Skip to main content
SpringerLink
Log in

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

  • Symposium
  • Published:
Annals of Finance Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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).

References

  • Algoet, P.H., Cover, T.M.: Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Ann Probab 16, 876–898 (1988)

    Article  Google Scholar 

  • Amir, R., Evstigneev, I.V., Hens, T., Schenk-Hoppé, K.R.: Market selection and survival of investment strategies. J Math Econ 41, 105–122 (2005)

    Article  Google Scholar 

  • Amir, R., Evstigneev, I.V., Hens, T., Xu, L.: Evolutionary finance and dynamic games. Math Financ Econ 5, 161–184 (2011)

    Article  Google Scholar 

  • Arkin, V.I., Evstigneev, I.V.: Stochastic Models of Control and Economic Dynamics. London: Academic Press (1987)

  • Bell, R.M., Cover, T.M.: Competitive optimality of logarithmic investment. Math Oper Res 5, 161–166 (1980)

    Article  Google Scholar 

  • Bell, R.M., Cover, T.M.: Game-theoretic optimal portfolios. Manag Sci 34, 724–733 (1988)

    Article  Google Scholar 

  • Bertsekas, D.P., Shreve, S.E.: Stochastic Optimal Control: The Discrete Time Case. New York: Academic Press (1978)

  • Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. New York: Springer (2011)

  • Blume, L., Easley, D.: Evolution and market behavior. J Econ Theory 58, 9–40 (1992)

    Article  Google Scholar 

  • Blume, L., Easley, D.: If you are so smart, why aren’t you rich? Belief selection in complete and incomplete markets. Econometrica 74, 929–966 (2006)

    Article  Google Scholar 

  • Blume, L., Easley, D.: Market selection and asset pricing. In: Hens, T., Schenk-Hoppé, K.R. (eds.) Handbook of Financial Markets: Dynamics and Evolution, chap. 7, pp. 403–437. Amsterdam: North-Holland (2009)

  • Borch, K.: A utility function derived from a survival game. Manag Sci 12, B287–B295 (1966)

    Article  Google Scholar 

  • Borel, E.: La théorie du jeu et les équations intégrales à noyau symétrique. CR Hebd Acad Sci 173, 1304–1308 (1921). English translation: Borel, E.: The theory of play and integral equations with skew symmetric kernels. Econometrica 21, 97–100 (1953)

    Google Scholar 

  • Borovkov, A.A.: Mathematical Statistics. Amsterdam: Gordon and Breach (1998)

  • Boulding, K.E.: Evolutionary Economics. London: Sage (1981)

  • Bouton, C.L.: Nim, a game with a complete mathematical theory. Ann Math 3, 35–39 (1901–1902)

    Google Scholar 

  • Breiman, L.: Optimal gambling systems for favorable games. In: Neyman, J. (ed.) Fourth Berkeley Symposium on Mathematical Statistics and Probability. Contributions to the Theory of Statistics, vol. 1, pp. 65–78. Berkeley: University of California Press (1961)

  • Cabrales, A.: Stochastic replicator dynamics. Int Econ Rev 41, 451–481 (2000)

    Article  Google Scholar 

  • Dorfman, R., Samuelson, P.A., Solow, R.M.: Linear Programming and Economic Analysis. New York: McGraw-Hill (1958)

  • Dynkin, E.B.: Game variant of a problem on optimal stopping. Sov Math Dokl 10, 270–274 (1969)

    Google Scholar 

  • Dynkin, E.B.: Stochastic concave dynamic programming. Math USSR Sbornik 16, 501–515 (1972)

    Article  Google Scholar 

  • Dynkin, E.B., Yushkevich, A.A.: Controlled Markov Processes and Their Applications. New York: Springer (1979)

  • Dubins, L., Savage, L.M.: How to Gamble if You Must. New York: McGraw-Hill (1965)

  • Evstigneev, I.V., Hens, T., Schenk-Hoppé, K.R.: Market selection of financial trading strategies: global stability. Math Financ 12, 329–339 (2002)

    Article  Google Scholar 

  • Evstigneev, I.V., Hens, T., Schenk-Hoppé, K.R.: Evolutionary stable stock markets. Econ Theor 27, 449–468 (2006)

    Article  Google Scholar 

  • Evstigneev, I.V., Hens, T., Schenk-Hoppé, K.R.: Globally evolutionarily stable portfolio rules. J Econ Theory 140, 197–228 (2008)

    Article  Google Scholar 

  • Evstigneev, I.V., Hens, T., Schenk-Hoppé, K.R.: Evolutionary finance. In: Hens, T., Schenk-Hoppé, K.R. (eds.) Handbook of Financial Markets: Dynamics and Evolution, chap. 9, pp. 507–566. Amsterdam: North-Holland (2009)

