arXiv subjects

Vijay V. Vazirani

Publications and source records attributed to Vijay V. Vazirani.

At least 19 recordsLinked to original sources

arXiv2026

Efficiently Restructuring Sovereign Debt via Arctic Auctions with Convex Costs

We study the problem of computing competitive equilibria in the Arctic product-mix auction, originally developed for the Icelandic government for exchanging blocked financial accounts, and more recently proposed by IMF staff for sovereign debt restructuring. From the buyers' perspective, the Arctic auction is equivalent to the quasi-linear Fisher market. However, unlike the standard Fisher model, the seller can express rich supply preferences through explicit supply-side costs and constraints. Despite extensive algorithmic literature on Fisher markets, the seller side has not received much attention, and no polynomial-time algorithm was previously known for computing competitive equilibrium when sellers face nontrivial costs. We examine the natural and expressive regime of separable, stepwise-increasing marginal costs that underlie the above-stated applications. Using polyhedral theory techniques, we first show that rational inputs lead to rational-valued competitive equilibria. Motivated by this result, we develop the first polynomial-time algorithm for this setting based on a non-trivial extension of classic primal-dual balanced-flow techniques for linear Fisher markets. Our work provides a robust computational foundation for auctions with sophisticated preferences, paving the way for flexible and institutionally feasible market designs in global finance.

cs.GT↗

arXiv2026

Arctic Auctions, Linear Fisher Markets, and Rational Convex Programs

This paper unifies two foundational constructs from economics and algorithmic game theory, the Arctic Auction and the linear Fisher market, to address the efficient allocation of differentiated goods in complex markets. Our main contributions are showing that an equilibrium for the Arctic Auction is captured by a Rational Convex Program, and deriving the first combinatorial polynomial-time algorithm for computing Arctic Auction equilibria.

cs.GT↗

arXiv2026

A Generalization of von Neumann's Reduction from the Assignment Problem to Zero-Sum Games

The equivalence between von Neumann's Minimax Theorem for zero-sum games and the LP Duality Theorem connects cornerstone problems of the two fields of game theory and optimization, respectively, and has been the subject of intense scrutiny for seven decades. Yet, as observed in this paper, the proof of the difficult direction of this equivalence is unsatisfactory: It does not assign distinct roles to the two players of the game, as is natural from the definition of a zero-sum game. In retrospect, a partial resolution to this predicament was provided in another brilliant paper of von Neumann, which reduced the assignment problem to zero-sum games. However, the underlying LP is highly specialized; all entries of its objective function vector are strictly positive, the constraint vector is all ones, and the constraint matrix is 0/1. We generalize von Neumann's result along two directions, each allowing negative entries in certain parts of the LP. Our reductions make explicit the roles of the two players of the reduced game, namely their maximin strategies are to play optimal solutions to the primal and dual LPs. Furthermore, unlike previous reductions, the value of the reduced game reveals the value of the given LP. Our generalizations encompass several basic economic scenarios.

cs.GT↗

arXiv2026

Robust Stable Matchings: Dealing with Changes in Preferences

We study stable matchings that are robust to preference changes in the two-sided stable matching setting of Gale and Shapley [GS62]. Given two instances $A$ and $B$ on the same set of agents, a matching is said to be robust if it is stable under both instances. This notion captures desirable robustness properties in matching markets where preferences may evolve, be misreported, or be subject to uncertainty. While the classical theory of stable matchings reveals rich lattice, algorithmic, and polyhedral structure for a single instance, it is unclear which of these properties persist when stability is required across multiple instances. Our work initiates a systematic study of the structural and computational behavior of robust stable matchings under increasingly general models of preference changes. We analyze robustness under a hierarchy of perturbation models: 1. a single upward shift in one agent's preference list, 2. an arbitrary permutation change by a single agent, and 3. arbitrary preference changes by multiple agents on both sides. For each regime, we characterize when: 1. the set of robust stable matchings forms a sublattice, 2. the lattice of robust stable matchings admits a succinct Birkhoff partial order enabling efficient enumeration, 3. worker-optimal and firm-optimal robust stable matchings can be computed efficiently, and 4. the robust stable matching polytope is integral (by studying its LP formulation). We provide explicit counterexamples demonstrating where these structural and geometric properties break down, and complement these results with XP-time algorithms running in $O(n^k)$ time, parameterized by $k$, the number of agents whose preferences change. Our results precisely delineate the boundary between tractable and intractable cases for robust stable matchings.

