Theory and Decision

, Volume 62, Issue 1, pp 47–96 | Cite as

Economic Darwinism: Who has the Best Probabilities?

Article

Abstract

Simulation evidence obtained within a Bayesian model of price-setting in a betting market, where anonymous gamblers queue to bet against a risk-neutral bookmaker, suggests that a gambler who wants to maximize future profits should trade on the advice of the analyst cum probability forecaster who records the best probability score, rather than the highest trading profits, during the preceding observation period. In general, probability scoring rules, specifically the log score and better known “Brier” (quadratic) score, are found to have higher probability of ranking rival analysts in predetermined “correct” order than either (i) the more usual method of counting categorical forecast errors (misclassifications), or (ii) an economic measure of forecasting success, described here as the “Kelly score” and defined as the trading profits accumulated by making log optimal bets (i.e. Kelly betting) against the market maker based on the probability forecasts of the analyst being assessed. This runs counter to the conventional wisdom that financial forecasts are more aptly evaluated in terms of their financial consequences than by an abstract non-monetary measure of statistical accuracy such as the number of misclassifications or a probability score.

Keywords

bid-ask spread economic forecast evaluation Kelly criterion probability forecasting probability scoring rules Kelly score 

Jel Classification

C11 C44 D40 D81 C52 G11 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alchain A. (1950) Uncertainty, evolution and economic theory. Journal of Political Economy 58:211–221CrossRefGoogle Scholar
  2. Bagehot W. (1971) The only game in town. Financial Analysts Journal 27:12–22CrossRefGoogle Scholar
  3. Barclay M.J., Warner J.B. (1993) Stealth trading and volatility Which trades move prices. Journal of Financial Economics 34:281–305CrossRefGoogle Scholar
  4. Bernardo, J.M. (1979), Expected information as expected utility, The Annals of Statistics 7, 686–690.Google Scholar
  5. Bernardo J.M. (1998) A decision analysis approach to multiple-choice examinations. In: Giron F.J. (eds) Applied Decision Analysis. Kluwer, Boston, pp. 195–207Google Scholar
  6. Bell R.M., Cover T.M. (1980) Competitive optimality of logarithmic investment. Mathematical Operations Research 5:161–166Google Scholar
  7. Blume L., Easley D. (1992) Evolution and market behavior. Journal of Economic Theory 58:9–40CrossRefGoogle Scholar
  8. Blume, L. and Easley, D. (2001), If You are so smart, why aren’t you rich? Belief selection in complete and incomplete markets, Cowles Foundation Discussion Paper No. 1319. Yale University.Google Scholar
  9. Blume L., Easley D. (2002) Optimality and natural selection in markets. Journal of Economic Theory 107:95–135CrossRefGoogle Scholar
  10. Breiman L. (1961) Optimal gambling systems for favorable games. In: Neyman J., Scott E. (eds) Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol 1. University of California Press, Berkeley, pp. 65–78Google Scholar
  11. Brier G.W. (1950) Verification of forecasts expressed in terms of probability. Monthly Weather Review 78:1–3CrossRefGoogle Scholar
  12. Cain M., Law D., Lindley D.V. (2000), The construction of a simple book. Journal of Risk and Uncertainty 20:119–140CrossRefGoogle Scholar
  13. Chakravarty S. (2001) Stealth trading: which traders’ trades move stock prices. Journal of Financial Economics 61:289–307CrossRefGoogle Scholar
  14. Clemon R.T. (1986) Calibration and the aggregation of probabilities. Management Science 32:312–314Google Scholar
  15. Clemon R.T., Winkler R.L (1986) Combining economic forecasts. Journal of Business and Economic Statistics 4:39–46CrossRefGoogle Scholar
  16. Clements M.P. (2004) Evaluating the bank of England density forecasts of inflation. Economic Journal 114:844–866CrossRefGoogle Scholar
  17. Copeland T., Galai D. (1983) Information effects and the bid-ask spread. Journal of Finance 38:1457–1469CrossRefGoogle Scholar
  18. Dawid A.P. (1982) The well-calibrated Bayesian. Journal of the American Statistical Association 77:605–613CrossRefGoogle Scholar
  19. Dawid A.P. (1986) Probability forecasting. In: Kotz S., Johnson N.L., Read C.B. (eds) Encyclopedia of Statistical Sciences, Vol. 4. Wiley, New York, pp. 228–236Google Scholar
  20. de Finetti B. (1962) Does it make sense to speak of ‘good probability appraisers’?. In: Good I.J. (eds) The Scientist Speculates: An Anthology of Partly Baked Ideas. Heinemann, London, pp. 357–364Google Scholar
