Izquierdo, Luis R. (2008): Advancing Learning and Evolutionary Game Theory with an Application to Social Dilemmas.

PDF
MPRA_paper_8664.pdf Download (9MB)  Preview 
Abstract
This thesis advances game theory by formally analysing the implications of replacing some of its most stringent assumptions with alternatives that –at least in certain contexts– have received greater empirical support. Specifically, this thesis makes two distinct contributions in the field of learning game theory and one in the field of evolutionary game theory. The method employed has been a symbiotic combination of computer simulation and mathematical analysis. Computer simulation has been used extensively to enhance our understanding of various formal systems beyond the current limits of mathematical tractability, and also to illustrate, complement and extend various analytical derivations.
The two extensions to learning game theory presented here abandon the orthodox assumption that players are fully rational, and assume instead that players follow one of two alternative decisionmaking processes –casebased reasoning or reinforcement learning– that have received strong support from cognitive science research. The formal results derived in this part of the thesis add to the growing body of work in learning game theory that supports the general principle that the stability of outcomes in games depends not only on how unilateral deviations affect the deviator but also, and crucially, on how they affect the nondeviators. Outcomes where unilateral deviations hurt the deviator (strict Nash) but not the nondeviators (protected) tend to be the most stable.
The contribution of this thesis to evolutionary game theory is a systematic study of the extent to which the assumptions made in mainstream evolutionary game theory for the sake of tractability are affecting its conclusions. Our results show that the type of strategies that are likely to emerge and be sustained in evolutionary contexts is strongly dependent on assumptions that traditionally have been thought to be unimportant or secondary (e.g. number of players, continuity of the strategy space, mutation rate, population structure…). This latter contribution is focused on the evolutionary emergence of cooperation.
Following the presentation of the main results and the discussion of their implications, this thesis provides some guidance on how the models analysed here could be parameterised and validated.
Item Type:  MPRA Paper 

Original Title:  Advancing Learning and Evolutionary Game Theory with an Application to Social Dilemmas 
Language:  English 
Keywords:  Game Theory; Learning Game Theory; Evolutionary Game Theory; Computer modelling; Stochastic Approximation; Slow Learning; Reinforcement Learning; Casebased Reasoning; Markov Chains; Stochastic Process; 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63  Computational Techniques ; Simulation Modeling D  Microeconomics > D8  Information, Knowledge, and Uncertainty > D83  Search ; Learning ; Information and Knowledge ; Communication ; Belief ; Unawareness C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C72  Noncooperative Games C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C73  Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games 
Item ID:  8664 
Depositing User:  Luis R. Izquierdo 
Date Deposited:  08. May 2008 19:19 
Last Modified:  06. May 2015 06:31 
