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In the $\varepsilon$-Consensus-Halving problem, a fundamental problem in fair division, there are $n$ agents with valuations over the interval $[0,1]$, and the goal is to divide the interval into pieces and assign a label "$+$" or "$-$" to each piece, such that every agent values the total amount of "$+$" and the total amount of "$-$" almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [2019] to be the first "natural" complete problem for the computational class PPA, answering a decade-old open question. In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [Filos-Ratsikas and Goldberg, 2019], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [Filos-Ratsikas and Goldberg, 2019]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving $\varepsilon$-Consensus-Halving for any $\varepsilon$, as well as a polynomial-time algorithm for $\varepsilon=1/2$. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-$1/k$-Division problem [Simmons and Su, 2003]. In particular, we prove that $\varepsilon$-Consensus-$1/3$-Division is PPAD-hard., Journal version. Preliminary version appeared at EC '20

We consider the $\varepsilon$-Consensus-Halving problem, in which a set of heterogeneous agents aim at dividing a continuous resource into two (not necessarily contiguous) portions that all of them simultaneously consider to be of approximately the same value (up to $\varepsilon$). This problem was recently shown to be PPA-complete, for $n$ agents and $n$ cuts, even for very simple valuation functions. In a quest to understand the root of the complexity of the problem, we consider the setting where there is only a constant number of agents, and we consider both the computational complexity and the query complexity of the problem. For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete. Similarly, we show that for two monotone agents the problem can be solved with polynomially-many queries, whereas for three or more agents, we provide exponential query complexity lower bounds. These results are enabled via an interesting connection to a monotone Borsuk-Ulam problem, which may be of independent interest. For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents., Comment: Journal version

We study thetruthful facility assignmentproblem, where a set of agents with private most-preferred points on a metric space have to be assigned to facilities that lie on the metric space, under capacity constraints on the facilities. The goal is to produce such an assignment that minimizes the social cost, i.e., the total distance between the most-preferred points of the agents and their corresponding facilities in the assignment, under the constraint of truthfulness, which ensures that agents do not misreport their most-preferred points. We propose aresource augmentation framework, where a truthful mechanism is evaluated by its worst-case performance on an instance with enhanced facility capacities against the optimal mechanism on the same instance with the original capacities. We study a well-known mechanism, Serial Dictatorship, and provide an exact analysis of its performance. Among other results, we prove that Serial Dictatorship has approximation ratio$$g/(g-2)$$g/(g-2)when the capacities are multiplied by any integer$$g \ge 3$$g≥3. Our results suggest that with a limited augmentation of the resources we can achieve exponential improvements on the performance of the mechanism and in particular, the approximation ratio goes to 1 as the augmentation factor becomes large. We complement our results with bounds on the approximation ratio of Random Serial Dictatorship, the randomized version of Serial Dictatorship, when there is no resource augmentation.

We consider the problem of computing a (pure) Bayes-Nash equilibrium in the first-price auction with continuous value distributions and discrete bidding space. We prove that when bidders have independent subjective prior beliefs about the value distributions of the other bidders, computing an $\varepsilon$-equilibrium of the auction is PPAD-complete, and computing an exact equilibrium is FIXP-complete. We also provide an efficient algorithm for solving a special case of the problem, for a fixed number of bidders and available bids., Journal version. Preliminary version appeared at EC '21

When a company undergoes a merger or transfers its ownership, the existing governing body has an opinion on which buyer should take over as the new owner. Similar situations occur while assigning the host of big sports tournaments, like the World Cup or the Olympics. In all these settings, the values of the external bidders are as important as the opinions of the internal experts. Motivated by such scenarios, we consider a social welfare maximizing approach to design and analyze truthful mechanisms inhybrid social choicesettings, where payments can be imposed to the bidders, but not to the experts. Since this problem is a combination of mechanism design with and without monetary transfers, classical solutions like VCG cannot be applied, making this a novel mechanism design problem. We consider the simple but fundamental scenario with one expert and two bidders, and provide tight approximation guarantees of the optimal social welfare. We distinguish between mechanisms that use ordinal and cardinal information, as well as between mechanisms that base their decisions on one of the two sides (either the bidders or the expert) or both. Our analysis shows that the cardinal setting is quite rich and admits several non-trivial randomized truthful mechanisms, and also allows for closer-to-optimal welfare guarantees.

