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Paths to Stability in the Assignment Problem

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  • Bettina Klaus
  • Frédéric Payot

Abstract

We study a labor market with finitely many heterogeneous workers and firms to illustrate the decentralized (myopic) blocking dynamics in two-sided one-to-one matching markets with continuous side payments (assignment problems, Shapley and Shubik, 1971). A labor market is unstable if there is at least one blocking pair, that is, a worker and a firm who would prefer to be matched to each other in order to obtain higher payoffs than the payoffs they obtain by being matched to their current partners. A blocking path is a sequence of outcomes (specifying matchings and payoffs) such that each outcome is obtained from the previous one by satisfying a blocking pair (i.e., by matching the two blocking agents and assigning new payoffs to them that are higher than the ones they received before). We are interested in the question if starting from any (unstable) outcome, there always exists a blocking path that will lead to a stable outcome. In contrast to discrete versions of the model (i.e., for marriage markets, one-to-one matching, or discretized assignment problems), the existence of blocking paths to stability cannot always be guaranteed. We identify a necessary and sufficient condition for an assignment problem (the existence of a stable outcome such that all matched agents receive positive payoffs) to guarantee the existence of paths to stability and show how to construct such a path whenever this is possible.

Suggested Citation

  • Bettina Klaus & Frédéric Payot, 2013. "Paths to Stability in the Assignment Problem," Cahiers de Recherches Economiques du Département d'économie 13.14, Université de Lausanne, Faculté des HEC, Département d’économie.
    • Handle: RePEc:lau:crdeep:13.14
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    References listed on IDEAS

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    Cited by:

    1. Biró, Péter & Kern, Walter & Paulusma, Daniël & Wojuteczky, Péter, 2018. "The stable fixtures problem with payments," Games and Economic Behavior, Elsevier, vol. 108(C), pages 245-268.
    2. Heinrich Nax & Bary Pradelski, 2015. "Evolutionary dynamics and equitable core selection in assignment games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(4), pages 903-932, November.
    3. Bolle Friedel & Otto Philipp E., 2016. "Matching as a Stochastic Process," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 236(3), pages 323-348, May.
    4. Newton, Jonathan & Sawa, Ryoji, 2015. "A one-shot deviation principle for stability in matching problems," Journal of Economic Theory, Elsevier, vol. 157(C), pages 1-27.
    5. Klaus, Bettina & Newton, Jonathan, 2016. "Stochastic stability in assignment problems," Journal of Mathematical Economics, Elsevier, vol. 62(C), pages 62-74.
    6. Nax, Heinrich H. & Pradelski, Bary S. R., 2015. "Evolutionary dynamics and equitable core selection in assignment games," LSE Research Online Documents on Economics 65428, London School of Economics and Political Science, LSE Library.
    7. Jonathan Newton, 2018. "Evolutionary Game Theory: A Renaissance," Games, MDPI, vol. 9(2), pages 1-67, May.

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    More about this item

    Keywords

    Assignment problem; competitive equilibria; core; decentralized market; random path; stability;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement

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