arXiv subjects

Bhaskar Ray Chaudhury

Publications and source records attributed to Bhaskar Ray Chaudhury.

At least 19 recordsLinked to original sources

arXiv2026

Equilibrium Pricing in Oligopolistic Data Markets

We study equilibrium pricing in oligopolistic data markets with budget-constrained buyers (e.g., machine learning companies purchasing data to improve model accuracy) and strategic data sellers. Sellers compete by setting prices for their datasets, giving rise to a pricing game whose pure Nash equilibria correspond to equilibrium prices. While equilibrium prices are guaranteed for rivalrous goods via competitive equilibrium, we show that the non-rivalry of data fundamentally alters this picture: an exact Nash equilibrium (NE) need not exist, and in fact, 1.363-approximate NE may also not exist under uniform pricing. We therefore investigate relaxed equilibrium notions. Allowing sellers to use beyond-uniform pricing---specifically, piecewise-linear convex pricing functions---guarantees approximate stability within a constant factor: there exists a pricing profile in which no seller can improve revenue by a factor of two by deviating to any uniform price (a 2-approximate NE against uniform deviations). Finally, our simulations demonstrate fast convergence and empirical approximation guarantees that outperform the worst-case bound of 2.

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arXiv2026

A Simple Polynomial-Time EFX Repair for Cancelable Valuations

The leximin++ proof of Plaut and Roughgarden for agents with identical monotone valuations gives a natural EFX-repair procedure: starting from an arbitrary partition, repeatedly transfer an eligible item to a minimum-valued bundle. The procedure terminates, but the standard argument gives no polynomial bound on the number of transfers, even for additive valuations. We show that a single deterministic tie-breaking rule makes this repair procedure polynomial for the broader class of cancelable valuations. Fix an ordering of the items consistent with their singleton values and always transfer the highest-ranked eligible item. Consecutive transferred items strictly decrease in this ordering, and hence the algorithm performs at most $m$ transfers, where $m$ is the number of items. Moreover, the repair procedure does not decrease the minimum bundle value or increase the maximum bundle value. As an application, for every fixed $\varepsilon>0$, we compute in polynomial time an allocation of restricted additive chores that is simultaneously EFX, $(1+\varepsilon)$-MMS, and a $2$-approximation to the optimal social cost. This improves upon the previous polynomial-time $4/3$-MMS guarantee. Finally, we exhibit a monotone cancelable ordering on five items with no additive representation, showing that the extension beyond additivity is genuine.

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arXiv2026

Fully Distributed T\^atonnement for Chores Markets

We study price-adjustment dynamics for computing competitive equilibria (CE) in Fisher markets with chores. Unlike in classical goods markets, prices in chores markets are payments for taking on undesirable tasks, and natural excess-demand dynamics can fail; even the na\"ive analogue of Walrasian t\^atonnement may diverge. Recent work of Chaudhury et al. [2025] overcomes this obstacle via relative t\^atonnement, which subtracts the average excess-demand signal from the excess demand vector. This recovers convergence, but at the cost of coupling the price updates across all chores. This leaves open whether such global coupling is inherent, or whether convergent t\^atonnement can be recovered through a genuinely local update in which each chore reacts only to its own excess demand. We answer this question affirmatively through multiplicative t\^atonnement, a fully distributed dynamics in which each chore price is updated using only its current price and its own excess-demand signal. Although the update contains no explicit normalization term, Walras' law and the multiplicative form of the update implicitly preserve the relevant aggregate price geometry. We prove that multiplicative t\^atonnement converges to a CE in any chores Fisher market with continuous, convex, and $1$-homogeneous (CCH) disutilities. For convex CES disutilities, we further prove an approximate-CE convergence rate with the same $O(1/\varepsilon^2)$ dependence as relative t\^atonnement, but with improved dependence on problem constants. Experiments on real-world and simulated instances show that multiplicative t\^atonnement is substantially faster in practice, often by an order of magnitude.