References:  Aamodt, A. and Plaza, E. (1994). "Casebased reasoning: foundational issues, methodological variations, and system approaches". AI Communications 7(1), pp. 3959. Akaike, H. (1969). "Fitting autoregressive models for prediction". Annals of the Institute of Statistical Mathematics 21, pp. 243247. Alexander, J. M. (2003). "Evolutionary Game Theory". In The Stanford Encyclopedia of Philosophy, Zalta, E. N. (ed.). Summer 2003 Edition. Arthur, W. B. (1991). "Designing economic agents that act like human agents: A behavioral approach to bounded rationality". American Economic Review 81(2), pp. 353359. Aumann, R. (1976). "Agreeing to disagree". Annals of Statistics 4(6), pp. 12361239. Aumann, R. J. and Hart, S. (1992). Handbook of Game Theory with Economic Applications. Amsterdam: NorthHolland. Aumann, R. J. and Sorin, S. (1989). "Cooperation and Bounded Recall". Games and Economic Behavior 1(1), pp. 539. Axelrod, R. (1984). The Evolution of Cooperation. Basic Books USA. Axelrod, R. (1987). "The evolution of strategies in the iterated prisoner's dilemma". In Genetic Algorithms and Simulated Annealing, Davis, L. (ed.), pp. 3241. Los Altos, CA: Morgan Kaufman. Beggs, A. (2002). "Stochastic evolution with slow learning". Economic Theory 19(2), pp. 379405. Beggs, A. W. (2005). "On the convergence of reinforcement learning". Journal of Economic Theory 122(1), pp. 136. Benaim, M. and Weibull, J. W. (2003). "Deterministic Approximation of Stochastic Evolution in Games". Econometrica 71(3), pp. 873903. Bendor, J., Diermeier, D. and Ting, M. (2004). "The Empirical Content of Adaptive Models". Stanford GSB Working Paper, 1877. Bendor, J., Diermeier, D. and Ting, M. (2007). "Comment: Adaptive Models in Sociology and the Problem of Empirical Content". The American Journal of Sociology 112(5), pp. 15341545. Bendor, J., Mookherjee, D. and Ray, D. (2001a). "AspirationBased Reinforcement Learning In Repeated Interaction Games: An Overview." International Game Theory Review 3(23), pp. 159174. Bendor, J., Mookherjee, D. and Ray, D. (2001b). "Reinforcement Learning in Repeated Interaction Games". Advances in Theoretical Economics 1(1), Article 3. Bendor, J. and Swistak, P. (1995). "Types of Evolutionary Stability and the Problem of Cooperation". Proceedings of the National Academy of Sciences of the United States of America 92(8), pp. 35963600. Bendor, J. and Swistak, P. (1997). "The evolutionary stability of cooperation". American Political Science Review 91(2), pp. 290307. Bendor, J. and Swistak, P. (1998). "Evolutionary equilibria: Characterization theorems and their implications". Theory and Decision 45(2), pp. 99159. Benveniste, A., Métivier, M. and Priouret, P. (1990). Adaptive Algorithms and Stochastic Approximations. Berlin: SpringerVerlag. Bernheim, B. D. (1984). "Rationalizable strategic behavior". Econometrica 52(4), pp. 10071028. Binmore, K. (1994). Playing Fair: Game Theory and the Social Contract. Cambridge, MA: MIT Press. Binmore, K. and Samuelson, L. (1993). "An Economist's Perspective on the Evolution of Norms". Journal of Institutional Theoretical Economics 150, pp. 4563. Binmore, K., Samuelson, L. and Vaughan, R. (1995). "Musical Chairs: Modeling Noisy Evolution". Games and Economic Behavior 11(1), pp. 135. Binmore, K. G. (1998). "Review of "The Complexity of Cooperation" by Robert Axelrod". Journal of Artificial Societies and Social Simulation 1(1). Börgers, T. and Sarin, R. (1997). "Learning through reinforcement and replicator dynamics". Journal of Economic Theory 77(1), pp. 114. Börgers, T. and Sarin, R. (2000). "Naive reinforcement learning with endogenous aspirations". International Economic Review 41(4), pp. 921950. Boylan, R. T. (1992). "Laws of large numbers for dynamical systems with randomly matched individuals". Journal of Economic Theory 57(2), pp. 473504. Boylan, R. T. (1995). "Continuous Approximation of Dynamical Systems with Randomly Matched Individuals". Journal of Economic Theory 66(2), pp. 615625. Brenan, G. and Lomasky, L. (1984). "Inefficient Unanimity". Journal of Applied Philosophy 1, pp. 151163. Brown, G. W. (1951). "Iterative Solutions of Games by Fictitious Play". In Activity Analysis of Production and Allocation, Koopmans, T. C. (ed.) New York: Wiley. Bush, R. R. and Mosteller, F. (1955). Stochastic Models for Learning. New York: John Wiliey & Son. Camerer, C. (2003). Behavioral Game Theory: Experiments on Strategic Interaction. New York: Russell Sage Foundation. Camerer, C. and Ho, T. H. (1999). "Experienceweighted attraction learning in normal form games". Econometrica 67(4), pp. 827874. Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. New York: Springer. Castellano, C., Marsili, M. and Vespignani, A. (2000). "Nonequilibrium phase transition in a model for social influence". Physical Review Letters 85(16), pp. 35363539. Colman, A. M. (1995). Game Theory and Its Applications in the Social and Biological Sciences. 2nd Edition. Oxford, UK: ButterworthHeinemann. Colman, A. M. (2003). "Cooperation, psychological game theory, and limitations of rationality in social interaction". Behavioral and Brain Sciences 26(2), pp. 139153. Cross, J. G. (1973). "A Stochastic Learning Model of Economic Behavior". The Quarterly Journal of Economics 87(2), pp. 239266. Chen, Y. and Tang, F. F. (1998). "Learning and IncentiveCompatible Mechanisms for Public Goods Provision: An Experimental Study". Journal of Political Economy 106(3), pp. 633662. Dawes, R. M. (1980). "Social Dilemmas". Annual Review of Psychology 31, pp. 169193. Dawes, R. M. and Thaler, R. H. (1988). "Anomalies: Cooperation". Journal of Economic Perspectives 2(3), pp. 187197. Dawkins, R. (1989). The Selfish Gene. 2nd edition. New York: Oxford University Press. Diekmann, A. (1985). "Volunteer's dilemma". Journal of Conflict Resolution 29, pp. 605610. Diekmann, A. (1986). "Volunteer's dilemma: A social trap without a dominant strategy and some empirical results". In Paradoxical Effects of Social Behavior: Essays in Honor of Anatol Rapoport, Diekmann, A. and Mitter, P. (eds.), pp. 187197. Heidelberg and Vienna: PhysicaVerlag. Doebeli, M. and Hauert, C. (2005). "Models of cooperation based on the Prisoner's Dilemma and the Snowdrift game". Ecology Letters 8, pp. 748766. Doran, J. (1997). "Analogical Problem Solving". In Artificial Intelligence Techniques: A Comprehensive Catalogue, Bundy, A. (ed.), p. 4. SpringerVerlag. Duffy, J. (2006). "AgentBased Models and Human Subject Experiments". In Handbook of Computational Economics II: AgentBased Computational Economics., Tesfatsion, L. and Judd, K. L. (eds.), pp. 9491011. Elsevier. Edmonds, B. (2000). "The Use of Models  Making MABS More Informative". In Lecture Notes in Computer Science 1979/2000: MultiAgentBased Simulation: Second International Workshop, MABS 2000, Boston, MA, USA, July. Revised and Additional Papers, Moss, S. and Davidsson, P. (eds.), pp. 269282. Edmonds, B. (2006). "The emergence of symbiotic groups resulting from skilldifferentiation and tags". Journal of Artificial Societies and Social Simulation 9(1), Article 10. Edmonds, B. and Hales, D. (2003). "Replication, replication and replication: Some hard lessons from model alignment". Journal of Artificial Societies and Social Simulation 6(4), Article 11. Edwards, M., Huet, S., Goreaud, F. and Deffuant, G. (2003). "Comparing an individualbased model of behaviour diffusion with its mean field aggregate approximation". Journal of Artificial Societies and Social Simulation 6(4). Elster, J. (1982). "Marxism, functionalism and game theory". Theory and Society 11(4), pp. 453482. Ellison, G. (2000). "Basins of Attraction, Long Run Equilibria, and the Speed of StepbyStep Evolution". Review of Economic Studies 67, pp. 1745. Erev, I., BerebyMeyer, Y. and Roth, A. E. (1999). "The effect of adding a constant to all payoffs: experimental investigation, and implications for reinforcement learning models". Journal of Economic Behavior & Organization 39(1), pp. 111128. Erev, I. and Roth, A. E. (1998). "Predicting How People Play Games: Reinforcement Learning in Experimental Games with Unique, Mixed Strategy Equilibria". American Economic Review 88(4), pp. 848881. Erev, I. and Roth, A. E. (2001). "Simple reinforcement learning models and reciprocation in the prisoner's dilemma game". In Bounded Rationality: The Adaptive Toolbox, Gigerenzer, G. and Selten, R. (eds.), pp. 216231. Cambridge, MA: MIT Press. Ericsson, K. A. and Simon, H. A. (1980). "Verbal reports as data". Psychological Review 87(3), pp. 215251. Eshel, I. and