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Register a new userOptimal Pricing Schemes for an Impatient Buyer
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Added by
Kangning Wang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A patient seller aims to sell a good to an impatient buyer (i.e., one who discounts utility over time). The buyer will remain in the market for a period of time $T$, and her private value is drawn from a publicly known distribution. What is the revenue-optimal pricing-curve (sequence of (price, time) pairs) for the seller? Is randomization of help here? Is the revenue-optimal pricing-curve computable in polynomial time? We answer these questions in this paper. We give an efficient algorithm for computing the revenue-optimal pricing curve. We show that pricing curves, that post a price at each point of time and let the buyer pick her utility maximizing time to buy, are revenue-optimal among a much broader class of sequential lottery mechanisms: namely, mechanisms that allow the seller to post a menu of lotteries at each point of time cannot get any higher revenue than pricing curves. We also show that the even broader class of mechanisms that allow the menu of lotteries to be adaptively set, can earn strictly higher revenue than that of pricing curves, and the revenue gap can be as big as the support size of the buyers value distribution.
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We consider a robust version of the revenue maximization problem, where a single seller wishes to sell $n$ items to a single unit-demand buyer. In this robust version, the seller knows the buyers marginal value distribution for each item separately, but not the joint distribution, and prices the items to maximize revenue in the worst case over all compatible correlation structures. We devise a computationally efficient (polynomial in the support size of the marginals) algorithm that computes the worst-case joint distribution for any choice of item prices. And yet, in sharp contrast to the additive buyer case (Carroll, 2017), we show that it is NP-hard to approximate the optimal choice of prices to within any factor better than $n^{1/2-epsilon}$. For the special case of marginal distributions that satisfy the monotone hazard rate property, we show how to guarantee a constant fraction of the optimal worst-case revenue using item pricing; this pricing equates revenue across all possible correlations and can be computed efficiently.
We study online pricing algorithms for the Bayesian selection problem with production constraints and its generalization to the laminar matroid Bayesian online selection problem. Consider a firm producing (or receiving) multiple copies of different product types over time. The firm can offer the products to arriving buyers, where each buyer is interested in one product type and has a private valuation drawn independently from a possibly different but known distribution. Our goal is to find an adaptive pricing for serving the buyers that maximizes the expected social-welfare (or revenue) subject to two constraints. First, at any time the total number of sold items of each type is no more than the number of produced items. Second, the total number of sold items does not exceed the total shipping capacity. This problem is a special case of the well-known matroid Bayesian online selection problem studied in [Kleinberg and Weinberg, 2012], when the underlying matroid is laminar. We give the first Polynomial-Time Approximation Scheme (PTAS) for the above problem as well as its generalization to the laminar matroid Bayesian online selection problem when the depth of the laminar family is bounded by a constant. Our approach is based on rounding the solution of a hierarchy of linear programming relaxations that systematically strengthen the commonly used ex-ante linear programming formulation of these problems and approximate the optimum online solution with any degree of accuracy. Our rounding algorithm respects the relaxed constraints of higher-levels of the laminar tree only in expectation, and exploits the negative dependency of the selection rule of lower-levels to achieve the required concentration that guarantees the feasibility with high probability.
We study revenue maximization by deterministic mechanisms for the simplest case for which Myersons characterization does not hold: a single seller selling two items, with independently distributed values, to a single additive buyer. We prove that optimal mechanisms are submodular and hence monotone. Furthermore, we show that in the IID case, optimal mechanisms are symmetric. Our characterizations are surprisingly non-trivial, and we show that they fail to extend in several natural ways, e.g. for correlated distributions or more than two items. In particular, this shows that the optimality of symmetric mechanisms does not follow from the symmetry of the IID distribution.
We consider the algorithmic question of choosing a subset of candidates of a given size $k$ from a set of $m$ candidates, with knowledge of voters ordinal rankings over all candidates. We consider the well-known and classic scoring rule for achieving diverse representation: the Chamberlin-Courant (CC) or $1$-Borda rule, where the score of a committee is the average over the voters, of the rank of the best candidate in the committee for that voter; and its generalization to the average of the top $s$ best candidates, called the $s$-Borda rule. Our first result is an improved analysis of the natural and well-studied greedy heuristic. We show that greedy achieves a $left(1 - frac{2}{k+1}right)$-approximation to the maximization (or satisfaction) version of CC rule, and a $left(1 - frac{2s}{k+1}right)$-approximation to the $s$-Borda score. Our result improves on the best known approximation algorithm for this problem. We show that these bounds are almost tight. For the dissatisfaction (or minimization) version of the problem, we show that the score of $frac{m+1}{k+1}$ can be viewed as an optimal benchmark for the CC rule, as it is essentially the best achievable score of any polynomial-time algorithm even when the optimal score is a polynomial factor smaller (under standard computational complexity assumptions). We show that another well-studied algorithm for this problem, called the Banzhaf rule, attains this benchmark. We finally show that for the $s$-Borda rule, when the optimal value is small, these algorithms can be improved by a factor of $tilde Omega(sqrt{s})$ via LP rounding. Our upper and lower bounds are a significant improvement over previous results, and taken together, not only enable us to perform a finer comparison of greedy algorithms for these problems, but also provide analytic justification for using such algorithms in practice.
We consider the problem of posting prices for unit-demand buyers if all $n$ buyers have identically distributed valuations drawn from a distribution with monotone hazard rate. We show that even with multiple items asymptotically optimal welfare can be guaranteed. Our main results apply to the case that either a buyers value for different items are independent or that they are perfectly correlated. We give mechanisms using dynamic prices that obtain a $1 - Theta left( frac{1}{log n}right)$-fraction of the optimal social welfare in expectation. Furthermore, we devise mechanisms that only use static item prices and are $1 - Theta left( frac{logloglog n}{log n}right)$-competitive compared to the optimal social welfare. As we show, both guarantees are asymptotically optimal, even for a single item and exponential distributions.
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