arXiv subjects

S. Matthew Weinberg

Publications and source records attributed to S. Matthew Weinberg.

At least 55 records · Page 3Linked to original sources

arXiv2021

Prior-free Dynamic Mechanism Design With Limited Liability

We study the problem of repeatedly auctioning off an item to one of $k$ bidders where: a) bidders have a per-round individual rationality constraint, b) bidders may leave the mechanism at any point, and c) the bidders' valuations are adversarially chosen (the prior-free setting). Without these constraints, the auctioneer can run a second-price auction to "sell the business" and receive the second highest total value for the entire stream of items. We show that under these constraints, the auctioneer can attain a constant fraction of the "sell the business" benchmark, but no more than $2/e$ of this benchmark. In the course of doing so, we design mechanisms for a single bidder problem of independent interest: how should you repeatedly sell an item to a (per-round IR) buyer with adversarial valuations if you know their total value over all rounds is $V$ but not how their value changes over time? We demonstrate a mechanism that achieves revenue $V/e$ and show that this is tight.

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arXiv2021

Approximately Strategyproof Tournament Rules in the Probabilistic Setting

We consider the manipulability of tournament rules which map the results of $\binom{n}{2}$ pairwise matches and select a winner. Prior work designs simple tournament rules such that no pair of teams can manipulate the outcome of their match to improve their probability of winning by more than $1/3$, and this is the best possible among any Condorcet-consistent tournament rule (which selects an undefeated team whenever one exists) [Schneider et al., 2017, Schvartzman et al., 2020]. These lower bounds require the manipulators to know precisely the outcome of all future matches. We take a beyond worst-case view and instead consider tournaments which are "close to uniform": the outcome of all matches are independent, and no team is believed to win any match with probability exceeding $1/2+\varepsilon$. We show that Randomized Single Elimination Bracket [Schneider et al., 2017] and a new tournament rule we term Randomized Death Match have the property that no pair of teams can manipulate the outcome of their match to improve their probability of winning by more than $\varepsilon/3 + 2\varepsilon^2/3$, for all $\varepsilon$, and this is the best possible among any Condorcet-consistent tournament rule. Our main technical contribution is a recursive framework to analyze the manipulability of certain forms of tournament rules. In addition to our main results, this view helps streamline previous analysis of Randomized Single Elimination Bracket, and may be of independent interest.

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arXiv2020

Separating the Communication Complexity of Truthful and Non-Truthful Combinatorial Auctions

We provide the first separation in the approximation guarantee achievable by truthful and non-truthful combinatorial auctions with polynomial communication. Specifically, we prove that any truthful mechanism guaranteeing a $(\frac{3}{4}-\frac{1}{240}+\varepsilon)$-approximation for two buyers with XOS valuations over $m$ items requires $\exp(Ω(\varepsilon^2 \cdot m))$ communication, whereas a non-truthful algorithm by Dobzinski and Schapira [SODA 2006] and Feige [2009] is already known to achieve a $\frac{3}{4}$-approximation in $poly(m)$ communication. We obtain our separation by proving that any {simultaneous} protocol ({not} necessarily truthful) which guarantees a $(\frac{3}{4}-\frac{1}{240}+\varepsilon)$-approximation requires communication $\exp(Ω(\varepsilon^2 \cdot m))$. The taxation complexity framework of Dobzinski [FOCS 2016] extends this lower bound to all truthful mechanisms (including interactive truthful mechanisms).

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arXiv2020

Decentralized Reinforcement Learning: Global Decision-Making via Local Economic Transactions

This paper seeks to establish a framework for directing a society of simple, specialized, self-interested agents to solve what traditionally are posed as monolithic single-agent sequential decision problems. What makes it challenging to use a decentralized approach to collectively optimize a central objective is the difficulty in characterizing the equilibrium strategy profile of non-cooperative games. To overcome this challenge, we design a mechanism for defining the learning environment of each agent for which we know that the optimal solution for the global objective coincides with a Nash equilibrium strategy profile of the agents optimizing their own local objectives. The society functions as an economy of agents that learn the credit assignment process itself by buying and selling to each other the right to operate on the environment state. We derive a class of decentralized reinforcement learning algorithms that are broadly applicable not only to standard reinforcement learning but also for selecting options in semi-MDPs and dynamically composing computation graphs. Lastly, we demonstrate the potential advantages of a society's inherent modular structure for more efficient transfer learning.

