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Evolutionary dynamics and equitable core selection in assignment games

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Abstract

We study evolutionary dynamics in assignment games where many agents interact anonymously at virtually no cost. The process is decentralized, very little information is available and trade takes place at many different prices simultaneously. We propose a completely uncoupled learning process that selects a subset of the core of the game with a natural equity interpretation. This happens even though agents have no knowledge of other agents’ strategies, payoffs, or the structure of the game, and there is no central authority with such knowledge either. In our model, agents randomly encounter other agents, make bids and offers for potential partnerships and match if the partnerships are profitable. Equity is favored by our dynamics because it is more stable, not because of any ex ante fairness criterion.

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Notes

  1. There is an extensive literature in psychology and experimental game theory on trial and error and aspiration adjustment. See in particular the learning models of Thorndike (1898), Hoppe (1931), Estes (1950), Bush and Mosteller (1955), Herrnstein (1961), and aspiration adjustment and directional learning dynamics of Heckhausen (1955), Sauermann and Selten (1962), Selten and Stoecker (1986), Selten (1998).

  2. This idea was introduced by Foster and Young (2006) and is a refinement of the concept of uncoupled learning due to Hart and Mas-Colell (2003, 2006). Recent work has shown that there exist completely uncoupled rules that lead to pure Nash equilibrium in generic noncooperative games with pure Nash equilibria (Germano and Lugosi 2007; Marden et al. 2009; Young 2009; Pradelski and Young 2012).

  3. Important subsequent papers include Crawford and Knoer (1981), Kelso and Crawford (1982), Demange and Gale (1985), and Demange et al. (1986).

  4. In a recent paper Pradelski (2014) discusses the differences to our set-up in more detail. He then investigates the convergence rate properties of a process closely related to ours.

  5. See Roth and Sotomayor (1992) for a textbook on two-sided matching.

  6. The two sides of the market could also, for example, represent buyers and sellers, or men and women in a (monetized) marriage market.

  7. Note that \({{\mathbb P}} [P_{ij}^t=0]>0\) and \({{\mathbb P}} [Q_{ij}^t=0]>0\) are reasonable assumptions, since we can adjust \(p_{ij}^+\) and \(q_{ij}^-\) in order for it to hold. This would alter the underlying game but then allow us to proceed as suggested.

  8. In this sense any alternative match that may block a current assignment because it is profitable (as defined earlier) is a strict blocking pair.

  9. See, for example, Roth and Sotomayor (1992), Balinski and Gale (1987), Sotomayor (2003).

  10. Note that the excess for coalitions, \(e^t(S)\), is usually defined with a reversed sign. In order to make it consistent with definition (6) we chose to reverse the sign.

  11. See Shapley and Shubik (1963, 1966) for the underlying idea of the least core, the strong \(\epsilon \)-core. See Maschler et al. (1979), Driessen (1999) and Llerena and Nunez (2011) for geometric interpretations of these concepts.

  12. \(\mathbf{{N}}\in {\mathbf{K}}\) is shown by Schmeidler (1969) for general cooperative games. Similarly \(\mathbf{{N}}\in {\mathbf{L}}\) is shown by Maschler et al. (1979). Driessen (1998) shows for the assignment game that \({\mathbf{K}} \subseteq {\mathbf{C}}.\) \( \mathbf{{L}}\subseteq {\mathbf{C}}\) follows directly from the definitions.

  13. The Poisson clocks’ arrival rates may depend on the agents’ themselves or on their position in the game. Single agents, for example, may be activated faster than matched agents.

  14. Note that the states \(Z^{t+2}\) and \(Z^{t+3}\) are both in the core, but \(Z^{t+3}\) is absorbing whereas \(Z^{t+2}\) is not.

  15. Note that this claim describes an absorbing state in the core. It may well be that the core is reached while a single’s aspiration level is more than zero. The latter state, however, is transient and will converge to the corresponding absorbing state.

  16. For simplicity we propose this specific distribution. But note that any probability distribution can be assumed as long as there exists a parameter \(\epsilon \) such that \({{\mathbb P}}[k+1]=\epsilon \cdot O({{\mathbb P}}[k])\) for all \(k \in {{\mathbb N}}_0\).

  17. Generically the optimal matching is unique. In particular this holds if the weights of the edges are independent, continuous random variables. Then, with probability 1, the optimal matching is unique.

  18. Note that the sequence naturally needs to alternate between firms and workers in order to make players single along the way.

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Acknowledgments

Foremost, we thank Peyton Young for his guidance. He worked with us throughout large parts of this project and provided invaluable guidance. Further, we thank Itai Arieli, Peter Biró, Gabrielle Demange, Sergiu Hart, Gabriel Kreindler, Jonathan Newton, Tom Norman, Tamás Solymosi, Zaifu Yang and anonymous referees for suggesting a number of improvements to earlier versions. We are also grateful for comments by participants at the 23rd International Conference on Game Theory at Stony Brook, the Paris Game Theory Seminar, the AFOSR MUIR 2013 meeting at MIT, the 18th CTN Workshop at the University of Warwick, the Economics Department theory group at the University of York, the Theory Workshop at the Center for the Study of Rationality at the Hebrew University, and the Game Theory Seminar at the Technion. The research was supported by the United States Air Force Office of Scientific Research Grant FA9550-09-1-0538 and the Office of Naval Research Grant N00014-09-1-0751. This paper supersedes the working paper “The evolution of core stability in decentralized matching markets”. Theorem 1 was reported without proof in the conference proceeding (Nax et al. 2013). Heinrich H. Nax acknowledges support by the European Commission through the ERC Advanced Investigator Grant ‘Momentum’ (Grant No. 324247). Bary S. R. Pradelski acknowledges support of the Oxford-Man Institute of Quantitative Finance.

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Authors and Affiliations

  1. Department of Social Sciences, ETH Zürich, Clausiusstrasse 50, 8092, Zürich, Switzerland

    Heinrich H. Nax

  2. Oxford-Man Institute, University of Oxford, Eagle House, Walton Well Road, Oxford, OX2 6ED, UK

    Bary S. R. Pradelski

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  1. Heinrich H. Nax
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Correspondence to Bary S. R. Pradelski.

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Nax, H.H., Pradelski, B.S.R. Evolutionary dynamics and equitable core selection in assignment games. Int J Game Theory 44, 903–932 (2015). https://doi.org/10.1007/s00182-014-0459-1

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