CavalliSforza, L. L. (1982). "Assortment of Encounters and Evolution of Cooperativeness". Proceedings of the National Academy of Sciences of the United States of America 79(4), pp. 13311335. Feltovich, N. (2000). "Reinforcementbased vs. Beliefbased learning models in experimental asymmetricinformation games". Econometrica 68(3), pp. 605641. Flache, A. and Hegselmann, R. (1999). "Rationality vs. Learning in the Evolution of Solidarity Networks: A Theoretical Comparison". Computational & Mathematical Organization Theory 5(2), pp. 97127. Flache, A. and Macy, M. W. (2002). "Stochastic collusion and the power law of learning: A general reinforcement learning model of cooperation". Journal of Conflict Resolution 46(5), pp. 629653. Foster and Young (1990). "Stochastic evolutionary game dynamics". Theoretical Population Biology 38, pp. 219232. Fudenberg, D. and Kreps, D. (1993). "Learning mixed equilibria". Games and Economic Behavior 5(3), pp. 320367. Fudenberg, D. and Levine, D. K. (1998). The theory of learning in games. Cambridge, MA: MIT Press. Galán, J. M. and Izquierdo, L. R. (2005). "Appearances can be deceiving: Lessons learned reimplementing Axelrod's 'evolutionary approach to norms'". Journal of Artificial Societies and Social Simulation 8(3), Article 2. Gayer, G., Gilboa, I. and Lieberman, O. (2007). "RuleBased and CaseBased Reasoning in Housing Prices". The B.E. Journal of Theoretical Economics 7(1), Article 10. Gibbons, R. (1992). A Primer in Game Theory. Harlow (England): FT Prentice Hall. Gilboa, I., Lieberman, O. and Schmeidler, D. (2006). "Empirical Similarity". The Review of Economics and Statistics 88(3), pp. 433444. Gilboa, I. and Schmeidler, D. (1995). "Casebased decision theory". Quarterly Journal of Economics 110(3), pp. 605639. Gilboa, I. and Schmeidler, D. (2001). A Theory of CaseBased Decisions. Cambridge, UK: Cambridge University Press. Gintis, H. (2000). Game Theory Evolving: A ProblemCentered Introduction to Modeling Strategic Interaction. Princeton, New Jersey: Princeton University Press. Gotts, N. M., Polhill, J. G. and Adam, W. J. (2003a). "Simulation and analysis in agentbased modelling of land use change". Online Proceedings of the First Conference of the European Social Simulation Association, Groningen, The Netherlands, 1821 September 2003. Gotts, N. M., Polhill, J. G. and Law, A. N. R. (2003b). "Agentbased simulation in the study of social dilemmas". Artificial Intelligence Review 19(1), pp. 392. Hales, D. (2000). "Cooperation without Memory or Space: Tags, Groups and the Prisoner's Dilemma". In Lecture Notes in Computer Science 1979/2000: MultiAgentBased Simulation: Second International Workshop, MABS 2000, Boston, MA, USA, July. Revised and Additional Papers, Moss, S. and Davidsson, P. (eds.), pp. 157166. Hales, D. and Edmonds, B. (2005). "Applying a socially inspired technique (tags) to improve cooperation in P2P networks". IEEE Transactions on Systems, Man, and Cybernetics Part A:Systems and Humans. 35(3), pp. 385395. Hamilton, W. D. (1967). "Extraordinary sex ratios". Science 156(3774), pp. 477488. Hardin, G. (1968). "The tragedy of the commons. The population problem has no technical solution; it requires a fundamental extension in morality". Science 162(859), pp. 12431248. Hargreaves Heap, S. P. and Varoufakis, Y. (1995). Game Theory: A Critical Introduction. Routledge. Harsanyi, J. C. (1967a). "Games with Incomplete Information Played by "Bayesian" Players IIII. Part I: The basic model." Management Science 14(3), pp. 159182. Harsanyi, J. C. (1967b). "Games with Incomplete Information Played by "Bayesian" Players IIII. Part II: Bayesian equilibrium points." Management Science 14(5), pp. 320334. Harsanyi, J. C. (1968). "Games with Incomplete Information Played by "Bayesian" Players IIII. Part III: The basic probability distribution of the game." Management Science 14(7), pp. 486502. Hauert, C. and Doebeli, M. (2004). "Spatial structure often inhibits the evolution of cooperation in the snowdrift game". Nature 