cs.GT↗

arXiv2025

Stable Matching: Dealing with Changes in Preferences

We study stable matchings that are robust to preference changes in the two-sided stable matching setting of Gale and Shapley[GS62]. Given two instances $A$ and $B$ on the same set of agents, a matching is said to be robust if it is stable under both instances. While prior work has considered the case where a single agent changes preferences between $A$ and $B$, we allow multiple agents on both sides to update their preferences and ask whether three central properties of stable matchings extend to robust stable matchings: (i) Can a robust stable matching be found in polynomial time? (ii) Does the set of robust stable matchings form a lattice? (iii) Is the fractional robust stable matching polytope integral? We show that all three properties hold when any number of agents on one side change preferences, as long as at most one agent on the other side does. For the case where two or more agents on both sides change preferences, we construct examples showing that both the lattice structure and polyhedral integrality fail-identifying this setting as a sharp threshold. We also present an XP-time algorithm for the general case, which implies a polynomial-time algorithm when the number of agents with changing preferences is constant. While these results establish the tractability of these regimes, closing the complexity gap in the fully general setting remains an interesting open question.

cs.DM↗

arXiv2025

Fair Core Imputations for the Assignment Game: New Solution Concepts and Efficient Algorithms

The assignment game is a classical model for profit sharing and a cornerstone of cooperative game theory. While an imputation in its core guarantees fairness among coalitions, it provides no fairness guarantee at the level of individual agents: single agents or one sided coalitions have zero standalone value and may receive arbitrarily small payoffs. Motivated by the growing focus on individual level fairness, we ask: Can one select a core imputation that is also fair to individuals? We introduce three individual fairness driven solution concepts, each promoting equity in a different way. The leximin and leximax core imputations extend max min and min max fairness to uplift the least advantaged and constrain the most advantaged agents, respectively. The min spread core imputation minimizes the gap between the largest and smallest positive payoffs, promoting equitable profit distribution. For all three solution concepts, we develop combinatorial, strongly polynomial algorithms. The leximin and leximax algorithms are based on a novel adaptation of the primal dual paradigm, while the min spread algorithm combines partial executions of the first two. We expect our work to revive innovation on the potent primal dual paradigm as well as promote further work on the algorithmic study of fairness and stability.

cs.GT↗

arXiv2025

Fair Rent Division: New Budget and Rent Constraints

We study the classical rent division problem, where $n$ agents must allocate $n$ indivisible rooms and split a fixed total rent $R$. The goal is to compute an envy-free (EF) allocation, where no agent prefers another agent's room and rent to their own. This problem has been extensively studied under standard assumptions, where efficient algorithms for computing EF allocations are known. We extend this framework by introducing two practically motivated constraints: (i) lower and upper bounds on room rents, and (ii) room-specific budget for agents. We develop efficient combinatorial algorithms that either compute a feasible EF allocation or certify infeasibility. We further design algorithms to optimize over EF allocations using natural fairness objectives such as maximin utility, leximin utility, and minimum utility spread. Our approach unifies both constraint types within a single algorithmic framework, advancing the applicability of fair division methods in real-world platforms such as Spliddit.

