Outline
E GALITARIAN A LLOCATIONS OF I NDIVISIBLE
R ESOURCES : T HEORY AND C OMPUTATION
Paul-Amaury Matt
Francesca Toni
Logic and Artificial Intelligence Group, Department of Computing,
South Kensington Campus, Imperial College London - UK
Email: {pmatt, ft}@imperial.ac.uk
10th International Workshop on
Cooperative Information Agents (CIA 2006),
University of Edinburgh, September 11-13, 2006
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Outline
OUTLINE
1 INTRODUCTION
Motivation
Problem illustration
2 PROBLEM STATEMENT
3 COMPUTATION OF EGALITARIAN ALLOCATIONS
Problem decomposition
Iterated dichotomous updates
Search for social consensus
4 OPTIMISED FEATURES
Simplifying the search space
Optimizing search trees
Negotiation order
5 CONCLUSION
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
Motivation
Problem illustration
MOTIVATION
• Goal: fair distributions of resources to agents.
• Motivation: the proper distribution of wealth is an important
aspect of justice in society.
• Applications: service agents, satellite earth observation,
holonic manufacturing, markets allocations
• Approach: maximisation of a social measure of worth:
egalitarian social welfare (cf. social choice theories and
welfare economics)
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
Motivation
Problem illustration
PROBLEM ILLUSTRATION


u
r1
r2
r3
r4
r5
a1 : 0.1
 a1 : 0.7 0.3 0.7 1.0 0.6 

W =  a2 : 0.4  U = 
 a2 : 0.8 0.3 0.9 0.2 0.1 
a3 : 0.2
a3 : 0.8 0
0 0.4 0.1


Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
Motivation
Problem illustration
PROBLEM ILLUSTRATION




u
r1
r2
r3
r4
r5
a1 : 0.1 → 1.1
 a1 : 0.7 0.3 0.7 1.0 0.6 

 a2 : 0.4 → 1.3 
U=
 a2 : 0.8 0.3 0.9 0.2 0.1  W =
a3 : 0.2 → 1.1
a3 : 0.8 0
0 0.4 0.1
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
PROBLEM CONSTRUCTION
• Problem: given a set of resources, maximise the
egalitarian social welfare in a society of agents.
• Mathematical criterion to optimise: the minimum worth of
an agent.
• Assumptions:
• resources cannot be shared (indivisible resources)
• utilities of resources add-up (semi-linearity)
• agents fully cooperate
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
RESULTING MATHEMATICAL PROBLEM
GOAL : FIND AN EGALITARIAN ALLOCATION
A∗
• A∗ = argmax swe (A)
• swe (A) = minni=1 wi (A)
P
• wi (A) = Wi + m
j=1 ui,j .Ai,j
CONSTRAINTS
• Ai,j ∈ {0, 1}
•
Pn
i=1 Ai,j
Paul-Amaury Matt, Francesca Toni
≤1
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
Problem decomposition
Iterated dichotomous updates
Search for social consensus
PROBLEM DECOMPOSITION
• Two problems must be solved:
• a) find the optimal egalitarian welfare swe (A∗ )
• b) find an optimal allocation (egalitarian allocation) A∗
• Solution:
• a) can be solved by iterated dichotomous updates
• b) the allocation construction is seen as a search for social
consensus between agents and can be solved by
incremental negotiation
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
Problem decomposition
Iterated dichotomous updates
Search for social consensus
ITERATED DICHOTOMOUS UPDATES
• Idea: bound the value of swe (A∗ ) by an interval I=[a,b] and
divide its length by two at each step.
• Step: exist allocation A such that swe (A) ≥ (a + b)/2 ? y/n
• Illustration
A
( A + B )/2 ≈ swe (A∗ )
B
EXIST
A?
• 0.1
• 1.5
• 0.8
• yes
• 0.8
• 1.5
• 1.15
• no
• 0.8
• 1.15
• 0.975
• yes
• 0.975
• 1.15
• 1.0625
• yes
• Solution: swe (A∗ ) = 1.1!
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
Problem decomposition
Iterated dichotomous updates
Search for social consensus
SEARCH FOR SOCIAL CONSENSUS
• Mechanism: distributed and incremental construction of a
consensus via top-down exploration of allocation space
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
Simplifying the search space
Optimizing search trees
Negotiation order
SIMPLIFYING THE SEARCH SPACE
• Problem: rapidly too many search trees and allocations
• Frugal reduction: agents opportunistically eliminate
allocations that in comparison over-consume resources
i.e. are not minimal wrt ≤:
P
P
• Def. A ≤ B iff ∀j : ni=1 Ai,j ≤ ni=1 Bi,j
• Illustration:

1 0 1

{ 0 1 0
0 0 0

0 1
→ { 0 0
0 0

0
 0
0

0
1 
0
1
0
0
0
0
1
0
1
0
1
0
0



0 1 0
0 0 1
  0 0 1   0 1 0 }
1 0 0
1 0 0

0
1 }
0
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
Simplifying the search space
Optimizing search trees
Negotiation order
STATISTICAL OPTIMISATION OF THE NEGOTIATION TIME
• A good node splitting strategy reduces the search trees
size: ++ give priority to most useful resources
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
Simplifying the search space
Optimizing search trees
Negotiation order
NEGOTIATION ORDER ( COORDINATION )
• The order in which the agents join in for negotiating
impacts on the length of negotiations. A good heuristic
consists in following the initial social order
Wi1 ≤ Wi2 ≤ ... ≤ Win . Agents notably abort earlier
negotiations the cannot lead to a consensus.
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources
Introduction
Problem statement
Computation of egalitarian allocations
Optimised features
Conclusion
CONCLUSION AND RELATED WORK
• Distributed negotiation mechanism for finding efficiently
egalitarian allocations of indivisible resources to agents.
• Combined heuristics divide negotiation time by 30.
• Other approaches: minimizing envy (Lipton 2004),
approximate max-min fair allocations (Bezakova and Dani
2005), allocations to activities (Luss 1999 and Yu 1996) or
tasks (Shehory and Kraus 1998)
• Related work: negotiation of socially optimal allocations
(Endriss, Maudet, Sadri and Toni 2006), complexity results
(Golovin 2005 and Bouveret 2005)
Paul-Amaury Matt, Francesca Toni
Egalitarian Allocations of Indivisible Resources