Allocating resources to individuals in a fair manner has been a topic of interest since the ancient times, with most of the early rigorous mathematical work on the problem focusing on infinitely divisible resources. Recently, there has been a surge of papers studying computational questions regarding various different notions of fairness for the indivisible case, like maximin share fairness (MMS) and envy-freeness up to any good (EFX). We survey the most important results in the discrete fair division literature, focusing on the case of additive valuation functions and paying particular attention to the progress made in the last 10 years.

We consider a single-facility location problem, where agents are positioned on the real line and are partitioned into multiple disjoint districts. The goal is to choose a location (where a public facility is to be built) so as to minimize the total distance of the agents from it. This process is distributed: the positions of the agents in each district are first aggregated into a representative location for the district, and then one of the district representatives is chosen as the facility location. This indirect access to the positions of the agents inevitably leads to inefficiency, which is captured by the notion of distortion. We study the discrete version of the problem, where the set of alternative locations is finite, as well as the continuous one, where every point of the line is an alternative, and paint an almost complete picture of the distortion landscape of both general and strategyproof distributed mechanisms.

The notion of distortion in social choice problems has been defined to measure the loss in efficiency -- typically measured by the utilitarian social welfare, the sum of utilities of the participating agents -- due to having access only to limited information about the preferences of the agents. We survey the most significant results of the literature on distortion from the past 15 years, and highlight important open problems and the most promising avenues of ongoing and future work., Survey

Aggregating the preferences of individuals into a collective decision is the core subject of study of social choice theory. In 2006, Procaccia and Rosenschein considered a utilitarian social choice setting, where the agents have explicit numerical values for the alternatives, yet they only report their linear orderings over them. To compare different aggregation mechanisms, Procaccia and Rosenschein introduced the notion of distortion, which quantifies the inefficiency of using only ordinal information when trying to maximize the social welfare, i.e., the sum of the underlying values of the agents for the chosen outcome. Since then, this research area has flourished and bounds on the distortion have been obtained for a wide variety of fundamental scenarios. However, the vast majority of the existing literature is focused on the case where nothing is known beyond the ordinal preferences of the agents over the alternatives. In this paper, we take a more expressive approach, and consider mechanisms that are allowed to further ask a few cardinal queries in order to gain partial access to the underlying values that the agents have for the alternatives. With this extra power, we design new deterministic mechanisms that achieve significantly improved distortion bounds and, in many cases, outperform the best-known randomized ordinal mechanisms. We paint an almost complete picture of the number of queries required by deterministic mechanisms to achieve specific distortion bounds.

We consider a social choice setting with agents that are partitioned into disjoint groups, and have metric preferences over a set of alternatives. Our goal is to choose a single alternative aiming to optimize various objectives that are functions of the distances between agents and alternatives in the metric space, under the constraint that this choice must be made in a distributed way: The preferences of the agents within each group are first aggregated into a representative alternative for the group, and then these group representatives are aggregated into the final winner. Deciding the winner in such a way naturally leads to loss of efficiency, even when complete information about the metric space is available. We provide a series of (mostly tight) bounds on the distortion of distributed mechanisms for variations of well-known objectives, such as the (average) total cost and the maximum cost, and also for new objectives that are particularly appropriate for this distributed setting and have not been studied before.

We initiate the study of the heterogeneous facility location problem with limited resources. We mainly focus on the fundamental case where a set of agents are positioned in the line segment [0,1] and have approval preferences over two available facilities. A mechanism takes as input the positions and the preferences of the agents, and chooses to locate a single facility based on this information. We study mechanisms that aim to maximize the social welfare (the total utility the agents derive from facilities they approve), under the constraint of incentivizing the agents to truthfully report their positions and preferences. We consider three different settings depending on the level of agent-related information that is public or private. For each setting, we design deterministic and randomized strategyproof mechanisms that achieve a good approximation of the optimal social welfare, and complement these with nearly-tight impossibility results.

We resolve the computational complexity of three problems known as Necklace Splitting, Consensus-Halving, and Discrete Ham sandwich, showing that they are PPA-complete. For NECKLACE SPLITTING, this result is specific to the important special case in which two thieves share the necklace. These are the first PPA-completeness results for problems whose definition does not contain an explicit circuit, thus settling the status of PPA as a class that captures the complexity of such “natural' problems.

