This short note compares and contrasts two forms of learning which are present in most agent-based financial markets. First, passive learning refers to a form of “as if rationality” where wealth accumulates on strategies which have done relatively well. Second active learning refers to the active switching of agents across strategies. Most heterogeneous agent markets contain some form of both these types of learning. From what we know so far the dynamics of each may be quite different, and may yield a rich and complex joint dynamic.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
This is not a survey of learning, or heterogeneous agent models finance. This is well beyond the scope of this short paper. On heterogeneous agent models many excellent surveys exist including, Chiarella et al. , Hommes , LeBaron , and Lux . On learning models finance in general a recent survey of this large literature can be found in Pastor and Veronesi .
Another early theoretical derivation is in Breiman . A nice summary of this is in Markowitz . Blume and Easley  and Blume and Easley  state the problem in the context of a utility maximizing portfolio decision. The latter paper proves that in a complete market world the convergence to true beliefs will occur regardless of preference parameters. However, the authors point out that in an incomplete market world this convergence is not guaranteed. Evstigneev et al.  look at an incomplete markets world with endogenous prices. In their framework, the growth optimal strategy will dominate any other competing strategy in terms of acquiring all wealth in the long run.
Often this can be calibrated to some actual macro series.
There is one important class of models where passive learning is inactive. Models with Constant Absolute Risk Aversion (CARA) utility and adaptive rule selection generally have no passive learning component. Two very different examples of this are Brock and Hommes  and Arthur et al. . Price formation depends on the fraction of traders in a given strategy, and not on their wealth.
The best-known case would be log utility.
See Pastor and Stambaugh  for a more complete treatment of systems of this form in finance.
The consumption fraction, λ, is irrelevant for wealth races of this form where it is considered to be the same across all agents. Each period all agents consume the same fraction of wealth, so the relative performance is not affected by λ.
This is the risk-free return, which would generate the same utility as the return on the risky asset.
The evidence in support of various forms of active learning extends beyond casual introspection. Laboratory evidence shows some support for various forms of active learning. Some of this work in financial markets is surveyed in Hommes .
This is where Sims's  critique of deviations from rationality is in full force.
It reminds one of Fisher Black's discussions in Black .
Anufriev, M., and P. Dindo . 2010. Wealth-driven Selection in a Financial Market with Heterogeneous Agents. Journal of Economic Dynamics and Control, 73: 327–358.
Arthur, W.B., J. Holland, B. LeBaron, R. Palmer, and P. Tayler . 1997. Asset Pricing under Endogenous Expectations in an Artificial Stock Market, in The Economy as an Evolving Complex System II, edited by W.B. Arthur, S. Durlauf and D. Lane. Reading, MA: Addison-Wesley, 15–44.
Berrada, T. 2009. Bounded Rationality and Asset Pricing. Review of Finance, 13: 693–725.
Black, F. 1986. Noise. Journal of Finance, 41: 529–543.
Blume, L., and D. Easley . 1990. Evolution and Market Behavior. Journal of Economic Theory, 58: 9–40.
Blume, L., and D. Easley . 2006. If You’re so Smart, Why aren’t you Rich? Belief Selection in Complete and Incomplete Markets. Econometrica, 74: 929–966.
Boswijk, H.P., C.H. Hommes, and S. Manzan . 2007. Behavioral Heterogeneity in Stock Prices. Journal of Economic Dynamics and Control, 31 (6): 1938–1970.
Breiman, L. 1961. Optimal Gambling Systems for Favorable Games, in Proceedings of the Fourth Berkeley Symposium of Math Statistics, and Probability, Vol. 1, edited by J. Newyman and E. Scott, Berkely, CA: University of California Berkely Press.
Brock, W.A., and C.H. Hommes . 1997. A Rational Route to Randomness. Econometrica, 65: 1059–1097.
Brock, W.A., and C.H. Hommes . 1998. Heterogeneous Beliefs and Routes to Chaos in a Simple Asset Pricing Model. Journal of Economic Dynamics and Control, 22 (8–9): 1235–1274.
Campbell, J.Y., and L.M. Viceira . 2002. Strategic Asset Allocation. Oxford, UK: Oxford University Press.
Chen, S.-H., and C.-H. Yeh . 2001. Evolving Traders and the Business School with Genetic Programming: A New Architecture of the Agent-based Artificial Stock market. Journal of Economic Dynamics and Control, 25: 363–394.
Chiarella, C., R. Dieci, and X.-Z. He . 2009. Heterogeneity, Market Mechanisms, and Asset Price Dynamics, in Handbook of Financial Markets: Dynamics and Evolution, edited by T. Hens and K.R. Schenk-Hoppe. USA: Elsevier, 277–344.
