Abstract
A stochastic model is developed to describe behavioral changes by imitative pair interactions of individuals. ‘Microscopic’ assumptions on the specific form of the imitative processes lead to a stochastic version of the game dynamical equations, which means that the approximate mean value equations of these equations are the game dynamical equations of evolutionary game theory.
The stochastic version of the game dynamical equations allows the derivation of covariance equations. These should always be solved along with the ordinary game dynamical equations. On the one hand, the average behavior is affected by the covariances so that the game dynamical equations must be corrected for increasing covariances; otherwise they may become invalid in the course of time. On the other hand, the covariances are a measure of the reliability of game dynamical descriptions. An increase of the covariances beyond a critical value indicates a phase transition, i.e. a sudden change in the properties of the social system under consideration.
The applicability and use of the equations introduced are illustrated by computational results for the social self-organization of behavioral conventions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arthur, W.B.: 1988, ‘Competing Technologies: An Overview’, in: Dosi, R. et al. (eds.), Technical Change and Economic Theory, Pinter Publishers, London and New York.
Arthur, W.B.: 1989, ‘Competing Technologies, Increasing Returns, and Lock-In by Historical Events’, The Economic Journal 99, 116–131.
Axelrod, R.: 1984, The Evolution of Cooperation, Basic Books, New York.
Boltzmann, L.: 1964, Lectures on Gas Theory, University of California, Berkeley.
Domencich, T.A. and McFadden, D.: 1975, Urban Travel Demand. A Behavioral Analysis, North-Holland, Amsterdam, pp. 61–69.
Durlauf, S.: 1989, ‘Locally Interacting Systems, Coordination Failure, and the Long Run Behavior of Aggregate Activity’, Working Paper No. 3719, National Bureau of Economic Research, Cambridge, MA.
Durlauf, S.: 1991, ‘Nonergodic Economic Growth’, mimeo Stanford University.
Eigen, M.: 1971, ‘The Selforganization of Matter and the Evolution of Biological Macromolecules’, Naturwissenschaften 58, 465.
Eigen, M. and Schuster, P.: 1979, The Hypercycle, Springer, Berlin.
Feistel, R. and Ebeling, W.: 1989, Evolution of Complex Systems, Kluwer Academic, Dordrecht.
Fisher, R.A.: 1930, The Genetical Theory of Natural Selection, Oxford University, Oxford.
Fokker, A.D.: 1914, Annalen der Physik 43, 810ff.
Föllmer, H.: 1974, ‘Random Economics with Many Interacting Agents’, Journal of Mathematical Economics 1, 51–62.
Gardiner, C.W.: 1983, Handbook of Stochastic Methods, Springer, Berlin.
Glance, N.S. and Huberman, B.A.: 1992, ‘Dynamics with Expectations’, Physics Letters A 165, 432–440.
Haag, G., Hilliges, M., and Teichmann, K.: 1993, ‘Towards a Dynamic Disequilibrium Theory of Economy’, in: Nijkamp, P. and Reggiani, A. (eds.), Nonlinear Evolution of Spatial Economic Systems, Springer, Berlin.
Haken, H.: 1975, ‘Cooperative Phenomena in Systems Far from Thermal Equilibrium and in Nonphysical Systems’, Reviews of Modern Physics 47, 67–121.
Haken, H.: 1979, Synergetics. An Introduction, Springer, Berlin.
Haken, H.: 1983, Advanced Synergetics, Springer, Berlin.
Hauk, M.: 1994, Evolutorische Ökonomik und private Transaktionsmedien, Lang, Frankfurt/Main.
Helbing, D.: 1991, ‘A Mathematical Model for the Behavior of Pedestrians’, Behavioral Science 36, 298–310.
Helbing, D.: 1992, Stochastische Methoden, nichtlineare Dynamik und quantitative Modelle sozialer Prozesse, Ph.D. thesis, University of Stuttgart. Published (1993) by Shaker, Aachen. Corrected and enlarged English edition: Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes, published (1995) by Kluwer Academic, Dordrecht.
Helbing, D.: 1992a, ‘Interrelations between Stochastic Equations for Systems with Pair Interactions’, Physica A 181, 29–52.
Helbing, D.: 1992b, ‘A Mathematical Model for the Behavior of Individuals in a Social Field’, Journal of Mathematical Sociology 19, 189–219.
