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Two prominent objectives in social choice are utilitarian - maximizing the sum of agents' utilities, and leximin - maximizing the smallest agent's utility, then the second-smallest, etc. Utilitarianism is typically computationally easier to attain but is generally viewed as less fair. This paper presents a general reduction scheme that, given a utilitarian solver, produces a distribution over outcomes that is leximin in expectation. Importantly, the scheme is robust in the sense that, given an approximate utilitarian solver, it produces an outcome that is approximately-leximin (in expectation) - with the same approximation factor. We apply our scheme to several social choice problems: stochastic allocations of indivisible goods, giveaway lotteries, and fair lotteries for participatory budgeting.

We study the problem of aggregating distributions, such as budget proposals, into a collective distribution. An ideal aggregation mechanism would be Pareto efficient, strategyproof, and fair. Most previous work assumes that agents evaluate budgets according to the $\ell_1$ distance to their ideal budget. We investigate and compare different models from the larger class of star-shaped utility functions - a multi-dimensional generalization of single-peaked preferences. For the case of two alternatives, we extend existing results by proving that under very general assumptions, the uniform phantom mechanism is the only strategyproof mechanism that satisfies proportionality - a minimal notion of fairness introduced by Freeman et al. (2021). Moving to the case of more than two alternatives, we establish sweeping impossibilities for $\ell_1$ and $\ell_\infty$ disutilities: no mechanism satisfies efficiency, strategyproofness, and proportionality. We then propose a new kind of star-shaped utilities based on evaluating budgets by the ratios of shares between a given budget and an ideal budget. For these utilities, efficiency, strategyproofness, and fairness become compatible. In particular, we prove that the mechanism that maximizes the Nash product of individual utilities is characterized by group-strategyproofness and a core-based fairness condition.

We investigate the problem of fairly dividing a divisible heterogeneous resource, also known as a cake, among a set of agents who may have different entitlements. We characterize the existence of a connected strongly-proportional allocation -- one in which every agent receives a contiguous piece worth strictly more than their proportional share. The characterization is supplemented with an algorithm that determines its existence using O(n * 2^n) queries. We devise a simpler characterization for agents with strictly positive valuations and with equal entitlements, and present an algorithm to determine the existence of such an allocation using O(n^2) queries. We provide matching lower bounds in the number of queries for both algorithms. When a connected strongly-proportional allocation exists, we show that it can also be computed using a similar number of queries. We also consider the problem of deciding the existence of a connected allocation of a cake in which each agent receives a piece worth a small fixed value more than their proportional share, and the problem of deciding the existence of a connected strongly-proportional allocation of a pie., Comment: Full version of paper accepted to ECAI 2024. Added pie allocation, stronger-than-strongly-proportional allocation, and fixed typographic issues

Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into a minimum number of bins such that the sum of item sizes in a bin does not exceed the capacity. We define a new variant called \emph{$k$-times bin-packing ($k$BP)}, where the goal is to pack the items such that each item appears exactly $k$ times, in $k$ different bins. We generalize some existing approximation algorithms for bin-packing to solve $k$BP, and analyze their performance ratio. The study of $k$BP is motivated by the problem of \emph{fair electricity distribution}. In many developing countries, the total electricity demand is higher than the supply capacity. We prove that every electricity division problem can be solved by $k$-times bin-packing for some finite $k$. We also show that $k$-times bin-packing can be used to distribute the electricity in a fair and efficient way. Particularly, we implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve $k$BP, and apply the generalizations to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem., Comment: 38 pages, 7 figures, 3 tables. This article is a full version of our accepted paper at "17th Symposium on Algorithmic Game Theory (SAGT 2024)"

