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Preference-Based Resource Allocation: Using Heuristics to Solve Two-Sided Matching Problems with Indifferences
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Preference-Based Resource Allocation: Using Heuristics to Solve Two-Sided Matching Problems with Indifferences

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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 …

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.

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  • 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

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