arXiv subjects

S. Matthew Weinberg

Publications and source records attributed to S. Matthew Weinberg.

At least 37 records · Page 2Linked to original sources

arXiv2024

Undetectable Selfish Mining

Seminal work of Eyal and Sirer (2014) establishes that a strategic Bitcoin miner may strictly profit by deviating from the intended Bitcoin protocol, using a strategy now termed *selfish mining*. More specifically, any miner with $>1/3$ of the total hashrate can earn bitcoin at a faster rate by selfish mining than by following the intended protocol (depending on network conditions, a lower fraction of hashrate may also suffice). One convincing critique of selfish mining in practice is that the presence of a selfish miner is *statistically detectable*: the pattern of orphaned blocks created by the presence of a selfish miner cannot be explained by natural network delays. Therefore, if an attacker chooses to selfish mine, users can detect this, and this may (significantly) negatively impact the value of BTC. So while the attacker may get slightly more bitcoin by selfish mining, these bitcoin may be worth significantly less USD. We develop a selfish mining variant that is provably *statistically undetectable*: the pattern of orphaned blocks is statistically identical to a world with only honest miners but higher network delay. Specifically, we consider a stylized model where honest miners with network delay produce orphaned blocks at each height independently with probability $β'$. We propose a selfish mining strategy that instead produces orphaned blocks at each height independently with probability $β> β'$. We further show that our strategy is strictly profitable for attackers with $38.2\% \ll 50\%$ of the total hashrate (and this holds for all natural orphan rates $β'$).

cs.GT↗

arXiv2024

Contracting with a Learning Agent

Many real-life contractual relations differ completely from the clean, static model at the heart of principal-agent theory. Typically, they involve repeated strategic interactions of the principal and agent, taking place under uncertainty and over time. While appealing in theory, players seldom use complex dynamic strategies in practice, often preferring to circumvent complexity and approach uncertainty through learning. We initiate the study of repeated contracts with a learning agent, focusing on agents who achieve no-regret outcomes. Optimizing against a no-regret agent is a known open problem in general games; we achieve an optimal solution to this problem for a canonical contract setting, in which the agent's choice among multiple actions leads to success/failure. The solution has a surprisingly simple structure: for some $α> 0$, initially offer the agent a linear contract with scalar $α$, then switch to offering a linear contract with scalar $0$. This switch causes the agent to ``free-fall'' through their action space and during this time provides the principal with non-zero reward at zero cost. Despite apparent exploitation of the agent, this dynamic contract can leave \emph{both} players better off compared to the best static contract. Our results generalize beyond success/failure, to arbitrary non-linear contracts which the principal rescales dynamically. Finally, we quantify the dependence of our results on knowledge of the time horizon, and are the first to address this consideration in the study of strategizing against learning agents.

cs.GT↗

arXiv2023

Simple and Optimal Online Contention Resolution Schemes for $k$-Uniform Matroids

We provide a simple $(1-O(\frac{1}{\sqrt{k}}))$-selectable Online Contention Resolution Scheme for $k$-uniform matroids against a fixed-order adversary. If $A_i$ and $G_i$ denote the set of selected elements and the set of realized active elements among the first $i$ (respectively), our algorithm selects with probability $1-\frac{1}{\sqrt{k}}$ any active element $i$ such that $|A_{i-1}| + 1 \leq (1-\frac{1}{\sqrt{k}})\cdot \mathbb{E}[|G_i|]+\sqrt{k}$. This implies a $(1-O(\frac{1}{\sqrt{k}}))$ prophet inequality against fixed-order adversaries for $k$-uniform matroids that is considerably simpler than previous algorithms [Ala14, AKW14, JMZ22]. We also prove that no OCRS can be $(1-Ω(\sqrt{\frac{\log k}{k}}))$-selectable for $k$-uniform matroids against an almighty adversary. This guarantee is matched by the (known) simple greedy algorithm that accepts every active element with probability $1-Θ(\sqrt{\frac{\log k}{k}})$ [HKS07].

