Rohit Vaish

AAMAS Conference 2025 Conference Paper

Approximating One-Sided and Two-Sided Nash Social Welfare With Capacities

• Salil Gokhale
• Harshul Sagar
• Rohit Vaish
• Jatin Yadav

We study the problem of maximizing Nash social welfare, which is the geometric mean of agents’ utilities, in two well-known models. The first model involves one-sided preferences, where a set of indivisible items is allocated among a group of agents (commonly studied in fair division). The second model deals with two-sided preferences, where a set of workers and firms, each having numerical valuations for the other side, are matched with each other (commonly studied in matching-under-preferences literature). We study these models under capacity constraints, which restrict the number of items (respectively, workers) that an agent (respectively, a firm) can receive. We contribute constant-factor approximation algorithms for both problems under a broad class of valuations. Specifically, our main results are the following: (a) For any 𝜀 > 0, a (6 + 𝜀)-approximation algorithm for the one-sided problem when agents have submodular valuations, and (b) a 1. 33-approximation algorithm for the two-sided problem when the firms have subadditive valuations. The former result provides the first constant-factor approximation algorithm for Nash welfare in the one-sided problem with submodular valuations and capacities, while the latter result significantly improves upon an existing √ OPT-approximation algorithm for additive valuations. Our result for the two-sided setting also establishes a computational separation between the Nash and utilitarian welfare objectives. We also complement our algorithms with hardness-of-approximation results.

AAAI Conference 2025 Conference Paper

Fair and Efficient Completion of Indivisible Goods

• Vishwa Prakash HV
• Ayumi Igarashi
• Rohit Vaish

We formulate the problem of fair and efficient completion of indivisible goods, defined as follows: Given a partial allocation of indivisible goods among agents, does there exist an allocation of the remaining goods (i.e., a completion) that satisfies fairness and economic efficiency guarantees of interest? We study the computational complexity of the completion problem for prominent fairness and efficiency notions such as envy-freeness up to one good (EF1), proportionality up to one good (Prop1), maximin share (MMS), and Pareto optimality (PO), and focus on the class of additive valuations as well as its subclasses such as binary additive and lexicographic valuations. We find that while the completion problem is significantly harder than the standard fair division problem (wherein the initial partial allocation is empty), the consideration of restricted preferences facilitates positive algorithmic results for threshold-based fairness notions (Prop1 and MMS). On the other hand, the completion problem remains computationally intractable for envy-based notions such as EF1 and EF1+PO even under restricted preferences.

AAMAS Conference 2024 Conference Paper

Capacity Modification in the Stable Matching Problem

• Salil Gokhale
• Samarth Singla
• Shivika Narang
• Rohit Vaish

We study the problem of capacity modification in the many-to-one stable matching of workers and firms. Our goal is to systematically study how the set of stable matchings changes when some seats are added to or removed from the firms. We make three main contributions: First, we examine whether firms and workers can improve or worsen upon changing the capacities under worker-proposing and firm-proposing deferred acceptance algorithms. Second, we study the computational problem of adding or removing seats to either match a fixed worker-firm pair in some stable matching or make a fixed matching stable with respect to the modified problem. We develop polynomial-time algorithms for these problems when only the overall change in the firms’ capacities is restricted, and show NP-hardness when there are additional constraints for individual firms. Lastly, we compare capacity modification with the classical model of preference manipulation by firms and identify scenarios under which one mode of manipulation outperforms the other. We find that a threshold on a given firm’s capacity, which we call its peak, crucially determines the effectiveness of different manipulation actions.

AAMAS Conference 2024 Conference Paper

Charging Electric Vehicles Fairly and Efficiently

• Ramsundar Anandanarayanan
• Swaprava Nath
• Rohit Vaish

Motivated by electric vehicle (EV) charging, we formulate the problem of fair and efficient allocation of a divisible resource among agents that arrive and depart over time and consume the resource at different rates. The agents (EVs) derive utility from the amount of charge gained, which depends on their own charging rate as well as that of the charging outlet. The goal is to allocate charging time at different outlets among the EVs such that the final allocation is envyfree, pareto optimal, and in certain contexts, group-strategyproof. The differences in the charging rates of the outlets and the EVs, and a continuous time-window where the arrivals and departures occur make this a non-trivial combinatorial optimization problem. We show possibilities and impossibilities of achieving a combination of properties such as envy-freeness, pareto optimality, leximin, and group-strategyproofness under different operational settings, e. g. , when the EVs have (dis)similar charging technology, or when there are one or more dissimilar charging outlets. We complement the positive existence results with polynomial-time algorithms.

