1,033 research outputs found
An Improved Envy-Free Cake Cutting Protocol for Four Agents
We consider the classic cake-cutting problem of producing envy-free allocations, restricted to the case of four agents. The problem asks for a partition of the cake to four agents, so that every agent finds her piece at least as valuable as every other agent's piece. The problem has had an interesting history so far. Although the case of three agents is solvable with less than 15 queries, for four agents no bounded procedure was known until the recent breakthroughs of Aziz and Mackenzie (STOC 2016, FOCS 2016). The main drawback of these new algorithms, however, is that they are quite complicated and with a very high query complexity. With four agents, the number of queries required is close to 600. In this work we provide an improved algorithm for four agents, which reduces the current complexity by a factor of 3.4. Our algorithm builds on the approach of Aziz and Mackenzie (STOC 2016) by incorporating new insights and simplifying several steps. Overall, this yields an easier to grasp procedure with lower complexity
Efficient Algorithms for Envy-Free Stick Division With Fewest Cuts
Given a set of n sticks of various (not necessarily different) lengths, what
is the largest length so that we can cut k equally long pieces of this length
from the given set of sticks? We analyze the structure of this problem and show
that it essentially reduces to a single call of a selection algorithm; we thus
obtain an optimal linear-time algorithm.
This algorithm also solves the related envy-free stick-division problem,
which Segal-Halevi, Hassidim, and Aumann (AAMAS, 2015) recently used as their
central primitive operation for the first discrete and bounded envy-free cake
cutting protocol with a proportionality guarantee when pieces can be put to
waste.Comment: v3 adds more context about the proble
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
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 of the agents in each group,
and prove that the 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.Comment: Appears in the 27th International Joint Conference on Artificial
Intelligence and the 23rd European Conference on Artificial Intelligence
(IJCAI-ECAI), 201
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
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