  • Foster, D., Young, P.: Stochastic evolutionary game dynamics. Theor Popul Biol 38, 219–232 (1990)

    Article  Google Scholar 

  • Fudenberg, D., Harris, C.: Evolutionary dynamics with aggregate shocks. J Econ Theory 57, 420–441 (1992)

    Article  Google Scholar 

  • Gale, D.: On optimal development in a multi-sector economy. Rev Econ Stud 34, 1–18 (1967)

    Article  Google Scholar 

  • Gale, D., Stewart, F.M.: Infinite games with perfect information. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games II, Annals of Mathematical Studies, vol. 28, pp. 245–266. Princeton: Princeton University Press (1953)

  • Germano, F.: Stochastic evolution of rules for playing finite normal form games. Theor Decis 62, 311–333 (2007)

    Article  Google Scholar 

  • Grandmont, J.-M. (ed.): Temporary Equilibrium. Academic Press, San Diego (1988)

    Google Scholar 

  • Hakansson, N.H., Ziemba, W.T.: Capital growth theory. In: Jarrow, R.A., Maksimovic, V., Ziemba, W.T. (eds.) Handbooks in Operations Research and Management Science, vol. 9, pp. 65–86. Finance, chap. 3. Amsterdam: Elsevier (1995)

  • Hamilton, W.D.: Extraordinary sex ratios. Science 156, 477–488 (1967)

    Article  Google Scholar 

  • Haurie, A., Zaccour, G., Smeers, Y.: Stochastic equilibrium programming for dynamic oligopolistic markets. J Optim Theory Appl 66, 243–253 (1990)

    Article  Google Scholar 

  • Hodgeson, G.M.: Economics and Evolution: Bringing Life Back into Economics. Ann Arbor: University of Michigan Press (1993)

  • Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge: Cambridge University Press (1998)

  • Karni, E., Schmeidler, D.: Self-preservation as a foundation of rational behavior under risk. J Econ Behav Organ 7, 71–81 (1986)

    Article  Google Scholar 

  • Kelly, J.L.: A new interpretation of information rate. Bell Syst Tech J 35, 917–926 (1956)

    Google Scholar 

  • Kifer, Yu.: Game options. Financ Stoch 4, 443–463 (2000)

    Google Scholar 

  • Kojima, F.: Stability and instability of the unbeatable strategy in dynamic processes. Int J Econ Theory 2, 41–53 (2006)

    Article  Google Scholar 

  • Luce R.D., Raiffa H.: Games and Decisions, 2nd edn. New York: Dover (1989)

  • MacLean, L.C., Thorp, E.O., Ziemba, W.T. (eds.): The Kelly Capital Growth Investment Criterion: Theory and Practice. Singapore: World Scientific (2011)

  • Magill, M., Quinzii, M.: Theory of Incomplete Markets. Cambridge: MIT Press (1996)

  • Maitra, A.P., Sudderth, W.D.: Discrete Gambling and Stochastic Games. New York: Springer (1996)

  • Martin, D.A.: Borel determinacy. Ann Math 102, 363–371 (1975)

    Google Scholar 

  • Maynard, Smith J.: Evolution and the Theory of Games. Cambridge: Cambridge University Press (1982)

  • Maynard, Smith J., Price, G.: The logic of animal conflicts. Nature 246, 15–18 (1973)

    Article  Google Scholar 

  • McKenzie, L.W.: Optimal economic growth turnpike theorems and comparative dynamics. In: Arrow, K.J., Intrilligator, M.D. (eds.) Handbook of Mathematical Economics III, pp. 1281–1355. Amsterdam: North-Holland (1986)

  • Milnor, J., Shapley, L.S.: On games of survival. In: Dresher, M., Tucker, A.W., Wolfe, P. (eds.) Contributions to the Theory of Games III, Annals of Mathematical Studies, vol. 39, pp. 15–45. Princeton: Princeton University Press (1957)

  • Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge (1991)

    Google Scholar 

  • Nelson, R.R., Winter, S.G.: An Evolutionary Theory of Economic Change. Cambridge: Harvard University Press (1982)

  • Neveu, J.: Mathematical Foundations of the Calculus of Probability Theory. San Francisco: Holden Day (1965)

  • Neyman, A., Sorin, S. (eds.): Stochastic Games and Applications (NATO ASI series). Dordrecht: Kluwer (2003)

  • Nikaido, H.: Convex Structures and Economic Theory. New York: Academic Press (1968)

  • Rockafellar, R.T., Wets, R.: Stochastic convex programming: basic duality. Pac J Math 62, 173–195 (1976)

    Article  Google Scholar 

  • Saks, S.: The Theory of Integral. New York: Dover (1964)