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arXiv2026

Competitive Equilibrium in Labor Economies through the Lens of Goods and Chores Fisher Markets

In this paper, we study a two-sided labor market that couples the classical Fisher market with goods and the Fisher market with bads into a single unified framework. In our model, users demand tasks in order to derive utility, while workers supply labor to perform these tasks in exchange for earnings. Each task thus plays a dual role: it is a good for the user side of the market and a chore for the worker side. Given prices for tasks, users choose utility-maximizing bundles subject to budgets, while workers choose disutility-minimizing task bundles subject to earning requirements; the resulting choices induce demand and supply endogenously for each task, and a CE corresponds to prices at which these coincide. We show that such markets are guaranteed to admit a CE in a very general setting, and the first and second welfare theorems hold for our labor market model. We next study the computation of equilibria under linear preferences. We show that, similar to the chores setting, equilibria correspond to KKT points of an Eisenberg-Gale-like non-convex program. Despite the non-convex characterization, we go on to show a set of surprisingly positive results. First, we show that there exists a polynomial-time combinatorial algorithm for computing CE, which relies on a natural Walrasian scheme for updating prices. In the "CEEI-like" case, this yields a strongly polynomial-time algorithm. We next show that our market admits a natural dual program, and this non-convex labor-market program admits a change of variables that transforms it into a linear program (albeit with irrational coefficients). Finally, leveraging this LP, we give yet another polynomial-time algorithm while deriving an approach for addressing the irrational coefficients in an efficient manner. We note that, even for goods-only linear Fisher markets, obtaining such an LP formulation remains open.

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arXiv2026

Revenue-Optimal Pricing for Budget-Constrained Buyers in Data Markets

We study revenue-optimal pricing in data markets with rational, budget-constrained buyers. Such a market offers multiple datasets for sale, and buyers aim to improve the accuracy of their prediction tasks by acquiring data bundles. The market's objective is to price datasets to maximize total revenue, considering that buyers with quasi-linear utilities choose their bundles optimally under budget constraints. We allow the buyers to purchase fractions of datasets, and the amount they pay is proportional to the fraction they receive. Although competitive equilibrium gives revenue-optimal pricing in rivalrous markets with quasi-linear buyers, we show that revenue maximization in data markets is APX-hard. Despite the hardness, we design a 2-approximation algorithm when datasets arrive online, and a $(1-1/e)^{-1}$-approximation algorithm for the offline setting.

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arXiv2025

T\^atonnement Dynamics for Fisher Markets with Chores

In this paper, we initiate the study of t\^atonnement dynamics in markets with chores. T\^atonnement is a fundamental market dynamics, capturing how prices evolve when they are adjusted in proportion of their excess demand. While its convergence to a competitive equilibrium (CE) is well understood in goods markets for broad classes of utilities, no analogous results are known for chore markets. Analyzing t\^atonnement in the chores market presents new challenges. Several elegant structural properties that facilitate convergence in goods markets-such as convexity of the equilibrium price set and monotonicity of excess demand under the t\^atonnement price updates-fail to hold in the chore setting. Consistent with these difficulties, we first show that na\"ive t\^atonnement diverges. To overcome this, we propose a modified process called relative t\^atonnement, where prices are updated according to normalized excess demand. We prove its convergence to a CE under suitable step-size choices for a broad class of disutility functions, namely continuous, convex, and 1-homogeneous (CCH) disutilities. This class includes many standard forms such as linear and convex CES disutilities. Our proof proceeds by introducing a nonsmooth, nonconvex yet regular objective function-a generalization of the objective in the Eisenberg-Gale-type dual program introduced by [CKMN24]. For convex CES disutilities, where disutility is the weighted $p$-norm of the individual chore disutilities for $p \in (1, \infty)$, we show that relative t\^atonnement converges to an $\varepsilon$-CE in $O(1/\varepsilon^2)$ iterations. This quadratic convergence rate is established by leveraging the polar gauge (or gauge dual) of the disutility function. Finally, following the framework of [AH58], we analyze the stability of CE and provide a complete characterization of local stability.