Given a function f: [a,b] -> R, if f(a) < 0 and f(b)> 0 and f is continuous, the Intermediate Value Theorem implies that f has a root in [a,b]. Moreover, given a value-oracle for f, an approximate root of f can be computed using the bisection method, and the number of required evaluations is polynomial in the number of accuracy digits. The goal of this note is to identify conditions under which this polynomiality result extends to a multi-dimensional function that satisfies the conditions of Miranda's theorem -- the natural multi-dimensional extension of the Intermediate Value Theorem. In general, finding an approximate root might require an exponential number of evaluations even for a two-dimensional function. We show that, if f is two-dimensional and satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. For any fixed dimension d, if f is a d-dimensional function that satisfies all d^2-d ``ex-diagonal'' monotonicity conditions (that is, component i of f is monotonically decreasing with respect to variable j for all i!=j), then the number of required evaluations is polynomial in the accuracy. But if f satisfies only d^2-d-2 ex-diagonal conditions, then the number of required evaluations may be exponential in the accuracy. The case of d^2-d-1 ex-diagonal conditions remains unsolved. As an example application, we show that computing approximate roots of monotone functions can be used for approximate envy-free cake-cutting., Comment: We solved all the open cases, except the case when f has 3 or more dimensions, and satisfies all monotonicity conditions except one. Any ideas?

We consider the allocation of indivisible objects among agents with different valuations, which can be positive or negative. An egalitarian allocation is an allocation that maximizes the smallest value given to an agent; finding such an allocation is NP-hard. We present a simple polynomial-time algorithm that finds an allocation that is Pareto-efficient and almost-egalitarian: each agent's value is at least his value in an egalitarian allocation, minus the absolute value of a single object. The main tool is an algorithm for rounding a fractional allocation to a discrete allocation, by which each agent loses at most one good or gains at most one chore. Our algorithm generalizes and simplifies three previous algorithms. We discuss several aspects and observations about the algorithm and the problem at hand that open doors for efficient and robust implementations.

Charity is typically done either by individual donors, who donate money to the charities that they support, or by centralized organizations such as governments or municipalities, which collect the individual contributions and distribute them among a set of charities. On the one hand, individual charity respects the will of the donors but may be inefficient due to a lack of coordination. On the other hand, centralized charity is potentially more efficient but may ignore the will of individual donors. We present a mechanism that combines the advantages of both methods by distributing the contribution of each donor in an efficient way such that no subset of donors has an incentive to redistribute their donations. Assuming Leontief utilities (i.e., each donor is interested in maximizing an individually weighted minimum of all contributions across the charities), our mechanism is group-strategyproof, preference-monotonic, contribution-monotonic, maximizes Nash welfare, and can be computed using convex programming., Comment: Appears in the 24th ACM Conference on Economics and Computation (EC), 2023

Leximin is a common approach to multi-objective optimization, frequently employed in fair division applications. In leximin optimization, one first aims to maximize the smallest objective value; subject to this, one maximizes the second-smallest objective; and so on. Often, even the single-objective problem of maximizing the smallest value cannot be solved accurately. What can we hope to accomplish for leximin optimization in this situation? Recently, Henzinger et al. (2022) defined a notion of \emph{approximate} leximin optimality. Their definition, however, considers only an additive approximation. In this work, we first define the notion of approximate leximin optimality, allowing both multiplicative and additive errors. We then show how to compute, in polynomial time, such an approximate leximin solution, using an oracle that finds an approximation to a single-objective problem. The approximation factors of the algorithms are closely related: an $(\alpha,\epsilon)$-approximation for the single-objective problem (where $\alpha \in (0,1]$ and $\epsilon \geq 0$ are the multiplicative and additive factors respectively) translates into an $\left(\frac{\alpha^2}{1-\alpha + \alpha^2}, \frac{\epsilon}{1-\alpha +\alpha^2}\right)$-approximation for the multi-objective leximin problem, regardless of the number of objectives.

Maximum bipartite matching is a fundamental algorithmic problem which can be solved in polynomial time. We consider a natural variant in which there is a separation constraint: the vertices on one side lie on a path or a grid, and two vertices that are close to each other are not allowed to be matched simultaneously. We show that the problem is hard to approximate even for paths, and provide constant-factor approximation algorithms for both paths and grids.

