Fully absorbing dynamic compromise

I consider a repeated divide-the-dollar voting model with rejections leading to the implementation of the previous period's allocation (see Kalandrakis [14]). I show that if proposals can be non-exhaustive, then equal division can be achieved as an absorbing steady state from any initial allocation given a large enough discount factor as a part of a Markov Perfect equilibrium. This result is robust to changes in voting thresholds and persistence in proposal power outside of unanimity or total persistence.