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arXiv2025

Human-AI Collaboration with Misaligned Preferences

In many real-life settings, algorithms play the role of assistants, while humans ultimately make the final decision. Often, algorithms specifically act as curators, narrowing down a wide range of options into a smaller subset that the human picks between: consider content recommendation or chatbot responses to questions with multiple valid answers. Crucially, humans may not know their own preferences perfectly either, but instead may only have access to a noisy sampling over preferences. Algorithms can assist humans by curating a smaller subset of items, but must also face the challenge of misalignment: humans may have different preferences from each other (and from the algorithm), and the algorithm may not know the exact preferences of the human they are facing at any point in time. In this paper, we model and theoretically study such a setting. Specifically, we show instances where humans benefit by collaborating with a misaligned algorithm. Surprisingly, we show that humans gain more utility from a misaligned algorithm (which makes different mistakes) than from an aligned algorithm. Next, we build on this result by studying what properties of algorithms maximize human welfare when the goals could be either utilitarian welfare or ensuring all humans benefit. We conclude by discussing implications for designers of algorithmic tools and policymakers.

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arXiv2025

On the Existence and Complexity of Core-Stable Data Exchanges

The rapid growth of data-driven technologies and the emergence of various data-sharing paradigms have underscored the need for efficient and stable data exchange protocols. In any such exchange, agents must carefully balance the benefit of acquiring valuable data against the cost of sharing their own. Ensuring stability in these exchanges is essential to prevent agents -- or groups of agents -- from departing and conducting local (and potentially more favorable) exchanges among themselves. To address this, we study a model where agents participate in a data exchange. Each agent has an associated payoff for the data acquired from other agents and a cost incurred during sharing its own data. The net utility of an agent is payoff minus the cost. We adapt the classical notion of core-stability from cooperative game theory to data exchange. A data exchange is core-stable if no subset of agents has any incentive to deviate to a different exchange. We show that a core-stable data exchange is guaranteed to exist when agents have concave payoff functions and convex cost functions -- a setting typical in domains like PAC learning and random discovery models. We show that relaxing either of the foregoing conditions may result in the nonexistence of core-stable data exchanges. Then, we prove that finding a core-stable exchange is PPAD-hard, even when the potential blocking coalitions are restricted to constant size. To the best of our knowledge, this provides the first known PPAD-hardness result for core-like guarantees in data economics. Finally, we show that data exchange can be modelled as a balanced $n$-person game. This immediately gives a pivoting algorithm via Scarf's theorem \cite{Scarf1967core}. We show that the pivoting algorithm works well in practice through our empirical results.

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arXiv2024

On the Theoretical Foundations of Data Exchange Economies

The immense success of ML systems relies heavily on large-scale, high-quality data. The high demand for data has led to many paradigms that involve selling, exchanging, and sharing data, motivating the study of economic processes with data as an asset. However, data differs from classical economic assets in terms of free duplication: there is no concept of limited supply since it can be replicated at zero marginal cost. This distinction introduces fundamental differences between economic processes involving data and those concerning other assets. We study a parallel to exchange (Arrow-Debreu) markets where data is the asset. Here, agents with datasets exchange data fairly and voluntarily, aiming for mutual benefit without monetary compensation. This framework is particularly relevant for non-profit organizations that seek to improve their ML models through data exchange, yet are restricted from selling their data for profit. We propose a general framework for data exchange, built on two core principles: (i) fairness, ensuring that each agent receives utility proportional to their contribution to others; contributions are quantifiable using standard credit-sharing functions like the Shapley value, and (ii) stability, ensuring that no coalition of agents can identify an exchange among themselves which they unanimously prefer to the current exchange. We show that fair and stable exchanges exist for all monotone continuous utility functions. Next, we investigate the computational complexity of finding approximate fair and stable exchanges. We present a local search algorithm for instances with monotone submodular utility functions, where each agent contributions are measured using the Shapley value. We prove that this problem lies in CLS under mild assumptions. Our framework opens up several intriguing theoretical directions for research in data economics.