We consider allocating indivisible chores among agents with different cost functions, such that all agents receive a cost of at most a constant factor times their maximin share. The state-of-the-art was presented in In EC 2021 by Huang and Lu. They presented a non-polynomial-time algorithm, called HFFD, that attains an 11/9 approximation, and a polynomial-time algorithm that attains a 5/4 approximation. In this paper, we show that HFFD can be reduced to an algorithm called MultiFit, developed by Coffman, Garey and Johnson in 1978 for makespan minimization in job scheduling. Using this reduction, we prove that the approximation ratio of HFFD is in fact equal to that of MultiFit, which is known to be 13/11 in general, 20/17 for n at most 7, and 15/13 for n=3. Moreover, we develop an algorithm for (13/11+epsilon)-maximin-share allocation for any epsilon>0, with run-time polynomial in the problem size and 1/epsilon. For n=3, we can improve the algorithm to find a 15/13-maximin-share allocation with run-time polynomial in the problem size. Thus, we have practical algorithms that attain the best known approximation to maximin-share chore allocation., Comment: Full version of paper accepted to EC 2023. Made minor corrections to images and notation

Cake cutting is a classic fair division problem, with the cake serving as a metaphor for a heterogeneous divisible resource. Recently, it was shown that for any number of players with arbitrary preferences over a cake, it is possible to partition the players into groups of any desired size and divide the cake among the groups so that each group receives a single contiguous piece and every player is envy-free. For two groups, we characterize the group sizes for which such an assignment can be computed by a finite algorithm, showing that the task is possible exactly when one of the groups is a singleton. We also establish an analogous existence result for chore division, and show that the result does not hold for a mixed cake.

In many developing countries, the total electricity demand is larger than the limited generation capacity of power stations. Many countries adopt the common practice of routine load shedding - disconnecting entire regions from the power supply - to maintain a balance between demand and supply. Load shedding results in inflicting hardship and discomfort on households, which is even worse and hence unfair to those whose need for electricity is higher than that of others during load shedding hours. Recently, Oluwasuji et al. [2020] presented this problem and suggested several heuristic solutions. In this work, we study the electricity distribution problem as a problem of fair division, model it using the related literature on cake-cutting problems, and discuss some insights on which parts of the time intervals are allocated to each household. We consider four cases: identical demand, uniform utilities; identical demand, additive utilities; different demand, uniform utilities; different demand, additive utilities. We provide the solution for the first two cases and discuss the novel concept of q-times bin packing in relation to the remaining cases. We also show how the fourth case is related to the consensus k-division problem. One can study objectives and constraints using utilitarian and egalitarian social welfare metrics, as well as trying to keep the number of cuts as small as possible. A secondary objective can be to minimize the maximum utility-difference between agents., Comment: 16 pages, 1 figure

We consider the problem of fairly and efficiently allocating indivisible items (goods or bads) under capacity constraints. In this setting, we are given a set of categorized items. Each category has a capacity constraint (the same for all agents), that is an upper bound on the number of items an agent can receive from each category. Our main result is a polynomial-time algorithm that solves the problem for two agents with additive utilities over the items. When each category contains items that are all goods (positively evaluated) or all chores (negatively evaluated) for each of the agents, our algorithm finds a feasible allocation of the items, which is both Pareto-optimal and envy-free up to one item. In the general case, when each item can be a good or a chore arbitrarily, our algorithm finds an allocation that is Pareto-optimal and envy-free up to one good and one chore., Comment: 13 pages, 3 figures

