Optimal Mechanism for Selling Two Goods

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractWe solve for the optimal mechanism for selling two goods when the buyer's demand characteristics are unobservable. In the case of substitutable goods, the seller has an incentive to offer lotteries over goods in order to charge the buyers with large differences in the valuations a higher price for obtaining their desired good with certainty. However, the seller also has a countervailing incentive to make the allocation of the goods among the participating buyers more efficient in order to increase the overall demand. In the case when the buyer can consume both goods, the seller has an incentive to underprovide one of the goods in order to charge the buyers with large valuations a higher price for the bundle of both goods. As in the case of substitutable goods, the seller also has a countervailing incentive to lower the price of the bundle in order to increase the overall demand.

On the optimality of pure bundling for a monopolist

“…12 We note that the optimal pure bundling price P * is larger than v 1 + v 2 , as is shown by Armstrong (1996), even when J m 1 ≥ 0 and J m 2 ≥ 0 hold. In fact, if we let G and g denote the c.d.f.…”

“…First, since v 1 , v 2 are identically distributed, we can focus on determining the optimal q 1 , q 2 , T for (v 1 , v 2 ) such that v 2 ≥ v 1 ; symmetric results are obtained for v 2 < v 1 . Second, Pavlov (2011) shows that under the condition…”

“…Precisely, suppose that f 1 is increasing, and that J m 1 ≥ 0, which is equivalent to v 1 f 1 (v 1 ) ≥ 1. Then consider the density g 1 with support [v 1 ,v 1 ] and 12 Notice that we have used only the assumption J m 1 ≥ 0, and not J m 2 ≥ 0, because the initial mixed bundling schedule is such that…”

“…Although this is typically a complicated problem, the results in Pavlov (2011) give some insights on the optimal mechanism. First, since v 1 , v 2 are identically distributed, we can focus on determining the optimal q 1 , q 2 , T for (v 1 , v 2 ) such that v 2 ≥ v 1 ; symmetric results are obtained for v 2 < v 1 .…”

“…2 In general, many different mechanisms can be optimal as the distribution of valuations varies, and typically not much is known about the optimal mechanism for a given distribution, except for a few specific settings. For instance, when the valuations are independently and identically distributed, Manelli and Vincent (2006), Pavlov (2011), and Giannakopoulos (2014) solve the problem for uniform or exponential distributions; Hart and Nisan (2014) prove that pure bundling is optimal if the density for each valuation decreases quickly. Hart and Nisan (2014) also try to bound from below the fraction of the optimal profit that can be obtained by selling the objects separately (i.e., posting a suitable price for each object, as in two unrelated one-object settings), or by selling them in a bundle.…”