We derive conditions under which a peer-consistency mechanism can be used to elicit truthful data from non-trusted rational agents when an aggregate statistic of the collected data affects the amount of their incentives to lie. Furthermore, we discuss the relative saving that can be achieved by the mechanism, compared to the rational outcome, if no such mechanism was implemented. Our work is motivated by distributed platforms, where decentralized data oracles collect information about real-world events, based on the aggregate information provided by often self-interested participants. We compare our theoretical observations with numerical simulations on two public real datasets.

Voting can abstractly model any decision-making scenario and as such it has been extensively studied over the decades. Recently, the related literature has focused on quantifying the impact of utilizing only limited information in the voting process on the societal welfare for the outcome, by bounding the distortion of voting rules. Even though there has been significant progress towards this goal, almost all previous works have so far neglected the fact that in many scenarios (like presidential elections) voting is actually a distributed procedure. In this paper, we consider a setting in which the voters are partitioned into disjoint districts and vote locally therein to elect local winning alternatives using a voting rule; the final outcome is then chosen from the set of these alternatives. We prove tight bounds on the distortion of well-known voting rules for such distributed elections both from a worst-case perspective as well as from a best-case one. Our results indicate that the partition of voters into districts leads to considerably higher distortion, a phenomenon which we also experimentally showcase using real-world data.

We resolve the computational complexity of two problems known as NECKLACE-SPLITTING and DISCRETE HAM SANDWICH, showing that they are PPA-complete. For NECKLACE SPLITTING, this result is specific to the important special case in which two thieves share the necklace. We do this via a PPA-completeness result for an approximate version of the CONSENSUS-HALVING problem, strengthening our recent result that the problem is PPA-complete for inverse-exponential precision. At the heart of our construction is a smooth embedding of the high-dimensional M\"obius strip in the CONSENSUS-HALVING problem. These results settle the status of PPA as a class that captures the complexity of "natural" problems whose definitions do not incorporate a circuit., Comment: 58 pages, 20 figures

We present a novel anytime heuristic (ALMA), inspired by the human principle of altruism, for solving the assignment problem. ALMA is decentralized, completely uncoupled, and requires no communication between the participants. We prove an upper bound on the convergence speed that is polynomial in the desired number of resources and competing agents per resource; crucially, in the realistic case where the aforementioned quantities are bounded independently of the total number of agents/resources, the convergence time remains constant as the total problem size increases. We have evaluated ALMA under three test cases: (i) an anti-coordination scenario where agents with similar preferences compete over the same set of actions, (ii) a resource allocation scenario in an urban environment, under a constant-time constraint, and finally, (iii) an on-line matching scenario using real passenger-taxi data. In all of the cases, ALMA was able to reach high social welfare, while being orders of magnitude faster than the centralized, optimal algorithm. The latter allows our algorithm to scale to realistic scenarios with hundreds of thousands of agents, e.g., vehicle coordination in urban environments.

We study the trade-off between the Price of Anarchy (PoA) and the Price of Stability (PoS) in mechanism design, in the prototypical problem of unrelated machine scheduling. We give bounds on the space of feasible mechanisms with respect to the above metrics, and observe that two fundamental mechanisms, namely the First-Price (FP) and the Second-Price (SP), lie on the two opposite extrema of this boundary. Furthermore, for the natural class of anonymous task-independent mechanisms, we completely characterize the PoA/PoS Pareto frontier; we design a class of optimal mechanisms $\mathcal{SP}_\alpha$ that lie exactly on this frontier. In particular, these mechanisms range smoothly, with respect to parameter $\alpha\geq 1$ across the frontier, between the First-Price ($\mathcal{SP}_1$) and Second-Price ($\mathcal{SP}_\infty$) mechanisms. En route to these results, we also provide a definitive answer to an important question related to the scheduling problem, namely whether non-truthful mechanisms can provide better makespan guarantees in the equilibrium, compared to truthful ones. We answer this question in the negative, by proving that the Price of Anarchy of all scheduling mechanisms is at least $n$, where $n$ is the number of machines., Comment: To be published in Mathematics of Operations Research