The $n$-way number partitioning problem is a classic problem in combinatorial optimization, with applications to diverse settings such as fair allocation and machine scheduling. All these problems are NP-hard, but various approximation algorithms are known. We consider three closely related kinds of approximations. The first two variants optimize the partition such that: in the first variant some fixed number $s$ of items can be \emph{split} between two or more bins and in the second variant we allow at most a fixed number $t$ of \emph{splittings}. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number $u$ times the largest item size. When the number of bins $n$ is unbounded, we show that every variant is strongly {\sf NP}-complete. When the number of bins $n$ is fixed, the running time depends on the fixed parameters $s,t,u$. For each variant, we give a complete picture of its running time. For $n=2$, the running time is easy to identify. Our main results consider any fixed integer $n \geq 3$. Using a two-way polynomial-time reduction between the first and the third variant, we show that $n$-way number-partitioning with $s$ split items can be solved in polynomial time if $s \geq n-2$, and it is {\sf NP}-complete otherwise. Also, $n$-way number-partitioning with $t$ splittings can be solved in polynomial time if $t \geq n-1$, and it is {\sf NP}-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if $u \geq (n-2)/n$, and it is {\sf NP}-complete otherwise. Our positive results for the optimization problems consider both min-max and max-min versions. Using the same reduction, we provide a fully polynomial-time approximation scheme for the case where the number of split items is lower than $n-2$.

We study the problem of fairly allocating a set of m indivisible chores (items with non-positive value) to n agents. We consider the desirable fairness notion of 1-out-of-d maximin share (MMS) -- the minimum value that an agent can guarantee by partitioning items into d bundles and receiving the least valued bundle -- and focus on ordinal approximation of MMS that aims at finding the largest d <= n for which 1-out-of-d MMS allocation exists. Our main contribution is a polynomial-time algorithm for 1-out-of-floor(2n/3) MMS allocation, and a proof of existence of 1-out-of-floor(3n/4) MMS allocation of chores. Furthermore, we show how to use recently-developed algorithms for bin-packing to approximate the latter bound up to a logarithmic factor in polynomial time., Comment: Full version of paper accepted to AAMAS 2022

We revisit the setting of fairly allocating indivisible items when agents have different weights representing their entitlements. First, we propose a parameterized family of relaxations for weighted envy-freeness and the same for weighted proportionality; the parameters indicate whether smaller-weight or larger-weight agents should be given a higher priority. We show that each notion in these families can always be satisfied, but any two cannot necessarily be fulfilled simultaneously. We then introduce an intuitive weighted generalization of maximin share fairness and establish the optimal approximation of it that can be guaranteed. Furthermore, we characterize the implication relations between the various weighted fairness notions introduced in this and prior work, and relate them to the lower and upper quota axioms from apportionment., Comment: Appears in the 36th AAAI Conference on Artificial Intelligence (AAAI), 2022

An algorithm for number-partitioning is called value-monotone if whenever one of the input numbers increases, the objective function (the largest sum or the smallest sum of a subset in the output) weakly increases. This note proves that the List Scheduling algorithm and the Longest Processing Time algorithm are both value-monotone. This is in contrast to another algorithm -- MultiFit -- which is not value-monotone., Comment: I needed this lemma for something else, and could not find it anywhere, so I wrote a short proof. Is it already known?

In fair division of indivisible goods, $\ell$-out-of-$d$ maximin share (MMS) is the value that an agent can guarantee by partitioning the goods into $d$ bundles and choosing the $\ell$ least preferred bundles. Most existing works aim to guarantee to all agents a constant fraction of their 1-out-of-$n$ MMS. But this guarantee is sensitive to small perturbation in agents' cardinal valuations. We consider a more robust approximation notion, which depends only on the agents' \emph{ordinal} rankings of bundles. We prove the existence of $\ell$-out-of-$\lfloor(\ell+\frac{1}{2})n\rfloor$ MMS allocations of goods for any integer $\ell\geq 1$, and present a polynomial-time algorithm that finds a $1$-out-of-$\lceil\frac{3n}{2}\rceil$ MMS allocation when $\ell = 1$. We further develop an algorithm that provides a weaker ordinal approximation to MMS for any $\ell > 1$., Comment: Accepted to JAIR

This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of \emph{proportionality} is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the \emph{ordinal maximin share approximation}, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions., Comment: Appears in the 30th International Joint Conference on Artificial Intelligence (IJCAI), 2021