AAMAS Conference 2024 Conference Paper

Fair Scheduling of Indivisible Chores

• Yatharth Kumar
• Sarfaraz Equbal
• Rohit Gurjar
• Swaprava Nath
• Rohit Vaish

We study the problem of fairly assigning a set of discrete tasks (or chores) among a set of agents with additive valuations. Each chore is associated with an arrival time and a deadline, and each agent can perform at most one chore at any given time. The goal is to find a fair and efficient schedule of the chores, where fairness pertains to satisfying envy-freeness up to one chore (EF1) and efficiency pertains to maximality (i. e. , no unallocated chore can be feasibly assigned to any agent). Our main result is a polynomialtime algorithm for computing an EF1 and maximal schedule for two agents under monotone valuations when the conflict constraints constitute an arbitrary interval graph. The algorithm uses a coloring technique in interval graphs that may be of independent interest. For an arbitrary number of agents, we provide an algorithm for finding a fair schedule under identical dichotomous valuations when the constraints constitute a path graph. We also rule out the existence of schedules satisfying stronger fairness and efficiency properties, including envy-freeness up to any chore (EFX) together with maximality and EF1 together with Pareto optimality.

AAAI Conference 2024 Conference Paper

Maximizing Nash Social Welfare under Two-Sided Preferences

• Pallavi Jain
• Rohit Vaish

The maximum Nash social welfare (NSW)---which maximizes the geometric mean of agents' utilities---is a fundamental solution concept with remarkable fairness and efficiency guarantees. The computational aspects of NSW have been extensively studied for *one-sided* preferences where a set of agents have preferences over a set of resources. Our work deviates from this trend and studies NSW maximization for *two-sided* preferences, wherein a set of workers and firms, each having a cardinal valuation function, are matched with each other. We provide a systematic study of the computational complexity of maximizing NSW for many-to-one matchings under two-sided preferences. Our main negative result is that maximizing NSW is NP-hard even in a highly restricted setting where each firm has capacity 2, all valuations are in the range {0,1,2}, and each agent positively values at most three other agents. In search of positive results, we develop approximation algorithms as well as parameterized algorithms in terms of natural parameters such as the number of workers, the number of firms, and the firms' capacities. We also provide algorithms for restricted domains such as symmetric binary valuations and bounded degree instances.

AAMAS Conference 2024 Conference Paper

Tight Approximations for Graphical House Allocation

• Hadi Hosseini
• Andrew McGregor
• Rik Sengupta
• Rohit Vaish
• Vignesh Viswanathan

The Graphical House Allocation problem asks: how can 𝑛 houses (each with a fixed non-negative value) be assigned to the 𝑛 vertices of an undirected graph 𝐺, so as to minimize the “aggregate local envy”, i. e. , the sum of absolute differences along the edges of 𝐺? This problem generalizes the classical Minimum Linear Arrangement problem, as well as the well-known House Allocation Problem from Economics, the latter of which has notable practical applications in organ exchanges. Recent work has studied the computational aspects of Graphical House Allocation and observed that the problem is NP-hard and inapproximable even on particularly simple classes of graphs, such as vertex disjoint unions of paths. However, the dependence of any approximations on the structural properties of the underlying graph had not been studied. In this work, we give a complete characterization of the approximability of Graphical House Allocation. We present algorithms to approximate the optimal envy on general graphs, trees, planar graphs, bounded-degree graphs, bounded-degree planar graphs, and bounded-degree trees. For each of these graph classes, we then prove matching lower bounds, showing that in each case, no significant improvement can be attained unless P = NP. We also present general approximation ratios as a function of structural parameters of the underlying graph, such as treewidth; these match the aforementioned tight upper bounds in general, and are significantly better approximations for many natural subclasses of graphs. Finally, we present constant factor approximation schemes for the special classes of complete binary trees and random graphs. Some of the technical highlights of our work are the use of expansion properties of Ramanujan graphs in the context of a classical resource allocation problem, and approximating optimal cuts in binary trees by analyzing the behavior of consecutive runs in bitstrings.

