12,505 research outputs found
An Algorithmic Framework for Strategic Fair Division
We study the paradigmatic fair division problem of allocating a divisible
good among agents with heterogeneous preferences, commonly known as cake
cutting. Classical cake cutting protocols are susceptible to manipulation. Do
their strategic outcomes still guarantee fairness?
To address this question we adopt a novel algorithmic approach, by designing
a concrete computational framework for fair division---the class of Generalized
Cut and Choose (GCC) protocols}---and reasoning about the game-theoretic
properties of algorithms that operate in this model. The class of GCC protocols
includes the most important discrete cake cutting protocols, and turns out to
be compatible with the study of fair division among strategic agents. In
particular, GCC protocols are guaranteed to have approximate subgame perfect
Nash equilibria, or even exact equilibria if the protocol's tie-breaking rule
is flexible. We further observe that the (approximate) equilibria of
proportional GCC protocols---which guarantee each of the agents a
-fraction of the cake---must be (approximately) proportional. Finally, we
design a protocol in this framework with the property that its Nash equilibrium
allocations coincide with the set of (contiguous) envy-free allocations
Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations
Cake cutting is one of the most fundamental settings in fair division and
mechanism design without money. In this paper, we consider different levels of
three fundamental goals in cake cutting: fairness, Pareto optimality, and
strategyproofness. In particular, we present robust versions of envy-freeness
and proportionality that are not only stronger than their standard
counter-parts but also have less information requirements. We then focus on
cake cutting with piecewise constant valuations and present three desirable
algorithms: CCEA (Controlled Cake Eating Algorithm), MEA (Market Equilibrium
Algorithm) and CSD (Constrained Serial Dictatorship). CCEA is polynomial-time,
robust envy-free, and non-wasteful. It relies on parametric network flows and
recent generalizations of the probabilistic serial algorithm. For the subdomain
of piecewise uniform valuations, we show that it is also group-strategyproof.
Then, we show that there exists an algorithm (MEA) that is polynomial-time,
envy-free, proportional, and Pareto optimal. MEA is based on computing a
market-based equilibrium via a convex program and relies on the results of
Reijnierse and Potters [24] and Devanur et al. [15]. Moreover, we show that MEA
and CCEA are equivalent to mechanism 1 of Chen et. al. [12] for piecewise
uniform valuations. We then present an algorithm CSD and a way to implement it
via randomization that satisfies strategyproofness in expectation, robust
proportionality, and unanimity for piecewise constant valuations. For the case
of two agents, it is robust envy-free, robust proportional, strategyproof, and
polynomial-time. Many of our results extend to more general settings in cake
cutting that allow for variable claims and initial endowments. We also show a
few impossibility results to complement our algorithms.Comment: 39 page
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.Comment: Revised versio
Redividing the Cake
A heterogeneous resource, such as a land-estate, is already divided among
several agents in an unfair way. It should be re-divided among the agents in a
way that balances fairness with ownership rights. We present re-division
protocols that attain various trade-off points between fairness and ownership
rights, in various settings differing in the geometric constraints on the
allotments: (a) no geometric constraints; (b) connectivity --- the cake is a
one-dimensional interval and each piece must be a contiguous interval; (c)
rectangularity --- the cake is a two-dimensional rectangle or rectilinear
polygon and the pieces should be rectangles; (d) convexity --- the cake is a
two-dimensional convex polygon and the pieces should be convex.
Our re-division protocols have implications on another problem: the
price-of-fairness --- the loss of social welfare caused by fairness
requirements. Each protocol implies an upper bound on the price-of-fairness
with the respective geometric constraints.Comment: Extended IJCAI 2018 version. Previous name: "How to Re-Divide a Cake
Fairly
Cake Division with Minimal Cuts: Envy-Free Procedures for 3 Person, 4 Persons, and Beyond
The minimal number of parallel cuts required to divide a cake into n pieces is n-1. A new 3-person procedure, requiring 2 parallel cuts, is given that produces an envy- free division, whereby each person thinks he or she receives at least a tied- for- largest piece. An extension of this procedure leads to a 4-person division, us ing 3 parallel cuts, that makes at most one player envious. Finally, a 4-person envy-free procedure is given, but it requires up to 5 parallel cuts, and some pieces may be disconnected. All these procedures improve on extant procedures by using fewer moving knives, making fewer people envious, or using fewer cuts. While the 4-person, 5-cut procedure is complex, endowing people with more information about others' preferences, or allowing them to do things beyond stopping moving knives, may yield simpler procedures for making envy- free divisions with minimal cuts, which are known always to existFAIR DIVISION; CAKE CUTTING; ENVY-FREENESS; MAXIMIN
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