Appendix of “ Optimal Multiperiod Pricing with Service Guarantees ”

  • Published 2014

Abstract

A. Online Appendix of “Optimal Multiperiod Pricing with Service Guarantees” A.1. Appendix to Section 3 Proof of Lemma 1 First note that we can add the constraint 0 p 1 to the problem without loss of optimality, since customer valuations are bounded by 1. Consequently, it follows that for a given fixed ranking R, the set of consistent and feasible prices defines a closed and bounded set. Since the objective function is continuous in prices (for a fixed ranking), we conclude that optimal prices exist for any given ranking R. By maximizing over the finitely many possible rankings, we conclude that an optimal solution of OPT-2 exists. For the second claim, observe that if p is a feasible solution of OPT-1, then (p,R(p)) is a feasible solution of OPT2 with the same objective value. Thus, the maximum of OPT-2 is an upper bound on the supremum of OPT-1. Given an optimal solution (p,R) of OPT-2, and any ✏> 0, p+ ✏R is a feasible vector of prices that is consistent with the ranking R, and hence (p+ ✏R,R) is a feasible solution of OPT-2. This is because, if R t <R t0 , then pt  pt0 , and consequently pt + ✏R t < pt0 + ✏R t0 . Moreover, this inequality also implies that in p + ✏R no price is repeated, and hence the only consistent ranking with this price vector is R. This implies that R is the customer-preferred ranking corresponding to p + ✏R, and thus this price vector is feasible in OPT-1 with the same objective value. Since the objective of OPT-2 is continuous in prices for a fixed ranking R, the value of (p + ✏R,R) approaches to that of (p,R), as ✏ goes to 0. Thus for ✏ > 0, p + ✏R is a feasible solution of OPT-1, value of which converges to maximum of OPT-2 as ✏ goes to 0. Since maximum of OPT-2 is an upper bound on the supremum of OPT-1, it follows that these values are equal, and p + ✏R converges to the supremum of OPT-1, as claimed. If p is an optimal solution of OPT-1, then its value equals to the supremum value. However, as explained earlier this value equals to the maximum of OPT-2, and (p,R(p)) is a feasible solution of this problem with the same value. Thus, the claim follows. A.2. Appendix to Section 4 Proof of Proposition 1 Since the valuations are bounded by 1, it is not beneficial to set a price above 1. Now, suppose (p,R) is a feasible and consistent price ranking. Let p0 be the price vector such that pt = max{pM , pt}. We claim that (p0,R) is both consistent and feasible. For consistency, note that if Rt < Rt0 then pt  pt0 . Hence, max{pM , pt} max{pM , pt0}. Therefore, (p0,R) is consistent. Moreover, because we have (weakly) increased the prices, it is a feasible solution. Finally, observe that the revenue obtained from (p0,R) is at least equal to the revenue of (p,R). The reason is ⇢t(R) does not change, but the uncapacitated revenue function, p(1 F (p)), increases. Namely,

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@inproceedings{2014AppendixO, title={Appendix of “ Optimal Multiperiod Pricing with Service Guarantees ”}, author={}, year={2014} }