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arXiv2020

Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms when Demand Queries are NP-hard

State-of-the-art posted-price mechanisms for submodular bidders with $m$ items achieve approximation guarantees of $O((\log \log m)^3)$ [Assadi and Singla, 2019]. Their truthfulness, however, requires bidders to compute an NP-hard demand-query. Some computational complexity of this form is unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an $m^{1/2-\varepsilon}$-approximation for any $\varepsilon > 0$ [Dobzinski and Vondrák, 2016]. Together, these establish a stark distinction between computationally-efficient and communication-efficient truthful mechanisms. We show that this distinction disappears with a mild relaxation of truthfulness, which we term implementation in advised strategies, and that has been previously studied in relation to "Implementation in Undominated Strategies" [Babaioff et al, 2009]. Specifically, advice maps a tentative strategy either to that same strategy itself, or one that dominates it. We say that a player follows advice as long as they never play actions which are dominated by advice. A poly-time mechanism guarantees an $α$-approximation in implementation in advised strategies if there exists poly-time advice for each player such that an $α$-approximation is achieved whenever all players follow advice. Using an appropriate bicriterion notion of approximate demand queries (which can be computed in poly-time), we establish that (a slight modification of) the [Assadi and Singla, 2019] mechanism achieves the same $O((\log \log m)^3)$-approximation in implementation in advised strategies.

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arXiv2020

A Simple and Approximately Optimal Mechanism for an Additive Buyer

We consider a monopolist seller with $n$ heterogeneous items, facing a single buyer. The buyer has a value for each item drawn independently according to (non-identical) distributions, and her value for a set of items is additive. The seller aims to maximize his revenue. We suggest using the a-priori better of two simple pricing methods: selling the items separately, each at its optimal price, and bundling together, in which the entire set of items is sold as one bundle at its optimal price. We show that for any distribution, this mechanism achieves a constant-factor approximation to the optimal revenue. Beyond its simplicity, this is the first computationally tractable mechanism to obtain a constant-factor approximation for this multi-parameter problem. We additionally discuss extensions to multiple buyers and to valuations that are correlated across items.

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arXiv2020

On the (in)-approximability of Bayesian Revenue Maximization for a Combinatorial Buyer

We consider a revenue-maximizing single seller with $m$ items for sale to a single buyer whose value $v(\cdot)$ for the items is drawn from a known distribution $D$ of support $k$. A series of works by Cai et al. establishes that when each $v(\cdot)$ in the support of $D$ is additive or unit-demand (or $c$-demand), the revenue-optimal auction can be found in $\operatorname{poly}(m,k)$ time. We show that going barely beyond this, even to matroid-based valuations (a proper subset of Gross Substitutes), results in strong hardness of approximation. Specifically, even on instances with $m$ items and $k \leq m$ valuations in the support of $D$, it is not possible to achieve a $1/m^{1-\varepsilon}$-approximation for any $\varepsilon>0$ to the revenue-optimal mechanism for matroid-based valuations in (randomized) poly-time unless NP $\subseteq$ RP (note that a $1/k$-approximation is trivial). Cai et al.'s main technical contribution is a black-box reduction from revenue maximization for valuations in class $\mathcal{V}$ to optimizing the difference between two values in class $\mathcal{V}$. Our main technical contribution is a black-box reduction in the other direction (for a wide class of valuation classes), establishing that their reduction is essentially tight.