cs.GT↗

arXiv2025

Equitable Core Imputations for Max-Flow, MST and $b$-Matching Games

We study fair allocation of profit (or cost) for three central problems from combinatorial optimization: Max-Flow, MST and $b$-matching. The essentially unequivocal choice of solution concept for this purpose would be the core, because of its highly desirable properties. However, recent work [Vaz24] observed that for the assignment game, an arbitrary core imputation makes no fairness guarantee at the level of individual agents. To rectify this deficiency, special core imputations, called equitable core imputations, were defined - there are two such imputations, leximin and leximax - and efficient algorithms were given for finding them. For all three games, we start by giving examples to show that an arbitrary core imputation can be excessively unfair to certain agents. This led us to seek equitable core imputations for our three games as well. However, the ubiquitous tractable vs intractable schism separates the assignment game from our three games, making our task different from that of [Vaz24]. As is usual in the presence of intractability, we resorted to defining the Owen set for each game and algorithmically relating it to the set of optimal dual solutions of the underlying combinatorial problem. We then give polynomial time algorithms for finding equitable imputations in the Owen set. The motivation for this work is two-fold: the emergence of automated decision-making, with a special emphasis on fairness, and the plethora of industrial applications of our three games.

cs.GT↗

arXiv2025

On the Core of the $b$-Matching Game

The core is a quintessential solution concept for profit sharing in cooperative game theory. An imputation allocates the worth of the given game among its agents. The imputation lies in the core of the game if, for each sub-coalition, the amount allocated to its agents is at least the worth of this sub-coalition. Hence, under a core imputation, each of exponentially many sub-coalitions gets satisfied. The following computational question has received much attention: Given an imputation, does it lie in the core? Clearly, this question lies in co-NP, since a co-NP certificate for this problem would be a sub-coalition which is not satisfied under the imputation. This question is in P for the assignment game [SS71] and has been shown to be co-NP-hard for several natural games, including max-flow [FZCD02] and MST [FKFH97]. The one natural game for which this question has remained open is the b-matching game when the number of times an edge can be matched is unconstrained; in case each edge can be matched at most once, it is co-NP-hard [BKPW18]. At the outset, it was not clear which way this open question would resolve: on the one hand, for all but one game, this problem was shown co-NP-hard and on the other hand, proximity to the assignment problem and the deep structural properties of matching could lead to a positive result. In this paper, we show that the problem is indeed co-NP-hard.

cs.GT↗

arXiv2025

Cardinal-Utility Matching Markets: The Quest for Envy-Freeness, Pareto-Optimality, and Efficient Computability

Unlike ordinal-utility matching markets, which are well-developed from the viewpoint of both theory and practice, recent insights from a computer science perspective have left cardinal-utility matching markets in a state of flux. The celebrated pricing-based mechanism for one-sided cardinal-utility matching markets due to Hylland and Zeckhauser, which had long eluded efficient algorithms, was finally shown to be intractable; the problem of computing an approximate equilibrium is PPAD-complete. This led us to ask the question: is there an alternative, polynomial time, mechanism for one-sided cardinal-utility matching markets which achieves the desirable properties of HZ, i.e. (ex-ante) envy-freeness (EF) and Pareto-optimality (PO)? We show that the problem of finding an EF+PO lottery in a one-sided cardinal-utility matching market is by itself already PPAD-complete. However, a $(2 + ε)$-approximately envy-free and (exactly) Pareto-optimal lottery can be found in polynomial time using the Nash-bargaining-based mechanism of Hosseini and Vazirani. Moreover, the mechanism is also $(2 + ε)$-approximately incentive compatible. We also present several results on two-sided cardinal-utility matching markets, including non-existence of EF+PO lotteries as well as existence of justified-envy-free and weak Pareto-optimal lotteries.

cs.GT↗

arXiv2024

Matching Markets with Chores

The fair division of chores, as well as mixed manna (goods and chores), has received substantial recent attention in the fair division literature; however, ours is the first paper to extend this research to matching markets. Indeed, our contention is that matching markets are a natural setting for this purpose, since the manna that fit into the limited number of hours available in a day can be viewed as one unit of allocation. We extend several well-known results that hold for goods to the settings of chores and mixed manna. In addition, we show that the natural notion of an earnings-based equilibrium, which is more natural in the case of all chores, is equivalent to the pricing-based equilibrium given by Hylland and Zeckhauser for the case of goods.