We study the recently introduced cake-cutting setting in which the cake is represented by an undirected graph. This generalizes the canonical interval cake and allows for modeling the division of road networks. We show that when the graph is a forest, an allocation satisfying the well-known criterion of maximin share fairness always exists. Our result holds even when separation constraints are imposed, in which case no multiplicative approximation of proportionality can be guaranteed. Furthermore, while maximin share fairness is not always achievable for general graphs, we prove that ordinal relaxations can be attained., Comment: Appears in the 30th International Joint Conference on Artificial Intelligence (IJCAI), 2021

We study the problem of fairly allocating a divisible resource, also known as cake cutting, with an additional requirement that the shares that different agents receive should be sufficiently separated from one another. This captures, for example, constraints arising from social distancing guidelines. While it is sometimes impossible to allocate a proportional share to every agent under the separation requirement, we show that the well-known criterion of maximin share fairness can always be attained. We then provide algorithmic analysis of maximin share fairness in this setting -- for instance, the maximin share of an agent cannot be computed exactly by any finite algorithm, but can be approximated with an arbitrarily small error. In addition, we consider the division of a pie (i.e., a circular cake) and show that an ordinal relaxation of maximin share fairness can be achieved. We also prove that an envy-free or equitable allocation that allocates the maximum amount of resource exists under separation., Comment: Appears in the 35th AAAI Conference on Artificial Intelligence (AAAI), 2021

We analyze the run-time complexity of computing allocations that are both fair and maximize the utilitarian social welfare, defined as the sum of agents' utilities. We focus on two tractable fairness concepts: envy-freeness up to one item (EF1) and proportionality up to one item (PROP1). We consider two computational problems: (1) Among the utilitarian-maximal allocations, decide whether there exists one that is also fair; (2) among the fair allocations, compute one that maximizes the utilitarian welfare. We show that both problems are strongly NP-hard when the number of agents is variable, and remain NP-hard for a fixed number of agents greater than two. For the special case of two agents, we find that problem (1) is polynomial-time solvable, while problem (2) remains NP-hard. Finally, with a fixed number of agents, we design pseudopolynomial-time algorithms for both problems. We extend our results to the stronger fairness notions envy-freeness up to any item (EFx) and proportionality up to any item (PROPx)., Comment: Added experiments comparing dynamic programming to integer linear programming

A d-partite hypergraph is called *fractionally balanced* if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall's theorem we prove lower bounds on the matching number of such hypergraphs. These bounds yield rainbow versions of the KKM theorem for products of simplices, which in turn are used to obtain some results on multiple-cake division, and on rainbow matchings in families of d-intervals.

We study fair allocation of indivisible goods among additive agents with feasibility constraints. In these settings, every agent is restricted to get a bundle among a specified set of feasible bundles. Such scenarios have been of great interest to the AI community due to their applicability to real-world problems. Following some impossibility results, we restrict attention to matroid feasibility constraints that capture natural scenarios, such as the allocation of shifts to medical doctors, and the allocation of conference papers to referees. We focus on the common fairness notion of envy-freeness up to one good (EF1). Previous algorithms for finding EF1 allocations are either restricted to agents with identical feasibility constraints, or allow free disposal of items. An open problem is the existence of EF1 complete allocations among heterogeneous agents, where the heterogeneity is both in the agents' feasibility constraints and in their valuations. In this work, we make progress on this problem by providing positive and negative results for different matroid and valuation types. Among other results, we devise polynomial-time algorithms for finding EF1 allocations in the following settings: (i) $n$ agents with heterogeneous partition matroids and heterogeneous binary valuations, (ii) 2 agents with heterogeneous partition matroids and heterogeneous additive valuations, and (iii) at most 3 agents with heterogeneous binary valuations and identical base-orderable matroid constraints., Comment: Contains all proofs omitted from the AAAI'21 paper, an uptodate literature survey, a more general non-existence result in Subsection 3.3, a simpler proof of Theorem 5, and simpler algorithms and proofs in Section 8

A fundamental result in cake cutting states that for any number of players with arbitrary preferences over a cake, there exists a division of the cake such that every player receives a single contiguous piece and no player is left envious. We generalize this result by showing that it is possible to partition the players into groups of any desired sizes and divide the cake among the groups, so that each group receives a single contiguous piece and no player finds the piece of another group better than that of the player's own group.