Chiarella, C., and X. -Z. He . 2001. Asset Pricing and Wealth Dynamics under Heterogeneous Expectations. Quantitative Finance, 1: 509–526.
Chiarella, C., and X.-Z. He . 2008. An Adaptive Model on Asset Pricing and Wealth Dynamics with Heterogeneous Trading Strategies, in Handbook of Information Technology in Finance, edited by D. Seese, C. Weinhardt and F. Schlottmann. Heidelberg, Germany: Springer-Verlag.
Evstigneev, I.V., T. Hens, and K.R. Schenk-Hoppe . 2006. Evolutionary Stable Stock Markets. Economic Theory, 27: 449–468.
Evstigneev, I.V., T. Hens, and K.R. Schenk-Hoppe . 2009. Evolutionary Finance, in Handbook of Financial Markets: Dynamics and Evolution, edited by T. Hens and K.R. Schenk-Hoppe. Amsterdam, the Netherlands: Handbooks in Finance, North-Holland, 509–564.
Friedman, M. 1953. Essays in Positive Economics. Chicago, IL: University of Chicago Press.
Goldbaum, D., and B. Mizrach . 2008. Estimating the Intensity of Choice in a Dynamic Mutual Fund Allocation Decision. Journal of Economic Dynamics and Control, 32: 3866–3876.
Hakansson, N.H. 1971. Multi-period Mean-variance Analysis: Toward a General Theory of Portfolio Choice. Journal of Finance, 26: 857–884.
Hommes, C.H. 2006. Heterogeneous Agent Models in Economics and Finance, in Handbook of Computational Economics, edited by K.L. Judd and L. Tesfatsion. Amsterdam, the Netherlands: Elsevier.
Hommes, C.H . 2010. The Heterogeneous Expectations Hypothesis: Some Evidence from the Lab, Technical Report, CeNDEF, University of Amsterdam.
Kelley, J.L. 1956. A New Interpretation of Information Rate. Bell Systems Technical Journal, 35: 917–926.
LeBaron, B. 2001. Evolution and Time Horizons in an Agent-based Stock Market. Macroeconomic Dynamics, 5 (2): 225–254.
LeBaron, B. . 2006. Agent-based Computational Finance, in Handbook of Computational Economics, edited by K.L. Judd and L. Tesfatsion. Amsterdam, the Netherlands: Elsevier, 1187–1233.
LeBaron, B. . 2007. Wealth Evolution and Distorted Financial Forecasts, Technical Report, International Business School, Brandeis University.
LeBaron, B. . 2010. Heterogenous Gain Learning and the Dynamics of Asset Prices, Technical Report, International Business School, Brandeis University, Waltham, MA.
Levy, M., H. Levy, and S. Solomon . 1994. A Microscopic Model of the Stock Market: Cycles, Booms, and Crashes. Economics Letters, 45: 103–111.
Lux, T. 2009. Stochastic Behavioral Asset Pricing Stochastic Behavioral Asset Pricing Models and the Stylized Facts, in Handbook of Financial Markets: Dynamics and Evolution, edited by T. Hens and K.R. Schenk-Hoppe. North-Holland.
Markowitz, H. 1976. Investment for the Long Run: New Evidence for an Old Rule. Journal of Finance, 31: 1273–1286.
Pastor, L., and P. Veronesi . 2009. Learning in Financial Markets. Annual Review of Financial Economics, 1: 361–381.
Pastor, L., and R.F. Stambaugh . 2009. Predictive Systems: Living with Imperfect Predictors. Journal of Finance, 64: 1583–1628.
Radner, R. 1998. Economic Survival, in Frontiers of Research in Economic Theory, edited by D.P. Jacobs, E. Kalai and M.I. Kamien. Econometric Society Monographs, Cambridge, UK: Cambridge University Press, 183–209.
Samuelson, P. 1971. The “fallacy” of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling. Proceedings of the National Academy of Science, 68: 2493–2496.
Sims, C.A. 1980. Macroeconomics and Reality. Econometrica, 48: 1–48.
Winter, S.G . 1987. Competition and Selection, in The New Palgrave: A Dictionary of Economic, edited by J. Eatwell, M. Milgate and P. Newman. Basingstoke: Palgrave Macmillan, 545–548.
Yan, H. 2008. Natural Selection in Financial Markets: Does It Work? Management Science, 54 (11): 1935–1950.
About this article
Cite this article
LeBaron, B. Active and Passive Learning in Agent-based Financial Markets. Eastern Econ J 37, 35–43 (2011). https://doi.org/10.1057/eej.2010.53