AAMAS Conference 2023 Conference Paper

Fairly Dividing Mixtures of Goods and Chores under Lexicographic Preferences

• Hadi Hosseini
• Sujoy Sikdar
• Rohit Vaish
• Lirong Xia

We study fair allocation of indivisible goods and chores among agents with lexicographic preferences—a subclass of additive valuations. In sharp contrast to the goods-only setting, we show that an allocation satisfying envy-freeness up to any item (EFX) could fail to exist for a mixture of objective goods and chores. To our knowledge, this negative result provides the first counterexample for EFX over (any subdomain of) additive valuations. To complement this non-existence result, we identify a class of instances with (possibly subjective) mixed items where an EFX and Pareto optimal allocation always exists and can be efficiently computed. When the fairness requirement is relaxed to maximin share (MMS), we show positive existence and computation for any mixed instance. More broadly, our work examines the existence and computation of fair and efficient allocations both for mixed items as well as chores-only instances, and highlights the additional difficulty of these problems vis-à-vis their goods-only counterparts.

AAMAS Conference 2023 Conference Paper

Graphical House Allocation

• Hadi Hosseini
• Justin Payan
• Rik Sengupta
• Rohit Vaish
• Vignesh Viswanathan

The classical house allocation problem involves assigning 𝑛 houses (or items) to𝑛 agents according to their preferences. A key criteria in such problems is satisfying some fairness constraints such as envyfreeness. We consider a generalization of this problem wherein the agents are placed along the vertices of a graph (corresponding to a social network), and each agent can only experience envy towards its neighbors. Our goal is to minimize the aggregate envy among the agents as a natural fairness objective, i. e. , the sum of the envy value over all edges in a social graph. When agents have identical and evenly-spaced valuations, our problem reduces to the well-studied problem of linear arrangements. For identical valuations with possibly uneven spacing, we show a number of deep and surprising ways in which our setting is a departure from this classical problem. More broadly, we contribute several structural and computational results for various classes of graphs, including NP-hardness results for disjoint unions of paths, cycles, stars, or cliques; we also obtain fixed-parameter tractable (and, in some cases, polynomial-time) algorithms for paths, cycles, stars, cliques, and their disjoint unions. Additionally, a conceptual contribution of our work is the formulation of a structural property for disconnected graphs that we call separability which results in efficient parameterized algorithms for finding optimal allocations.

AAMAS Conference 2023 Conference Paper

Semi-Popular Matchings and Copeland Winners

• Telikepalli Kavitha
• Rohit Vaish

Given a graph 𝐺 = (𝑉, 𝐸) where every vertex has a weak ranking over its neighbors, we consider the problem of computing an optimal matching as per agent preferences. Classical notions of optimality such as stability and its relaxation popularity could fail to exist when 𝐺 is non-bipartite. In light of the non-existence of a popular matching, we consider its relaxations that satisfy universal existence. We find a positive answer in the form of semi-popularity. A matching 𝑀 is semi-popular if for a majority of the matchings 𝑁 in 𝐺, 𝑀 does not lose a head-to-head election against 𝑁. We show that a semi-popular matching always exists in any graph 𝐺 and complement this existence result with a fully polynomial-time randomized approximation scheme (FPRAS). A special subclass of semi-popular matchings is the set of Copeland winners—the notion of Copeland winner is classical in social choice theory and a Copeland winner always exists in any voting instance. We study the complexity of computing a matching that is a Copeland winner and show there is no polynomial-time algorithm for this problem unless P = NP.

IJCAI Conference 2022 Conference Paper

Two for One & One for All: Two-Sided Manipulation in Matching Markets

• Hadi Hosseini
• Fatima Umar
• Rohit Vaish

Strategic behavior in two-sided matching markets has been traditionally studied in a "one-sided" manipulation setting where the agent who misreports is also the intended beneficiary. Our work investigates "two-sided" manipulation of the deferred acceptance algorithm where the misreporting agent and the manipulator (or beneficiary) are on different sides. Specifically, we generalize the recently proposed accomplice manipulation model (where a man misreports on behalf of a woman) along two complementary dimensions: (a) the two for one model, with a pair of misreporting agents (man and woman) and a single beneficiary (the misreporting woman), and (b) the one for all model, with one misreporting agent (man) and a coalition of beneficiaries (all women). Our main contribution is to develop polynomial-time algorithms for finding an optimal manipulation in both settings. We obtain these results despite the fact that an optimal one for all strategy fails to be inconspicuous, while it is unclear whether an optimal two for one strategy satisfies the inconspicuousness property. We also study the conditions under which stability of the resulting matching is preserved. Experimentally, we show that two-sided manipulations are more frequently available and offer better quality matches than their one-sided counterparts.

