Market sentiments and convergence dynamics in decentralized assignment economies

Abstract

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

Notes

  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.

References

  1. Ackermann H, Goldberg P, Mirrokni V, Roeglin H, Voecking B (2011) Uncoordinated two-sided matching markets. SIAM J Comput 40(1):92–106

    Article  Google Scholar 

  2. Arieli I, Young HP (2016) Stochastic learning dynamics and speed of convergence in population games. Econometrica 84:627–676

    Article  Google Scholar 

  3. Bayati M, Borgs C, Chayes J, Kanoria Y, Montanari A (2015) Bargaining dynamics in exchange networks. J Econ Theory 156:417–454

    Article  Google Scholar 

  4. Biró P, Bomhoff M, Golovach PA, Kern W, Paulusma D (2012) Solutions for the stable roommates problem with payments. Lect Notes Comput Sci 7551:69–80

    Article  Google Scholar 

  5. Chen B, Fujishige S, Yang Z (2016) Random decentralized market processes for stable job matchings with competitive salaries. J Econ Theory 165:25–36

    Article  Google Scholar 

  6. Edmonds J (1965) Paths, trees, and flowers. Can J Math 17:49–467

    Article  Google Scholar 

  7. Feller W (1950) An introduction to probability theory and ist applications. Wiley, Oxford

    Google Scholar 

  8. Gale D, Shapley LS (1962) College admissions and stability of marriage. Am Math Mon 69:9–15

    Article  Google Scholar 

  9. Georgakopoulos A, Wagner S (2017) Hitting times, cover cost, and the Wiener index of a tree. J Gr Theory 84(3):311–326

    Article  Google Scholar 

  10. Klaus B, Payot F (2015) Paths to stability in the assignment problem. J Dyn Games 2:257–287

    Article  Google Scholar 

  11. Leshno JD, Pradelski BSR (2018) Efficient price discovery and information in the decentralized assignment game. Working paper

  12. Lewis D (1969) Convention: a philosophical study. Harvard University Press, Harvard

    Google Scholar 

  13. Nax HH, Pradelski BSR (2015) Evolutionary dynamics and equitable core selection in assignment games. Int J Game Theory 44(4):903–932

    Article  Google Scholar 

  14. Nax HH, Pradelski BSR (2016) Core stability and core selection in a decentralized labor matching market. Games 7:10

    Article  Google Scholar 

  15. Newton J, Angus SD (2015) Coalitions, tipping points and the speed of evolution. J Econ Theory 157:172–187

    Article  Google Scholar 

  16. Núñez MNO, Rafels C (2015) A survey on assignment markets. J Dyn Games 2(3–4):227–256

    Google Scholar 

  17. Pradelski BSR (2015) Decentralized dynamics and fast convergence in the assignment game. In: Proceedings of the 16th ACM conference on economics and computation (EC15) Portland Oregon, p 43

  18. Roth AE, Vande Vate H (1990) Random paths to stability in two-sided matching. Econometrica 58:1475–1480

    Article  Google Scholar 

  19. Shapley LS, Shubik M (1972) The assignment game 1: the core. Int J Game Theory 1(1):111–130

    Article  Google Scholar 

  20. Tetali P, Winkler P (1991) On a random walk problem arising in self-stabilizing token management. In: Proceedings of the tenth annual ACM symposium on principles of distributed computing, ACM, New York, NY, USA, PODC ’91, pp 273–280

  21. von Böhm-Bawerk E (1891) The positive theory of capital. Macmillan and Company, London

    Google Scholar 

Download references

Acknowledgements

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).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Heinrich H. Nax.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pradelski, B.S.R., Nax, H.H. Market sentiments and convergence dynamics in decentralized assignment economies. Int J Game Theory 49, 275–298 (2020). https://doi.org/10.1007/s00182-019-00694-0

Download citation

Keywords

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