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Asset market games of survival: a synthesis of evolutionary and dynamic games

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Abstract

The paper examines a game-theoretic model of a financial market in which asset prices are determined endogenously in terms of a short-run equilibrium. Investors use general, adaptive strategies (portfolio rules) depending on the exogenous states of the world and the observed history of the game. The main goal is to identify portfolio rules, allowing an investor to “survive,” i.e., to possess a positive, bounded away from zero, share of market wealth over an infinite time horizon. The model under consideration combines a strategic framework characteristic for stochastic dynamic games with an evolutionary solution concept (survival strategies), thereby linking two fundamental paradigms of game theory.

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Notes

  1. See Magill and Quinzii (1996) for a textbook treatment of the subject.

  2. Tversky and Kahneman (1991), Shleifer (2000), Shiller (2003), and Thaler (2005).

  3. See a survey in Evstigneev et al. (2009).

  4. The state of the art in the area of research related to the Kelly investment criterion is surveyed in MacLean et al. (2011).

  5. For more recent developments, see Neyman and Sorin (2003). In view of this paper’s concern with long-run survival only, the class of stochastic games most closely related would be the one with undiscounted rewards (Vieille 2000a, b, 2000c)

  6. For textbook treatments of this class of games, see Luce and Raiffa (1989, Section A8.4) and Maitra and Sudderth (1996, Section 7.16). For more recent research on similar classes of games see Secchi and Sudderth (2001) and references therein. Related questions are discussed in Borch (1966), Shubik and Thompson (1959) and Karni and Schmeidler (1986).

  7. The evolutionary process may involve random noise (Foster and Young 1990; Fudenberg and Harris 1992; Cabrales 2000) and the underlying game may be random (Germano 2007).

  8. In Sect. 6 we will show that the goal of survival in the present context is equivalent to the objective of winning a certain game associated with the original one (“in order to survive you have to win”).

  9. A stochastic kernel \(p_{t}(s^{t},\varGamma )\) is a function of \(s^{t}\in S^{t}\) and a measurable set \(\varGamma \subseteq S\) such that \(p_{t}(s^{t},\varGamma )\) is a probability measure with respect to \(\varGamma \) for each \(s^{t}\) and a measurable function with respect to \(s^{t}\) for each \(\varGamma \).

  10. An unconditional variant of the Maynard Smith’s ESS was considered by Kojima (2006).

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Acknowledgments

The authors are grateful to Yuri Kifer for fruitful discussions.

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Authors and Affiliations

  1. Department of Economics, University of Arizona, Tucson, AZ, 85721-0108, USA

    Rabah Amir

  2. Economics Department, School of Social Sciences, University of Manchester, Manchester, M13 9PL, UK

    Igor V. Evstigneev

  3. Leeds University Business School and School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Klaus Reiner Schenk-Hoppé

  4. Department of Finance and Management Science, NHH—Norwegian School of Economics, 5045 , Bergen, Norway

    Klaus Reiner Schenk-Hoppé

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  1. Rabah Amir
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  2. Igor V. Evstigneev
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Correspondence to Klaus Reiner Schenk-Hoppé .

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Financial support from the Swiss National Center of Competence in Research “Financial Valuation and Risk Management” (project “Behavioural and Evolutionary Finance”), the Finance Market Fund, Norway (project “Stochastic Dynamics of Financial Markets”) and the European Commission under the Marie Curie Intra-European Fellowship Programme (grant agreement PIEF-GA-2010-274454) is gratefully acknowledged.

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Amir, R., Evstigneev, I.V. & Schenk-Hoppé , K.R. Asset market games of survival: a synthesis of evolutionary and dynamic games. Ann Finance 9, 121–144 (2013). https://doi.org/10.1007/s10436-012-0210-5

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