The Outside Option Threat

Players 1 and 2 bargain over one dollar with alternating offers: player 1 proposes in round 1, player 2 in round 2, and so on without a deadline. Each unit of delay discounts both players' payoffs by $\delta = 9/10$ (an agreement in round $t$ giving money $x$ is worth $\delta^{t-1}x$ to the player receiving it). Whenever player 2 **rejects** one of player 1's offers, he may, instead of continuing, take an outside option and end the game; taking it after rejecting the round-$t$ offer is worth $\delta^{t-1}\cdot 3/5$ to him (so $3/5$ if taken in round 1), in which case player 1 gets 0. Player 1 has no outside option. A player who is indifferent accepts, and player 2 prefers to continue bargaining when the two courses are equal. In the subgame-perfect equilibrium, what share of the dollar does player 1 receive? Give a reduced fraction.

• induction
• nash equilibrium

Source: Alternating-offer bargaining with an outside option, after Shaked and Sutton (1984) and Binmore, Shaked and Sutton (1989). Statement written for AxiomIQ.