Rental Harmony is the problem of assigning rooms in a rented house to tenants with different preferences, and simultaneously splitting the rent among them, such that no tenant envies the bundle (room+price) given to another tenant. Different papers have studied this problem under two incompatible assumptions: the miserly tenants assumption is that each tenant prefers a free room to a non-free room; the quasilinear tenants assumption is that each tenant attributes a monetary value to each room, and prefers a room of which the difference between value and price is maximum. This note shows how to adapt the main technique used for rental harmony with miserly tenants, using Sperner's lemma, to a much more general class of preferences, that contains both miserly and quasilinear tenants as special cases. This implies that some recent results derived for miserly tenants apply to this more general preference class too., Comment: Generalized all results to "compensable tenants" - a class that contains both miserly and quasilinear tenants

Given a finite set $X$ and an ordering $\succeq$ over its subsets, the $l$-out-of-$d$ maximin-share of $X$ is the maximal (by $\succeq$) subset of $X$ that can be constructed by partitioning $X$ into $d$ parts and picking the worst union of $l$ parts. A pair of integers $(l,d)$ dominates a pair $(l',d')$ if, for any set $X$ and ordering $\succeq$, the $l$-out-of-$d$ maximin-share of $X$ is at least as good (by $\succeq$) as the $l'$-out-of-$d'$ maximin-share of $X$. This note presents a necessary and sufficient condition for deciding whether a given pair of integers dominates another pair, and an algorithm for finding all non-dominated pairs. It compares the $l$-out-of-$d$ maximin-share to some other criteria for fair allocation of indivisible objects among people with different entitlements., Comment: First draft

A set of objects is to be divided fairly among agents with different tastes, modeled by additive utility-functions. An agent is allowed to share a bounded number of objects between two or more agents in order to attain fairness. The paper studies various notions of fairness, such as proportionality, envy-freeness, equitability, and consensus. We analyze the run-time complexity of finding a fair allocation with a given number of sharings under several restrictions on the agents' valuations, such as: binary generalized-binary and non-degenerate., Comment: Full version of a paper accepted to SAGT 2024

In two-sided markets, Myerson and Satterthwaite's impossibility theorem states that one can not maximize the gain-from-trade while also satisfying truthfulness, individual-rationality and no deficit. Attempts have been made to circumvent Myerson and Satterthwaite's result by attaining approximately-maximum gain-from-trade: the double-sided auctions of McAfee (1992) is truthful and has no deficit, and the one by Segal-Halevi et al. (2016) additionally has no surplus --- it is strongly-budget-balanced. They consider two categories of agents --- buyers and sellers, where each trade set is composed of a single buyer and a single seller. The practical complexity of applications such as supply chain require one to look beyond two-sided markets. Common requirements are for: buyers trading with multiple sellers of different or identical items, buyers trading with sellers through transporters and mediators, and sellers trading with multiple buyers. We attempt to address these settings. We generalize Segal-Halevi et al. (2016)'s strongly-budget-balanced double-sided auction setting to a multilateral market where each trade set is composed of any number of agent categories. Our generalization refines the notion of competition in multi-sided auctions by introducing the concepts of external competition and trade reduction. We also show an obviously-truthful implementation of our auction using multiple ascending prices., Comment: Preliminary version accepted to AAAI 2020. This version adds (1) External competition auction for arbitrary recipe vectors; (2) Obvious-truthfulness proof; (3) Simulation experiments

In cake-cutting, strategy-proofness is a very costly requirement in terms of fairness: for n=2 it implies a dictatorial allocation, whereas for n > 2 it requires that one agent receives no cake. We show that a weaker version of this property recently suggested by Troyan and Morril, called non-obvious manipulability, is compatible with the strong fairness property of proportionality, which guarantees that each agent receives 1/n of the cake. Both properties are satisfied by the leftmost leaves mechanism, an adaptation of the Dubins - Spanier moving knife procedure. Most other classical proportional mechanisms in literature are obviously manipulable, including the original moving knife mechanism. Non-obvious manipulability explains why leftmost leaves is manipulated less often in practice than other proportional mechanisms.

