Summary
A learning process for 2-person games in normal form is introduced. The game is assumed to be played repeatedly by two large populations, one for player 1 and one for player 2. Every individual plays against changing opponents in the other population. Mixed strategies are adapted to experience. The process evolves in discrete time.
All individuals in the same population play the same mixed strategy. The mixed strategies played in one period are publicly known in the next period. The payoff matrices of both players are publicly known.
In a preliminary version of the model, the individuals increase and decrease probabilities of pure strategies directly in response to payoffs against last period’s observed opponent strategy. In this model, the stationary points are the equilibrium points, but genuinely mixed equilibrium points fail to be locally stable.
On the basis of the preliminary model an anticipatory learning process is defined, where the individuals first anticipate the opponent strategies according to the preliminary model and then react to these anticipated strategies in the same way as to the observed strategies in the preliminary model. This means that primary learning effects on the other side are anticipated, but not the secondary effects due to anticipations in the opponent population.
Local stability of the anticipatory learning process is investigated for regular games, i.e., for games where all equilibrium points are regular. Astability criterion is derived which is necessary and sufficient for sufficiently small adjustment speeds. This criterion requires that the eigenvalues of a matrix derived from both payoff matrices are negative.
It is shown that the stability criterion is satisfied for 2x 2-games without pure strategy equilibrium points, for zero-sum games and for games where one player’s payoff matrix is the unit matrix and the other player's payoff matrix is negative definite. Moreover, the addition of constants to rows or columns of payoff matrices does not change stability.
The stability criterion is related to an additive decomposition of payoffs reminiscent of a two way analysis of variance. Payoffs are decomposed into row effects, column effects and interaction effects. Intuitively, the stability criterion requires a preponderance of negative covariance between the interaction effects in both players’ payoffs.
The anticipatory learning process assumes that the effects of anticipations on the other side remain unanticipated. At least for completely mixed equilibrium points the stability criterion remains unchanged, if anticipations of anticipation effects are introduced.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bomze, I.M. (1986). Non-Cooperative Two-Person Games in Biology: A Classification. International Journal of Game Theory 15: 31–57.
Brown, G. (1951). Iterative Solution of Games by Fictitious Play. In: T. Koopmanns (Ed.): “Activity Analysis of Production and Allocation”. pp. 374–376. New York: Wiley.
Bush, R.R., and F. Mosteller (1955). Stochastic Models for Learning. New York: Wiley.
Crawford, V.P, (1985). Learning Behavior and Mixed Strategy Nash Equilibria. Journal of Economic Behavior and Organization 6: 69–78.
Cross, J.G. (1983). A Theory of Adaptive Economic Behavior. Cambridge: Cambridge University Press.
Darlington, R.B. (1975). Radicals and Squares. New York: Logan Hill Press.
Gabisch, G., and H.W. Lorenz (1987). Business Cycle Theory. Lecture Notes in Economics and Mathematical Systems 283. Berlin: Springer-Verlag.
Harley, C.B. (1981). Learning the Evolutionarily Stable Strategy. Journal of Theoretical Biology 89: 611–633.
Harsanyi, J.G. (1973). Games with Randomly Disturbed Payoffs. A New Rationale for Mixed-Strategy Equilibrium Points. Internat. Journal of Game Theory 2: 235–250.
Maynard Smith, J. (1982). Evolution and the Theory of Games. Cambridge: Cambridge University Press.
Maynard Smith, J., and G.A. Price (1973). The Logic of Animal Conflict. Nature 246: 15–18.
Milinski, M. (1979). An Evolutionarily Stable Feeding Strategy in Sticklebacks. Zeitschrift für Tierpsychologie 51: 36–40.
Miyasawa, K. (1961). On the Convergence of the Learning Process in a 2x2 Non-Zero-Sum Two-Person Game. Economic Research Program, Research Memorandum No. 33. Princeton, N.J.: Princeton University.
Nash, J. (1951). Non-Cooperative Games. Annals of Mathematics 54: 286–295.
O’Neill, B. (1987). Nonmetric Test of the Minimax Theory of Two-Person Zerosum Games. Proceedings of the National Academy of Sciences, U.S.A., Vol. 84: 2106–2109.
Robinson, J. (1951). An Iterative Method of Solving a Game. Annals of Mathematics 54: 296–301.
Rosenmüller, J. (1971). Über Periodizitätseigenschaften spieltheoretischer
Lernprozesse. Zeitschr. Wahrsch. Verw. Gebiete 17: 259–308.
Selten, R. (1980). A Note on Evolutionarily Stable Strategies in Asymmetric Animal Conflicts. Journal of Theoretical Biology 83: 93–101.
Selten, R., and P. Hammerstein (1984). Gaps in Harley’s Argument on Evolutionarily Stable Learning Rules and in the Logic of “Tit for Tat”. The Behavioral and Brain Sciences 7: 115–116.
Selten, R., and R. Stoecker (1986). End Behavior in Sequences of Finite Prisoner’s Dilemma Supergames. Journal of Economic Behavior and Organization 7: 47–70.
Shapley, L.S. (1964). Some Topics in Two-Person Games. In: M. Dresher, L. Shapley, and A. Tucker, “Advances in Game Theory.” Annals of Mathem. Studies, No. 52. pp. 1–28. Princeton, N.J.: Princeton University Press.
Taylor, P.D., and L.B. Jonker (1978). Evolutionarily Stable Strategies and Game Dynamics. Mathem. Biosci. 40: 145–156.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Selten, R. (1991). Anticipatory Learning in Two-Person Games. In: Selten, R. (eds) Game Equilibrium Models I. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02674-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-02674-8_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08108-8
Online ISBN: 978-3-662-02674-8
eBook Packages: Springer Book Archive