Backward induction and unacceptable offers

How to establish the existence of subgame perfect equilibrium (SPE) in bargaining models if no stationary SPEs (SSPEs) exist? The backward-induction technique of Shaked and Sutton (1984, Econometrica) applies to the cyclical structure of SPE payoffs and provides recursive dynamics on the bounds of SPE payoffs. Acceptable and unacceptable offers have to be incorporated for these dynamics to be necessary and sufficient for extreme SPEs. In this paper, we demonstrate how these recursive dynamics are directly applicable to establish the existence of SPE in a model with no SSPE. Also from these dynamics, the extreme SPE strategy profiles can easily be recovered.