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arXiv2020

Asynchronous Majority Dynamics in Preferential Attachment Trees

We study information aggregation in networks where agents make binary decisions (labeled incorrect or correct). Agents initially form independent private beliefs about the better decision, which is correct with probability $1/2+δ$. The dynamics we consider are asynchronous (each round, a single agent updates their announced decision) and non-Bayesian (agents simply copy the majority announcements among their neighbors, tie-breaking in favor of their private signal). Our main result proves that when the network is a tree formed according to the preferential attachment model \cite{BarabasiA99}, with high probability, the process stabilizes in a correct majority within $O(n \log n/ \log\log n)$ rounds. We extend our results to other tree structures, including balanced $M$-ary trees for any $M$.

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arXiv2020

Credible, Truthful, and Two-Round (Optimal) Auctions via Cryptographic Commitments

We consider the sale of a single item to multiple buyers by a revenue-maximizing seller. Recent work of Akbarpour and Li formalizes \emph{credibility} as an auction desideratum, and prove that the only optimal, credible, strategyproof auction is the ascending price auction with reserves (Akbarpour and Li, 2019). In contrast, when buyers' valuations are MHR, we show that the mild additional assumption of a cryptographically secure commitment scheme suffices for a simple \emph{two-round} auction which is optimal, strategyproof, and credible (even when the number of bidders is only known by the auctioneer). We extend our analysis to the case when buyer valuations are $α$-strongly regular for any $α> 0$, up to arbitrary $\varepsilon$ in credibility. Interestingly, we also prove that this construction cannot be extended to regular distributions, nor can the $\varepsilon$ be removed with multiple bidders.

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arXiv2020

When to Limit Market Entry under Mandatory Purchase

We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with private insurers to provide plans for their employees. The market regulator can choose to do nothing, running a Free Market, or can exercise her regulatory power by limiting the entry of providers (decreasing consumer welfare by limiting options, but also decreasing revenue via enhanced competition). We investigate whether limiting entry increases or decreases the utility (welfare minus revenue) of the consumers who purchase from the providers, specifically in settings where the outside option of "purchasing nothing" is prohibitively undesirable. We focus primarily on the case where providers are symmetric. We propose a sufficient condition on the distribution of consumer values for (a) a unique symmetric equilibrium to exist in both markets and (b) utility to be higher with limited entry. (We also establish that these conclusions do not necessarily hold for all distributions, and therefore some condition is necessary.) Our techniques are primarily based on tools from revenue maximization, and in particular Myerson's virtual value theory. We also consider extensions to settings where providers have identical costs for providing plans, and to two providers with an asymmetric distribution.

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arXiv2020

Optimal Mechanism Design for Single-Minded Agents

We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting is the FedEx Problem, for which FGKK [2016] characterize the optimal mechanism for a single agent. We ask: how far can such characterizations go? In particular, we consider single-minded agents. A seller has heterogenous items. A buyer has a value v for a specific subset of items S, and obtains value v iff he gets (at least) all the items in S. We show: 1. Deterministic mechanisms are optimal for distributions that satisfy the "declining marginal revenue" (DMR) property; we give an explicit construction of the optimal mechanism. 2. Without DMR, the result depends on the structure of the directed acyclic graph (DAG) representing the partial order among types. When the DAG has out-degree at most 1, we characterize the optimal mechanism a la FedEx. 3. Without DMR, when the DAG has some node with out-degree at least 2, we show that in this case the menu complexity is unbounded: for any M, there exist distributions over (v,S) pairs such that the menu complexity of the optimal mechanism is at least M. 4. For the case of 3 types, we show that for all distributions there exists an optimal mechanism of finite menu complexity. This is in contrast to 2 additive heterogenous items or which the menu complexity could be uncountable [MV07; DDT15]. In addition, we prove that optimal mechanisms for Multi-Unit Pricing (without DMR) can have unbounded menu complexity. We also propose an extension where the menu complexity of optimal mechanisms can be countable but not uncountable. Together these results establish that optimal mechanisms in interdimensional settings are both much richer than single-dimensional settings, yet also vastly more structured than multi-dimensional settings.