We study the optimization of large-scale, real-time ridesharing systems and propose a modular design methodology, Component Algorithms for Ridesharing (CAR). We evaluate a diverse set of CARs (14 in total), focusing on the key algorithmic components of ridesharing. We take a multi-objective approach, evaluating 12 metrics related to global efficiency, complexity, passenger, driver, and platform incentives, in settings designed to closely resemble reality in every aspect, focusing on vehicles of capacity two. To the best of our knowledge, this is the largest and most comprehensive evaluation to date. We (i) identify CARs that perform well on global, passenger, driver or platform metrics, (ii) demonstrate that lightweight relocation schemes can significantly improve the Quality of Service by up to $50\%$, and (iii) highlight a practical, scalable, on-device CAR that works well across all metrics., Comment: A version of this paper has been published in the Artificial Intelligence Review (https://doi.org/10.1007/s10462-022-10145-0)

In this work, we study the computational complexity of the Minimum Distance Code Detection problem. In this problem, we are given a set of noisy codeword observations and we wish to find a code in a set of linear codes $\mathcal{C}$ of a given dimension $k$, for which the sum of distances between the observations and the code is minimized. We prove that, for the practically relevant case when the set $\mathcal{C}$ only contains a fixed number of candidate linear codes, the detection problem is NP-hard and we identify a number of interesting open questions related to the code detection problem., Accepted for presentation at IEEE ISIT 2019

In large e-commerce websites, sellers have been observed to engage in fraudulent behaviour, faking historical transactions in order to receive favourable treatment from the platforms, specifically through the allocation of additional buyer impressions which results in higher revenue for them, but not for the system as a whole. This emergent phenomenon has attracted considerable attention, with previous approaches focusing on trying to detect illicit practices and to punish the miscreants. In this paper, we employ the principles of reinforcement mechanism design, a framework that combines the fundamental goals of classical mechanism design, i.e. the consideration of agents' incentives and their alignment with the objectives of the designer, with deep reinforcement learning for optimizing the performance based on these incentives. In particular, first we set up a deep-learning framework for predicting the sellers' rationality, based on real data from any allocation algorithm. We use data from one of largest e-commerce platforms worldwide and train a neural network model to predict the extent to which the sellers will engage in fraudulent behaviour. Using this rationality model, we employ an algorithm based on deep reinforcement learning to optimize the objectives and compare its performance against several natural heuristics, including the platform's implementation and incentive-based mechanisms from the related literature.

We study the problem of allocating impressions to sellers in e-commerce websites, such as Amazon, eBay or Taobao, aiming to maximize the total revenue generated by the platform. We employ a general framework of reinforcement mechanism design, which uses deep reinforcement learning to design efficient algorithms, taking the strategic behaviour of the sellers into account. Specifically, we model the impression allocation problem as a Markov decision process, where the states encode the history of impressions, prices, transactions and generated revenue and the actions are the possible impression allocations in each round. To tackle the problem of continuity and high-dimensionality of states and actions, we adopt the ideas of the DDPG algorithm to design an actor-critic policy gradient algorithm which takes advantage of the problem domain in order to achieve convergence and stability. We evaluate our proposed algorithm, coined IA(GRU), by comparing it against DDPG, as well as several natural heuristics, under different rationality models for the sellers - we assume that sellers follow well-known no-regret type strategies which may vary in their degree of sophistication. We find that IA(GRU) outperforms all algorithms in terms of the total revenue.

We show that the computational problem CONSENSUS-HALVING is PPA-complete, the first PPA-completeness result for a problem whose definition does not involve an explicit circuit. We also show that an approximate version of this problem is polynomial-time equivalent to NECKLACE SPLITTING, which establishes PPAD-hardness for NECKLACE SPLITTING, and suggests that it is also PPA-complete.

Strategic behaviour from sellers on e-commerce websites, such as faking transactions and manipulating the recommendation scores through artificial reviews, have been among the most notorious obstacles that prevent websites from maximizing the efficiency of their recommendations. Previous approaches have focused almost exclusively on machine learning-related techniques to detect and penalize such behaviour. In this paper, we tackle the problem from a different perspective, using the approach of the field of mechanism design. We put forward a game model tailored for the setting at hand and aim to construct truthful mechanisms, i.e. mechanisms that do not provide incentives for dishonest reputation-augmenting actions, that guarantee good recommendations in the worst-case. For the setting with two agents, we propose a truthful mechanism that is optimal in terms of social efficiency. For the general case of m agents, we prove both lower and upper bound results on the efficiency of truthful mechanisms and propose truthful mechanisms that yield significantly better results, when compared to an existing mechanism from a leading e-commerce site on real data.