A collection of objects, some of which are good and some are bad, is to be divided fairly among agents with different tastes, modeled by additive utility functions. If the objects cannot be shared, so that each of them must be entirely allocated to a single agent, then a fair division may not exist. What is the smallest number of objects that must be shared between two or more agents in order to attain a fair and efficient division? In this paper, fairness is understood as proportionality or envy-freeness, and efficiency, as fractional Pareto-optimality. We show that, for a generic instance of the problem (all instances except a zero-measure set of degenerate problems), a fair fractionally Pareto-optimal division with the smallest possible number of shared objects can be found in polynomial time, assuming that the number of agents is fixed. The problem becomes computationally hard for degenerate instances, where agents' valuations are aligned for many objects., Comment: Accepted to Operations Research

We consider a multi-agent resource allocation setting that models the assignment of papers to reviewers. A recurring issue in allocation problems is the compatibility of welfare/efficiency and fairness. Given an oracle to find a welfare-achieving allocation, we embed such an oracle into a flexible algorithm called the Constrained Round Robin (CRR) algorithm, that achieves the required welfare level. Our algorithm also allows the system designer to lower the welfare requirements in order to achieve a higher degree of fairness. If the welfare requirement is lowered enough, a strengthening of envy-freeness up to one item is guaranteed. Hence, our algorithm can be viewed as a computationally efficient way to interpolate between welfare and approximate envy-freeness in allocation problems.

A matching in a bipartite graph with parts X and Y is called envy-free if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do not. We prove that every bipartite graph has a unique partition such that all envy-free matchings are contained in one of the partition sets. Using this structural theorem, we provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum total weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources ("cakes") or discrete ones. In particular, we propose a symmetric algorithm for proportional cake-cutting, an algorithm for 1-out-of-(2n-2) maximin-share allocation of discrete goods, and an algorithm for 1-out-of-floor(2n/3) maximin-share allocation of discrete bads among n agents., Comment: Appeared in Information Sciences, 587:164--187. But during the production, the main theorem text was deleted. The arXiv version is the correct one

In the classic problem of fair cake-cutting, a single interval ("cake") has to be divided among n agents with different value measures, giving each agent a single sub-interval with a value of at least 1/n of the total. This paper studies a generalization in which the cake is made of m disjoint intervals, and each agent should get at most k sub-intervals. The paper presents a polynomial-time algorithm that guarantees to each agent at least min(1/n, k/(m+n-1)) of the total value, and shows that this is the largest fraction that can be guaranteed. The algorithm simultaneously guarantees to each agent at least 1/n of the value that the agent can get without partners to share with. The main technical tool is envy-free matching in a bipartite graph. Some of the results remain valid even with additional fairness constraints such as envy-freeness. Besides the natural application of the algorithm to simultaneous division of multiple land-estates, the paper shows an application to a geometric problem -- fair division of a two-dimensional land estate shaped as a rectilinear polygon, where each agent should receive a rectangular piece., Comment: Previous name: "Fair Division of an Archipelago". This version proves the archipelago conjecture for any k, n and m

We investigate whether fairness is compatible with efficiency in economies with multi-self agents, who may not be able to integrate their multiple objectives into a single complete and transitive ranking. We adapt envy-freeness, egalitarian-equivalence and the fair-share guarantee in two different ways. An allocation is unambiguously-fair if it satisfies the chosen criterion of fairness according to every objective of any agent; it is aggregate-fair if it satisfies the criterion for some aggregation of each agent's objectives. While efficiency is always compatible with the unambiguous fair-share guarantee, it is incompatible with unambiguous envy-freeness in economics with at least three agents. Two agents are enough for efficiency and unambiguous egalitarian-equivalence to clash. Efficiency and the unambiguous fair-share guarantee can be attained together with aggregate envy-freeness, or aggregate egalitarian-equivalence., Comment: Previous title: "Fair and Efficient Division among Families"

Using a lab experiment, we investigate the real-life performance of envy-free and proportional cake-cutting procedures with respect to fairness and preference manipulation. We find that envy-free procedures, in particular Selfridge-Conway, are fairer and also are perceived as fairer than their proportional counterparts, despite the fact that agents very often manipulate them. Our results support the practical use of the celebrated Selfridge-Conway procedure, and more generally, of envy-free cake-cutting mechanisms. We also find that subjects learn their opponents' preferences after repeated interaction and use this knowledge to improve their allocated share of the cake. Learning reduces truth-telling behavior, but also reduces envy.