  • Samuelson, L.: Evolutionary Games and Equilibrium Selection. Cambridge: MIT Press (1997)

  • Sandholm, W.H.: Population Games and Evolutionary Dynamics. Cambridge: MIT Press (2010)

  • Sandroni, A.: Do markets favor agents able to make accurate predictions? Econometrica 68, 1303–1341 (2000)

    Article  Google Scholar 

  • Schaffer, M.: Evolutionarily stable strategies for a finite population and a variable contest size. J Theor Biol 132, 469–478 (1988)

    Article  Google Scholar 

  • Schaffer, M.: Are profit-maximizers the best survivors? J Econ Behav Organ 12, 29–45 (1989)

    Article  Google Scholar 

  • Schumpeter, J.A.: Theorie der wirtschaftlichen Entwicklung. Duncker& Humblot, Leipzig (1911)

    Google Scholar 

  • Schwalbe, U., Walker, P.: Zermelo and the early history of game theory. Game Econ Behav 34, 123–137 (2001)

    Article  Google Scholar 

  • Sciubba, E.: Asymmetric information and survival in financial markets. Econ Theor 25, 353–379 (2005)

    Article  Google Scholar 

  • Secchi, P., Sudderth, W.D.: Stay-in-a-set games. Int J Game Theory 30, 479–490 (2001)

    Article  Google Scholar 

  • Shapley, L.S.: Stochastic games. Proc Natl Acad Sci USA 39, 1095–1100 (1953)

    Article  Google Scholar 

  • Shapley, L.S., Shubik, M.: Trade using one commodity as a means of payment. J Polit Econ 85, 937–968 (1977)

    Article  Google Scholar 

  • Shiller, R.J.: From efficient markets theory to behavioral finance. J Econ Perspect 17, 83–104 (2003)

    Article  Google Scholar 

  • Shleifer, A.: Inefficient Markets: An Introduction to Behavioral Finance. Oxford: Oxford University Press (2000)

  • Shubik, M., Thompson, G.: Games of economic survival. Nav Res Logist Q 6, 111–123 (1959)

    Article  Google Scholar 

  • Shubik, M., Whitt, W.: Fiat money in an economy with one durable good and no credit. In: Blaquiere, A. (ed.) Topics in Differential Games, pp. 401–448. North-Holland, Amsterdam (1973)

  • Telgársky, R.: Topological games: on the 50th anniversary of the Banach–Mazur game. Rocky Mt J Math 17, 227–276 (1987)

    Article  Google Scholar 

  • Thaler, R.H. (ed.): Advances in Behavioral Finance II. Princeton University Press, Princeton (2005)

    Google Scholar 

  • Thorp, E.O.: The Kelly criterion in Blackjack sports betting and the stock market. In: Zenios, S.A., Ziemba, W.T. (eds.) Handbook of Asset and Liability Management, vol. 1, pp. 387–428. Amsterdam: Elsevier (2006)

  • Tversky, A., Kahneman, D.: Loss aversion in riskless choice: a reference-dependent model. Q J Econ 106, 1039–1061 (1991)

    Article  Google Scholar 

  • Vega-Redondo, F.: Evolution, Games, and Economic Behavior. Oxford: Oxford University Press (1996)

  • Vieille, N.: Two-player stochastic games I: a reduction. Isr J Math 119, 55–91 (2000a)

    Article  Google Scholar 

  • Vieille, N.: Two-player stochastic games II: the case of recursive games. Isr J Math 119, 93–126 (2000b)

    Article  Google Scholar 

  • Vieille, N.: Small perturbations and stochastic games. Isr J Math 119, 127–142 (2000c)

    Article  Google Scholar 

  • von Neumann, J.: Zur Theorie der Gesellschaftsspiele. Math Ann 100, 295–320 (1928)

    Article  Google Scholar 

  • Wallace, S.W. (ed.): Applications of Stochastic Programming. Philadelphia: SIAM—Mathematical Programming Society Series on Optimization (2005)

  • Weibull, J.: Evolutionary Game Theory. Cambridge: MIT Press (1995)

  • Zermelo, E.: Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. In: Hobson, E.W., Love, A.E.H. (eds.) Proceedings of the Fifth International Congress of Mathematicians (Cambridge 1912), vol. 2, pp. 501–504. Cambridge: Cambridge University Press (1913)

Download references

Acknowledgments

The authors are grateful to Yuri Kifer for fruitful discussions.

Author information

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

Authors
  1. Rabah Amir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor V. Evstigneev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klaus Reiner Schenk-Hoppé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Klaus Reiner Schenk-Hoppé .

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10436-012-0210-5

Keywords

JEL Classification

Search

Navigation