428(6983), pp. 643646. Hegselmann, R. and Flache, A. (2000). "Rational and Adaptive Playing". Analyse & Kritik 22(1), pp. 7597. Hofbauer, J., Schuster, P. and Sigmund, K. (1979). "A note on evolutionary stable strategies and game dynamics". Journal of Theoretical Biology 81(3), pp. 609612. Holt, C. A. and Roth, A. E. (2004). "The Nash equilibrium: A perspective". Proceedings of the National Academy of Sciences of the United States of America 101(12), pp. 39994002. Holland, J. (1993). "The Effect of Labels (Tags) on Social Interactions". Santa Fe Institute Working Paper, 9310064. Santa Fe, NM Hopkins, E. (2002). "Two competing models of how people learn in games". Econometrica 70(6), pp. 21412166. Hopkins, E. and Posch, M. (2005). "Attainability of boundary points under reinforcement learning". Games and Economic Behavior 53(1), pp. 110125. Huet, S., Edwards, M. and Deffuant, G. (2007). "Taking into Account the Variations of Neighbourhood Sizes in the MeanField Approximation of the Threshold Model on a Random Network". Journal of Artificial Societies and Social Simulation 10(1), Article 10. Ianni, A. (2001). "Reinforcement learning and the power law of practice: Some analytic results". Mimeo. University of Southampton. Imhof, L. A., Fudenberg, D. and Nowak, M. A. (2005). "Evolutionary cycles of cooperation and defection". Proceedings of the National Academy of Sciences of the United States of America 102(31), pp. 1079710800. Intriligator, M. D., Bodkin, R. G. and Hsiao, C. (1996). Econometric Models, Techniques and Applications. 2nd edition. Prentice Hall. Izquierdo, L. R., Gotts, N. M. and Polhill, J. G. (2003). "CaseBased Reasoning and Social Dilemmas: An AgentBased Simulation". Online Proceedings of the First Conference of the European Social Simulation Association, Groningen, The Netherlands, 1821 September 2003. Izquierdo, L. R., Gotts, N. M. and Polhill, J. G. (2004). "Casebased reasoning, social dilemmas, and a new equilibrium concept". Journal of Artificial Societies and Social Simulation 7(3), Article 1. Izquierdo, L. R. and Polhill, J. G. (2006). "Is your model susceptible to floatingpoint errors?" Journal of Artificial Societies and Social Simulation 9(4), Article 4. Janssen, J. and Manca, R. (2006). Applied SemiMarkov Processes. New York, NY, USA: Springer. Joyce, D., Kennison, J., Densmore, O., Guerin, S., Barr, S., Charles, E. and Thompson, N. S. (2006). "My Way or the Highway: a More Naturalistic Model of Altruism Tested in an Iterative Prisoners' Dilemma". Journal of Artificial Societies and Social Simulation 9(2), Article 4. Kahneman, D., Slovic, P. and Tversky, A. (1982). Judgment under Uncertainty: Heuristics and Biases. Cambridge University Press. Kalai, E. and Lehrer, E. (1993a). "Rational Learning Leads to Nash Equilibrium". Econometrica 61(5), pp. 10191045. Kalai, E. and Lehrer, E. (1993b). "Subjective Equilibrium in Repeated Games". Econometrica 61(5), pp. 12311240. Karandikar, R., Mookherjee, D., Ray, D. and VegaRedondo, F. (1998). "Evolving Aspirations and Cooperation". Journal of Economic Theory 80(2), pp. 292331. Kim, Y. (1999). "Satisficing and optimality in 2x2 common interest games". Economic Theory 13(2), pp. 365375. Kirchkamp, O. (1999). "Simultaneous evolution of learning rules and strategies". Journal of Economic Behavior & Organization 40(3), pp. 295312. Kirchkamp, O. (2000). "Evolution of learning rules in space". In Tools and Techniques for Social Science Simulation, Suleiman, R., Troitzsch, K. G. and Gilbert, G. N. (eds.), pp. 179195. Berlin: PhysicaVerlag. Kleijnen, J. P. C. (1995). "Verification and validation of simulation models". European Journal of Operational Research 82(1), pp. 145162. Klein, G. A. and Calderwood, R. (1988). "How do people use analogues to make decisions?" Proceedings of the DARPA Workshop on Casebased Reasoning, Calif., USA. Morgan Kaufmann. Kleindorfer, G. B., O'Neill, L. and Ganeshan, R. (1998). "Validation in simulation: Various positions in the philosophy