IJCAI Conference 2021 Conference Paper

Accomplice Manipulation of the Deferred Acceptance Algorithm

• Hadi Hosseini
• Fatima Umar
• Rohit Vaish

The deferred acceptance algorithm is an elegant solution to the stable matching problem that guarantees optimality and truthfulness for one side of the market. Despite these desirable guarantees, it is susceptible to strategic misreporting of preferences by the agents on the other side. We study a novel model of strategic behavior under the deferred acceptance algorithm: manipulation through an accomplice. Here, an agent on the proposed-to side (say, a woman) partners with an agent on the proposing side---an accomplice---to manipulate on her behalf (possibly at the expense of worsening his match). We show that the optimal manipulation strategy for an accomplice comprises of promoting exactly one woman in his true list (i. e. , an inconspicuous manipulation). This structural result immediately gives a polynomial-time algorithm for computing an optimal accomplice manipulation. We also study the conditions under which the manipulated matching is stable with respect to the true preferences. Our experimental results show that accomplice manipulation outperforms self manipulation both in terms of the frequency of occurrence as well as the quality of matched partners.

AAAI Conference 2021 Conference Paper

Fair and Efficient Allocations under Lexicographic Preferences

• Hadi Hosseini
• Sujoy Sikdar
• Rohit Vaish
• Lirong Xia

Envy-freeness up to any good (EFX) provides a strong and intuitive guarantee of fairness in the allocation of indivisible goods. But whether such allocations always exist or whether they can be efficiently computed remains an important open question. We study the existence and computation of EFX in conjunction with various other economic properties under lexicographic preferences–a well-studied preference model in artificial intelligence and economics. In sharp contrast to the known results for additive valuations, we not only prove the existence of EFX and Pareto optimal allocations, but in fact provide an algorithmic characterization of these two properties. We also characterize the mechanisms that are, in addition, strategyproof, non-bossy, and neutral. When the efficiency notion is strengthened to rank-maximality, we obtain non-existence and computational hardness results, and show that tractability can be restored when EFX is relaxed to another well-studied fairness notion called maximin share guarantee (MMS).

AAAI Conference 2021 Conference Paper

Representative Proxy Voting

• Elliot Anshelevich
• Zack Fitzsimmons
• Rohit Vaish
• Lirong Xia

We study a model of proxy voting where the candidates, voters, and proxies are all located on the real line, and instead of voting directly, each voter delegates its vote to the closest proxy. The goal is to find a set of proxies that is θrepresentative, which entails that for any voter located anywhere on the line, its favorite candidate is within a distance θ of the favorite candidate of its closest proxy. This property guarantees a strong form of representation as the set of voters is not required to be fixed in advance, or even be finite. We show that for candidates located on a line, an optimal proxy arrangement can be computed in polynomial time. Moreover, we provide upper and lower bounds on the number of proxies required to form a θ-representative set, thus showing that a relatively small number of proxies is enough to capture the preferences of any set of voters. An additional beneficial property of a θ-representative proxy arrangement is that for strict-Condorcet voting rules, the outcome of proxy voting is similarly close to the outcome of direct voting.

AIJ Journal 2021 Journal Article

Stable fractional matchings

• Ioannis Caragiannis
• Aris Filos-Ratsikas
• Panagiotis Kanellopoulos
• Rohit Vaish

AAAI Conference 2020 Conference Paper

Fair Division Through Information Withholding

• Hadi Hosseini
• Sujoy Sikdar
• Rohit Vaish
• Hejun Wang
• Lirong Xia

Envy-freeness up to one good (EF1) is a well-studied fairness notion for indivisible goods that addresses pairwise envy by the removal of at most one good. In the worst case, each pair of agents might require the (hypothetical) removal of a different good, resulting in a weak aggregate guarantee. We study allocations that are nearly envy-free in aggregate, and define a novel fairness notion based on information withholding. Under this notion, an agent can withhold (or hide) some of the goods in its bundle and reveal the remaining goods to the other agents. We observe that in practice, envyfreeness can be achieved by withholding only a small number of goods overall. We show that finding allocations that withhold an optimal number of goods is computationally hard even for highly restricted classes of valuations. In contrast to the worst-case results, our experiments on synthetic and realworld preference data show that existing algorithms for finding EF1 allocations withhold a close-to-optimal amount of information.

