The allocation of resources between providers to consumers is a well-known problem and has received significant attention, typically using notions of monetary exchanges. In this paper, we study resource matching in settings without monetary transactions by using a two-sided matching approach, e.g., in social and collaborative environments where users define preferences for with whom they may be matched. Whereas two-sided matching for strict and complete preference rankings (i.e., without indifferences) has been extensively studied, it is known that the matching problem is NP-hard for more realistic preference structures. We study, via simulation, the applicability of a heuristic procedure in settings with indiffernces in preferences, and compare its performance to existing algorithms. We study performance metrics like fairness and welfare in addition to the classic stability objective. Our results show interesting trade-offs between performance metrics and promising performance of the heuristic.

1. Preference‐Based Resource Allocation:
Using Heuristics to Solve Two‐Sided Matching
Problems with Indifferences
KARLSRUHE SERVICE RESEARCH INSTITUTE (KSRI)
Christian Haas, Steven Kimbrough, Simon Caton, Christof Weinhardt
GECON 2013
Zaragoza, Spain
19th September 2013
KIT – University of the State of Baden-Württemberg and
National Research Center of the Helmholtz Association
www.kit.edu

2. Agenda
1
2
22.05.2013
Heuristics to Solve Two-Sided Matching with Indifferences
3
2
Two-Sided Matching: Concepts and Challenges
Outlook and Future Work
Haas et al. – Heuristics for Preference-Based Resource Allocation
Karlsruhe Service Research Institute
www.ksri.kit.edu

3. Scenario
A
A
 Users provide and request resources
A
B
B
A
A
C
C
D
D
Requests
E
C
Offers
Allocation
A-Z: Resource Types
3
22.05.2013
 Resource exchange without
monetary transactions
 How can we allocate resources
while still retaining certain
allocation properties (e.g. welfare)?
 Preference‐Based Matching
Haas et al. – Heuristics for Preference-Based Resource Allocation
Karlsruhe Service Research Institute
www.ksri.kit.edu

4. Two Sided Matching: Concepts
Two‐Sided Market
•
•
•
•
Two sides with n members each which have to be matched
Both sides have preferences with whom they want to be matched
Matching consists of pairs, one member of each side
Examples:
Preferences
3≻4≻2≻1
• Preferences are given as ordered lists
• Complete vs. Incomplete lists:
All members of the other side ranked and
acceptable?
• Strict vs. Indifferences:
Preferences strictly ordered, or are ties allowed?
• Most algorithms consider strict and complete preferences
4
22.05.2013
Haas et al. – Heuristics for Preference-Based Resource Allocation
Karlsruhe Service Research Institute
www.ksri.kit.edu

5. Matching Objectives and Related Approaches
Objectives
Stability
Welfare
Fairness
No incentive to deviate
from solution
Average rank of matched user
Welfare distribution
between the two sides
16
5
16: 20 ≻ 5
Unstable pair:
3
20
20: 16 ≻ 3
16
20
Indicates how close average
user is matched to most
preferred partner
Ideally, both sides are
treated equally
Related Algorithms (developed for strict preferences)
Deferred Acceptance
(DA)1
•
•
Always yields stable
solutions
Particularly unfair solution
Welfare-Optimal (WO)2
Fairness-Equal (FE)3
•
•
Yields stable solution with
the best welfare in case of
strict preferences
•
Stable solution with
balanced welfare distribution
Approximation (Problem
NP-hard)
1: [Gale and Shapley 1962]; 2: [Irving 1986]; 3: [Iwama 2010]
5
22.05.2013
Haas et al. – Heuristics for Preference-Based Resource Allocation
Karlsruhe Service Research Institute
www.ksri.kit.edu

6. The Effect of Introducing Indifferences
Indifferences in Preferences
• In realistic preferences, users might be indifferent between certain options
• Previous algorithms can still be applied, after artificially breaking ties
• However, they cannot guarantee solution quality anymore
Preferences
Complete, strict
Complete, indifferences
Stability & Fairness
Stability & Welfare
Polynomial
Scenario 1
NP-hard
NP-hard1
Scenario 1
NP-hard1
1: Also hard to approximate; [Halldorsson et al. 2003, 2007]
Research Question: Efficiency of Heuristics
For preferences with indifferences, are heuristic procedures able to yield solutions for the
two‐sided matching problem that are superior to the solutions of the standard algorithms?
6
22.05.2013
Haas et al. – Heuristics for Preference-Based Resource Allocation
Karlsruhe Service Research Institute
www.ksri.kit.edu

7. Heuristic: Genetic Algorithm
Genetic Algorithm1
• Population and Chromosomes
• GA has several chromosomes which are encoded potential solutions
• Mutation
• Randomly change two matched pairs
• Crossover
• Cycle crossover combines two chromosomes to 2 new, valid solutions
• Powerful in sampling large search spaces
• Able to accommodate various objective functions
1: Goldberg 1989, Holland 1990
Evaluation
• For 100 repetitions:
• Create Preferences
• Run GA and standard algorithms (after randomly breaking ties)
• Compare solution quality for different problem sizes
7
22.05.2013
Haas et al. – Heuristics for Preference-Based Resource Allocation
Karlsruhe Service Research Institute
www.ksri.kit.edu

8. Evaluation – Stable Solution with Welfare
Optimization
1
1
1: based on 50 repetitions
GA with welfare objective significantly better than average DA and WO solution
Welfare could further be increased if small number of unstable pairs would be permitted
8
22.05.2013
Haas et al. – Heuristics for Preference-Based Resource Allocation
Karlsruhe Service Research Institute
www.ksri.kit.edu

9. Evaluation – Stable Solution with Fairness
Optimization
1
1
1: based on 50 repetitions
DA yields most unfair solutions
GA with fairness objective yields better results than average FE solution
9
22.05.2013
Haas et al. – Heuristics for Preference-Based Resource Allocation
Karlsruhe Service Research Institute
www.ksri.kit.edu

10. Conclusion and Outlook
• Indifferences can occur in realistic preferences
Summary
• In this case, standard algorithms cannot guarantee solution quality
• GAs yields (significantly) better solutions than standard algorithms
in case indifferences are allowed in preferences
Outlook
• Styilzed settings considered for SMTI
• Extend evaluation to incomplete preferences
• For real datasets (large tie-lengths), shift• Break is not scalable!
Compare GA with other heuristic approaches
• GA-TA yields at least as good solutions on
• average, while preserving scalability
Include more complex preferences (correlation, real data, etc.)
• Study robustness against strategic manipulation of preferences
Thank you!
Christian Haas
Karlsruhe Service Research Institute
ch.haas@kit.edu
10
22.05.2013
Haas et al. – Heuristics for Preference-Based Resource Allocation
Karlsruhe Service Research Institute
www.ksri.kit.edu