Patience
Players 1 and 2 bargain over one dollar. In round 1 player 1 proposes a split $(x,1-x)$ and player 2 accepts or rejects. If player 2 rejects, then in round 2 player 2 proposes a split and player 1 accepts or rejects, and so on without any deadline, alternating forever. An agreement reached in round $t$ gives player $i$ a payoff of $\delta_i^{\,t-1}$ times the money received, with $\delta_1 = 4/5$ and $\delta_2 = 3/5$. If no agreement is ever reached, both get 0. A player who is indifferent between accepting and rejecting accepts. In the unique subgame-perfect equilibrium, agreement is reached immediately. What share of the dollar does player 1 receive? Give a reduced fraction.
Related problems
Source: Rubinstein (1982), 'Perfect equilibrium in a bargaining model', Econometrica, with different discount factors for the two players. Statement written for AxiomIQ.