of science". Management Science 44(8), pp. 10871099. Kolodner, J. L. (1993). CaseBased Reasoning. San Mateo, USA: Morgan Kaufman Publishers. Kreps, D., Milgrom, P., Roberts, J. and Wilson, R. (1982). "Rational cooperation in the finitely repeated prisoner's dilemma". Journal of Economic Theory 27(2), pp. 245252. Kuhn, S. (2001). "Prisoner's dilemma". In The Stanford Encyclopedia of Philosophy, Zalta, E. N. (ed.). Winter 2001 Edition. Kulkarni, V. G. (1995). Modeling and Analysis of Stochastic Systems. Chapman & Hall/CRC. Kushner, H. J. and Yin, G. G. (1997). Stochastic Approximation Algorithms and Applications. New York: SpringerVerlag. Laslier, J. F., Topol, R. and Walliser, B. (2001). "A Behavioral Learning Process in Games". Games and Economic Behavior 37(2), pp. 340366. Laslier, J. F. and Walliser, B. (2005). "A reinforcement learning process in extensive form games". International Journal of Game Theory 33(2), pp. 219227. Ledyard, J. O. (1995). "Public goods: A survey of experimental research". In The Handbook of Experimental Economics, Kagel, J. H. and Roth, A. E. (eds.), pp. 111194. Princeton University Press. Lewontin, R. C. (1961). "Evolution and the Theory of Games". Journal of Theoretical Biology 1, pp. 382403. Linster, B. G. (1992). "Evolutionary stability in the infinitely repeated prisoner's dilemma played by twostate Moore machines". Southern Economic Journal 58(4), pp. 880903. Ljung, L. (1999). System Identification. Theory for the User. 2nd edition. Englewood Cliffs, NJ: PrenticeHall. Loui, R. (1999). "CaseBased Reasoning and Analogy". In The MIT Encyclopedia of the Cognitive Sciences, Wilson, R. A. and Keil, F. C. (eds.), pp. 99101. Luce, R. D. and Raiffa, H. (1957). Games and Decisions: Introduction and Critical Survey. New York Wiley. Macy, M. and Flache, A. (2007). "Reply: Collective Action and the Empirical Content of Stochastic Learning Models". The American Journal of Sociology 112(5), pp. 15461554. Macy, M. W. (1995). "PAVLOV and the Evolution of Cooperation. An Experimental Test". Social Psychology Quarterly 58(2), pp. 7487. Macy, M. W. (1998). "Social Order in Artificial Worlds". Journal of Artificial Societies and Social Simulation 1(1), Article 4. Macy, M. W. and Flache, A. (2002). "Learning dynamics in social dilemmas". Proceedings of the National Academy of Sciences of the United States of America 99, pp. 72297236. Maier, N. R. F. and Schneirla, T. C. (1964). Principles of Animal Psychology. New York: Dover Publications. Maynard Smith, J. and Price, G. R. (1973). "The logic of animal conflict". Nature 246(5427), pp. 1518. McAllister, P. (1991). "Adaptive approaches to stochastic programming". Annals of Operations Research 30(1), pp. 4562. Mohler, R. R. (1991). Nonlinear Systems, Volume I: Dynamics and Control. Englewood Cliffs: Prentice Hall. Mookherjee, D. and Sopher, B. (1994). "Learning Behavior in an Experimental Matching Pennies Game". Games and Economic Behavior 7(1), pp. 6291. Mookherjee, D. and Sopher, B. (1997). "Learning and Decision Costs in Experimental Constant Sum Games". Games and Economic Behavior 19(1), pp. 97132. Nash, J. F. (1951). "Noncooperative games". Annals of Mathematics 54(2), pp. 286295. Németh, A. and Takács, K. (2007). "The Evolution of Altruism in Spatially Structured Populations". Journal of Artificial Societies and Social Simulation 10(3), Article 4. Nicolov, N. (1997). "Casebased Reasoning". In Artificial Intelligence Techniques: A Comprehensive Catalogue, Bundy, A. (ed.), pp. 1314. SpringerVerlag. Norman, M. F. (1968). "Some convergence theorems for stochastic learning models with distance diminishing operators". Journal of Mathematical Psychology 5(1), pp. 61101. Norman, M. F. (1972). Markov Processes and Learning Models. New York: Academic Press. Nowak, M. (1990). "Stochastic strategies in the prisoner's dilemma". Theor. Pop. Biol. 38, pp. 93112. Nowak, M. and Sigmund, K. (1990). "The evolution of stochastic strategies in the Prisoner's Dilemma". Acta