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arXiv2019

Approximation Schemes for a Unit-Demand Buyer with Independent Items via Symmetries

We consider a revenue-maximizing seller with $n$ items facing a single buyer. We introduce the notion of symmetric menu complexity of a mechanism, which counts the number of distinct options the buyer may purchase, up to permutations of the items. Our main result is that a mechanism of quasi-polynomial symmetric menu complexity suffices to guarantee a $(1-\varepsilon)$-approximation when the buyer is unit-demand over independent items, even when the value distribution is unbounded, and that this mechanism can be found in quasi-polynomial time. Our key technical result is a polynomial time, (symmetric) menu-complexity-preserving black-box reduction from achieving a $(1-\varepsilon)$-approximation for unbounded valuations that are subadditive over independent items to achieving a $(1-O(\varepsilon))$-approximation when the values are bounded (and still subadditive over independent items). We further apply this reduction to deduce approximation schemes for a suite of valuation classes beyond our main result. Finally, we show that selling separately (which has exponential menu complexity) can be approximated up to a $(1-\varepsilon)$ factor with a menu of efficient-linear $(f(\varepsilon) \cdot n)$ symmetric menu complexity.

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arXiv2019

Approximately Strategyproof Tournament Rules: On Large Manipulating Sets and Cover-Consistence

We consider the manipulability of tournament rules, in which $n$ teams play a round robin tournament and a winner is (possibly randomly) selected based on the outcome of all $\binom{n}{2}$ matches. Prior work defines a tournament rule to be $k$-SNM-$α$ if no set of $\leq k$ teams can fix the $\leq \binom{k}{2}$ matches among them to increase their probability of winning by $>α$ and asks: for each $k$, what is the minimum $α(k)$ such that a Condorcet-consistent (i.e. always selects a Condorcet winner when one exists) $k$-SNM-$α(k)$ tournament rule exists? A simple example witnesses that $α(k) \geq \frac{k-1}{2k-1}$ for all $k$, and [Schneider et al., 2017] conjectures that this is tight (and prove it is tight for $k=2$). Our first result refutes this conjecture: there exists a sufficiently large $k$ such that no Condorcet-consistent tournament rule is $k$-SNM-$1/2$. Our second result leverages similar machinery to design a new tournament rule which is $k$-SNM-$2/3$ for all $k$ (and this is the first tournament rule which is $k$-SNM-$(<1)$ for all $k$). Our final result extends prior work, which proves that single-elimination bracket with random seeding is $2$-SNM-$1/3$([Schneider et al., 2017]), in a different direction by seeking a stronger notion of fairness than Condorcet-consistence. We design a new tournament rule, which we call Randomized-King-of-the-Hill, which is $2$-SNM-$1/3$ and \emph{cover-consistent} (the winner is an uncovered team with probability $1$).

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arXiv2019

New Query Lower Bounds for Submodular Function MInimization

We consider submodular function minimization in the oracle model: given black-box access to a submodular set function $f:2^{[n]}\rightarrow \mathbb{R}$, find an element of $\arg\min_S \{f(S)\}$ using as few queries to $f(\cdot)$ as possible. State-of-the-art algorithms succeed with $\tilde{O}(n^2)$ queries [LeeSW15], yet the best-known lower bound has never been improved beyond $n$ [Harvey08]. We provide a query lower bound of $2n$ for submodular function minimization, a $3n/2-2$ query lower bound for the non-trivial minimizer of a symmetric submodular function, and a $\binom{n}{2}$ query lower bound for the non-trivial minimizer of an asymmetric submodular function. Our $3n/2-2$ lower bound results from a connection between SFM lower bounds and a novel concept we term the cut dimension of a graph. Interestingly, this yields a $3n/2-2$ cut-query lower bound for finding the global mincut in an undirected, weighted graph, but we also prove it cannot yield a lower bound better than $n+1$ for $s$-$t$ mincut, even in a directed, weighted graph.