cs.GT↗

arXiv2024

Time-Efficient Algorithms for Nash-Bargaining-Based Matching Market Models

In the area of matching-based market design, existing models using cardinal utilities suffer from two deficiencies: First, the Hylland-Zeckhauser (HZ) mechanism, which has remained a classic in economics for one-sided matching markets, is intractable; computation of even an approximate equilibrium is PPAD-complete. Second, there is an extreme paucity of such models. This led Hosseini and Vazirani (2022) to define a rich collection of Nash-bargaining-based models for one-sided and two-sided matching markets, in both Fisher and Arrow-Debreu settings, together with very fast implementations using available solvers and very encouraging experimental results. In this paper, we give fast algorithms with proven running times for the models introduced by Hosseini and Vazirani, using the techniques of multiplicative weights update (MWU) and conditional gradient descent (CGD). Additionally, we make the following contributions: (1) By Tröbst and Vazirani (2024), a linear one-sided Nash-bargaining-based matching market satisfies envy-freeness within factor two. We show that the other models satisfy approximate equal-share fairness, where the exact factor depends on the utility function being used in the particular model. (2) We define a Nash-bargaining-based model for non-bipartite matching markets and give fast algorithms for it using conditional gradient descent.

cs.GT↗

arXiv2023

The Investment Management Game: Extending the Scope of the Notion of Core

The core is a dominant solution concept in economics and cooperative game theory; it is predominantly used for profit, equivalently cost or utility, sharing. This paper demonstrates the versatility of this notion by proposing a completely different use: in a so-called investment management game, which is a game against nature rather than a cooperative game. This game has only one agent whose strategy set is all possible ways of distributing her money among investment firms. The agent wants to pick a strategy such that in each of exponentially many future scenarios, sufficient money is available in the right firms so she can buy an optimal investment for that scenario. Such a strategy constitutes a core imputation under a broad interpretation, though traditional formal framework, of the core. Our game is defined on perfect graphs, since the maximum stable set problem can be solved in polynomial time for such graphs. We completely characterize the core of this game, analogous to Shapley and Shubik characterization of the core of the assignment game. A key difference is the following technical novelty: whereas their characterization follows from total unimodularity, ours follows from total dual integrality

econ.TH↗

arXiv2023

Towards a Practical, Budget-Oblivious Algorithm for the Adwords Problem under Small Bids

Motivated by recent insights into the online bipartite matching problem (\textsc{OBM}), our goal was to extend the optimal algorithm for it, namely \textsc{Ranking}, all the way to the special case of adwords problem, called \textsc{Small}, in which bids are small compared to budgets; the latter has been of considerable practical significance in ad auctions \cite{MSVV}. The attractive feature of our approach was that it would yield a {\em budget-oblivious algorithm}, i.e., the algorithm would not need to know budgets of advertisers and therefore could be used in autobidding platforms. We were successful in obtaining an optimal, budget-oblivious algorithm for \textsc{Single-Valued}, under which each advertiser can make bids of one value only. However, our next extension, to \textsc{Small}, failed because of a fundamental reason, namely failure of the {\em No-Surpassing Property}. Since the probabilistic ideas underlying our algorithm are quite substantial, we have stated them formally, after assuming the No-Surpassing Property, and we leave the open problem of removing this assumption. With the help of two undergrads, we conducted extensive experiments on our algorithm on randomly generated instances. Our findings are that the No-Surpassing Property fails less than $2\%$ of the time and that the performance of our algorithms for \textsc{Single-Valued} and \textsc{Small} are comparable to that of \cite{MSVV}. If further experiments confirm this, our algorithm may be useful as such in practice, especially because of its budget-obliviousness.

cs.DS↗

arXiv2023

Nash-Bargaining-Based Models for Matching Markets: One-Sided and Two-Sided; Fisher and Arrow-Debreu

