Dagstuhl Seminar Proceedings, Volume 7261

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07261 Abstracts Collection – Fair Division

Authors: Steven J. Brams and Kirk Pruhs

Abstract

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.

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Steven J. Brams and Kirk Pruhs. 07261 Abstracts Collection – Fair Division. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{brams_et_al:DagSemProc.07261.1,
  author =	{Brams, Steven J. and Pruhs, Kirk},
  title =	{{07261 Abstracts Collection – Fair Division}},
  booktitle =	{Fair Division},
  pages =	{1--16},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.1},
  URN =		{urn:nbn:de:0030-drops-12444},
  doi =		{10.4230/DagSemProc.07261.1},
  annote =	{Keywords: Economics, Fairness, Allocation, Political Science}
}

Document

DOI: 10.4230/DagSemProc.07261.2

07261 Summary – Fair Division

Authors: Steven J. Brams and Kirk Pruhs

Abstract

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.

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Steven J. Brams and Kirk Pruhs. 07261 Summary – Fair Division. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{brams_et_al:DagSemProc.07261.2,
  author =	{Brams, Steven J. and Pruhs, Kirk},
  title =	{{07261 Summary  – Fair Division}},
  booktitle =	{Fair Division},
  pages =	{1--3},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.2},
  URN =		{urn:nbn:de:0030-drops-12434},
  doi =		{10.4230/DagSemProc.07261.2},
  annote =	{Keywords: Economics, Fairness, Allocation, Political Science}
}

Document

DOI: 10.4230/DagSemProc.07261.3

A Pie That Can't Be Cut Fairly (revised for DSP)

Authors: Walter Stromquist

Abstract

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.

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Walter Stromquist. A Pie That Can't Be Cut Fairly (revised for DSP). In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{stromquist:DagSemProc.07261.3,
  author =	{Stromquist, Walter},
  title =	{{A Pie That Can't Be Cut Fairly (revised for DSP)}},
  booktitle =	{Fair Division},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.3},
  URN =		{urn:nbn:de:0030-drops-12192},
  doi =		{10.4230/DagSemProc.07261.3},
  annote =	{Keywords: Cake cutting, pie cutting, envy free}
}

Document

DOI: 10.4230/DagSemProc.07261.4

Approximating min-max k-clustering

Authors: Asaf Levin

Abstract

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.

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Asaf Levin. Approximating min-max k-clustering. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{levin:DagSemProc.07261.4,
  author =	{Levin, Asaf},
  title =	{{Approximating min-max k-clustering}},
  booktitle =	{Fair Division},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.4},
  URN =		{urn:nbn:de:0030-drops-12282},
  doi =		{10.4230/DagSemProc.07261.4},
  annote =	{Keywords: Approximation algorithms}
}

Document

DOI: 10.4230/DagSemProc.07261.5

Better Ways to Cut a Cake - Revisited

Authors: Steven J. Brams, Michael A. Jones, and Christian Klamler

Abstract

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.

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Steven J. Brams, Michael A. Jones, and Christian Klamler. Better Ways to Cut a Cake - Revisited. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{brams_et_al:DagSemProc.07261.5,
  author =	{Brams, Steven J. and Jones, Michael A. and Klamler, Christian},
  title =	{{Better Ways to Cut a Cake - Revisited}},
  booktitle =	{Fair Division},
  pages =	{1--24},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.5},
  URN =		{urn:nbn:de:0030-drops-12278},
  doi =		{10.4230/DagSemProc.07261.5},
  annote =	{Keywords: Fair division, cake-cutting, envy-freeness, strategy-proofness}
}

Document

DOI: 10.4230/DagSemProc.07261.6

Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Procedure

Authors: Steven J. Brams, Michael A. Jones, and Christian Klamler

Abstract

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.

Cite as

Steven J. Brams, Michael A. Jones, and Christian Klamler. Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Procedure. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{brams_et_al:DagSemProc.07261.6,
  author =	{Brams, Steven J. and Jones, Michael A. and Klamler, Christian},
  title =	{{Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Procedure}},
  booktitle =	{Fair Division},
  pages =	{1--31},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.6},
  URN =		{urn:nbn:de:0030-drops-12211},
  doi =		{10.4230/DagSemProc.07261.6},
  annote =	{Keywords: Cake-cutting, proportionality, envy-freeness, efficiency, strategy-proofness}
}

Document

DOI: 10.4230/DagSemProc.07261.7

Efficient cost sharing with a cheap residual claimant

Authors: Hervé Moulin

Abstract

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.

