We consider the problem of a single seller repeatedly selling a single item to a single buyer (specifically, the buyer has a value drawn fresh from known distribution $D$ in every round). Prior work assumes that the buyer is fully rational and will perfectly reason about how their bids today affect the seller's decisions tomorrow. In this work we initiate a different direction: the buyer simply runs a noregret learning algorithm over possible bids. We provide a fairly complete characterization of optimal auctions for the seller in this domain. Specifically:
1) If the buyer bids according to EXP3 (or any ``meanbased'' learning algorithm), then the seller can extract expected revenue arbitrarily close to the expected welfare. This auction is independent of the buyer's valuation $D$, but somewhat unnatural as it is sometimes in the buyer's interest to overbid.
2) There exists a learning algorithm $\mathcal{A}$ such that if the buyer bids according to $\mathcal{A}$ then the optimal strategy for the seller is simply to post the Myerson reserve for $D$ every round.
3) If the buyer bids according to EXP3 (or any ``meanbased'' learning algorithm), but the seller is restricted to ``natural'' auction formats where overbidding is dominated (e.g. Generalized FirstPrice or Generalized SecondPrice), then the optimal strategy for the seller is a payyourbid format with decreasing reserves over time. Moreover, the seller's optimal achievable revenue is characterized by a linear program, and can be unboundedly better than the best truthful auction yet simultaneously unboundedly worse than the expected welfare.
more »
« less
FixedPrice Approximations in Bilateral Trade
We consider the bilateral trade problem, in which two agents trade a single indivisible item. It is known that the only dominantstrategy truthful mechanism is the fixedprice mechanism: given commonly known distributions of the buyer's value B and the seller's value S, a price p is offered to both agents and trade occurs if S ≤ p ≤ B. The objective is to maximize either expected welfare or expected gains from trade .
We improve the approximation ratios for several welfare maximization variants of this problem. When the agents' distributions are identical, we show that the optimal approximation ratio for welfare is . With just one prior sample from the common distribution, we show that a 3/4approximation to welfare is achievable. When agents' distributions are not required to be identical, we show that a previously bestknown (1–1/e)approximation can be strictly improved, but 1–1/e is optimal if only the seller's distribution is known.
more »
« less
 Award ID(s):
 2127781
 NSFPAR ID:
 10352737
 Editor(s):
 Naor, Joseph; Buchbinder, Niv
 Date Published:
 Journal Name:
 Proceedings of the 2022 Annual ACMSIAM Symposium on Discrete Algorithms (SODA)
 Page Range / eLocation ID:
 2964  2985
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


We study gains from trade in multidimensional twosided markets. Specifically, we focus on a setting with n heterogeneous items, where each item is owned by a different seller i, and there is a constrainedadditive buyer with feasibility constraint ℱ. Multidimensional settings in onesided markets, e.g. where a seller owns multiple heterogeneous items but also is the mechanism designer, are wellunderstood. In addition, singledimensional settings in twosided markets, e.g. where a buyer and seller each seek or own a single item, are also wellunderstood. Multidimensional twosided markets, however, encapsulate the major challenges of both lines of work: optimizing the sale of heterogeneous items, ensuring incentivecompatibility among both sides of the market, and enforcing budget balance. We present, to the best of our knowledge, the first worstcase approximation guarantee for gains from trade in a multidimensional twosided market. Our first result provides an O(log(1/r))approximation to the firstbest gains from trade for a broad class of downwardclosed feasibility constraints (such as matroid, matching, knapsack, or the intersection of these). Here r is the minimum probability over all items that a buyer's value for the item exceeds the seller's cost. Our second result removes the dependence on r and provides an unconditional O(log n)approximation to the secondbest gains from trade. We extend both results for a general constrainedadditive buyer, losing another O(log n)factor enroute. The first result is achieved using a fixed posted price mechanism, and the analysis involves a novel application of the prophet inequality or a new concentration inequality. Our second result follows from a stitching lemma that allows us to upper bound the secondbest gains from trade by the firstbest gains from trade from the “likely to trade” items (items with trade probability at least 1/n) and the optimal profit from selling the “unlikely to trade” items. We can obtain an O(log n)approximation to the first term by invoking our O(log(1/r))approximation on the “likely to trade” items. We introduce a generalization of the fixed posted price mechanism—seller adjusted posted price—to obtain an O(log n)approximation to the optimal profit for the “unlikely to trade” items. Unlike fixed posted price mechanisms, not all seller adjusted posted price mechanisms are incentive compatible and budget balanced. We develop a new argument based on “allocation coupling” to show the seller adjusted posted price mechanism used in our approximation is indeed budget balanced and incentivecompatible.more » « less