cs.DS↗

arXiv2023

Optimal Stopping with Multi-Dimensional Comparative Loss Aversion

Despite having the same basic prophet inequality setup and model of loss aversion, conclusions in our multi-dimensional model differs considerably from the one-dimensional model of Kleinberg et al. For example, Kleinberg et al. gives a tight closed-form on the competitive ratio that an online decision-maker can achieve as a function of $λ$, for any $λ\geq 0$. In our multi-dimensional model, there is a sharp phase transition: if $k$ denotes the number of dimensions, then when $λ\cdot (k-1) \geq 1$, no non-trivial competitive ratio is possible. On the other hand, when $λ\cdot (k-1) < 1$, we give a tight bound on the achievable competitive ratio (similar to Kleinberg et al.). As another example, Kleinberg et al. uncovers an exponential improvement in their competitive ratio for the random-order vs. worst-case prophet inequality problem. In our model with $k\geq 2$ dimensions, the gap is at most a constant-factor. We uncover several additional key differences in the multi- and single-dimensional models.

cs.GT↗

arXiv2023

Selling to Multiple No-Regret Buyers

We consider the problem of repeatedly auctioning a single item to multiple i.i.d buyers who each use a no-regret learning algorithm to bid over time. In particular, we study the seller's optimal revenue, if they know that the buyers are no-regret learners (but only that their behavior satisfies some no-regret property -- they do not know the precise algorithm/heuristic used). Our main result designs an auction that extracts revenue equal to the full expected welfare whenever the buyers are "mean-based" (a property satisfied by standard no-regret learning algorithms such as Multiplicative Weights, Follow-the-Perturbed-Leader, etc.). This extends a main result of [BMSW18] which held only for a single buyer. Our other results consider the case when buyers are mean-based but never overbid. On this front, [BMSW18] provides a simple LP formulation for the revenue-maximizing auction for a single-buyer. We identify several formal barriers to extending this approach to multiple buyers.

cs.GT↗

arXiv2023

q-Partitioning Valuations: Exploring the Space Between Subadditive and Fractionally Subadditive Valuations

For a set $M$ of $m$ elements, we define a decreasing chain of classes of normalized monotone-increasing valuation functions from $2^M$ to $\mathbb{R}_{\geq 0}$, parameterized by an integer $q \in [2,m]$. For a given $q$, we refer to the class as $q$-partitioning. A valuation function is subadditive if and only if it is $2$-partitioning, and fractionally subadditive if and only if it is $m$-partitioning. Thus, our chain establishes an interpolation between subadditive and fractionally subadditive valuations. We show that this interpolation is smooth, interpretable , and non-trivial. We interpolate prior results that separate subadditive and fractionally subadditive for all $q \in \{2,\ldots, m\}$. Two highlights are the following:(i) An $Ω\left(\frac{\log \log q}{\log \log m}\right)$-competitive posted price mechanism for $q$-partitioning valuations. Note that this matches asymptotically the state-of-the-art for both subadditive ($q=2$) [DKL20], and fractionally subadditive ($q=m$) [FGL15]. (ii)Two upper-tail concentration inequalities on $1$-Lipschitz, $q$-partitioning valuations over independent items. One extends the state-of-the-art for $q=m$ to $q 2$. Our concentration inequalities imply several corollaries that interpolate between subadditive and fractionally subadditive, for example: $\mathbb{E}[v(S)]\le (1 + 1/\log q)\text{Median}[v(S)] + O(\log q)$. To prove this, we develop a new isoperimetric inequality using Talagrand's method of control by $q$ points, which may be of independent interest. We also discuss other probabilistic inequalities and game-theoretic applications of $q$-partitioning valuations, and connections to subadditive MPH-$k$ valuations [EFNTW19].

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arXiv2023

Tight Bounds on 3-Team Manipulations in Randomized Death Match

Consider a round-robin tournament on n teams, where a winner must be (possibly randomly) selected as a function of the results from the ${n \choose 2}$ pairwise matches. A tournament rule is said to be k-SNM-$α$ if no set of k teams can ever manipulate the ${k \choose 2}$ pairwise matches between them to improve 2 the joint probability that one of these k teams wins by more than $α$. Prior work identifies multiple simple tournament rules that are 2-SNM-1/3 (Randomized Single Elimination Bracket [SSW17], Randomized King of the Hill [SWZZ20], Randomized Death Match [DW21]), which is optimal for k = 2 among all Condorcet-consistent rules (that is, rules that select an undefeated team with probability 1). Our main result establishes that Randomized Death Match is 3-SNM-(31/60), which is tight (for Randomized Death Match). This is the first tight analysis of any Condorcet-consistent tournament rule and at least three manipulating teams. Our proof approach is novel in this domain: we explicitly find the most-manipulable tournament, and directly show that no other tournament can be more manipulable. In addition to our main result, we establish that Randomized Death Match disincentivizes Sybil attacks (where a team enters multiple copies of themselves into the tournament, and arbitrarily manipulates the outcomes of matches between their copies). Specifically, for any tournament, and any team u that is not a Condorcet winner, the probability that u or one of its Sybils wins in Randomized Death Match approaches 0 as the number of Sybils approaches $\infty$.