Applicandae Mathematicae 20(3), pp. 247265. Nowak, M. A. and May, R. M. (1992). "Evolutionary games and spatial chaos". Nature 359(6398), pp. 826829. Nowak, M. A. and May, R. M. (1993). "The spatial dilemmas of evolution". International Journal of Bifurcation and Chaos 3(1), pp. 3578. Nowak, M. A., Sasaki, A., Taylor, C. and Fudenherg, D. (2004). "Emergence of cooperation and evolutionary stability in finite populations". Nature 428(6983), pp. 646650. Nowak, M. A. and Sigmund, K. (1992). "Tit for tat in heterogeneous populations". Nature 355(6357), pp. 250253. Nowak, M. A. and Sigmund, K. (1995). "Invasion dynamics of the finitely repeated prisoner's dilemma". Games and Economic Behavior 11, pp. 364390. Nowak, M. A. and Sigmund, K. (2004). "Evolutionary Dynamics of Biological Games". 303(5659), pp. 793799. Palomino, F. and VegaRedondo, F. (1999). "Convergence of aspirations and (partial) cooperation in the prisoner's dilemma". International Journal of Game Theory 28(4), pp. 465488. Pazgal, A. (1997). "Satisficing leads to cooperation in mutual interests games". International Journal of Game Theory 26(4), pp. 439453. Pearce, D. G. (1984). "Rationalizable Strategic Behavior and the Problem of Perfection". Econometrica 52(4), pp. 10291050. Phansalkar, V. V., Sastry, P. S. and Thathachar, M. A. L. (1994). "Absolutely expedient algorithms for learning Nash equilibria". Proceedings of the Indian Academy of Sciences Mathematical Sciences 104(1), pp. 279294. Polhill, G. and Izquierdo, L. (2005). "Lessons learned from converting the artificial stock market to interval Arithmetic". Journal of Artificial Societies and Social Simulation 8(2), Article 2. Polhill, J. G. and Edmonds, B. (2007). "Open Access for Social Simulation". Journal of Artificial Societies and Social Simulation 10(3), Article 10. Polhill, J. G., Izquierdo, L. R. and Gotts, N. M. (2005). "The ghost in the model (and other effects of floating point arithmetic)". Journal of Artificial Societies and Social Simulation 8(1). Polhill, J. G., Izquierdo, L. R. and Gotts, N. M. (2006). "What every agentbased modeller should know about floating point arithmetic". Environmental Modelling and Software 21(3), pp. 283309. Posch, M. (1997). "Cycling in a stochastic learning algorithm for normal form games". Journal of Evolutionary Economics 7(2), pp. 193207. Probst, D. (1996). On evolution and learning in games. PhD thesis. University of Bonn. Probst, D. (1999). "Review of "Evolutionary Game Theory", by Jörgen Weibull." Journal of Artificial Societies and Social Simulation 2(1). Raub, W. (1988). "An Analysis of the Finitely Repeated Prisoners' Dilemma". European Journal of Political Economy 4(3), pp. 367380. Raub, W. and Voss, T. (1986). "Conditions for Cooperation in Problematic Social Situations". In Paradoxical Effects of Social Behavior: Essays in Honor of Anatol Rapoport, Diekmann, A. and Mitter, P. (eds.), pp. 85103. Heidelberg and Vienna: PhysicaVerlag. Reisberg, D. (1999). "Learning". In The MIT Encyclopedia of the Cognitive Sciences, Wilson, R. A. and Keil, F. C. (eds.), pp. 460461. Riolo, R. L. (1997). "The Effects and Evolution of TagMediated Selection of Partners in Populations Playing the Iterated Prisoner's Dilemma". Proceedings of the Seventh International Conference on Genetic Algorithms (ICGA97). Morgan Kaufmann: San Francisco. Riolo, R. L., Cohen, M. D. and Axelrod, R. (2001). "Evolution of cooperation without reciprocity". Nature 414(6862), pp. 441443. Rissanen, J. (1978). "Modeling by shortest data description". Automatica 14, pp. 465471. Roberts, G. and Sherratt, T. N. (2002). "Behavioural evolution (Communication arising): Does similarity breed cooperation?" Nature 418(6897), pp. 499500. Ross, B. H. (1989). "Some psychological results on casebased reasoning". CaseBased Reasoning Workshop, DARPA. Morgan Kaufmann, Inc. Ross, D. (2006). "Game Theory". In The Stanford Encyclopedia of Philosophy, Zalta, E. N. (ed.). Spring 2006 Edition. Roth, A. E. (1995). "Introduction