## Visiting address

Eilert Sundts hus

4th floor (map)

Moltke Moesvei 31

0851
OSLO

Norway

Lecturer: Professor J�rgen W. Weibull,

Stockholm School of Economics, Sweden

Dates: 6. - 10. August 2001

**Objectives**

The traditional interpretation of game theory is that the game in question is played exactly once by perfectly rational players who know the game, whose knowledge of the game and of each others' rationality is common or mutual knowledge, and who hold interpersonally consistent expectations of each others' play. Recently, evolutionary interpretations have been developed where the game is instead recurrently played by boundedly rational individuals who are randomly drawn from large populations. These individuals need not know much about the game, but adjust their strategy choice in the light of individual or social experience from play. Equilibrium may then arise in the long run, as dynamically or stochastically stable states at the population level. However, in many games the evolutionary predictions are not equilibrium points but sets containing equilibria. The course aims at explaining some of the key concepts and results in the evolutionary paradigm and to relate those to concepts in the rationalistic.

**Basic readings**

- van Damme E. (1987):
*Stability and Perfection of Nash Equilibria*, Springer Verlag. - Weibull J (1995):
*Evolutionary Game Theory*, MIT Press. - Young P. (1998):
*Individual Strategy and Social Structure*. Princeton University Press.

**Outline of lectures**

__Lecture 1: Introduction__

A brief history of game theory. Game theory as a research tool. Rationalistic and evolutionary interpretations. Extensive, normal and characteristic forms. Examples. The course outline. [Binmore (1986,1987), vanDamme (1987), vanDamme-Weibull (1995), Kandori (1996)]

__Lecture 2: Rationality__

Normal form games. Dominance relations. Iterated elimination of dominated strategies. Best replies and rationality. Rationalizability. Sets closed under rational behavior. Examples. [Pearce (1984), Basu-Weibull (1991), Aumann-Brandenburger (1995), Weibull (1995)]

__Lecture 3 Equilibrium__

Best replies and Nash equilibrium. The structure of the set of Nash equilibria. Perfection. Strategic stability. Essentiality. Sets closed under rational behavior. Examples. [Selten (1975), Kohlberg-Mertens (1986), vanDamme (1987), Weibull (1995), Ritzberger-Weibull (1995)]

__Lecture 4: Evolutionary stability__

Symmetric two-player games. Definition of ESS. Characterizations of ESS. Relation to perfection. Social efficiency in doubly symmetric games. Examples. [vanDamme (1987), Weibull(1995)]

__Lecture 5: Boundedly rational strategy choice__

Micro models of boundedly rational social learning. Satisficing and imitation. Comparison and imitation. Experimentation. Noisy and smooth myopic best replies. Induced stochastic population processes. [Binmore-Samuelson-Vaughn (1995), Bena�m-Weibull (2000)]

__Lecture 6: Tools for deterministic dynamic analysis__

Vector fields. Solution mappings. Trajectories and orbits. Invariance and stationarity. Stability concepts. Attractors.[Weibull (1995)]

__Lecture 7: Deterministic dynamic approximation of stochastic evolution__

The mean-field equation. Approximation over bounded time intervals. Bounds on exit times. Empirical visitation rates. Large deviations. The radius and co-radius of attractors. [Binmore-Samuelson-Vaughn (1995), Ellison (2000), Bena�m-Weibull (2000)]

__Lecture 8: Evolution, rationality and equilibrium__

The two versions of the replicator dynamics. Examples. General classes of selection dynamics. ''As if'' rationality, and ''as if'' knowledge of rationality, in the long run. Nash equilibrium. Sets closed under better replies. Coordination games. .[Weibull (1995), Ritzberger-Weibull (1995), Hofbauer-Weibull (1996), Bena�m-Weibull (2000)]

__Lecture 9: Perpetually perturbed best reply dynamics__

The evolution of social conventions. Stochastic stability. [Kandori-Mailath-Rob (1993), Young (1993, 1998), Bergin-Lipman (1996), vanDamme-Weibull (2001)]

__Lecture 10: Concluding discussion__

Contrasting rationalistic and evolutionary predictions. Robust set-valued predictors. Extensive-form analysis. Laboratory experiments. Testing.

**Readings**

- Aumann R. and A. Brandenburger (1995): "Epistemic conditions for Nash equilibrium,"
*Econometrica*63, 1161-1180. - Basu K. and J. Weibull (1991): "Strategy subsets closed under rational behavior",
*Economics Letters*36, 141-146. - Benaim M. and J. Weibull (2001): "Deterministic approximation of stochastic evolution in games,"
*mimeo*. - Bergin J. and B. Lipman (1996): "Evolution with state-dependent mutations,"
*Econometrica*64, 943-956. - Binmore K., L. Samuelson and R. Vaughn (1995): "Musical chairs: modelling noisy evolution,"
*Games and Economic Behavior*11, 1-35. - Binmore K. and L. Samuelson (1997): "Muddling through: noisy equililbrium selection,"
*Journal of Economic Theory*74, 235-265. - Borgers T. and R. Sarin (1997): "Learning through reinforcement and replicator dynamics,"
*Journal of Economic Theory*77, 1-14. - van Damme E. (1987):
*Stability and Perfection of Nash Equilibria*, Springer Verlag. - Ellison G. (2000): "Basins of attraction, long-run stochastic stability, and the speed of step-by-tesp evolution,"
*Review of Economic Studies*67, 17-45. - van Damme E. and J. Weibull (1995): "Equilibrium in strategic interaction: the contributions of John C. Harsanyi, John F. Nash and Reinhard Selten,"
*Scandinavian Journal of Economics*97. - van Damme E. and J. Weibull (2001): "Evolution and refinement in games with endogenous mistake probabilities,"
*mimeo*. - Gale J., K. Binmore and L. Samuelson (1995): "Learning to be imperfect: the ultimatum game,"
*Games and Economic Behavior*8, 56-90. - Hofbauer J. and J. Weibull (1996), "Evolutionary selection against dominated strategies,"
*Journal of Economic Theory*71, 558-573. - Kandori M., Mailath G. and R. Rob (1993): "Learning, mutation, and long run equilibria in games,"
*Econometrica*61, 29-56. - Kandori M. (1996): "Evolutionary game theory in economics," in D. Kreps and K. Wallis,
*Advances in Economics and Econometrics: Theory and Applications.*Econometric Society World Congress 7, Vol. I, Cambridge University Press, Cambridge UK. - Kohlberg E. and J.-F. Mertens (1986): "On the strategic stability of equilibria,"
*Econometrica*54, 1003-1037. - Pearce D. (1984): "Rationalizable strategic behavior and the problem of perfection,"
*Econometrica*52, 1029-1050. - Ritzberger K. and J. Weibull (1995), "Evolutionary selection in normal-form games,"
*Econometrica*63, 1371-1399. - Schlag K. (1998): "Why imitate, and if so, how? A bounded rationality approach to multi-armed bandit problems,"
*Journal of Economic Theory*78, 130-156. - Weibull J (1995):
*Evolutionary Game Theory*, MIT Press. - Young P. (1993): "The evolution of conventions,"
*Econometrica*61, 57-84. - Young P. (1998):
*Individual Strategy and Social Structure*. Princeton University Press (Princeton).

**The lecturer**

J�rgen W. Weibull is A.O. Wallenberg professor of economics at the Stockholm School of Economics. His main area of reseach is game theory, both concerning its foundations and its applications to economics, in particular to political economy modelling.