arXiv subjects

Erel Segal-Halevi

Publications and source records attributed to Erel Segal-Halevi.

At least 73 records · Page 4Linked to original sources

arXiv2019

Envy-Free Cake-Cutting among Families

This paper extends the classic cake-cutting problem to a situation in which the "cake" is divided among families. Each piece of cake is owned and used simultaneously by all members of the family. A typical example of such a cake is land. We examine three ways to assess the fairness of such a division, based on the classic no-envy criterion: (a) Average envy-freeness means that for each family, the average value of its share (averaged over all family members) is weakly larger than the average value of any other share; (b) Unanimous envy-freeness means that in each family, each member values the family's share weakly more than any other share; (c) Democratic envy-freeness means that in each family, at least half the members value the family's share weakly more than any other share. We study each of these definitions from both an existential and a computational perspective.

cs.GT↗

arXiv2019

The Constrained Round Robin Algorithm for Fair and Efficient Allocation

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.

cs.GT↗

arXiv2019

Fair Allocation based on Diminishing Differences

Ranking alternatives is a natural way for humans to explain their preferences. It is being used in many settings, such as school choice, course allocations and residency matches. In some cases, several `items' are given to each participant. Without having any information on the underlying cardinal utilities, arguing about fairness of allocation requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is larger than the difference between two items at the bottom. This assumption implies a preference extension which we call diminishing differences (DD), where X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possibly-proportional according to this assumption. Based on this characterization, we present a polynomial-time algorithm for finding a necessarily-DD-proportional allocation if it exists. Using simulations, we show that with high probability, a necessarily-proportional allocation does not exist but a necessarily-DD-proportional allocation exists, and moreover, that allocation is proportional according to the underlying cardinal utilities. We also consider chore allocation under the analogous condition --- increasing-differences.

cs.GT↗

arXiv2019

Democratic Fair Allocation of Indivisible Goods

We study the problem of fairly allocating indivisible goods to groups of agents. Agents in the same group share the same set of goods even though they may have different preferences. Previous work has focused on unanimous fairness, in which all agents in each group must agree that their group's share is fair. Under this strict requirement, fair allocations exist only for small groups. We introduce the concept of democratic fairness, which aims to satisfy a certain fraction of the agents in each group. This concept is better suited to large groups such as cities or countries. We present protocols for democratic fair allocation among two or more arbitrarily large groups of agents with monotonic, additive, or binary valuations. For two groups with arbitrary monotonic valuations, we give an efficient protocol that guarantees envy-freeness up to one good for at least $1/2$ of the agents in each group, and prove that the $1/2$ fraction is optimal. We also present other protocols that make weaker fairness guarantees to more agents in each group, or to more groups. Our protocols combine techniques from different fields, including combinatorial game theory, cake cutting, and voting.

cs.GT↗

arXiv2019

Fair Cake-Cutting among Families

We study the fair division of a continuous resource, such as a land-estate or a time-interval, among pre-specified groups of agents, such as families. Each family is given a piece of the resource and this piece is used simultaneously by all family members, while different members may have different value functions. Three ways to assess the fairness of such a division are examined. (a) Average Fairness means that each family's share is fair according to the "family value function", defined as the arithmetic mean of the value functions of the family members. (b) Unanimous Fairness means that all members in all families feel that their family's share is fair according to their personal value function. (c) Democratic Fairness means that in each family, at least a fixed fraction (e.g. a half) of the members feel that their family's share is fair. We compare these criteria based on the number of connected components in the resulting division and on their compatibility with Pareto-efficiency.