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arXiv2024

Competitive Equilibrium for Chores: from Dual Eisenberg-Gale to a Fast, Greedy, LP-based Algorithm

We study the computation of competitive equilibrium for Fisher markets with $n$ agents and $m$ divisible chores. Competitive equilibria for chores are known to correspond to the nonzero KKT points of a program that minimizes the product of agent disutilities, which is a non-convex program whose zero points foil iterative optimization methods. We introduce a dual-like analogue of this program, and show that a simple modification to our "dual" program avoids such zero points, while retaining the correspondence between KKT points and competitive equilibria. This allows, for the first time ever, application of iterative optimization methods over a convex region for computing competitive equilibria for chores. We next introduce a greedy Frank-Wolfe algorithm for optimization over our program and show a new state-of-the-art convergence rate to competitive equilibrium. Moreover, our method is significantly simpler than prior methods: each iteration of our method only requires solving a simple linear program. We show through numerical experiments that our method is extremely practical: it easily solves every instance we tried, including instances with hundreds of agents and up to 1000 chores, usually in 10-30 iterations, is simple to implement, and has no numerical issues.

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arXiv2023

Fair and Efficient Allocation of Indivisible Chores with Surplus

We study fair division of indivisible chores among $n$ agents with additive disutility functions. Two well-studied fairness notions for indivisible items are envy-freeness up to one/any item (EF1/EFX) and the standard notion of economic efficiency is Pareto optimality (PO). There is a noticeable gap between the results known for both EF1 and EFX in the goods and chores settings. The case of chores turns out to be much more challenging. We reduce this gap by providing slightly relaxed versions of the known results on goods for the chores setting. Interestingly, our algorithms run in polynomial time, unlike their analogous versions in the goods setting. We introduce the concept of $k$ surplus which means that up to $k$ more chores are allocated to the agents and each of them is a copy of an original chore. We present a polynomial-time algorithm which gives EF1 and PO allocations with $(n-1)$ surplus. We relax the notion of EFX slightly and define tEFX which requires that the envy from agent $i$ to agent $j$ is removed upon the transfer of any chore from the $i$'s bundle to $j$'s bundle. We give a polynomial-time algorithm that in the chores case for $3$ agents returns an allocation which is either proportional or tEFX. Note that proportionality is a very strong criterion in the case of indivisible items, and hence both notions we guarantee are desirable.

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arXiv2022

EFX Allocations: Simplifications and Improvements

The existence of EFX allocations is a fundamental open problem in discrete fair division. Given a set of agents and indivisible goods, the goal is to determine the existence of an allocation where no agent envies another following the removal of any single good from the other agent's bundle. Since the general problem has been illusive, progress is made on two fronts: $(i)$ proving existence when the number of agents is small, $(ii)$ proving existence of relaxations of EFX. In this paper, we improve results on both fronts (and simplify in one of the cases). We prove the existence of EFX allocations with three agents, restricting only one agent to have an MMS-feasible valuation function (a strict generalization of nice-cancelable valuation functions introduced by Berger et al. which subsumes additive, budget-additive and unit demand valuation functions). The other agents may have any monotone valuation functions. Our proof technique is significantly simpler and shorter than the proof by Chaudhury et al. on existence of EFX allocations when there are three agents with additive valuation functions and therefore more accessible. Secondly, we consider relaxations of EFX allocations, namely, approximate-EFX allocations and EFX allocations with few unallocated goods (charity). Chaudhury et al. showed the existence of $(1-ε)$-EFX allocation with $O((n/ε)^{\frac{4}{5}})$ charity by establishing a connection to a problem in extremal combinatorics. We improve their result and prove the existence of $(1-ε)$-EFX allocations with $\tilde{O}((n/ ε)^{\frac{1}{2}})$ charity. In fact, some of our techniques can be used to prove improved upper-bounds on a problem in zero-sum combinatorics introduced by Alon and Krivelevich.

