Optimal Value Commitment in Bilateral Bargaining
We propose a new model to study the role of commitment as a sourceof strategic bargaining power. Two impatient players bargain aboutthe division of a pie under a standard bargaining protocol indiscrete time with time-invariant recognition probabilities.Instantaneous utility is linear, but players discount the future bya constant factor. Before bargaining starts, a player can commit notto enter into any agreement which gives him less than some utilitylevel. This commitment is perfectly binding initially. However, onceso much time has passed that even receiving the entire pie wouldyield less than the committed level of utility, then the commitmentbecomes void. Intuitively, this simply means that no player canremain committed to something which has become impossible. We use aslight refinement of subgame-perfect equilibrium as a solutionconcept. If only one player can commit, then we find an immediateand efficient agreement on a division which gives the committedplayer (strictly) between one half and the entire pie, the exactallocation being determined uniquely by the recognitionprobabilities. If both players can commit sequentially before thebargaining starts, we find a unique equilibrium division with afirst--mover advantage. Finally, we consider a version of the gamewhere both players commit simultaneously before the bargainingstarts. In this case, there is a range of equilibrium divisions.However, in the limit as the discount factor goes to one, no playerobtains less than one third of the pie, even with arbitrarily smallproposal power. Somewhat surprisingly, the equal split emerges asthe only division supported by an equilibrium for any choice of thediscount factor and the recognition probabilities.
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