We study the menu complexity of optimal and approximatelyoptimal auctions in the context of the ``FedEx'' problem, a socalled ``oneandahalfdimensional'' setting where a single bidder has both a value and a deadline for receiving an item [FGKK 16]. The menu complexity of an auction is equal to the number of distinct (allocation, price) pairs that a bidder might receive [HN 13]. We show the following when the bidder has $n$ possible deadlines: 1) Exponential menu complexity is necessary to be exactly optimal: There exist instances where the optimal mechanism has menu complexity $\geq 2^n1$. This matches exactly the upper bound provided by Fiat et al.'s algorithm, and resolves one of their open questions [FGKK 16]. 2) Fully polynomial menu complexity is necessary and sufficient for approximation: For all instances, there exists a mechanism guaranteeing a multiplicative $(1\epsilon)$approximation to the optimal revenue with menu complexity $O(n^{3/2}\sqrt{\frac{\min\{n/\epsilon,\ln(v_{\max})\}}{\epsilon}}) = O(n^2/\epsilon)$, where $v_{\max}$ denotes the largest value in the support of integral distributions. \item There exist instances where any mechanism guaranteeing a multiplicative $(1O(1/n^2))$approximation to the optimal revenue requires menu complexity $\Omega(n^2)$. Our main technique is the polygon approximation of concave functions [Rote 91], and our results here should be of independent interest. We further show how our techniques can be used to resolve an open question of [DW 17] on the menu complexity of optimal auctions for a budgetconstrained buyer.more » « less

null (Ed.)We introduce the \emph{pipeline intervention} problem, defined by a layered directed acyclic graph and a set of stochastic matrices governing transitions between successive layers. The graph is a stylized model for how people from different populations are presented opportunities, eventually leading to some reward. In our model, individuals are born into an initial position (i.e. some node in the first layer of the graph) according to a fixed probability distribution, and then stochastically progress through the graph according to the transition matrices, until they reach a node in the final layer of the graph; each node in the final layer has a \emph{reward} associated with it. The pipeline intervention problem asks how to best make costly changes to the transition matrices governing people's stochastic transitions through the graph, subject to a budget constraint. We consider two objectives: social welfare maximization, and a fairnessmotivated maximin objective that seeks to maximize the value to the population (starting node) with the \emph{least} expected value. We consider two variants of the maximin objective that turn out to be distinct, depending on whether we demand a deterministic solution or allow randomization. For each objective, we give an efficient approximation algorithm (an additive FPTAS) for constant width networks. We also tightly characterize the "price of fairness" in our setting: the ratio between the highest achievable social welfare and the highest social welfare consistent with a maximin optimal solution. Finally we show that for polynomial width networks, even approximating the maximin objective to any constant factor is NP hard, even for networks with constant depth. This shows that the restriction on the width in our positive results is essential.more » « less

Vidick, T. (Ed.)We study auctions for carbon licenses, a policy tool used to control the social cost of pollution. Each identical license grants the right to produce a unit of pollution. Each buyer (i.e., firm that pollutes during the manufacturing process) enjoys a decreasing marginal value for licenses, but society suffers an increasing marginal cost for each license distributed. The seller (i.e., the government) can choose a number of licenses to put up for auction, and wishes to maximize the societal welfare: the total economic value of the buyers minus the social cost. Motivated by emission license markets deployed in practice, we focus on uniform price auctions with a price floor and/or price ceiling. The seller has distributional information about the market, and their goal is to tune the auction parameters to maximize expected welfare. The target benchmark is the maximum expected welfare achievable by any such auction under truthtelling behavior. Unfortunately, the uniform price auction is not truthful, and strategic behavior can significantly reduce (even below zero) the welfare of a given auction configuration. We describe a subclass of “safeprice” auctions for which the welfare at any BayesNash equilibrium will approximate the welfare under truthtelling behavior. We then show that the better of a safeprice auction, or a truthful auction that allocates licenses to only a single buyer, will approximate the target benchmark. In particular, we show how to choose a number of licenses and a price floor so that the worstcase welfare, at any equilibrium, is a constant approximation to the best achievable welfare under truthtelling after excluding the welfare contribution of a single buyer.more » « less