Pieter Kleer

We consider an optimal stopping problem with n correlated offers where the goal is to design a (randomised) stopping strategy that maximises the expected value of the offer at which we stop. Instead of assuming to know the complete correlation structure, which is unrealistic in practice, we only assume to have knowledge of the distribution of the maximum value X _max of the sequence, and want to analyse the worst-case correlation structure whose maximum follows this distribution. This can be seen as a trade-off between the setting in which no distributional information is known, and the Bayesian setting in which the (possibly correlated) distributions of all the individual offers are known. As our first main result we show that a deterministic threshold strategy using the monopoly price of the distribution of the maximum value is asymptotically optimal assuming that the expectation of the maximum value grows sublinearly in n . In our second main result, we further tighten this bound by deriving a tight quadratic convergence guarantee for sufficiently smooth distributions of the maximum value. Our results also give rise to a more fine-grained picture regarding prophet inequalities with correlated values, for which distribution-free bounds only yield a performance guarantee of the order 1/ n .

Problem definition: Traditional monopoly pricing assumes sellers have full information about consumer valuations. We consider monopoly pricing under limited information when a seller only knows the mean, variance, and support of the valuation distribution. The objective is to maximize expected revenue by selecting the optimal fixed price. Methodology/results: We adopt a distributionally robust framework, in which the seller considers all valuation distributions that comply with the limited information. We formulate a maximin problem that seeks to maximize expected revenue for the worst case valuation distribution. The minimization problem that identifies the worst case valuation distribution is solved using primal-dual methods and, in turn, leads to an explicitly solvable maximization problem. This yields a closed-form optimal pricing policy and a new fundamental principle prescribing when to use low and high robust prices. Managerial implications: We show that the optimal policy switches from low to high prices when variance becomes sufficiently large, yielding significant performance gains compared with existing robust prices that generally decay with market uncertainty. This presents guidelines for when the seller should switch from targeting mass markets to niche markets. Similar guidelines are obtained for delay-prone services with rational utility-maximizing customers, underlining the universality and wide applicability of the novel pricing policy. Funding: This research was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Vici [Grant 202.068]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2024.0952 .

We consider an optimal stopping problem with n correlated offers where the goal is to design a (randomized) stopping strategy that maximizes the expected value of the offer in the sequence at which we stop. Instead of assuming to know the complete correlation structure, which is unrealistic in practice, we only assume to have knowledge of the distribution of the maximum value of the sequence, and want to analyze the worst-case correlation structure whose maximum follows this distribution. This can be seen as a trade-off between the setting in which no distributional information is known, and the Bayesian setting in which the (possibly correlated) distributions of all the individual offers are known. As our first main result we show that a deterministic threshold strategy using the monopoly price of the distribution of the maximum value is asymptotically optimal assuming that the expectation of the maximum value grows sublinearly in n. In our second main result, we further tighten this bound by deriving a tight quadratic convergence guarantee for sufficiently smooth distributions of the maximum value. Our results also give rise to a more fine-grained picture regarding prophet inequalities with correlated values, for which distribution-free bounds often only yield a performance guarantee that is of the order 1/n.

We consider clustering games in which the players are embedded into a network and want to coordinate (or anti-coordinate) their strategy with their neighbors. The goal of a player is to choose a strategy that maximizes her utility given the strategies of her neighbors. Recent studies show that even very basic variants of these games exhibit a large Price of Anarchy: A large inefficiency between the total utility generated in centralized outcomes and equilibrium outcomes in which players selfishly maximize their utility. Our main goal is to understand how structural properties of the network topology impact the inefficiency of these games. We derive topological bounds on the Price of Anarchy for different classes of clustering games. These topological bounds provide a more informative assessment of the inefficiency of these games than the corresponding worst-case Price of Anarchy bounds. More specifically, depending on the type of clustering game, our bounds reveal that the Price of Anarchy depends on the maximum subgraph density or the maximum degree of the graph. Among others, these bounds enable us to derive bounds on the Price of Anarchy for clustering games on Erdős-Rényi random graphs. Depending on the graph density, these bounds stand in stark contrast to the known worst-case Price of Anarchy bounds. Additionally, we characterize the set of distribution rules that guarantee the existence of a pure Nash equilibrium or the convergence of best-response dynamics. These results are of a similar spirit as the work of Gopalakrishnan et al. [ 19 ] and complement work of Anshelevich and Sekar [ 4 ].

Logit dynamics is a form of randomized game dynamics in which players have a bias toward strategic deviations that give a higher improvement in cost. It is used extensively in practice. In congestion (or potential) games, the dynamics converge to the so-called Gibbs distribution over the set of all strategy profiles when interpreted as a Markov chain. In general, logit dynamics can converge slowly to the Gibbs distribution, but beyond that, not much is known about its algorithmic aspects, nor that of the Gibbs distribution. In this work, we are interested in the following two questions for congestion games: (i) Is there an efficient algorithm for sampling from the Gibbs distribution? (ii) If yes, does there also exist natural randomized dynamics that converge quickly to the Gibbs distribution? We first study these questions in extension parallel congestion games, a well-studied special case of symmetric network congestion games. As our main result, we show that there is a simple variation on the logit dynamics that converges quickly to the Gibbs distribution in such games. We also address the first question for the class of so-called capacitated uniform congestion games and the second question for max cut games played on a complete graph. To prove our results, we rely on the recent breakthrough work of Anari et al. (2019) regarding the approximate sampling of a base of a matroid according to a strongly log-concave probability distribution.

The price of anarchy is the most well-known measure for quantifying the inefficiency of equilibrium flows in traffic networks and routing games. In this work, we give unifying price of anarchy bounds for atomic and non-atomic parallel link routing games with polynomial cost functions under various cost objectives including the arithmetic mean, geometric mean and worst-case cost objective. We do this through the study of the generalized p-mean as cost objective, and obtain upper bounds on the price of anarchy in terms of this parameter p. Our bounds unify existing results from the literature, and, in particular, give alternative proofs for price of anarchy results in parallel link routing games with polynomial cost functions under the geometric mean objective obtained by Vinci et al. (ACM Trans Econ Comput 10(2):41, 2022). We recover those simply as a limiting case. To the best of our knowledge, these are the first price of anarchy bounds that capture multiple cost objectives simultaneously in a closed-form expression.