  21. de Finetti B. (1965) Methods for discriminating levels of partial knowledge concerning a test item. The British Journal of Mathematical and Statistical Psychology 18:87–123Google Scholar
  22. de Finetti B. (1970) Logical foundations and measurement of subjective probability. Acta Psychologica 34:129–145CrossRefGoogle Scholar
  23. de Finetti, B. (1974), Theory of Probability, Vol. 1, Wiley, New York.Google Scholar
  24. de Finetti, B. (1976), Probability: beware of falsifications, Scientia 111, 283–303. Reprinted in Kyburg, H.E. and Smokler, H.E. (1980), Studies in Subjective Probability, 2nd ed, pp. 194–224. Kreiger, New York.Google Scholar
  25. De Long J.B., Shleifer A., Summers L.H., Waldmann R.J. (1990), Noise trader risk in financial markets. Journal of Political Economy 98:703–737CrossRefGoogle Scholar
  26. De Long J.B., Shleifer A., Summers L.H., Waldmann R.J. (1991), The survival of noise traders in financial markets. Journal of Business 64:1–19CrossRefGoogle Scholar
  27. Diebold F.X., Hahn J.Y., Tay A.S. (1999) Multivariate density forecast evaluation and calibration in financial risk management: high frequency returns on foreign exchange. Review of Economics and Statistics 81:661–673CrossRefGoogle Scholar
  28. Dixon M.J., Pope P.F. (2004) The value of statistical forecasts in the UK association football betting market. International Journal of Forecasting 20:697–711CrossRefGoogle Scholar
  29. Easley D., O’Hara M. (1987) Price, trade size, and information in securities markets. Journal of Financial Economics 19:69–90CrossRefGoogle Scholar
  30. Finkelstein M., Whitley R. (1981) Optimal strategies for repeated games. Advances in Applied Probability 13:415–428CrossRefGoogle Scholar
  31. Forrest D., Goddard J., Simmons R. (2005) Odds-setters as forecasters: the case of english football. International Journal of Forecasting 21:551–564CrossRefGoogle Scholar
  32. Friedman M. (1953) Essays in Positive Economics. University of Chicago Press, ChicagoGoogle Scholar
  33. Glosten L., Milgrom P. (1985) Bid, ask, and transaction prices in a specialist market with heterogeneously informed traders. Journal of Financial Economics 13:71–100CrossRefGoogle Scholar
  34. Graham J.R. (1996) Is a group of economists better than one? Than none?. Journal of Business 69:193–232CrossRefGoogle Scholar
  35. Good I.J. (1952) Rational decisions. Journal of the Royal Statistical Society Series B 14:107–114Google Scholar
  36. Good I.J. (1983) Good Thinking: The Foundations of Probability and Its Applications. University of Minnesota Press, MinneapolisGoogle Scholar
  37. Granger C.W.J., Pesaran M.H. (2000a) Economic and statistical measures of forecast accuracy. Journal of Forecasting 19:537–560CrossRefGoogle Scholar
  38. Granger C.W.J., Pesaran M.H. (2000b) A decision-based approach to forecast evaluation. In: Chan W.S., Li W.K., Tang H. (eds)Statistics and Finance: An Interface. Imperial College Press, LondonGoogle Scholar
  39. Gray P.K., Gray S.F. (1997) Testing market efficiency: evidence from the sports betting market. Journal of Finance 52:1725–1737CrossRefGoogle Scholar
  40. Hakansson N.H. (1971) Capital growth and the mean-variance approach to portfolio selection. Journal of Financial and Quantitative Analysis 6:517–557CrossRefGoogle Scholar
  41. Hosmer D.W., Lemeshow S.L. (2000) Applied Logistic Regression, 2nd edn. Wiley, New YorkGoogle Scholar
  42. Kadane J.B., Winkler R.L (1988) Separating probability elicitation from utilities. Journal of the American Statistical Association 83:357–363CrossRefGoogle Scholar
  43. Kelly J.L. (1956) A new interpretation of the information rate. Bell System Technical Journal 35:917–926Google Scholar
  44. Kraus A., Litzenberger R.H. (1975) Market equilibrium in a multiperiod state preference model with logarithmic utility. Journal of Finance 30:1213–1227CrossRefGoogle Scholar
  45. Leitch G., Tanner J.E. (1981) Economic forecast evaluation: profits versus conventional error measures. American Economic Review 81:580–590Google Scholar
  46. Leitch G., Tanner J.E. (1995) Professional economic forecasts: are they worth their costs?. Journal of Forecasting 14:143–157CrossRefGoogle Scholar
  47. Levitt S.D (2004) Why are gambling markets organized so differently from financial market?. The Economic Journal 114:223–246CrossRefGoogle Scholar
  48. Lindley D.V. (1982) The Improvement of probability judgements. Journal of the Royal Statistical Society Series A 145:117–126CrossRefGoogle Scholar
  49. Lopez J.A. (2001) Evaluating the predictive accuracy of models. Journal of Forecasting 20:87–109CrossRefGoogle Scholar
  50. Luenberger D.G. (1998) Investment Science. Oxford University Press, OxfordGoogle Scholar