We study the Price of Anarchy of mechanisms for the well-known problem of one-sided matching, or house allocation, with respect to the social welfare objective. We consider both ordinal mechanisms, where agents submit preference lists over the items, and cardinal mechanisms, where agents may submit numerical values for the items being allocated. We present a general lower bound of \(\varOmega (\sqrt{n})\) on the Price of Anarchy, which applies to all mechanisms. We show that two well-known mechanisms, Probabilistic Serial, and Random Priority, achieve a matching upper bound. We extend our lower bound to the Price of Stability of a large class of mechanisms that satisfy a common proportionality property, and show stronger bounds on the Price of Anarchy of all deterministic mechanisms.

We study a mechanism design version of matching computationin graphs that models the game played by hospitals participatingin pairwise kidney exchange programs. We present a new randomizedmatching mechanism for two agents which is truthful in expectation andhas an approximation ratio of 3/2 to the maximum cardinality matching.This is an improvement over a recent upper bound of 2 [Ashlagi et al., EC2010] and, furthermore, our mechanism beats for the first time the lowerbound on the approximation ratio of deterministic truthful mechanisms.We complement our positive result with new lower bounds. Among otherstatements, we prove that the weaker incentive compatibility propertyof truthfulness in expectation in our mechanism is necessary; universallytruthful mechanisms that have an inclusion-maximality property havean approximation ratio of at least 2.

We study fairness and efficiency properties of randomized algorithms for barter exchanges with direct applications to kidney exchange problems. It is well documented that randomization can serve as a tool to ensure fairness among participants. However, in many applications, practical constraints often restrict the maximum allowed cycle-length of the exchange and for randomized algorithms, this imposes constraints of the cycle-length of every realized exchange in their decomposition. We prove that standard fairness properties such as envy-freeness or symmetry are incompatible with even the weakest notion of economic efficiency in this setting. On the plus side, we adapt some well-known matching mechanisms to incorporate the restricted cycle constraint and evaluate their performance experimentally on instances of the kidney exchange problem, showing tradeoffs between fairness and efficiency.

We consider the fundamental mechanism design problem of approximate social welfare maximization under general cardinal preferences on a finite number of alternatives and without money. The well-known range voting scheme can be thought of as a non-truthful mechanism for exact social welfare maximization in this setting. With m being the number of alternatives, we exhibit a randomized truthful-in-expectation ordinal mechanism implementing an outcome whose expected social welfare is at least an Omega(m^{-3/4}) fraction of the social welfare of the socially optimal alternative. On the other hand, we show that for sufficiently many agents and any truthful-in-expectation ordinal mechanism, there is a valuation profile where the mechanism achieves at most an O(m^{-{2/3}) fraction of the optimal social welfare in expectation. We get tighter bounds for the natural special case of m = 3, and in that case furthermore obtain separation results concerning the approximation ratios achievable by natural restricted classes of truthful-in-expectation mechanisms. In particular, we show that for m = 3 and a sufficiently large number of agents, the best mechanism that is ordinal as well as mixed-unilateral has an approximation ratio between 0.610 and 0.611, the best ordinal mechanism has an approximation ratio between 0.616 and 0.641, while the best mixed-unilateral mechanism has an approximation ratio bigger than 0.660. In particular, the best mixed-unilateral non-ordinal (i.e., cardinal) mechanism strictly outperforms all ordinal ones, even the non-mixed-unilateral ordinal ones.

In a multi-unit market, a seller brings multiple units of a good and tries to sell them to a set of buyers that have monetary endowments. While a Walrasian equilibrium does not always exist in this model, natural relaxations of the concept that retain its desirable fairness properties do exist. We study the dynamics of (Walrasian) envy-free pricing mechanisms in this environment, showing that for any such pricing mechanism, the best response dynamic starting from truth-telling converges to a pure Nash equilibrium with small loss in revenue and welfare. Moreover, we generalize these bounds to capture all the (reasonable) Nash equilibria for a large class of (monotone) pricing mechanisms. We also identify a natural mechanism, which selects the minimum Walrasian envy-free price, in which for n=2 buyers the best response dynamic converges from any starting profile. We conjecture convergence of the mechanism for any number of buyers and provide simulation results to support our conjecture.

