Bargaining Foundations for the Outside Option Principle

Abstract

We study a bargaining game in which a seller can trade with one of two buyers, who have values h and l ( ⁠ h > l ⁠ ). The outside option principle (OOP) predicts that, as players become patient, the seller trades with the high-value buyer with probability converging to 1 at a price converging to max ( h / 2 , l ) ⁠ . While this prediction is supported by the Markov perfect equilibrium (MPE), a wide range of trading outcomes may emerge in subgame perfect equilibria (SPEs): in the patient limit, the seller can obtain any price in the interval [ h / 2 , h ] (and no other); moreover, allocative inefficiency and costly delay are possible. We propose equilibrium refinements less restrictive than Markov behavior that guarantee trading outcomes consistent with the OOP. One refinement requires that a buyer’s relative probability of trade does not increase dramatically following a failed negotiation with that buyer. Another refinement posits that the seller does not approach a buyer hoping that negotiations fail. SPEs satisfying both refinements conform with the OOP (but need not be MPEs). Our benchmark model features strategic matching by the seller. We provide a parallel analysis for the random matching protocol. Under random matching, prices in SPEs may also rise above and fall below l , but have a narrower range. A refinement particular to this protocol that restores the OOP requires that a random mismatch should not impact the seller excessively.