IJCAI Conference 2019 Conference Paper

Equitable Allocations of Indivisible Goods

• Rupert Freeman
• Sujoy Sikdar
• Rohit Vaish
• Lirong Xia

In fair division, equitability dictates that each participant receives the same level of utility. In this work, we study equitable allocations of indivisible goods among agents with additive valuations. While prior work has studied (approximate) equitability in isolation, we consider equitability in conjunction with other well-studied notions of fairness and economic efficiency. We show that the Leximin algorithm produces an allocation that satisfies equitability up to any good and Pareto optimality. We also give a novel algorithm that guarantees Pareto optimality and equitability up to one good in pseudopolynomial time. Our experiments on real-world preference data reveal that approximate envy-freeness, approximate equitability, and Pareto optimality can often be achieved simultaneously.

IJCAI Conference 2019 Conference Paper

Minimizing Time-to-Rank: A Learning and Recommendation Approach

• Haoming Li
• Sujoy Sikdar
• Rohit Vaish
• Junming Wang
• Lirong Xia
• Chaonan Ye

Consider the following problem faced by an online voting platform: A user is provided with a list of alternatives, and is asked to rank them in order of preference using only drag-and-drop operations. The platform's goal is to recommend an initial ranking that minimizes the time spent by the user in arriving at her desired ranking. We develop the first optimization framework to address this problem, and make theoretical as well as practical contributions. On the practical side, our experiments on the Amazon Mechanical Turk platform provide two interesting insights about user behavior: First, that users' ranking strategies closely resemble selection or insertion sort, and second, that the time taken for a drag-and-drop operation depends linearly on the number of positions moved. These insights directly motivate our theoretical model of the optimization problem. We show that computing an optimal recommendation is NP-hard, and provide exact and approximation algorithms for a variety of special cases of the problem. Experimental evaluation on MTurk shows that, compared to a random recommendation strategy, the proposed approach reduces the (average) time-to-rank by up to 50%.

AAMAS Conference 2018 Conference Paper

Greedy Algorithms for Maximizing Nash Social Welfare

• Siddharth Barman
• Sanath Kumar Krishnamurthy
• Rohit Vaish

We study the problem of fairly allocating a set of indivisible goods among agents with additive valuations. The extent of fairness of an allocation is measured by its Nash social welfare, which is the geometric mean of the valuations of the agents for their bundles. While the problem of maximizing Nash social welfare is known to be APX-hard in general, we study the effectiveness of simple, greedy algorithms in solving this problem in two interesting special cases. First, we show that a simple, greedy algorithm provides a 1. 061approximation guarantee when agents have identical valuations, even though the problem of maximizing Nash social welfare remains NP-hard for this setting. Second, we show that when agents have binary valuations over the goods, an exact solution (i. e. , a Nash optimal allocation) can be found in polynomial time via a greedy algorithm. Our results in the binary setting extend to provide novel, exact algorithms for optimizing Nash social welfare under concave valuations. Notably, for the above mentioned scenarios, our techniques provide a simple alternative to several of the existing, more sophisticated techniques for this problem such as constructing equilibria of Fisher markets or using real stable polynomials.

UAI Conference 2017 Conference Paper

Analysis of Thompson Sampling for Stochastic Sleeping Bandits

• Aritra Chatterjee 0001
• Ganesh Ghalme
• Shweta Jain 0002
• Rohit Vaish
• Yadati Narahari

We study a variant of the stochastic multiarmed bandit problem where the set of available arms varies arbitrarily with time (also known as the sleeping bandit problem). We focus on the Thompson Sampling algorithm and consider a regret notion defined with respect to the best available arm. Our main result is an O(log T ) regret bound for Thompson Sampling, which generalizes a similar bound known for this algorithm from the classical bandit setting. Our bound also matches (up to constants) the best-known lower bound for the sleeping bandit problem. We show via simulations that Thompson Sampling outperforms the UCB-style AUER algorithm for the sleeping bandit problem.

