Dagstuhl Seminar Proceedings, Volume 7261
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
7261
2007
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-7261
07261 Abstracts Collection – Fair Division
From 24.06. to 29.06.2007, the Dagstuhl Seminar 07261 % generate automatically
``Fair Division'' % generate automatically
was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
Economics
Fairness
Allocation
Political Science
1-16
Regular Paper
Steven J.
Brams
Steven J. Brams
Kirk
Pruhs
Kirk Pruhs
10.4230/DagSemProc.07261.1
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07261 Summary – Fair Division
The problem of fair division – dividing goods or "bads" (e.g., costs) among entities
in an impartial and equitable way – is one of the most important problems that society
faces. A Google search on the phrase "fair allocation" returns over 100K links, referring
to the division of sports tickets, health resources, computer networking resources, voting
power, intellectual property licenses, costs of environmental improvements, etc.
Economics
Fairness
Allocation
Political Science
1-3
Regular Paper
Steven J.
Brams
Steven J. Brams
Kirk
Pruhs
Kirk Pruhs
10.4230/DagSemProc.07261.2
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A Pie That Can't Be Cut Fairly (revised for DSP)
David Gale asked (Math. Intel. 1993) whether, when a pie is to be divided among n claimants, it is always possible to find a division that is both envy free and undominated. The pie is cut along n radii and the claimants' preferences are described by separate measures.
We answer Gale's question in the negative for n=3 by exhibiting three measures on a pie such that, when players have these measures, no division of the pie can be both envy free and undominated. The measures assign positive values to pieces with positive area.
Cake cutting
pie cutting
envy free
1-10
Regular Paper
Walter
Stromquist
Walter Stromquist
10.4230/DagSemProc.07261.3
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Approximating min-max k-clustering
We consider the
problems
of set partitioning into $k$ clusters with minimum of the maximum cost of a cluster. The cost function is given by an oracle, and we assume that it satisfies some natural structural constraints. That is, we assume that the cost function is monotone, the cost of a singleton is zero, and we assume that for all $S cap S'
eq emptyset$ the following holds
$c(S) + c(S') geq c(S cup S')$. For this problem we present
a $(2k-1)$-approximation algorithm for $kgeq 3$, a
2-approximation algorithm for $k=2$, and we also show a lower
bound of $k$ on the performance guarantee of any
polynomial-time algorithm.
We then consider special cases of this problem arising in vehicle routing problems, and present improved results.
Approximation algorithms
1-5
Regular Paper
Asaf
Levin
Asaf Levin
10.4230/DagSemProc.07261.4
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Better Ways to Cut a Cake - Revisited
Procedures to divide a cake among n people with n-1 cuts (the minimum number) are analyzed and compared. For 2 persons, cut-and-choose, while envy-free and efficient, limits the cutter to exactly 50% if he or she is ignorant of the chooser's preferences, whereas the chooser can generally obtain more. By comparison, a new 2-person surplus procedure (SP'), which induces the players to be truthful in order to maximize their minimum allocations, leads to a proportionally equitable division of the surplus - the part that remains after each player receives 50% - by giving each person a certain proportion of the surplus as he or she values it.
For n geq 3 persons, a new equitable procedure (EP) yields a maximally equitable division of a cake. This division gives all players the highest common value that they can achieve and induces truthfulness, but it may not be envy-free. The applicability of SP' and EP to the fair division of a heterogeneous, divisible good, like land, is briefly discussed.
Fair division
cake-cutting
envy-freeness
strategy-proofness
1-24
Regular Paper
Steven J.
Brams
Steven J. Brams
Michael A.
Jones
Michael A. Jones
Christian
Klamler
Christian Klamler
10.4230/DagSemProc.07261.5
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Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Procedure
Properties of discrete cake-cutting procedures that use a minimal number of cuts (n-1 if there are n players) are analyzed. None is always envy-free or efficient, but divide-and-conquer (D&C) minimizes the maximum number of players that any single player may envy. It works by asking n ≥ 2 players successively to place marks on a cake that divide it into equal or approximately equal halves, then halves of these halves, and so on. Among other properties, D&C (i) ensures players of more than 1/n shares if their marks are different and (ii) is strategyproof for risk-averse players. However, D&C may not allow players to obtain proportional, connected pieces if they have unequal entitlements. Possible applications of D&C to land division are briefly discussed.
Cake-cutting
proportionality
envy-freeness
efficiency
strategy-proofness
1-31
Regular Paper
Steven J.
Brams
Steven J. Brams
Michael A.
Jones
Michael A. Jones
Christian
Klamler
Christian Klamler
10.4230/DagSemProc.07261.6
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Efficient cost sharing with a cheap residual claimant
For the cooperative production problem where the commons is a one dimensional convex cost function, I propose the residual mechanism to implement the efficient production level . In contrast to the familiar cost sharing methods such as serial, average and incremental, the residual mechanism may subsidize an agent with a null demand. IFor a large class of smooth cost functions, the residual mechanism generates a budget surplus that is, even in the worst case, vanishes as 1/logn where n is the number of participants. Compare with the serial, average and incremental mechanisms, of which the budget surplus, in the worst case, converges to the efficient surplus as n grows.
The second problem is the assignment among n agents of p identical objects and cash transfers to compensate the losers. We assume p<n, and compute the optimal competitive performance among all VCG mechanisms generating no budget deficit. It goes to zero exponentially fast in n if the number of objects is fixed; and as (n)^(1/2) uniformly in p. The mechanism generates envy, and net utilities are not co-monotonic to valuations. When p>n/2, it may even fail to achieve voluntary participation.