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arXiv2019

Persuasion and Incentives Through the Lens of Duality

Lagrangian duality underlies both classical and modern mechanism design. In particular, the dual perspective often permits simple and detail-free characterizations of optimal and approximately optimal mechanisms. This paper applies this same methodology to a close cousin of traditional mechanism design, one which shares conceptual and technical elements with its more mature relative: the burgeoning field of persuasion. The dual perspective permits us to analyze optimal persuasion schemes both in settings which have been analyzed in prior work, as well as for natural generalizations which we are the first to explore in depth. Most notably, we permit combining persuasion policies with payments, which serve to augment the persuasion power of the scheme. In both single and multi-receiver settings, as well as under a variety of constraints on payments, we employ duality to obtain structural insights, as well as tractable and simple characterizations of optimal policies.

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arXiv2019

Computing exact minimum cuts without knowing the graph

We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem: on query $S \subset V$, the oracle returns the size of the cut between $S$ and $V \setminus S$. We provide algorithms computing an exact minimum $s$-$t$ cut in $G$ with $\tilde{O}(n^{5/3})$ queries, and computing an exact global minimum cut of $G$ with only $\tilde{O}(n)$ queries (while learning the graph requires $\tildeΘ(n^2)$ queries).

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arXiv2019

Selling a Single Item with Negative Externalities

We consider the problem of regulating products with negative externalities to a third party that is neither the buyer nor the seller, but where both the buyer and seller can take steps to mitigate the externality. The motivating example to have in mind is the sale of Internet-of-Things (IoT) devices, many of which have historically been compromised for DDoS attacks that disrupted Internet-wide services such as Twitter. Neither the buyer (i.e., consumers) nor seller (i.e., IoT manufacturers) was known to suffer from the attack, but both have the power to expend effort to secure their devices. We consider a regulator who regulates payments (via fines if the device is compromised, or market prices directly), or the product directly via mandatory security requirements. Both regulations come at a cost---implementing security requirements increases production costs, and the existence of fines decreases consumers' values---thereby reducing the seller's profits. The focus of this paper is to understand the \emph{efficiency} of various regulatory policies. That is, policy A is more efficient than policy B if A more successfully minimizes negatives externalities, while both A and B reduce seller's profits equally. We develop a simple model to capture the impact of regulatory policies on a buyer's behavior. {In this model, we show that for \textit{homogeneous} markets---where the buyer's ability to follow security practices is always high or always low---the optimal (externality-minimizing for a given profit constraint) regulatory policy need regulate \emph{only} payments \emph{or} production.} In arbitrary markets, by contrast, we show that while the optimal policy may require regulating both aspects, there is always an approximately optimal policy which regulates just one.

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arXiv2018

Optimal (and Benchmark-Optimal) Competition Complexity for Additive Buyers over Independent Items

The Competition Complexity of an auction setting refers to the number of additional bidders necessary in order for the (deterministic, prior-independent, dominant strategy truthful) Vickrey-Clarke-Groves mechanism to achieve greater revenue than the (randomized, prior-dependent, Bayesian-truthful) optimal mechanism without the additional bidders. We prove that the competition complexity of $n$ bidders with additive valuations over $m$ independent items is at most $n(\ln(1+m/n)+2)$, and also at most $9\sqrt{nm}$. When $n \leq m$, the first bound is optimal up to constant factors, even when the items are i.i.d. and regular. When $n \geq m$, the second bound is optimal for the benchmark introduced in [EFFTW17a] up to constant factors, even when the items are i.i.d. and regular. We further show that, while the Eden et al. benchmark is not necessarily tight in the $n \geq m$ regime, the competition complexity of $n$ bidders with additive valuations over even $2$ i.i.d. regular items is indeed $ω(1)$. Our main technical contribution is a reduction from analyzing the Eden et al. benchmark to proving stochastic dominance of certain random variables.

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