Solution
The Outside Option Threat
Show the problem again
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.
Worked solution
The answer is 2/5. Without the option, the Rubinstein solution gives player 2, as responder, only $\delta/(1+\delta)=9/19\approx 0.474$, which is less than the option value $0.6$. So the option is a credible, binding threat: player 1 must offer player 2 at least $3/5$. Check that $2/5$ is consistent. Suppose player 1 keeps $2/5$ and offers $3/5$; player 2 is indifferent between accepting and taking the option, and accepts. For this to be the best player 1 can do, look at what player 2 gets by rejecting and continuing: in round 2 he proposes, and player 1 as responder will accept any offer of at least $\delta\cdot\frac25=0.36$, so player 2 keeps $0.64$, worth $0.9\cdot0.64=0.576$ in round-1 terms, less than $0.6$. So the option beats continuing, player 2's reservation value is exactly $3/5$, and player 1 gets $1-3/5=2/5$. A backward-induction computation over horizons up to 401 rounds converges to $0.4$. With a smaller option, say $2/5$, the threat would be empty and the answer would be the unconstrained $10/19$.
Source: Alternating-offer bargaining with an outside option, after Shaked and Sutton (1984) and Binmore, Shaked and Sutton (1989). Statement written for AxiomIQ.