Assignment
cost sharing
Vickrey-Clarke-Groves mechanisms
competitive analysis
1-7
Regular Paper
Hervé
Moulin
Hervé Moulin
10.4230/DagSemProc.07261.7
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Envy-free cake divisions cannot be found by finite protocols
No finite protocol (even if unbounded) can guarantee an envy-free division of a cake among three or more players, if each player is to receive a single connected piece.
Cake cutting
envy free
finite protocol
1-9
Regular Paper
Walter
Stromquist
Walter Stromquist
10.4230/DagSemProc.07261.8
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Equilibria for two parallel links: The strong price of anarchy versus the price of anarchy
Following recent interest in the "strong price of anarchy" SPOA),
we consider this measure, as well as the well known "price of
anarchy" (POA) for the job scheduling problem on two uniformly
related parallel machines (or links). The atomic players are the
jobs, and the delay of a job is the completion time of the
machine running it. The social goal is to minimize the maximum
delay of any job. Thus the cost (or social cost) in this case is
the makespan of the schedule. The selfish goal of each job is to
minimize its delay, i.e., the delay of the machine that it
chooses to run on.
A pure Nash equilibrium is a schedule where no job can obtain a
smaller delay by selfishly moving to a different configuration
(machine), while other jobs remain in their original positions. A
strong equilibrium is a schedule where no (non-empty) subset of
jobs exists, where all jobs in this subset can benefit from
changing their configuration. We say that all jobs in a subset
benefit from moving to a different machine if all of them have a
strictly smaller delay as a result of moving (while the other
jobs remain in their positions, and may possibly have a larger
delay as a result).
The SPOA is the worst case ratio between the social cost of a (pure)
strong equilibrium and the cost of an optimal assignment, that
is, the minimum achievable social cost. The POA is a standard
measure which takes into account not only strong equilibria but
any (pure) equilibrium. These two measures consolidate and give
the same results for some problems, whereas for other problems,
the SPOA gives much more meaningful results than the POA.
We study the behavior of the SPOA versus the behavior of the POA
for this scheduling problem and give tight results for both these
measures. We find the exact SPOA for any possible speed ratio
s geq 1 of the machines, and compare it to the exact POA which
we also find. We show that for a wide range of speeds ratios
these two measures are very different (1.618<s<2.247), whereas
for other values of $s$, these two measures give the exact same
bound. We extend all our results for cases where a machine may
have an initial load resulting from jobs that can only be
assigned to this machine, and show tight bounds on the SPOA and
the POA for three such variants as well.
Nash equilibrium
strong equilibrium
uniformly related machyines
1-8
Regular Paper
Leah
Epstein
Leah Epstein
10.4230/DagSemProc.07261.9
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Maximizing the Minimum Load for Selfisch Agents
We consider the problem of maximizing the minimum load for
machines that are controlled by selfish agents, who are only
interested in maximizing their own profit. Unlike the classical
load balancing problem, this problem
has not been considered for selfish agents until now.
For a constant number of machines, $m$, we show a
monotone polynomial time approximation scheme (PTAS) with running
time that is linear in the number of jobs. It uses a new
technique for reducing the number of jobs while remaining close
to the optimal solution. We also present an FPTAS for the classical
machine covering problem, i.e., where no selfish agents are involved
(the previous best result for this case was a PTAS)
and use this to give a monotone FPTAS.
Additionally, we give a monotone approximation algorithm with
approximation ratio $min(m,(2+eps)s_1/s_m)$ where $eps>0$ can
be chosen arbitrarily small and $s_i$ is the (real) speed of
machine $i$. Finally we give improved results for two machines.
Scheduling
algorithmic mechanism design
maximizing minimum load
1-0
Regular Paper
Leah
Epstein
Leah Epstein
Rob
van Stee
Rob van Stee
10.4230/DagSemProc.07261.10
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Some Recent Results on Pie Cutting
For cake cutting, cuts are parallel to an axis and yield rectangular pieces. As such, cutting a cake is viewed as dividing a line segment. For pie cutting, cuts are radial from the center of a disc to the circumference and yield sectors or wedge-shaped pieces. As such, cutting a pie is viewed as dividing a circle. There is clearly a relationship between cutting a cake and cutting a pie. Once a circular pie has a single cut, then it can be straightened out into a segment, looking like a cake. Isn't a cake just a pie that has been cut? Gale (1993) suggested that this topology was a significant difference. This note is to summarize and compare some of the recent results on pie cutting that appear in Barbanel and Brams (2007) and Brams, Jones, and Klamler (2007). The geometric framework presented in Barbanel and Brams (2007) is used to prove and to explain results in Brams, Jones, and Klamler (2007).
Pie cutting
envy-free
proportional
undominated
1-10
Regular Paper
Michael A.
Jones
Michael A. Jones
10.4230/DagSemProc.07261.11
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Strong Price of Anarchy for Machine Load Balancing
As defined by Aumann in 1959, a strong equilibrium is a Nash
equilibrium that is resilient to deviations by coalitions. We give
tight bounds on the strong price of anarchy for load balancing on
related machines. We also give tight bounds for $k$-strong
equilibria, where the size of a deviating coalition is at most
$k$, for unrelated machines.
Game theory
Strong Nash equilibria
Load balancing
Price of Anarchy
1-19
Regular Paper
Amos
Fiat
Amos Fiat
Meital
Levy
Meital Levy
Haim
Kaplan
Haim Kaplan
Svetlana
Olonetsky
Svetlana Olonetsky
10.4230/DagSemProc.07261.12
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