KARLSRUHE SERVICE RESEARCH INSTITUTE (KSRI)

www.kit.eduKIT – University of the State of Baden-Württemberg andNational Research Center of the Helmholtz Association

Preference‐Based Resource Allocation:Using Heuristics to Solve Two‐Sided MatchingProblems with Indifferences

Christian Haas, Steven Kimbrough, Simon Caton, Christof Weinhardt

GECON 2013Zaragoza, Spain19th September 2013

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Haas et al. – Heuristics for Preference-Based Resource Allocation22.05.20132

Agenda

Two-Sided Matching: Concepts and Challenges

Heuristics to Solve Two-Sided Matching with Indifferences

1

2

Outlook and Future Work3

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Scenario

B

A

C

A

A

C

A

C

DRequests

OffersE

D

B

AllocationA-Z: Resource Types

A

Users provide and request resources

Resource exchange without monetary transactions

How can we allocate resources while still retaining certain allocation properties (e.g. welfare)?

Preference‐Based Matching

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Two Sided Matching: Concepts

• 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:

Two‐Sided Market

• 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

Preferences

3 ≻ 4 ≻ 2 ≻ 1

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Matching Objectives and Related Approaches

Objectives

StabilityNo incentive to deviate

from solution

Welfare

Average rank of matched user

Indicates how close average user is matched to most

preferred partner

Fairness

Welfare distribution between the two sides

Ideally, both sides are treated equally

316 5 20

16: 20 ≻ 5 20: 16 ≻ 3

16 20Unstable pair:

Related Algorithms (developed for strict preferences)

Deferred Acceptance (DA)1

• Always yields stable solutions

• Particularly unfair solution

Welfare-Optimal (WO)2

• Yields stable solution with the best welfare in case of strict preferences

Fairness-Equal (FE)3

• Stable solution with balanced welfare distribution

• Approximation (Problem NP-hard)

1: [Gale and Shapley 1962]; 2: [Irving 1986]; 3: [Iwama 2010]

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The Effect of Introducing Indifferences

• 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

Indifferences in Preferences

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?

Complete, strict Scenario 1

Stability & Welfare Stability & Fairness

Scenario 1 NP-hard1

NP-hard

NP-hard1

Preferences

Polynomial

Complete, indifferences

1: Also hard to approximate; [Halldorsson et al. 2003, 2007]

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Heuristic: Genetic Algorithm

• 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 functions1: Goldberg 1989, Holland 1990

• For 100 repetitions: • Create Preferences• Run GA and standard algorithms (after randomly breaking ties)

• Compare solution quality for different problem sizes

Evaluation

Genetic Algorithm1

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Evaluation – Stable Solution with WelfareOptimization

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

1: based on 50 repetitions

1 1

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Evaluation – Stable Solution with Fairness Optimization

DA yields most unfair solutions

GA with fairness objective yields better results than average FE solution

1 1

1: based on 50 repetitions

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Conclusion and Outlook

Summary

• Styilzed settings considered for SMTI• For real datasets (large tie-lengths), shift-

Break is not scalable!• GA-TA yields at least as good solutions on

average, while preserving scalabilityOutlook

• Indifferences can occur in realistic preferences

• In this case, standard algorithms cannot guarantee solution quality

• GAs yields (significantly) better solutions than standard algorithms in case indifferences are allowed in preferences

• Extend evaluation to incomplete preferences• Compare GA with other heuristic approaches• Include more complex preferences (correlation, real data, etc.)• Study robustness against strategic manipulation of preferences

Thank you!

Christian HaasKarlsruhe Service Research Institute

ch.haas@kit.edu