How am I to set a price when buyers arrive over time and I have a good that will expire, such as a baseball ticket or an airplane seat? “Yield management” pricing is widespread in industries like these, but the standard methods tend to involve nonstrategic agents. But a lack of myopia can sometimes be very profitable. Consider a home sale. Buyers arrive slowly, and the seller doesn’t know the distribution of potential buyer values. It’s possible that if I report a high value when I arrive first, the seller will Bayesian update about the future and will not sell me the house, since they believe that other buyers also value the house highly. If I report a low value, however, I may get the house.
Consider the following numerical example from Gershkov and Moldovanu. There are two agents, one arriving now and one arriving tomorrow. The seller doesn’t know whether the agent values are IID in [0,1] or IID in [1,2], but puts 50 percent weight on each possibility. With complete information, the dynamically efficient thing to do would be to sell to the first agent if she reports a value in [.5,1]U[1.5,2]. With incomplete information, however, there is no transfer than can simultaneously get the first agent to tell the truth when her value is in [.5,1] and tell the truth when her value is in [1,1.5]. By the revelation principle, then, there can be no dynamically efficient pricing mechanism.
Consider a more general problem, with N goods with qualities q1,q2..qN, and one buyer arriving each period. The buyer has a value x(i) drawn from a distribution F, and he gets utility x(i)*q(j) if he receives good j. Incomplete information by itself turns out not to be a major problem, as long as the seller knows the distribution: just find the optimal history-dependent cutoffs using a well-known result from Operations Research, then choose VCG style payments to ensure each agent reports truthfully. If the distribution from which buyer values is unknown, as in the example above, then seller’s learn about what the optimal cutoffs should be from the buyer’s reports. Unsurprisingly, we will need something like the following: since cutoffs depend on my report, implementation depends on the maximal amount the cutoff can change having a derivative less than one in my type. If the derivative is less than one, then the multiplicative nature of buyer utilities means that there will be no incentive to lie about your valuation in order to alter the seller’s beliefs about the buyer value distribution.
http://www.econ2.uni-bonn.de/moldovanu/pdf/learning-about-the-future-and-dynamic-efficiency.pdf (IDEAS version). Final version published in the September 2009 AER. I previously wrote about a followup by the same authors for the case where the seller does not observe the arrival time of potential buyers, in addition to not knowing the buyer’s values.
A Fine Theorem 