cs.GT↗

arXiv2022

An Improved Lower Bound for Matroid Intersection Prophet Inequalities

We consider prophet inequalities subject to feasibility constraints that are the intersection of $q$ matroids. The best-known algorithms achieve a $Θ(q)$-approximation, even when restricted to instances that are the intersection of $q$ partition matroids, and with i.i.d.~Bernoulli random variables. The previous best-known lower bound is $Θ(\sqrt{q})$ due to a simple construction of [Kleinberg-Weinberg STOC 2012] (which uses i.i.d.~Bernoulli random variables, and writes the construction as the intersection of partition matroids). We establish an improved lower bound of $q^{1/2+Ω(1/\log \log q)}$ by writing the construction of [Kleinberg-Weinberg STOC 2012] as the intersection of asymptotically fewer partition matroids. We accomplish this via an improved upper bound on the product dimension of a graph with $p^p$ disjoint cliques of size $p$, using recent techniques developed in [Alon-Alweiss European Journal of Combinatorics 2020].

cs.GT↗

arXiv2022

Optimal Strategic Mining Against Cryptographic Self-Selection in Proof-of-Stake

Cryptographic Self-Selection is a subroutine used to select a leader for modern proof-of-stake consensus protocols, such as Algorand. In cryptographic self-selection, each round $r$ has a seed $Q_r$. In round $r$, each account owner is asked to digitally sign $Q_r$, hash their digital signature to produce a credential, and then broadcast this credential to the entire network. A publicly-known function scores each credential in a manner so that the distribution of the lowest scoring credential is identical to the distribution of stake owned by each account. The user who broadcasts the lowest-scoring credential is the leader for round $r$, and their credential becomes the seed $Q_{r+1}$. Such protocols leave open the possibility of a selfish-mining style attack: a user who owns multiple accounts that each produce low-scoring credentials in round $r$ can selectively choose which ones to broadcast in order to influence the seed for round $r+1$. Indeed, the user can pre-compute their credentials for round $r+1$ for each potential seed, and broadcast only the credential (among those with a low enough score to be the leader) that produces the most favorable seed. We consider an adversary who wishes to maximize the expected fraction of rounds in which an account they own is the leader. We show such an adversary always benefits from deviating from the intended protocol, regardless of the fraction of the stake controlled. We characterize the optimal strategy; first by proving the existence of optimal positive recurrent strategies whenever the adversary owns last than $38\%$ of the stake. Then, we provide a Markov Decision Process formulation to compute the optimal strategy.

cs.CR↗

arXiv2022

Credible, Strategyproof, Optimal, and Bounded Expected-Round Single-Item Auctions for all Distributions

We consider a revenue-maximizing seller with a single item for sale to multiple buyers with i.i.d. valuations. Akbarpour and Li (2020) show that the only optimal, credible, strategyproof auction is the ascending price auction with reserves which has unbounded communication complexity. Recent work of Ferreira and Weinberg (2020) circumvents their impossibility result assuming the existence of cryptographically secure commitment schemes, and designs a two-round credible, strategyproof, optimal auction. However, their auction is only credible when buyers' valuations are MHR or $α$-strongly regular: they show their auction might not be credible even when there is a single buyer drawn from a non-MHR distribution. In this work, under the same cryptographic assumptions, we identify a new single-item auction that is credible, strategyproof, revenue optimal, and terminates in constant rounds in expectation for all distributions with finite monopoly price.