This paper addresses two deficiencies of models in the area of matching-based market design. The first arises from the recent realization that the most prominent solution that uses cardinal utilities, namely the Hylland-Zeckhauser (HZ) mechanism, is intractable; computation of even an approximate equilibrium is PPAD-complete. The second is the extreme paucity of models that use cardinal utilities. Our paper addresses both these issues by proposing Nash-bargaining-based matching market models. Since the Nash bargaining solution is captured by a convex program, efficiency follows. In addition, it possesses several desirable game-theoretic properties. Our approach yields a rich collection of models: for one-sided as well as two-sided markets, for Fisher as well as Arrow-Debreu settings, and for a wide range of utility functions, all the way from linear to Leontief. We give very fast implementations for these models using Frank-Wolfe and Cutting Plane algorithms. These help solve large instances with several thousand agents and goods in a matter of minutes on a PC, even for a one-sided matching market under piecewise-linear concave utility functions and a two-sided matching market under linear utility functions. In contrast, using HZ, going beyond even $n = 10$ is prohibitive. Several new ideas were needed, beyond the standard methods, to obtain these implementations. In particular, we present several lower bounding schemes, which not only help improve the convergence of our solution methods but also shed light on fairness properties of the Nash-bargaining-based models.

cs.GT↗

arXiv2023

LP-Duality Theory and the Cores of Games

LP-duality theory has played a central role in the study of the core, right from its early days to the present time. However, despite the extensive nature of this work, basic gaps still remain. We address these gaps using the following building blocks from LP-duality theory: 1. Total unimodularity (TUM). 2. Complementary slackness conditions and strict complementarity. Our exploration of TUM leads to defining new games, characterizing their cores and giving novel ways of using core imputations to enforce constraints that arise naturally in applications of these games. The latter include: 1. Efficient algorithms for finding min-max fair, max-min fair and equitable core imputations. 2. Encouraging diversity and avoiding over-representation in a generalization of the assignment game. Complementarity enables us to prove new properties of core imputations of the assignment game and its generalizations.

cs.GT↗

arXiv2022

New Characterizations of Core Imputations of Matching and $b$-Matching Games

We give new characterizations of core imputations for the following games: * The assignment game. * Concurrent games, i.e., general graph matching games having non-empty core. * The unconstrained bipartite $b$-matching game (edges can be matched multiple times). * The constrained bipartite $b$-matching game (edges can be matched at most once). The classic paper of Shapley and Shubik \cite{Shapley1971assignment} showed that core imputations of the assignment game are precisely optimal solutions to the dual of the LP-relaxation of the game. Building on this, Deng et al. \cite{Deng1999algorithms} gave a general framework which yields analogous characterizations for several fundamental combinatorial games. Interestingly enough, their framework does not apply to the last two games stated above. In turn, we show that some of the core imputations of these games correspond to optimal dual solutions and others do not. This leads to the tantalizing question of understanding the origins of the latter. We also present new characterizations of the profits accrued by agents and teams in core imputations of the first two games. Our characterization for the first game is stronger than that for the second; the underlying reason is that the characterization of vertices of the Birkhoff polytope is stronger than that of the Balinski polytope.

cs.GT↗

arXiv2022

Cores of Games via Total Dual Integrality, with Applications to Perfect Graphs and Polymatroids

LP-duality theory has played a central role in the study of cores of games, right from the early days of this notion to the present time. The classic paper of Shapley and Shubik \cite{Shapley1971assignment} introduced the "right" way of exploiting the power of this theory, namely picking problems whose LP-relaxations support polyhedra having integral vertices. So far, the latter fact was established by showing that the constraint matrix of the underlying linear system is {\em totally unimodular}. We attempt to take this methodology to its logical next step -- {\em using total dual integrality} -- thereby addressing new classes of games which have their origins in two major theories within combinatorial optimization, namely perfect graphs and polymatroids. In the former, we address the stable set and clique games and in the latter, we address the matroid independent set game. For each of these games, we prove that the set of core imputations is precisely the set of optimal solutions to the dual LPs. Another novelty is the way the worth of the game is allocated among sub-coalitions. Previous works follow the {\em bottom-up process} of allocating to individual agents; the allocation to a sub-coalition is simply the sum of the allocations to its agents. The {\em natural process for our games is top-down}. The optimal dual allocates to "objects" in the grand coalition; a sub-coalition inherits the allocation of each object with which it has non-empty intersection.

cs.GT↗