cs.GT↗

arXiv2019

Envy-Free Division of Land

Classic cake-cutting algorithms enable people with different preferences to divide among them a heterogeneous resource (``cake''), such that the resulting division is fair according to each agent's individual preferences. However, these algorithms either ignore the geometry of the resource altogether, or assume it is one-dimensional. In practice, it is often required to divide multi-dimensional resources, such as land-estates or advertisement spaces in print or electronic media. In such cases, the geometric shape of the allotted piece is of crucial importance. For example, when building houses or designing advertisements, in order to be useful, the allotments should be squares or rectangles with bounded aspect-ratio. We thus introduce the problem of fair land division --- fair division of a multi-dimensional resource wherein the allocated piece must have a pre-specified geometric shape. We present constructive division algorithms that satisfy the two most prominent fairness criteria, namely envy-freeness and proportionality. In settings where proportionality cannot be achieved due to the geometric constraints, our algorithms provide a partially-proportional division, guaranteeing that the fraction allocated to each agent be at least a certain positive constant. We prove that in many natural settings the envy-freeness requirement is compatible with the best attainable partial-proportionality.

cs.GT↗

arXiv2019

Cake-Cutting with Different Entitlements: How Many Cuts are Needed?

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.

math.CO↗

arXiv2019

Obtaining Costly Unverifiable Valuations from a Single Agent

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.

cs.GT↗

arXiv2018

Fairly Dividing a Cake after Some Parts Were Burnt in the Oven

There is a heterogeneous resource that contains both good parts and bad parts, for example, a cake with some parts burnt, a land-estate with some parts heavily taxed, or a chore with some parts fun to do. The resource has to be divided fairly among $n$ agents with different preferences, each of whom has a personal value-density function on the resource. The value-density functions can accept any real value --- positive, negative or zero. Each agent should receive a connected piece and no agent should envy another agent. We prove that such a division exists for 3 agents and present preliminary positive results for larger numbers of agents.

math.CO↗

arXiv2018

Double Auctions in Markets for Multiple Kinds of Goods

Motivated by applications such as stock exchanges and spectrum auctions, there is a growing interest in mechanisms for arranging trade in two-sided markets. Existing mechanisms are either not truthful, or do not guarantee an asymptotically-optimal gain-from-trade, or rely on a prior on the traders' valuations, or operate in limited settings such as a single kind of good. We extend the random market-halving technique used in earlier works to markets with multiple kinds of goods, where traders have gross-substitute valuations. We present MIDA: a Multi Item-kind Double-Auction mechanism. It is prior-free, truthful, strongly-budget-balanced, and guarantees near-optimal gain from trade when market sizes of all goods grow to $\infty$ at a similar rate.

cs.GT↗

arXiv2018

Counting Blanks in Polygonal Arrangements

Inside a two dimensional region ("cake"), there are $m$ non-overlapping tiles of a certain kind ("toppings"). We want to expand the toppings while keeping them non-overlapping, and possibly add some blank pieces of the same "certain kind", such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axes-parallel rectangles. (4) The cake is an axes-parallel rectilinear polygon and the toppings are axes-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks.

cs.CG↗

arXiv2018

Monotonicity and Competitive Equilibrium in Cake-cutting

We study the monotonicity properties of solutions in the classic problem of fair cake-cutting --- dividing a heterogeneous resource among agents with different preferences. Resource- and population-monotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all of them are better-off (if there is more to share or fewer to share among) or all are worse-off (if there is less to share or more to share among). We formally introduce these concepts to the cake-cutting problem and examine whether they are satisfied by various common division rules. We prove that the Nash-optimal rule, which maximizes the product of utilities, is resource-monotonic and population-monotonic, in addition to being Pareto-optimal, envy-free and satisfying a strong competitive-equilibrium condition. Moreover, we prove that it is the only rule among a natural family of welfare-maximizing rules that is both proportional and resource-monotonic.

cs.GT↗

arXiv2018

Waste Makes Haste: Bounded Time Protocols for Envy-Free Cake Cutting with Free Disposal