cs.GT↗

arXiv2022

On Infinite Separations Between Simple and Optimal Mechanisms

We consider a revenue-maximizing seller with $k$ heterogeneous items for sale to a single additive buyer, whose values are drawn from a known, possibly correlated prior $\mathcal{D}$. It is known that there exist priors $\mathcal{D}$ such that simple mechanisms -- those with bounded menu complexity -- extract an arbitrarily small fraction of the optimal revenue. This paper considers the opposite direction: given a correlated distribution $\mathcal{D}$ witnessing an infinite separation between simple and optimal mechanisms, what can be said about $\mathcal{D}$? Previous work provides a framework for constructing such $\mathcal{D}$: it takes as input a sequence of $k$-dimensional vectors satisfying some geometric property, and produces a $\mathcal{D}$ witnessing an infinite gap. Our first main result establishes that this framework is without loss: every $\mathcal{D}$ witnessing an infinite separation could have resulted from this framework. Even earlier work provided a more streamlined framework. Our second main result establishes that this restrictive framework is not tight. That is, we provide an instance $\mathcal{D}$ witnessing an infinite gap, but which provably could not have resulted from the restrictive framework. As a corollary, we discover a new kind of mechanism which can witness these infinite separations on instances where the previous ''aligned'' mechanisms do not.

cs.GT↗

arXiv2021

Proof-of-Stake Mining Games with Perfect Randomness

Proof-of-Stake blockchains based on a longest-chain consensus protocol are an attractive energy-friendly alternative to the Proof-of-Work paradigm. However, formal barriers to "getting the incentives right" were recently discovered, driven by the desire to use the blockchain itself as a source of pseudorandomness \cite{brown2019formal}. We consider instead a longest-chain Proof-of-Stake protocol with perfect, trusted, external randomness (e.g. a randomness beacon). We produce two main results. First, we show that a strategic miner can strictly outperform an honest miner with just $32.5\%$ of the total stake. Note that a miner of this size {\em cannot} outperform an honest miner in the Proof-of-Work model. This establishes that even with access to a perfect randomness beacon, incentives in Proof-of-Work and Proof-of-Stake longest-chain protocols are fundamentally different. Second, we prove that a strategic miner cannot outperform an honest miner with $30.8\%$ of the total stake. This means that, while not quite as secure as the Proof-of-Work regime, desirable incentive properties of Proof-of-Work longest-chain protocols can be approximately recovered via Proof-of-Stake with a perfect randomness beacon. The space of possible strategies in a Proof-of-Stake mining game is {\em significantly} richer than in a Proof-of-Work game. Our main technical contribution is a characterization of potentially optimal strategies for a strategic miner, and in particular, a proof that the corresponding infinite-state MDP admits an optimal strategy that is positive recurrent.

cs.GT↗

arXiv2021

Formal Barriers to Simple Algorithms for the Matroid Secretary Problem

Babaioff et al. [BIK2007] introduced the matroid secretary problem in 2007, a natural extension of the classic single-choice secretary problem to matroids, and conjectured that a constant-competitive online algorithm exists. The conjecture still remains open despite substantial partial progress, including constant-competitive algorithms for numerous special cases of matroids, and an $O(\log \log \text{rank})$-competitive algorithm in the general case. Many of these algorithms follow principled frameworks. The limits of these frameworks are previously unstudied, and prior work establishes only that a handful of particular algorithms cannot resolve the matroid secretary conjecture. We initiate the study of impossibility results for frameworks to resolve this conjecture. We establish impossibility results for a natural class of greedy algorithms and for randomized partition algorithms, both of which contain known algorithms that resolve special cases.

cs.DS↗

arXiv2021

Optimal Multi-Dimensional Mechanisms are not Locally-Implementable

We introduce locality: a new property of multi-bidder auctions that formally separates the simplicity of optimal single-dimensional multi-bidder auctions from the complexity of optimal multi-dimensional multi-bidder auctions. Specifically, consider the revenue-optimal, Bayesian Incentive Compatible auction for buyers with valuations drawn from $\vec{D}:=\times_i D_i$, where each distribution has support-size $n$. This auction takes as input a valuation profile $\vec{v}$ and produces as output an allocation of the items and prices to charge, $Opt_{\vec{D}}(\vec{v})$. When each $D_i$ is single-dimensional, this mapping is locally-implementable: defining each input $v_i$ requires $Θ(\log n)$ bits, and $Opt_{\vec{D}}(\vec{v})$ can be fully determined using just $Θ(\log n)$ bits from each $D_i$. This follows immediately from Myerson's virtual value theory [Mye81]. Our main result establishes that optimal multi-dimensional mechanisms are not locally-implementable: in order to determine the output $Opt_{\vec{D}}(\vec{v})$ on one particular input $\vec{v}$, one still needs to know (essentially) the entire distribution $\vec{D}$. Formally, $Ω(n)$ bits from each $D_i$ is necessary: (essentially) enough to fully describe $D_i$, and exponentially more than the $Θ(\log n)$ needed to define the input $v_i$. We show that this phenomenon already occurs with just two bidders, even when one bidder is single-dimensional, and when the other bidder is barely multi-dimensional. More specifically, the multi-dimensional bidder is ``inter-dimensional'' from the FedEx setting with just two days [FGKK16]. Our techniques are fairly robust: we additionally establish that optimal mechanisms for single-dimensional buyers with budget constraints are not locally-implementable. This occurs with just two bidders, even when one has no budget constraint, and even when the other's budget is public.