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arXiv2022

Fairness in Federated Learning via Core-Stability

Federated learning provides an effective paradigm to jointly optimize a model benefited from rich distributed data while protecting data privacy. Nonetheless, the heterogeneity nature of distributed data makes it challenging to define and ensure fairness among local agents. For instance, it is intuitively "unfair" for agents with data of high quality to sacrifice their performance due to other agents with low quality data. Currently popular egalitarian and weighted equity-based fairness measures suffer from the aforementioned pitfall. In this work, we aim to formally represent this problem and address these fairness issues using concepts from co-operative game theory and social choice theory. We model the task of learning a shared predictor in the federated setting as a fair public decision making problem, and then define the notion of core-stable fairness: Given $N$ agents, there is no subset of agents $S$ that can benefit significantly by forming a coalition among themselves based on their utilities $U_N$ and $U_S$ (i.e., $\frac{|S|}{N} U_S \geq U_N$). Core-stable predictors are robust to low quality local data from some agents, and additionally they satisfy Proportionality and Pareto-optimality, two well sought-after fairness and efficiency notions within social choice. We then propose an efficient federated learning protocol CoreFed to optimize a core stable predictor. CoreFed determines a core-stable predictor when the loss functions of the agents are convex. CoreFed also determines approximate core-stable predictors when the loss functions are not convex, like smooth neural networks. We further show the existence of core-stable predictors in more general settings using Kakutani's fixed point theorem. Finally, we empirically validate our analysis on two real-world datasets, and we show that CoreFed achieves higher core-stability fairness than FedAvg while having similar accuracy.

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arXiv2022

Maximizing Nash Social Welfare in 2-Value Instances: Delineating Tractability

We study the problem of allocating a set of indivisible goods among a set of agents with \emph{2-value additive valuations}. In this setting, each good is valued either $1$ or $p/q$, for some fixed co-prime numbers $p,q\in \mathbb{N}$ such that $1\leq q < p$. Our goal is to find an allocation maximizing the \emph{Nash social welfare} (\NSW), i.e., the geometric mean of the valuations of the agents. In this work, we give a complete characterization of polynomial-time tractability of \NSW\ maximization that solely depends on the values of $q$. We start by providing a rather simple polynomial-time algorithm to find a maximum \NSW\ allocation when the valuation functions are \emph{integral}, that is, $q=1$. We then exploit more involved techniques to get an algorithm producing a maximum \NSW\ allocation for the \emph{half-integral} case, that is, $q=2$. Finally, we show it is \classNP-hard to compute an allocation with maximum \NSW\ whenever $q\geq3$.

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arXiv2022

Competitive Equilibrium with Chores: Combinatorial Algorithm and Hardness

We study the computational complexity of finding a competitive equilibrium (CE) with chores when agents have linear preferences. CE is one of the most preferred mechanisms for allocating a set of items among agents. CE with equal incomes (CEEI), Fisher, and Arrow-Debreu (exchange) are the fundamental economic models to study allocation problems, where CEEI is a special case of Fisher and Fisher is a special case of exchange. When the items are goods (giving utility), the CE set is convex even in the exchange model, facilitating several combinatorial polynomial-time algorithms (starting with the seminal work of Devanur, Papadimitriou, Saberi and Vazirani) for all of these models. In sharp contrast, when the items are chores (giving disutility), the CE set is known to be non-convex and disconnected even in the CEEI model. Further, no combinatorial algorithms or hardness results are known for these models. In this paper, we give two main results for CE with chores: 1) A combinatorial algorithm to compute a $(1-\varepsilon)$-approximate CEEI in time $\tilde{\mathcal{O}}(n^4m^2 / \varepsilon^2)$, where $n$ is the number of agents and $m$ is the number of chores. 2) PPAD-hardness of finding a $(1-1/\mathit{poly}(n))$-approximate CE in the exchange model under a sufficient condition. To the best of our knowledge, these results show the first separation between the CEEI and exchange models when agents have linear preferences, assuming PPAD $\neq $ P. Finally, we show that our new insight implies a straightforward proof of the existence of an allocation that is both envy-free up to one chore (EF1) and Pareto optimal (PO) in the discrete setting when agents have factored bivalued preferences.

