Optimal value commitment in bilateral bargaining

Two impatient players bargain about the division of a pie under a standard bargaining protocol in discrete time with time-invariant recognition probabilities. Instantaneous utility is linear, but players discount the future by a constant factor. Before bargaining starts, a player can commit to a utility level. This commitment is perfectly binding initially. However, once so much time has passed that even receiving the entire pie would yield less than the committed level of utility, then the commitment becomes void. Intuitively, this simply means that no player can remain committed to something which has become impossible. If only one player can commit, his subgame–perfect equilibrium payoff varies between one half and the entire pie, depending on the distribution of proposal power. If both players commit sequentially before the bargaining starts, we find a unique perfect equilibrium division. If both players commit simultaneously, there is a range of perfect equilibrium divisions. However, no player obtains less than one third of the pie, even with arbitrarily small proposal power. The equal split is the only division supported by a perfect equilibrium for any choice of the discount factor and the recognition probabilities.