cs.GT↗

arXiv2021

A Permutation-Equivariant Neural Network Architecture For Auction Design

Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. Theoretical approaches to the problem have hit some limits in the past decades and analytical solutions are known for only a few simple settings. Computational approaches to the problem through the use of LPs have their own set of limitations. Building on the success of deep learning, a new approach was recently proposed by Duetting et al. (2019) in which the auction is modeled by a feed-forward neural network and the design problem is framed as a learning problem. The neural architectures used in that work are general purpose and do not take advantage of any of the symmetries the problem could present, such as permutation equivariance. In this work, we consider auction design problems that have permutation-equivariant symmetry and construct a neural architecture that is capable of perfectly recovering the permutation-equivariant optimal mechanism, which we show is not possible with the previous architecture. We demonstrate that permutation-equivariant architectures are not only capable of recovering previous results, they also have better generalization properties.

cs.GT↗

arXiv2021

Auction learning as a two-player game

Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. While theoretical approaches to the problem have hit some limits, a recent research direction initiated by Duetting et al. (2019) consists in building neural network architectures to find optimal auctions. We propose two conceptual deviations from their approach which result in enhanced performance. First, we use recent results in theoretical auction design (Rubinstein and Weinberg, 2018) to introduce a time-independent Lagrangian. This not only circumvents the need for an expensive hyper-parameter search (as in prior work), but also provides a principled metric to compare the performance of two auctions (absent from prior work). Second, the optimization procedure in previous work uses an inner maximization loop to compute optimal misreports. We amortize this process through the introduction of an additional neural network. We demonstrate the effectiveness of our approach by learning competitive or strictly improved auctions compared to prior work. Both results together further imply a novel formulation of Auction Design as a two-player game with stationary utility functions.

cs.GT↗

arXiv2021

Binary Scoring Rules that Incentivize Precision

All proper scoring rules incentivize an expert to predict \emph{accurately} (report their true estimate), but not all proper scoring rules equally incentivize \emph{precision}. Rather than treating the expert's belief as exogenously given, we consider a model where a rational expert can endogenously refine their belief by repeatedly paying a fixed cost, and is incentivized to do so by a proper scoring rule. Specifically, our expert aims to predict the probability that a biased coin flipped tomorrow will land heads, and can flip the coin any number of times today at a cost of $c$ per flip. Our first main result defines an \emph{incentivization index} for proper scoring rules, and proves that this index measures the expected error of the expert's estimate (where the number of flips today is chosen adaptively to maximize the predictor's expected payoff). Our second main result finds the unique scoring rule which optimizes the incentivization index over all proper scoring rules. We also consider extensions to minimizing the $\ell^{th}$ moment of error, and again provide an incentivization index and optimal proper scoring rule. In some cases, the resulting scoring rule is differentiable, but not infinitely differentiable. In these cases, we further prove that the optimum can be uniformly approximated by polynomial scoring rules. Finally, we compare common scoring rules via our measure, and include simulations confirming the relevance of our measure even in domains outside where it provably applies.

cs.GT↗

arXiv2021

The Sample Complexity of Up-to-$\varepsilon$ Multi-Dimensional Revenue Maximization

We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of $n$ additive bidders whose values for $m$ heterogeneous items are drawn independently. For any such instance and any $\varepsilon>0$, we show that it is possible to learn an $\varepsilon$-Bayesian Incentive Compatible auction whose expected revenue is within $\varepsilon$ of the optimal $\varepsilon$-BIC auction from only polynomially many samples. Our fully nonparametric approach is based on ideas that hold quite generally, and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive, and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well-understood, our corollary for this case extends slightly the state-of-the-art.

cs.GT↗