We study the problem of locating a single facility on a real line based on the reports of self-interested agents, when agents have double-peaked preferences, with the peaks being on opposite sides of their locations. We observe that double-peaked preferences capture real-life scenarios and thus complement the well-studied notion of single-peaked preferences. As a motivating example, assume that the government plans to build a primary school along a street; an agent with single-peaked preferences would prefer having the school built exactly next to her house. However, while that would make it very easy for her children to go to school, it would also introduce several problems, such as noise or parking congestion in the morning. A 5-min walking distance would be sufficiently far for such problems to no longer be much of a factor and at the same time sufficiently close for the school to be easily accessible by the children on foot. There are two positions (symmetrically) in each direction and those would be the agent’s two peaks of her double-peaked preference. Motivated by natural scenarios like the one described above, we mainly focus on the case where peaks are equidistant from the agents’ locations and discuss how our results extend to more general settings. We show that most of the results for single-peaked preferences do not directly apply to this setting, which makes the problem more challenging. As our main contribution, we present a simple truthful-in-expectation mechanism that achieves an approximation ratio of (Formula presented.) for both the social and the maximum cost, where b is the distance of the agent from the peak and c is the minimum cost of an agent. For the latter case, we provide a 3 / 2 lower bound on the approximation ratio of any truthful-in-expectation mechanism. We also study deterministic mechanisms under some natural conditions, proving lower bounds and approximation guarantees. We prove that among a large class of reasonable strategyproof mechanisms, there is no deterministic mechanism that outperforms our truthful-in-expectation mechanism. In order to obtain this result, we first characterize mechanisms for two agents that satisfy two simple properties; we use the same characterization to prove that no mechanism in this class can be group-strategyproof.

Blockchain based systems allow various kinds of financial transactions to be executed in a decentralized manner. However, these systems often rely on a trusted third party (oracle) to get correct information about the real-world events, which trigger the financial transactions. In this paper, we identify two biggest challenges in building decentralized, trustless and transparent oracles. The first challenge is acquiring correct information about the real-world events without relying on a trusted information provider. We show how a peer-consistency incentive mechanism can be used to acquire truthful information from an untrusted and self-interested crowd, even when the crowd has outside incentives to provide wrong informations. The second is a system design and implementation challenge. For the first time, we show how to implement a trustless and transparent oracle in Ethereum. We discuss various non-trivial issues that arise in implementing peer-consistency mechanisms in Ethereum, suggest several optimizations to reduce gas cost and provide empirical analysis.

The Fisher market model is one of the most fundamental resource allocation models in economics. In a Fisher market, the prices and allocations of goods are determined according to the preferences and budgets of buyers to clear the market. In a Fisher market game, however, buyers are strategic and report their preferences over goods; the market-clearing prices and allocations are then determined based on their reported preferences rather than their real preferences. We show that the Fisher market game always has a pure Nash equilibrium, for buyers with linear, Leontief, and Cobb-Douglas utility functions, which are three representative classes of utility functions in the important Constant Elasticity of Substitution (CES) family. Furthermore, to quantify the social efficiency, we prove Price of Anarchy bounds for the game when the utility functions of buyers fall into these three classes respectively.

We consider the One-Sided Matching problem, where n agents have preferences over n items, and these preferences are induced by underlying cardinal valuation functions. The goal is to match every agent to a single item so as to maximize the social welfare. Most of the related literature, however, assumes that the values of the agents are not a priori known, and only access to the ordinal preferences of the agents over the items is provided. Consequently, this incomplete information leads to loss of efficiency, which is measured by the notion of distortion. In this paper, we further assume that the agents can answer a small number of queries, allowing us partial access to their values. We study the interplay between elicited cardinal information (measured by the number of queries per agent) and distortion for One-Sided Matching, as well as a wide range of well-studied related problems. Qualitatively, our results show that with a limited number of queries, it is possible to obtain significant improvements over the classic setting, where only access to ordinal information is given.

We consider the classic problem of fairly allocating indivisible goods among agents with additive valuation functions and explore the connection between two prominent fairness notions: maximum Nash welfare (MNW) and envy-freeness up to any good (EFX). We establish that an MNW allocation is always EFX as long as there are at most two possible values for the goods, whereas this implication is no longer true for three or more distinct values. As a notable consequence, this proves the existence of EFX allocations for these restricted valuation functions. While the efficient computation of an MNW allocation for two possible values remains an open problem, we present a novel algorithm for directly constructing EFX allocations in this setting. Finally, we study the question of whether an MNW allocation implies any EFX guarantee for general additive valuation functions under a natural new interpretation of approximate EFX allocations.