to Experimental Economics". In Handbook of Experimental Economics, Kagel, J. H. and Roth, A. E. (eds.), pp. 3109. Princeton University Press. Roth, A. E. and Erev, I. (1995). "Learning in extensiveform games: Experimental data and simple dynamic models in the intermediate term". Games and Economic Behavior 8(1), pp. 164212. Rustichini, A. (1999). "Optimal Properties of Stimulus  Response Learning Models". Games and Economic Behavior 29(12), pp. 244273. Santos, F. C., Pacheco, J. M. and Lenaerts, T. (2006). "Evolutionary dynamics of social dilemmas in structured heterogeneous populations". Proceedings of the National Academy of Sciences of the United States of America 103(9), pp. 34903494. Sastry, P. S., Phansalkar, V. V. and Thathachar, M. A. L. (1994). "Decentralized learning of Nash equilibria in multiperson stochastic games with incomplete information". IEEE Transactions on Systems, Man and Cybernetics 24(5), pp. 769777. Schank, R. C. (1982). Dynamic Memory: A Theory of Reminding and Learning in Computers and People. Cambridge, UK: Cambridge University Press Schank, R. C. and Abelson, R. P. (1977). Scripts, Plans, Goals and Understanding. Hillsdale, New Jersey: Lawrence Erlbaum Associates. Selten, R. (1975). "Reexamination of the perfectness concept for equilibrium points in extensive games". International Journal of Game Theory 4(1), pp. 2555. Sethi, R. and Somanathan, E. (1996). "The Evolution of Social Norms in Common Property Resource Use". The American Economic Review 86(4), pp. 766788. Simon, H. A. (1957). Models of Man: Social and Rational; Mathematical Essays on Rational Human Behavior in a Social Setting. New York: John Wiley and Sons. Simon, H. A. (1982). Models of Bounded Rationality. Cambridge, MA: The MIT Press. Söderström, T. and Stoica, R. (1989). System Identification. Hemel Hempstead, UK: Prentice Hall International. Taylor, C., Fudenberg, D., Sasaki, A. and Nowak, M. A. (2004). "Evolutionary game dynamics in finite populations". Bulletin of Mathematical Biology 66(6), pp. 16211644. Taylor, P. D. and Jonker, L. B. (1978). "Evolutionarily stable strategies and game dynamics". Mathematical Biosciences 40(12), pp. 145156. Thorndike, E. L. (1898). Animal intelligence: An experimental study of the associative processes in animals. New York MacMillan. Thorndike, E. L. (1911). Animal Intelligence: Experimental Studies. New York: The Macmillan Company. Traulsen, A., Nowak, M. A. and Pacheco, J. M. (2006). "Stochastic dynamics of invasion and fixation". Physical Review E  Statistical, Nonlinear, and Soft Matter Physics 74(1), Article 011909. Van Damme, E. (1987). Stability and Perfection of Nash Equilibria. 2nd edition. Berlin Springer Verlag. Vanderschraaf, P. and Sillari, G. (2007). "Common Knowledge". In The Stanford Encyclopedia of Philosophy Zalta, E. N. (ed.). Fall 2007 Edition. VegaRedondo, F. (2003). Economics and the Theory of Games. Cambridge University Press. Von Neumann, J. and Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press. Watson, I. (1997). Applying casebased reasoning: Techniques for enterprise systems. Morgan Kaufman Publishers. Weibull, J. W. (1995). Evolutionary Game Theory. Cambridge, MA: MIT Press. Weibull, J. W. (1998). "Evolution, rationality and equilibrium in games". European Economic Review 42(35), pp. 641649. Weibull, J. W. (2002). "What have we learned from evolutionary game theory so far?" Stockholm School of Economics and the Research Institute of Industrial Economics. Working Paper. Wilensky, U. (1999). NetLogo. Evanston, IL Center for Connected Learning and ComputerBased Modeling, Northwestern University. Wustmann, G., Rein, K., Wolf, R. and Heisenberg, M. (1996). "A new paradigm for operant conditioning of Drosophila melanogaster". Journal of Comparative Physiology  A Sensory, Neural, and Behavioral Physiology 179(3), pp. 429436. Young, H. P. (1993). "The Evolution of Conventions". Econometrica 61(1), pp. 5784. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/8664 