Hofbauer, J., Schuster, P., and Sigmund, K.: 1979, ‘A Note on Evolutionarily Stable Strategies and Game Dynamics’, J. Theor. Biology 81, 609–612.
Hofbauer, J., Schuster, P., Sigmund, K., and Wolff, R.: 1980, ‘Dynamical Systems under Constant Organization’, J. Appl. Math. 38, 282–304.
Hofbauer, J. and Sigmund, K.: 1988, The Theory of Evolution and Dynamical Systems, Cambridge University, Cambridge.
Ising, E.: 1925, Zeitschrift für Physik 31, 253ff.
Kramers, H.A.: 1940, Physica 7, 284ff.
Langevin, R: 1908, Comptes Rendues 146, 530ff.
Luce, R.D. and Raiffa, H.: 1957, Games and Decisions, Wiley, New York.
Luce, R.D.: 1959, Individual Choice Behavior, Wiley, New York, Ch. 2.A: ‘Fechner's Problem’.
Moyal, J.E.: 1949, J. Royal Stat. Soc. 11, 151–210.
von Neumann, J. and Morgenstern, O.: 1944, Theory of Games and Economic Behavior, Princeton University, Princeton.
Nicolis, G. and Prigogine, I.: 1977, Self-Organization in Nonequilibrium Systems, Wiley, New York.
Orléan, A.: 1992, ‘Contagion des Opinions et Fonctionnement des Marchés Financiers’, Revue Économique 43, 685–698.
Orléan, A.: 1993, Decentralized Collective Learning and Imitation: A Quantitative Approach', mimeo CREA.
Orléan, A. and Robin, J.-M.: 1992, ‘Variability of Opinions and Speculative Dynamics on the Market of a Storable Goods’, mimeo CREA.
Pauli, H.: 1928, in: Debye, P. (ed.), Probleme der Modernen Physik, Hirzel, Leipzig.
Planck, M.: 1917, in Sitzungsber. Preuss. Akad. Wiss., pp. 324ff.
Prigogine, I.: 1976, ‘Order through Fluctuation: Self-Organization and Social System’, in: Jantsch, E. and Waddington, C.H. (eds.), Evolution and Consciousness. Human Systems in Transition, Addison-Wesley, Reading, MA.
Rapoport, A. and Chammah, A.M.: 1965, Prisoner's Dilemma. A Study in Conflict and Cooperation, University of Michigan Press, Ann Arbor.
Schnabl, W., Stadler, P.F., Forst, C., and Schuster, P.: 1991, ‘Full Characterization of a Strange Attractor’, Physica D 48, 65–90.
Schuster, P., Sigmund, K., Hofbauer, J., and Wolff, R.: 1981, ‘Selfregulation of Behavior in Animal Societies’, Biological Cybernetics 40, 1–25.
Stratonovich, R.L.: 1963, 1967, Topics in the Theory of Random Noise, Vols. 1 and 2, Gordon and Breach, New York.
Taylor, P. and Jonker, L.: 1978, ‘Evolutionarily Stable Strategies and Game Dynamics’, Math. Biosciences 40, 145–156.
Topol, R.: 1991, ‘Bubbles and Volatility of Stock Prices: Effect of Mimetic Contagion’, The Economic Journal 101, 786–800.
Weidlich, W: 1971, ‘The Statistical Description of Polarization Phenomena in Society’, Br. J. Math. Stat. Psychol. 24, 51ff.
Weidlich, W.: 1972, ‘The Use of Statistical Models in Sociology’, Collective Phenomena 1, 51–59.
Weidlich, W.: 1991, ‘Physics and Social Science - The Approach of Synergetics’, Physics Reports 204, 1–163.
Weidlich, W. and Braun, M.: 1992, ‘The Master Equation Approach to Nonlinear Economics’, Journal of Evolutionary Economics 2, 233–265.
Weidlich, W. and Haag, G.: 1983, Concepts and Models of a Quantitative Sociology. The Dynamics of Interacting Populations, Springer, Berlin.
Zeeman, E.C.: 1980, ‘Population Dynamics from Game Theory’, in: Global Theory of Dynamical Systems, Lecture Notes in Mathematics Vol. 819.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Helbing, D. A stochastic behavioral model and a ‘Microscopic’ foundation of evolutionary game theory. Theor Decis 40, 149–179 (1996). https://doi.org/10.1007/BF00133171
Issue Date:
DOI: https://doi.org/10.1007/BF00133171