Market sentiments and convergence dynamics in decentralized assignment economies


In two-sided markets with transferable utility (‘assignment games’), we study the dynamics of trade arrangements and price adjustments as agents from the two market sides stochastically match, break up, and re-match in their pursuit of better opportunities. The underlying model of individual adjustments is based on the behavioral theories of adaptive learning and aspiration adjustment. Dynamics induced by this model converge to approximately optimal and stable market outcomes, but this convergence may be (exponentially) slow. We introduce the notion of a ‘market sentiment’ that governs which of the two market sides is temporarily more or less amenable to price adjustments, and show that such a feature may significantly speed up convergence.

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  1. 1.

    As an example of such a market, Shapley and Shubik discuss the famous ‘Böhm-Bawerk (horse) market’ (von Böhm-Bawerk 1891).

  2. 2.

    Note that this implies that the market sentiment changes as slow as \({\varTheta }(N^{2})\) or slower but not exponentially slow.

  3. 3.

    Such \(\delta \) always exists since the rational numbers are dense in the real numbers. Hence, one can choose \(\delta \) small enough to approximate the match values to any desired degree and thus sustain tie-freeness.

  4. 4.

    We say that a matching is perfect when all players in a given component are matched.

  5. 5.

    We define deg(x) to be the degree of node x in T and neigh(x) to be the set of neighbors of node x in T.

  6. 6.

    In line with Tetali and Winkler (1991, Conjecture 2) we conjecture that this bound is not optimal and the total meeting time of k random walks on truncated trees is of the same order as the two-token meeting time. Together with the necessary activation of single tokens this would yield the bound \(O(N^3)\).

  7. 7.

    See also Leshno and Pradelski (2018) who show that there exist assignment games for which any decentralized dynamic with no additional memory or information (such as through market sentiment as is proposed here) takes exponential time to converge.


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Pradelski thanks Pierre Tarrès and Peyton Young for their guidance and support throughout and beyond his PhD which shaped this research. Further, this paper has benefitted from generous comments and criticism by Yakov Babichenko, Doyne Farmer, Satoru Fujishige, Olivier Gossner, Sergiu Hart, Michael Krivelevich, Jacob Leshno, Noam Nisan, Thomas Norman, Bill Sandholm, Jay Sethuraman, Eran Shmaya, Gregory Sorkin and Bernhard von Stengel, as well as from anonymous referees. All of their comments are gratefully acknowledged. Comments by participants at seminars at Hebrew University (Israel), Technion (Israel), University of Oxford (UK), MEDS, Northwestern (Evanston), DRO, Columbia Business School, Microsoft Research, New England, EC’ 15, Portland, GAMES 2016, Maastricht, IMA, OR Society Conference on Mathematics of Operational Research 2017, Paris Game Theory Seminar, and Ecole Polytechnique were also important, and we thank all of them. This paper builds on the extended abstract published by Pradelski (2015).

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Correspondence to Heinrich H. Nax.

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A prior version of this manuscript was circulated as ‘Decentralized Dynamics and Fast Convergence in the Assignment Game’ authored solely by Pradelski, of which an extended abstract appeared in the Proceedings of the 16th ACM Conference on Electronic Commerce in 2015 (see Pradelski (2015)). We acknowledge the generous support of the Oxford-Man Institute of Quantitative Finance, the Office of Naval Research grant (Grant no. N00014-09-1-0751), the Air Force Office of Scientific Research grant (Grant no. FA9550-09-1-0538), and the European Commission through the ERC Advanced Investigator Grant ‘Momentum’ (Grant no. 324247).

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Pradelski, B.S.R., Nax, H.H. Market sentiments and convergence dynamics in decentralized assignment economies. Int J Game Theory 49, 275–298 (2020).

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  • Assignment games
  • Core
  • Evolutionary game theory
  • Matching markets
  • Convergence time
  • Market psychology