Prophet inequalities made easy: stochastic optimization by pricing nonstochastic inputs

We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approxima- tion guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms and is used to derive new and improved results for combinatorial markets (with and without complements), multidimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees

Making Auctions Robust to Aftermarkets

A prevalent assumption in auction theory is that the auctioneer has full control over the market and that the allocation she dictates is final. In practice, however, agents might be able to resell acquired items in an aftermarket. A prominent example is the market for carbon emission allowances. These allowances are commonly allocated by the government using uniform-price auctions, and firms can typically trade these allowances among themselves in an aftermarket that may not be fully under the auctioneer's control. While the uniform-price auction is approximately efficient in isolation, we show that speculation and resale in aftermarkets might result in a significant welfare loss. Motivated by this issue, we consider three approaches, each ensuring high equilibrium welfare in the combined market. The first approach is to adopt smooth auctions such as discriminatory auctions. This approach is robust to correlated valuations and to participants acquiring information about others' types. However, discriminatory auctions have several downsides, notably that of charging bidders different prices for identical items, resulting in fairness concerns that make the format unpopular. Two other approaches we suggest are either using posted-pricing mechanisms, or using uniform-price auctions with anonymous reserves. We show that when using balanced prices, both these approaches ensure high equilibrium welfare in the combined market. The latter also inherits many of the benefits from uniform-price auctions such as price discovery, and can be introduced with a minor modification to auctions currently in use to sell carbon emission allowances

Market Pricing for Matroid Rank Valuations

In this paper, we study the problem of maximizing social welfare in combinatorial markets through pricing schemes. We consider the existence of prices that are capable to achieve optimal social welfare without a central tie-breaking coordinator. In the case of two buyers with rank valuations, we give polynomial-time algorithms that always find such prices when one of the matroids is a simple partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by D\"uetting and V\'egh in 2017. We further formalize a weighted variant of the conjecture of D\"uetting and V\'egh, and show that the weighted variant can be reduced to the unweighted one based on the weight-splitting theorem for weighted matroid intersection by Frank. We also show that a similar reduction technique works for M${}^\natural$-concave functions, or equivalently, gross substitutes functions

A dual approach for dynamic pricing in multi-demand markets

Dynamic pricing schemes were introduced as an alternative to posted-price mechanisms. In contrast to static models, the dynamic setting allows to update the prices between buyer-arrivals based on the remaining sets of items and buyers, and so it is capable of maximizing social welfare without the need for a central coordinator. In this paper, we study the existence of optimal dynamic pricing schemes in combinatorial markets. In particular, we concentrate on multi-demand valuations, a natural extension of unit-demand valuations. The proposed approach is based on computing an optimal dual solution of the maximum social welfare problem with distinguished structural properties. Our contribution is twofold. By relying on an optimal dual solution, we show the existence of optimal dynamic prices in unit-demand markets and in multi-demand markets up to three buyers, thus giving new interpretations of results of Cohen-Addad et al. and Berger et al., respectively. Furthermore, we provide an optimal dynamic pricing scheme for bi-demand valuations with an arbitrary number of buyers. In all cases, our proofs also provide efficient algorithms for determining the optimal dynamic prices.Comment: 17 pages, 8 figure

Pricing Multi-Unit Markets

We study the power and limitations of posted prices in multi-unit markets, where agents arrive sequentially in an arbitrary order. We prove upper and lower bounds on the largest fraction of the optimal social welfare that can be guaranteed with posted prices, under a range of assumptions about the designer's information and agents' valuations. Our results provide insights about the relative power of uniform and non-uniform prices, the relative difficulty of different valuation classes, and the implications of different informational assumptions. Among other results, we prove constant-factor guarantees for agents with (symmetric) subadditive valuations, even in an incomplete-information setting and with uniform prices

Multiple sequences Prophet Inequality Under Observation Constraints

In our problem, we are given access to a number of sequences of nonnegative i.i.d. random variables, whose realizations are observed sequentially. All sequences are of the same finite length. The goal is to pick one element from each sequence in order to maximize a reward equal to the expected value of the sum of the selections from all sequences. The decision on which element to pick is irrevocable, i.e., rejected observations cannot be revisited. Furthermore, the procedure terminates upon having a single selection from each sequence. Our observation constraint is that we cannot observe the current realization of all sequences at each time instant. Instead, we can observe only a smaller, yet arbitrary, subset of them. Thus, together with a stopping rule that determines whether we choose or reject the sample, the solution requires a sampling rule that determines which sequence to observe at each instant. The problem can be solved via dynamic programming, but with an exponential complexity in the length of the sequences. In order to make the solution computationally tractable, we introduce a decoupling approach and determine each stopping time using either a single-sequence dynamic programming, or a Prophet Inequality inspired threshold method, with polynomial complexity in the length of the sequences. We prove that the decoupling approach guarantees at least 0.745 of the optimal expected reward of the joint problem. In addition, we describe how to efficiently compute the optimal number of samples for each sequence, and its' dependence on the variances.Comment: 6 pages, 1 figur

Max-Min Greedy Matching

A bipartite graph G(U,V;E) that admits a perfect matching is given. One player imposes a permutation pi over V, the other player imposes a permutation sigma over U. In the greedy matching algorithm, vertices of U arrive in order sigma and each vertex is matched to the highest (under pi) yet unmatched neighbor in V (or left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals pi, and the second (min) player responds with the worst possible sigma for pi, does there exist a permutation pi ensuring to match strictly more than a half of the vertices? Can such a permutation be computed in polynomial time? The main result of this paper is an affirmative answer for these questions: we show that there exists a polytime algorithm to compute pi for which for every sigma at least rho > 0.51 fraction of the vertices of V are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian graphs. Our solution solves an open problem regarding the welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations

Market Pricing for Matroid Rank Valuations

In this paper, we study the problem of maximizing social welfare in combinatorial markets through pricing schemes. We consider the existence of prices that are capable to achieve optimal social welfare without a central tie-breaking coordinator. In the case of two buyers with rank valuations, we give polynomial-time algorithms that always find such prices when one of the matroids is a simple partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by D\"uetting and V\'egh in 2017. We further formalize a weighted variant of the conjecture of D\"uetting and V\'egh, and show that the weighted variant can be reduced to the unweighted one based on the weight-splitting theorem for weighted matroid intersection by Frank. We also show that a similar reduction technique works for M${}^\natural$-concave functions, or equivalently, gross substitutes functions