17 research outputs found
Chore Cutting: Envy and Truth
We study the fair division of divisible bad resources with strategic agents
who can manipulate their private information to get a better allocation. Within
certain constraints, we are particularly interested in whether truthful
envy-free mechanisms exist over piecewise-constant valuations. We demonstrate
that no deterministic truthful envy-free mechanism can exist in the
connected-piece scenario, and the same impossibility result occurs for hungry
agents. We also show that no deterministic, truthful dictatorship mechanism can
satisfy the envy-free criterion, and the same result remains true for
non-wasteful constraints rather than dictatorship. We further address several
related problems and directions.Comment: arXiv admin note: text overlap with arXiv:2104.07387 by other author
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
Deterministic, Strategyproof, and Fair Cake Cutting
We study the classic cake cutting problem from a mechanism design
perspective, in particular focusing on deterministic mechanisms that are
strategyproof and fair. We begin by looking at mechanisms that are non-wasteful
and primarily show that for even the restricted class of piecewise constant
valuations there exists no direct-revelation mechanism that is strategyproof
and even approximately proportional. Subsequently, we remove the non-wasteful
constraint and show another impossibility result stating that there is no
strategyproof and approximately proportional direct-revelation mechanism that
outputs contiguous allocations, again, for even the restricted class of
piecewise constant valuations. In addition to the above results, we also
present some negative results when considering an approximate notion of
strategyproofness, show a connection between direct-revelation mechanisms and
mechanisms in the Robertson-Webb model when agents have piecewise constant
valuations, and finally also present a (minor) modification to the well-known
Even-Paz algorithm that has better incentive-compatible properties for the
cases when there are two or three agents.Comment: A shorter version of this paper will appear at IJCAI 201
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
Communication Complexity of Cake Cutting
We study classic cake-cutting problems, but in discrete models rather than
using infinite-precision real values, specifically, focusing on their
communication complexity. Using general discrete simulations of classical
infinite-precision protocols (Robertson-Webb and moving-knife), we roughly
partition the various fair-allocation problems into 3 classes: "easy" (constant
number of rounds of logarithmic many bits), "medium" (poly-logarithmic total
communication), and "hard". Our main technical result concerns two of the
"medium" problems (perfect allocation for 2 players and equitable allocation
for any number of players) which we prove are not in the "easy" class. Our main
open problem is to separate the "hard" from the "medium" classes.Comment: Added efficient communication protocol for the monotone crossing
proble
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