Cite as

Hervé Moulin. Efficient cost sharing with a cheap residual claimant. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{moulin:DagSemProc.07261.7,
  author =	{Moulin, Herv\'{e}},
  title =	{{Efficient cost sharing with a cheap residual claimant}},
  booktitle =	{Fair Division},
  pages =	{1--7},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.7},
  URN =		{urn:nbn:de:0030-drops-12312},
  doi =		{10.4230/DagSemProc.07261.7},
  annote =	{Keywords: Assignment, cost sharing, Vickrey-Clarke-Groves mechanisms, competitive analysis}
}

Document

DOI: 10.4230/DagSemProc.07261.8

Envy-free cake divisions cannot be found by finite protocols

Authors: Walter Stromquist

Abstract

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.

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Walter Stromquist. Envy-free cake divisions cannot be found by finite protocols. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{stromquist:DagSemProc.07261.8,
  author =	{Stromquist, Walter},
  title =	{{Envy-free cake divisions cannot be found by finite protocols}},
  booktitle =	{Fair Division},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.8},
  URN =		{urn:nbn:de:0030-drops-12208},
  doi =		{10.4230/DagSemProc.07261.8},
  annote =	{Keywords: Cake cutting, envy free, finite protocol}
}

Document

DOI: 10.4230/DagSemProc.07261.9

Equilibria for two parallel links: The strong price of anarchy versus the price of anarchy

Authors: Leah Epstein

Abstract

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.

Cite as

Leah Epstein. Equilibria for two parallel links: The strong price of anarchy versus the price of anarchy. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{epstein:DagSemProc.07261.9,
  author =	{Epstein, Leah},
  title =	{{Equilibria for two parallel links: The strong price of anarchy versus the price of anarchy}},
  booktitle =	{Fair Division},
  pages =	{1--8},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.9},
  URN =		{urn:nbn:de:0030-drops-12228},
  doi =		{10.4230/DagSemProc.07261.9},
  annote =	{Keywords: Nash equilibrium, strong equilibrium, uniformly related machyines}
}

Document

DOI: 10.4230/DagSemProc.07261.10

Maximizing the Minimum Load for Selfisch Agents

Authors: Leah Epstein and Rob van Stee

Abstract

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.

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Leah Epstein and Rob van Stee. Maximizing the Minimum Load for Selfisch Agents. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{epstein_et_al:DagSemProc.07261.10,
  author =	{Epstein, Leah and van Stee, Rob},
  title =	{{Maximizing the Minimum Load for Selfisch Agents}},
  booktitle =	{Fair Division},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.10},
  URN =		{urn:nbn:de:0030-drops-12427},
  doi =		{10.4230/DagSemProc.07261.10},
  annote =	{Keywords: Scheduling, algorithmic mechanism design, maximizing minimum load}
}

Document

DOI: 10.4230/DagSemProc.07261.11

Some Recent Results on Pie Cutting

Authors: Michael A. Jones

Abstract

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).

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Michael A. Jones. Some Recent Results on Pie Cutting. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{jones:DagSemProc.07261.11,
  author =	{Jones, Michael A.},
  title =	{{Some Recent Results on Pie Cutting}},
  booktitle =	{Fair Division},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.11},
  URN =		{urn:nbn:de:0030-drops-12246},
  doi =		{10.4230/DagSemProc.07261.11},
  annote =	{Keywords: Pie cutting, envy-free, proportional, undominated}
}

Document

DOI: 10.4230/DagSemProc.07261.12

Strong Price of Anarchy for Machine Load Balancing

Authors: Amos Fiat, Meital Levy, Haim Kaplan, and Svetlana Olonetsky

Abstract

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.

Cite as

Amos Fiat, Meital Levy, Haim Kaplan, and Svetlana Olonetsky. Strong Price of Anarchy for Machine Load Balancing. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{fiat_et_al:DagSemProc.07261.12,
  author =	{Fiat, Amos and Levy, Meital and Kaplan, Haim and Olonetsky, Svetlana},
  title =	{{Strong Price of Anarchy  for  Machine Load Balancing}},
  booktitle =	{Fair Division},
  pages =	{1--19},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.12},
  URN =		{urn:nbn:de:0030-drops-12256},
  doi =		{10.4230/DagSemProc.07261.12},
  annote =	{Keywords: Game theory, Strong Nash equilibria, Load balancing, Price of Anarchy}
}

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