January 28, 2001. March 3, 2002. October 30, 2003. Eric Rasmusen notes, erasmuse@indiana.edu Http://Php.Indiana.edu/~erasmuse 54. Ariel Rubinstein, ``Perfect Equilibrium in a Bargaining Model, '' Econometrica, 50: 97-109 (January 1982). John Sutton has an easier proof of the main theorem in John Sutton, "Non-Cooperative Bargaining Theory: An Introduction," The Review of Economic Studies, Vol. 53, No. 5. (October 1986), pp. 709-724, and, originally I think (but I haven't checked), in Avner Shaked and John Sutton, "Involuntary Unemployment as a Perfect Equilibrium in a Bargaining Model," Econometrica, Vol. 52, No. 6. (November 1984), pp. 1351-1364. I think I used that proof in Games and Information, but I don't remember for sure. The proof in Rubinstein's paper is difficult because Rubinstein is making it general-- for any bargaining costs satisfying A1 to A5. But in the end what is most commonly used in models are the two special cases I and II, and for those, simpler proofs work (but see my other note, at the end of this file). The Sutton (1986) and, I think, Shaked and Sutton (1984) discuss exit options-- the possibility that a player can exit the game when it is his turn to make an offer and take an alternative payoff instead. This has a peculiarly different effect than the conventional threat point. p. 299. N is 1, 2,3, etc., representing the period at which the bargaining ends. A player prefers an earlier ending ordinarily. A1. A bigger share of the pie is better. A2. Agreement sooner is better than agreement later, if the player gets a positive share of the pie. A3. If r in period t_1 is preferred by i to s a period later, then the same is true in period t_2, for any two periods t_1 and t_2. (t_1