  51. MacLean L.C., Ziemba W.T., Blazenko G. (1992) Growth versus security in dynamic investment analysis. Management Science 11:1562–1585Google Scholar
  52. Markowitz H.M. (1976) Investment for the long run: new evidence for an old rule. Journal of Finance 31:1273–1286CrossRefGoogle Scholar
  53. Murphy A.H. (1977) The value of climatological, categorical and probabilistic forecasts in the cost-loss ratio situation. Monthly Weather Review 105:803–816CrossRefGoogle Scholar
  54. Murphy A.H. (1993) What is a good forecast? an essay on the nature of goodness in weather forecasting. Weather and Forecasting 8:281–293CrossRefGoogle Scholar
  55. Nau R.F. (2001) De Finetti was right: probability does not exist. Theory and Decision 51:89–124CrossRefGoogle Scholar
  56. Nau R.F. (2002) The aggregation of imprecise probabilities. Journal of Statistical Planning and Inference 105:265–282CrossRefGoogle Scholar
  57. O’Hara M. (1995) Market Microstructure Theory. Blackwell, Cambridge, MAGoogle Scholar
  58. Ottaviani, M. and Sorensen, P.N. (2005), Professional advice: the theory of reputational cheap talk, Working Paper: Institute of Economics, University of Copenhagen.Google Scholar
  59. Pesaran M.H., Skouras S. (2002) Decision-based methods for forecast evaluation. In: Clements M.P., Hendry D.F. (eds) A Companion to Economic Forecasting. Blackwells, Oxford, pp. 241–267Google Scholar
  60. Pesaran M.H., Timmerman A. (1994) Forecasting stock returns: an examination of stock market trading in the presence of transaction costs. Journal of Forecasting 13:330–365CrossRefGoogle Scholar
  61. Pesaran M.H., Timmerman A. (1995) The robustness and economic significance of predictability of stock market returns. Journal of Finance 50:1201–1228CrossRefGoogle Scholar
  62. Roll R. (1973) Evidence on the ‘growth optimum’ model. Journal of Finance 28:551–567CrossRefGoogle Scholar
  63. Ross S. (2003) An Elementary Introduction to Mathematical Finance: Options and Other Topics. Cambridge University Press, CambridgeGoogle Scholar
  64. Rubinstein M. (1976) The strong case for the generalized logarithmic utility model as the premier model of financial markets. Journal of Finance 31:551–571CrossRefGoogle Scholar
  65. Sandroni A. (2000) Do markets favor agents able to make accurate predictions?. Econometrica 68:1303–1341CrossRefGoogle Scholar
  66. Savage L.J. (1971) Elicitation of personal probabilities and expectations. Journal of the American Statistical Association 66:783–801CrossRefGoogle Scholar
  67. Thorp E.O. (1966) Beat the Dealer, 2nd edn. Vintage, New YorkGoogle Scholar
  68. Thorp E.O. (1969) Optimal gambling systems for favorable games. International Statistical Review 37:273–293CrossRefGoogle Scholar
  69. Thorp, E.O. (1971), Portfolio choice and the Kelly criterion, Proceedings of the Business and Economics Section of the American Statistical Association. pp. 215–224. (Also in Ziemba, W.T and Vickson, R.G. (1975). Stochastic Optimization Models in Finance, Academic Press, New York, pp. 599–619.)Google Scholar
  70. Winkler R.L. (1967) The quantification of judgment: some methodological suggestions. Journal of the American Statistical Association 62:1105–1120CrossRefGoogle Scholar
  71. Winkler R.L. (1969) Scoring rules and the evaluation of probability assessors. Journal of the American Statistical Association 64:1073–1078CrossRefGoogle Scholar
  72. Winkler R.L. (1981) Combining probabilities from dependent information sources. Management Science 27:479–488CrossRefGoogle Scholar
  73. Winkler R.L. (1986) On “good probability appraisers”. In: Goel P.K., Zellner A. (eds) Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti. Elsevier Science Publishers, Amsterdam, pp. 265–278Google Scholar
  74. Winkler R.L. (1996) Scoring rules and the evaluation of probabilities (with discussion). Test 5:1–60CrossRefGoogle Scholar
  75. Winkler R.L. (1999) Evaluation of probabilities: a level playing field. In: Shanteau J., Mellers B.A., Schum D.A. (eds) Decision Science and Technology: Reflections on the Contributions of Ward Edwards. Kluwer, Boston, pp.155–70Google Scholar
  76. Winkler R.L., Murphy A.H. (1968) “Good” probability assessors. Journal of Applied Meteorology 7:751–758CrossRefGoogle Scholar
  77. Winkler R.L., Murphy A.H. (1970) Nonlinear utility and the probability score. Journal of Applied Meteorology 9:143–148CrossRefGoogle Scholar
  78. Ziemba, W.T. and Hausch, D. (1985), Betting at the Racetrack. Dr Z Investments, Los AngelesGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.National Australia Bank Professor of Finance, Discipline of FinanceUniversity of SydneySydneyAustralia

Personalised recommendations