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arXiv2021

Nash Social Welfare for 2-value Instances

This paper is merged with arXiv:2107.08965v2. We refer the reader to the full and updated version. We study the problem of allocating a set of indivisible goods among agents with 2-value additive valuations. Our goal is to find an allocation with maximum Nash social welfare, i.e., the geometric mean of the valuations of the agents. We give a polynomial-time algorithm to find a Nash social welfare maximizing allocation when the valuation functions are integrally 2-valued, i.e., each agent has a value either $1$ or $p$ for each good, for some positive integer $p$. We then extend our algorithm to find a better approximation factor for general 2-value instances.

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arXiv2021

Maximizing Nash Social Welfare in 2-Value Instances

We consider the problem of maximizing the Nash social welfare when allocating a set $\mathcal{G}$ of indivisible goods to a set $\mathcal{N}$ of agents. We study instances, in which all agents have 2-value additive valuations: The value of every agent $i \in \mathcal{N}$ for every good $j \in \mathcal{G}$ is $v_{ij} \in \{p,q\}$, for $p,q \in \mathbb{N}$, $p \le q$. Maybe surprisingly, we design an algorithm to compute an optimal allocation in polynomial time if $p$ divides $q$, i.e., when $p=1$ and $q \in \mathbb{N}$ after appropriate scaling. The problem is \classNP-hard whenever $p$ and $q$ are coprime and $p \ge 3$. In terms of approximation, we present positive and negative results for general $p$ and $q$. We show that our algorithm obtains an approximation ratio of at most 1.0345. Moreover, we prove that the problem is \classAPX-hard, with a lower bound of $1.000015$ achieved at $p/q = 4/5$.

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arXiv2021

Polynomial Time Algorithms to Find an Approximate Competitive Equilibrium for Chores

Competitive equilibrium with equal income (CEEI) is considered one of the best mechanisms to allocate a set of items among agents fairly and efficiently. In this paper, we study the computation of CEEI when items are chores that are disliked (negatively valued) by agents, under 1-homogeneous and concave utility functions which includes linear functions as a subcase. It is well-known that, even with linear utilities, the set of CEEI may be non-convex and disconnected, and the problem is PPAD-hard in the more general exchange model. In contrast to these negative results, we design FPTAS: A polynomial-time algorithm to compute $ε$-approximate CEEI where the running-time depends polynomially on $1/ε$. Our algorithm relies on the recent characterization due to Bogomolnaia et al.~(2017) of the CEEI set as exactly the KKT points of a non-convex minimization problem that have all coordinates non-zero. Due to this non-zero constraint, naive gradient-based methods fail to find the desired local minima as they are attracted towards zero. We develop an exterior-point method that alternates between guessing non-zero KKT points and maximizing the objective along supporting hyperplanes at these points. We show that this procedure must converge quickly to an approximate KKT point which then can be mapped to an approximate CEEI; this exterior point method may be of independent interest. When utility functions are linear, we give explicit procedures for finding the exact iterates, and as a result show that a stronger form of approximate CEEI can be found in polynomial time. Finally, we note that our algorithm extends to the setting of un-equal incomes (CE), and to mixed manna with linear utilities where each agent may like (positively value) some items and dislike (negatively value) others.

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