We consider the problem of a principal who needs to elicit the true worth of an object she owns from an agent who has a unique ability to compute this information. The correctness of the information cannot be verified by the principal, so it is important to incentivize the agent to report truthfully. Previous works coped with this unverifiability by employing two or more information agents and awarding them according to the correlation between their reports. In this paper we show that even with only one information agent truthful information can be elicited, as long as the object is valuable for the agent too. In particular the paper introduces a mechanism that, under mild realistic assumptions, is proved to elicit the information truthfully, even when computing the information is costly for the agent. Moreover, using this mechanism, the principal obtains the truthful information incurring an arbitrarily small expense beyond whatever unavoidable costs the setting dictates., Comment: Full version of paper accepted to AAMAS 2019. Previous title: "Making an Appraiser Work for You"

A cake has to be divided fairly among $n$ agents. When all agents have equal entitlements, it is known that such a division can be implemented with $n-1$ cuts. When agents may have different entitlements, the paper shows that at least $2 n -2$ cuts may be necessary, and $O(n \log(n))$ cuts are always sufficient., Comment: Shorten the proof of the upper bound

Off-chain transaction channels represent one of the leading techniques to scale the transaction throughput in cryptocurrencies. However, the economic effect of transaction channels on the system has not been explored much until now. We study the economics of Bitcoin transaction channels, and present a framework for an economic analysis of the lightning network and its effect on transaction fees on the blockchain. Our framework allows us to reason about different patterns of demand for transactions and different topologies of the lightning network, and to derive the resulting fees for transacting both on and off the blockchain. Our initial results indicate that while the lightning network does allow for a substantially higher number of transactions to pass through the system, it does not necessarily provide higher fees to miners, and as a result may in fact lead to lower participation in mining within the system., Comment: An earlier version of the paper was presented at Scaling Bitcoin 2017

In a seminal paper, McAfee (1992) presented a truthful mechanism for double auctions, attaining asymptotically-optimal gain-from-trade without any prior information on the valuations of the traders. McAfee's mechanism handles single-parametric agents, allowing each seller to sell a single unit and each buyer to buy a single unit. This paper presents a double-auction mechanism that handles multi-parametric agents and allows multiple units per trader, as long as the valuation functions of all traders have decreasing marginal returns. The mechanism is prior-free, ex-post individually-rational, dominant-strategy truthful and strongly-budget-balanced. Its gain-from-trade approaches the optimum when the market size is sufficiently large., Comment: Accepted to the AAAI2018 conference

A seminal theorem of Myerson and Satterthwaite (1983) proves that, in a game of bilateral trade between a single buyer and a single seller, no mechanism can be simultaneously individually-rational, budget-balanced, incentive-compatible and socially-efficient. However, the impossibility disappears if the price is fixed exogenously and the social-efficiency goal is subject to individual-rationality at the given price. We show that the impossibility comes back if there are multiple units of the same good, or multiple types of goods, even when the prices are fixed exogenously. Particularly, if there are $M$ units of the same good or $M$ kinds of goods, for some $M\geq 2$, then no truthful mechanism can guarantee more than $1/M$ of the optimal gain-from-trade. In the single-good multi-unit case, if both agents have submodular valuations (decreasing marginal returns), then no truthful mechanism can guarantee more than $1/H_M$ of the optimal gain-from-trade, where $H_M$ is the $M$-th harmonic number ($H_M\approx \ln{M}+1/2$). All upper bounds are tight.