IJCAI Conference 2017 Conference Paper

Manipulating Gale-Shapley Algorithm: Preserving Stability and Remaining Inconspicuous

• Rohit Vaish
• Dinesh Garg

We study the problem of manipulation of the men-proposing Gale-Shapley algorithm by a single woman via permutation of her true preference list. Our contribution is threefold: First, we show that the matching induced by an optimal manipulation is stable with respect to the true preferences. Second, we identify a class of optimal manipulations called inconspicuous manipulations which, in addition to preserving stability, are also nearly identical to the true preference list of the manipulator (making the manipulation hard to be detected). Third, for optimal inconspicuous manipulations, we strengthen the stability result by showing that the entire stable lattice of the manipulated instance is contained inside the original lattice. ​

SODA Conference 2017 Conference Paper

Opting Into Optimal Matchings

• Avrim Blum
• Ioannis Caragiannis
• Nika Haghtalab
• Ariel D. Procaccia
• Eviatar B. Procaccia
• Rohit Vaish

We revisit the problem of designing optimal, individually rational matching mechanisms (in a general sense, allowing for cycles in directed graphs), where each player—who is associated with a subset of vertices—matches as many of his own vertices when he opts into the matching mechanism as when he opts out. We offer a new perspective on this problem by considering an arbitrary graph, but assuming that vertices are associated with players at random. Our main result asserts that, under certain conditions, any fixed optimal matching is likely to be individually rational up to lower-order terms. We also show that a simple and practical mechanism is (fully) individually rational, and likely to be optimal up to lower-order terms. We discuss the implications of our results for market design in general, and kidney exchange in particular.

AAMAS Conference 2016 Conference Paper

On the Computational Hardness of Manipulating Pairwise Voting Rules

• Rohit Vaish
• Neeldhara Misra
• Shivani Agarwal
• Avrim Blum

Standard voting rules usually assume that the preferences of voters are provided in the form of complete rankings over a fixed set of alternatives. This assumption does not hold in applications like recommendation systems where the set of alternatives is extremely large and only partial preferences can be elicited. In this paper, we study the problem of strategic manipulation of voting rules that aggregate voter preferences provided in the form of pairwise comparisons between alternatives. Our contributions are twofold: first, we show that any onto pairwise voting rule is manipulable in principle. Next, we analyze how the computational complexity of manipulation of such rules varies with the structure of the graph induced by the pairs of alternatives that the manipulator is allowed to vote over and the type of the preference relation. Building on natural connections between the pairwise manipulation and sports elimination problems (including a mixed-elimination variant that we introduce in this paper), we show that manipulating pairwise voting rules can be computationally hard even in the single-manipulator setting, a setting where most standard voting rules are known to be easy to manipulate. General Terms Algorithms, Economics, Theory

NeurIPS Conference 2014 Conference Paper

On the Statistical Consistency of Plug-in Classifiers for Non-decomposable Performance Measures

• Harikrishna Narasimhan
• Rohit Vaish
• Shivani Agarwal

We study consistency properties of algorithms for non-decomposable performance measures that cannot be expressed as a sum of losses on individual data points, such as the F-measure used in text retrieval and several other performance measures used in class imbalanced settings. While there has been much work on designing algorithms for such performance measures, there is limited understanding of the theoretical properties of these algorithms. Recently, Ye et al. (2012) showed consistency results for two algorithms that optimize the F-measure, but their results apply only to an idealized setting, where precise knowledge of the underlying probability distribution (in the form of the true' posterior class probability) is available to a learning algorithm. In this work, we consider plug-in algorithms that learn a classifier by applying an empirically determined threshold to a suitable estimate' of the class probability, and provide a general methodology to show consistency of these methods for any non-decomposable measure that can be expressed as a continuous function of true positive rate (TPR) and true negative rate (TNR), and for which the Bayes optimal classifier is the class probability function thresholded suitably. We use this template to derive consistency results for plug-in algorithms for the F-measure and for the geometric mean of TPR and precision; to our knowledge, these are the first such results for these measures. In addition, for continuous distributions, we show consistency of plug-in algorithms for any performance measure that is a continuous and monotonically increasing function of TPR and TNR. Experimental results confirm our theoretical findings.