Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1609.03938

Envy-Free Division of Land

Erel Segal-Halevi·Shmuel Nitzan·Avinatan Hassidim·Yonatan Aumann
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

cs.GTcs.CG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Erel Segal-Halevi, Shmuel Nitzan, Avinatan Hassidim, Yonatan Aumann. 2019-03-09. Envy-Free Division of Land. https://doi.org/10.1287/moor.2019.1016

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

MMS Allocation for Chores with Online Agent Arrivals

We study the fair allocation of $m$ indivisible chores to $n$ agents with subadditive cost functions arriving online in an arbitrary order. Upon an agent's arrival, we are informed of her cost function and must irrevocably assign her a set of chores. We focus on the Maximin Share (MMS) fairness notion and aim to compute an allocation in which all items are assigned, and no agent incurs a cost more than $\alpha$ times her MMS. Without any prior information about the instance (other than $n$ and $m$), we design an algorithm with a competitive ratio of $O(\min\{n, k\log^{1+\epsilon}k, \log m\})$ for any constant $\epsilon > 0$, where $k$ denotes the number of cost function types. Our bound matches the best known offline approximation guarantees for MMS under subadditive costs and is nearly optimal with respect to all three parameters: we show that even for binary additive cost functions, no online algorithm can achieve a competitive ratio of $o(\min\{n, k\log k, \log m\})$. We then consider the setting in which the $k$ cost function types are known in advance (though the realized types of arriving agents are not). For additive cost functions, we provide an algorithm with a competitive ratio of $O(\min\{\log k, \log(kn)/\log\log(kn)\})$, and show that constant-competitive algorithms do not exist for general $k$, even for the binary additive setting. For binary additive functions when $k \le n$, we propose a $3$-competitive algorithm and establish a lower bound of $2$.

cs.GT↗
arXiv2026

Truncated Noisy Best-Response Algorithms: Toward Game Theoretic Learning with Safety Guarantees

We consider a game theoretic approach to solve multi-agent coordination problems with submodular maximization objectives. It is known for such problems that the Nash equilibria for the corresponding game are always within 50% of the optimal, but that the equilibria which achieve this worst-case bound are not stable. To exploit this instability, we propose a family of algorithms which we call Truncated Noisy Best-Response (TNBR) Algorithms. These algorithms are flexibly characterized by agents asynchronously and stochastically selecting actions from a neighbourhood of their best response payoffs. We compute bounds on the recurrent classes of TNBR algorithms' associated Markov chains. Our bounds fall into two categories: first, "Performance" bounds ensure that TNBR algorithms always have a high-value recurrent state; second, "Safety" bounds ensure that TNBR algorithms never have arbitrarily-bad recurrent states. Furthermore, these two types of bounds are linked by a waterbed-like effect: every game with a poor Safety guarantee necessarily has a favorable Performance guarantee.

cs.GT↗
arXiv2026

Existence of the Core in Approval-Based Committee Elections

We settle the main open question in the theory of approval-based multi-winner elections: we show that there always exists a committee in the core. The core is a stability and group fairness concept. The proof introduces a new voting rule that optimizes an entropy-like objective function over committees and payment systems. All local optima of this objective function lie in the core, which implies that a core committee can be found in polynomial time.

cs.GT↗