15 research outputs found
A Discrete and Bounded Envy-free Cake Cutting Protocol for Four Agents
We consider the well-studied cake cutting problem in which the goal is to
identify a fair allocation based on a minimal number of queries from the
agents. The problem has attracted considerable attention within various
branches of computer science, mathematics, and economics. Although, the elegant
Selfridge-Conway envy-free protocol for three agents has been known since 1960,
it has been a major open problem for the last fifty years to obtain a bounded
envy-free protocol for more than three agents. We propose a discrete and
bounded envy-free protocol for four agents
On the Complexity of Chore Division
We study the proportional chore division problem where a protocol wants to
divide an undesirable object, called chore, among different players. The
goal is to find an allocation such that the cost of the chore assigned to each
player be at most of the total cost. This problem is the dual variant of
the cake cutting problem in which we want to allocate a desirable object.
Edmonds and Pruhs showed that any protocol for the proportional cake cutting
must use at least queries in the worst case, however,
finding a lower bound for the proportional chore division remained an
interesting open problem. We show that chore division and cake cutting problems
are closely related to each other and provide an lower bound
for chore division
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
Non-Exploitable Protocols for Repeated Cake Cutting
We introduce the notion of exploitability in cut-and-choose protocols for repeated cake cutting. If a cut-and-choose protocol is repeated, the cutter can possibly gain information about the chooser from her previous actions, and exploit this information for her own gain, at the expense of the chooser. We define a generalization of cut-and-choose protocols - forced-cut protocols - in which some cuts are made exogenously while others are made by the cutter, and show that there exist non-exploitable forced-cut protocols that use a small number of cuts per day: When the cake has at least as many dimensions as days, we show a protocol that uses a single cut per day. When the cake is 1-dimensional, we show an adaptive non-exploitable protocol that uses 3 cuts per day, and a non-adaptive protocol that uses n cuts per day (where n is the number of days). In contrast, we show that no non-adaptive non-exploitable forced-cut protocol can use a constant number of cuts per day. Finally, we show that if the cake is at least 2-dimensional, there is a non-adaptive non-exploitable protocol that uses 3 cuts per day
Fair Division of Mixed Divisible and Indivisible Goods
We study the problem of fair division when the resources contain both
divisible and indivisible goods. Classic fairness notions such as envy-freeness
(EF) and envy-freeness up to one good (EF1) cannot be directly applied to the
mixed goods setting. In this work, we propose a new fairness notion
envy-freeness for mixed goods (EFM), which is a direct generalization of both
EF and EF1 to the mixed goods setting. We prove that an EFM allocation always
exists for any number of agents. We also propose efficient algorithms to
compute an EFM allocation for two agents and for agents with piecewise
linear valuations over the divisible goods. Finally, we relax the envy-free
requirement, instead asking for -envy-freeness for mixed goods
(-EFM), and present an algorithm that finds an -EFM
allocation in time polynomial in the number of agents, the number of
indivisible goods, and .Comment: Appears in the 34th AAAI Conference on Artificial Intelligence
(AAAI), 202
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