This paper derives polynomial-time approximation schemes for several NP-hard stochastic optimization problems from the algorithmic mechanism design and operations research literatures. The problems we consider involve a principal or seller optimizing with respect to a subsequent choice by an agent or buyer. These include posted pricing for a unit-demand buyer with independent values (Chawla et al. [19], Cai and Daskalakis [16]), assortment optimization with independent utilities (Talluri and van Ryzin [53]), and delegated choice (Khodabakhsh et al. [36]). Our results advance the state of the art for each of these problems. For unit-demand pricing with discrete distributions, our multiplicative PTAS improves on the additive PTAS of Cai and Daskalakis [16], and we additionally give a PTAS for the unbounded regular case, improving on the latter paper’s QPTAS. For assortment optimization, no constant approximation was previously known. For delegated choice, we improve on both the 3 -approximation for the case with no outside option and the super-constant-approximation with an outside option.A key technical insight driving our results is an economically meaningful property we term utility alignment. Informally, a problem is utility aligned if, at optimality, the principal derives most of their utility from realizations where the agent’s utility is also high. Utility alignment allows the algorithm designer to focus on maximizing performance on realizations with high agent utility, which is often an algorithmically simpler task. We prove utility alignment results for all the problems mentioned above, including strong results for unit-demand pricing and delegation, as well as a weaker but very broad guarantee that holds for many other problems under very mild conditions.

This paper studies delegation in a model of discrete choice. In the delegation problem, an uninformed principal must consult an informed agent to make a decision. Both the agent and principal have preferences over the decided-upon action which vary based on the state of the world, and which may not be aligned. The principal may commit to a mechanism, which maps reports of the agent to actions. When this mechanism is deterministic, it can take the form of a menu of actions, from which the agent simply chooses upon observing the state. In this case, the principal is said to have delegated the choice of action to the agent.

Rotating savings and credit associations (roscas) are informal financial organizations common in settings where communities have reduced access to formal financial institutions. In a rosca, a fixed group of participants regularly contribute sums of money to a pot. This pot is then allocated periodically using lottery, aftermarket, or auction mechanisms. Roscas are empirically well-studied in economics. They are, however, challenging to study theoretically due to their dynamic nature. Typical economic analyses of roscas stop at coarse ordinal welfare comparisons to other credit allocation mechanisms, leaving much of roscas' ubiquity unexplained. In this work, we take an algorithmic perspective on the study of roscas. Building on techniques from the price of anarchy literature, we present worst-case welfare approximation guarantees. We further experimentally compare the welfare of outcomes as key features of the environment vary. These cardinal welfare analyses further rationalize the prevalence of roscas. We conclude by discussing several other promising avenues.

We study the problem of learning Bayesian-optimal revenue-maximizing auctions. The classical approach to maximizing revenue requires a known prior distribution on the demand of the bidders, although recent work has shown how to replace the knowledge of a prior distribution with a polynomial sample. However, in an online setting, when buyers can participate in multiple rounds, standard learning techniques are susceptible to \emph{strategic overfitting}: bidders can improve their long-term wellbeing by manipulating the trajectory of the learning algorithm in earlier rounds. For example, they may be able to strategically adjust their behavior in earlier rounds to achieve lower, more favorable future prices. Such non-truthful behavior can hinder learning and harm revenue. In this paper, we combine tools from differential privacy, mechanism design, and sample complexity to give a repeated auction that (1) learns bidder demand from past data, (2) is approximately revenue-optimal, and (3) strategically robust, as it incentivizes bidders to behave truthfully.

This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents’ values are independently distributed. The previous best bound was 1−1/e≈.63, derived using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the worst known example, the first price auction achieves a ≈.869-fraction of the optimal welfare, far better than the theoretical guarantee. Despite this large gap, it was unclear whether the 1−1/e bound was tight. We prove that it is not. Our analysis eschews smoothness, and instead uses the independence assumption on agents’ value distributions to give a more careful accounting of the welfare contribution of agents who win despite not having the highest value.

In a market with repeated sales of a single item to a single buyer, prior work has established the existence of a zero revenue perfect Bayesian equilibrium in the absence of a commitment device for the seller. This counter-intuitive outcome is the result of strategic purchasing decisions, where the buyer worries that the seller will update future prices in response to past purchasing behavior. We first show that in fact almost any revenue can be achieved in equilibrium, but the zero revenue equilibrium uniquely survives natural refinements. This establishes that single buyer markets without commitment are subject to market failure. However, our main result shows that this market failure depends crucially on the assumption of a single buyer. If there are multiple buyers, the seller can approximate the revenue that is possible with commitment. We construct an intuitive equilibrium for multiple buyers that survives our refinements, in which the seller learns from past purchasing behavior and obtains a constant factor of the per-round Myerson optimal revenue. The seller's pricing policy has a natural explore-exploit structure, where the seller starts with low prices that gradually ascend to learn buyers' values, and in later rounds exploits the surviving high-valued buyers. The result resembles an ascending-price auction, implemented over time. This relates to the intuition from the Coase conjecture in the durable goods literature [Coase 1972] which states that in the absence of commitment, one should expect the VCG outcome (which, for multiple buyers, yields non-trivial revenue for the seller).

This paper develops tools for welfare and revenue analyses of Bayes-Nash equilibria in asymmetric auctions with single-dimensional agents. We employ these tools to derive price of anarchy results for social welfare and revenue. Our approach separates the standard smoothness framework [e.g., Syrgkanis and Tardos 2013] into two distinct parts. The first part, value covering, employs best-response analysis to individually relate each agent's expected price for allocation and welfare in any Bayes-Nash equilibrium. The second part, revenue covering, uses properties of an auction's rules and feasibility constraints to relate the revenue of the auction to the agents' expected prices for allocation (not necessarily in equilibrium). Because value covering holds for any equilibrium, proving an auction is revenue covered is a sufficient condition for approximating optimal welfare, and under the right conditions, the optimal revenue. In mechanisms with reserve prices, our welfare results show approximation with respect to the optimal mechanism with the same reserves.

Given a simple, connected graph, a pebbling configuration is a function from its vertex set to the nonnegative integers. A pebbling move between adjacent vertices removes two pebbles from one vertex and adds one pebble to the other. A vertex $r$ is said to be reachable from a configuration if there exists a sequence of pebbling moves that places one pebble on $r$. A configuration is solvable if every vertex is reachable. We prove tight bounds on the number of vertices with two and three pebbles that an unsolvable configuration on a diameter two graph can have in terms of the size of the graph. We also prove that determining reachability of a vertex is NP-complete, even in graphs of diameter two.