Admati and Perry (1987) derive the equilibrium in a bargaining game between a seller and buyer when the buyer's valuation is private information. They show that, for some parameter values, trade occurs at the Rubinstein (1982) prices given the buyer's true valuation (pl if the buyer has a low valuation and ph if the buyer's valuation is high), but the low valuation buyer delays trade to signal her type. Trade with the high valuation buyer occurs immediately. In this model, there are no outside options. It is easy to see, however, that giving the seller an outside option can disrupt this equilibrium. This is even the case if this outside option is worth less than the Rubinstein price for the low valuation buyer. Say in the Admati and Perry equilibrium, trade with the low valuation buyer occurs after a delay which reduces the value of trade to d<1 of what it would have been had trade occurred immediately. Now, say the seller can sell the good on the spot market at a price of p>dpl. Then, if the buyer does not accept the seller's offer of ph in the first round, the seller is better off selling the good in the spot market than waiting for the low valuation buyer to make her counter-offer. Thus, the Admati and Perry equilibrium cannot be an equilibrium when the seller has an outside option. In this paper, I will derive the equilibrium for this bargaining game with asymmetric information and outside options. It turns out that giving the seller an outside option can reduce his expected equilibrium payoffs. The reason is that the seller cannot get as high a price from a high valuation buyer because trade with the low valuation buyer must happen sooner. The outside option reduces the credible threat to delay trade if a high valuation buyer mimics a low valuation buyer. One might think that the seller can use the outside option to get a higher price from the low valuation buyer, thereby making mimicking a low valuation buyer less attractive for any given delay. That is not possible, however, since when trade with the low valuation buyer actually occurs, the outside option is no longer binding. The seller also cannot get a high valuation buyer to pay ph by using the threat of actually taking the outside option. Such a threat is not credible if the buyer can make one last offer as the seller is leaving since that offer will always exceed the outside option payoff. The fact that having an outside option can actually make a seller worse off means that seller's of specialized products might have an incentive to engineer these products to make them less valuable on spot markets, even if doing so comes with a cost. Introducing outside options also changes the impact of asymmetric information. Without outside options, the asymmetric information solution approaches the complete information solution as the probability of low valuation goes to zero. With outside options, however, the expected revenue of the seller is discontinuous (it does not approach the complete information revenue as the probability of the buyer having a low valuation goes to zero).