We consider the classic problem of envy-free division of a heterogeneous good ("cake") among several agents. It is known that, when the allotted pieces must be connected, the problem cannot be solved by a finite algorithm for 3 or more agents. The impossibility result, however, assumes that the entire cake must be allocated. In this paper we replace the entire-allocation requirement with a weaker \emph{partial-proportionality} requirement: the piece given to each agent must be worth for it at least a certain positive fraction of the entire cake value. We prove that this version of the problem is solvable in bounded time even when the pieces must be connected. We present simple, bounded-time envy-free cake-cutting algorithms for: (1) giving each of $n$ agents a connected piece with a positive value; (2) giving each of 3 agents a connected piece worth at least 1/3; (3) giving each of 4 agents a connected piece worth at least 1/7; (4) giving each of 4 agents a disconnected piece worth at least 1/4; (5) giving each of $n$ agents a disconnected piece worth at least $(1-ε)/n$ for any positive $ε$.

cs.DS↗

arXiv2017

MUDA: A Truthful Multi-Unit Double-Auction Mechanism

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.

cs.GT↗

arXiv2017

Flexible Level-1 Consensus Ensuring Stable Social Choice: Analysis and Algorithms

Level-1 Consensus is a property of a preference-profile. Intuitively, it means that there exists a preference relation which induces an ordering of all other preferences such that frequent preferences are those that are more similar to it. This is a desirable property, since it enhances the stability of social choice by guaranteeing that there exists a Condorcet winner and it is elected by all scoring rules. In this paper, we present an algorithm for checking whether a given preference profile exhibits level-1 consensus. We apply this algorithm to a large number of preference profiles, both real and randomly-generated, and find that level-1 consensus is very improbable. We support these empirical findings theoretically, by showing that, under the impartial culture assumption, the probability of level-1 consensus approaches zero when the number of individuals approaches infinity. Motivated by these observations, we show that the level-1 consensus property can be weakened while retaining its stability implications. We call this weaker property Flexible Consensus. We show, both empirically and theoretically, that it is considerably more probable than the original level-1 consensus. In particular, under the impartial culture assumption, the probability for Flexible Consensus converges to a positive number when the number of individuals approaches infinity.

cs.GT↗

arXiv2017

Truthful Bilateral Trade is Impossible even with Fixed Prices

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.

cs.GT↗

arXiv2017

Resource-monotonicity and Population-monotonicity in Connected Cake-cutting

In the classic cake-cutting problem (Steinhaus, 1948), a heterogeneous resource has to be divided among n agents with different valuations in a proportional way --- giving each agent a piece with a value of at least 1/n of the total. In many applications, such as dividing a land-estate or a time-interval, it is also important that the pieces are connected. We propose two additional requirements: resource-monotonicity (RM) and population-monotonicity (PM). When either the cake or the set of agents changes and the cake is re-divided using the same rule, the utility of all remaining agents must change in the same direction. Classic cake-cutting protocols are neither RM nor PM. Moreover, we prove that no Pareto-optimal proportional division rule can be either RM or PM. Motivated by this negative result, we search for division rules that are weakly-Pareto-optimal --- no other division is strictly better for all agents. We present two such rules. The relative-equitable rule, which assigns the maximum possible relative value equal for all agents, is proportional and PM. The so-called rightmost-mark rule, which is an improved version of the Cut and Choose protocol, is proportional and RM for two agents.

cs.GT↗

arXiv2016

Concentration Inequalities for Random Sets

In a large, possibly infinite population, each subject is colored red with probability $p$, independently of the others. Then, a finite sub-population is selected, possibly as a function of the coloring. The imbalance in the sub-population is defined as the difference between the number of reds in it and p times its size. This paper presents high-probability upper bounds (tail-bounds) on this imbalance. To present the upper bounds we define the *UI dimension* --- a new measure for the richness of a set-family. We present three simple rules for upper-bounding the UI dimension of a set-family. Our upper bounds on the imbalance in a sub-population depend only on the size of the sub-population and on the UI dimension of its support. We relate our results to known concepts from machine learning, particularly the VC dimension and Rademacher complexity.

math.PR↗