Forbidden zones for the expectations of measurement data and problems of behavioral economics

Harin, Alexander (2019): Forbidden zones for the expectations of measurement data and problems of behavioral economics.

Abstract

A theorem, applied mathematical method and qualitative mathematical models are introduced in the present article. The method and models are based on the forbidden zones of the theorem and suppose that people decide as if there were some biases of the expectations of measurement data, e.g., under influence of noise. The article is motivated by the need for theoretical support for the practical analysis that was performed for the purposes of behavioral economics.

• Item Type: MPRA Paper
• Language: English
• Keywords: variance; expectation; noise; bias; measurement; utility; prospect theory; behavioral economics; psychology; social sciences
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D84 - Expectations ; Speculations

References

[1] Aczél, J., and D. R. Luce (2007). A behavioral condition for Prelec’s weighting function on the positive line without assuming W(1)=1. J. Math. Psychol. 51, 126–129.

[2] Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: critique des postulats et axiomes de l'école Américaine. Econometrica 21, 503–46.

[3] Applebaum, D., and M. Siakalli (2010). Stochastic stabilization of dynamical systems using Lévy noise. Stoch. Dynam. 10(04), 509–527.

[4] Arnold, L., H. Crauel, and V. Wihstutz (1983). Stabilization of linear systems by noise, SIAM J. Control Optim. 21, 451–461.

[5] Barbu, V. (2009). The internal stabilization by noise of the linearized Navier-Stokes equation. Contr. Op. Ca. Va. 17 (1), 117–130.

[6] Becherer, D. (2006). Bounded solutions to backward SDEs with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16(4), 2027–2054.

[7] Bhatia, R. and C. Davis (2000). A better bound on the variance, Am. Math. Mon. 107, 353–357.

[8] Biagini, S., and M. Frittelli (2008). A unified framework for utility maximization problems: an Orlicz space approach. Ann. Appl. Probab. 18, 929–966.

[9] Butler, D., and G. Loomes (2007). Imprecision as an Account of the Preference Reversal Phenomenon. Am. Econ. Rev. 97, 277–297.

[10] Cacoullos, T. (1982). On Upper and Lower Bounds for the Variance of a Function of a Random Variable. Ann. Probab. 10(3), 799–809.

[11] Cerrai, S. (2005). Stabilization by noise for a class of stochastic reaction-diffusion equations. Probab. Theory Rel. 133(2), 190–214.

[12] Cheraghchi, M. (2013). Noise-resilient group testing: Limitations and constructions. Discrete Appl. Math. 161(1), 81–95.

[13] Chernoff, H. (1981). The identification of an element of a large population in the presence of noise. Ann. Probab. 9, 533–536.

[14] Choulli, T., and J. Ma (2017). Explicit description of HARA forward utilities and their optimal portfolios. Theory Probab. Appl. 61(1), 57–93.

[15] Debreu, G. (1960). Topological methods in cardinal utility theory. In Mathematical Methods in Social Sciences, Stanford University Press, 16–26.

[16] Dokov, S. P., and D.P. Morton (2005). Second-Order Lower Bounds on the Expectation of a Convex Function. Math. Oper. Res. 30(3), 662–677.

[17] Egozcue, M., García, L.F., Wong, W.-K., and R. Zitikis, (2011). The covariance sign of transformed random variables with applications to economics and finance. IMA J. Manag. Math. 22(3), 291–300.

[18] Ellsberg, D. (1961). Risk, Ambiguity and the Savage Axioms. Q. J. Econ. 75, 643–669.

[19] Flandoli, F., B. Gess, M. Scheutzow (2017). Synchronization by noise. Probab. Theory Rel. 168(3–4), 511–556.

[20] Giacomin, G., and C. Poquet (2015). Noise, interaction, nonlinear dynamics and the origin of rhythmic behaviors. Braz. J. Prob. Stat. 29(2), 460–493.

[21] Harin, A. (2012). Data dispersion in economics (II) – Inevitability and Consequences of Restrictions. Review of Economics & Finance 2, 24–36.

[22] Harin, A. (2014). The random-lottery incentive system. Can p~1 experiments deductions be correct? 16th conference on the Foundations of Utility and Risk, Rotterdam.

[23] Harin, A. (2018). Forbidden zones for the expectation. New mathematical results for behavioral and social sciences. preprint, MPRA Paper No. 86650.

[24] Hey, J., and C. Orme (1994). Investigating Generalizations of Expected Utility Theory Using Experimental Data. Econometrica 62, 1291–1326.

[25] Hu, Y., J. Huang, D. Nualart, and S. Tindel (2015). Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab. 20 (2015), no. 55, 1–50.

[26] Hua, G., M. Liub, X. Maoc, and M. Songb (2009). Noise suppresses exponential growth under regime switching. J. Math. Anal. Appl. 355(2), 783–795.

[27] Kahneman, D., and R. Thaler (2006). Anomalies: Utility Maximization and Experienced Utility, J Econ. Perspect. 20(1), 221–234.

[28] Kahneman, D., and A. Tversky (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica 47, 263–291.

[29] Kwiecińska, A. A. (1999). Stabilization of partial differential equations by noise. Stoch. Proc. Appl. 79(2), 179–184.

[30] Madansky, A. (1959). Bounds on the expectation of a convex function of a multivariate random variable. Ann. Math. Stat. 30, 743–746.

[31] Moriguti, S. (1952). A lower bound for a probability moment of any absolutely continuous distribution with finite variance. Ann. Math. Stat. 23(2), 286–289.

[32] Pinelis, I. (2011). Exact lower bounds on the exponential moments of truncated random variables. J Appl. Probab. 48(2), 547–560.

[33] Prékopa, A. (1990). The discrete moment problem and linear programming, Discrete Appl. Math. 27(3), 235–254.

[34] Prelec, D. (1998). The Probability Weighting Function. Econometrica 66, 497–527.

[35] Scheutzow, M. (1985). Noise can create periodic behavior and stabilize nonlinear diffusions. Stoch. Proc. Appl. 20(2), 323–331.

[36] Schoemaker, P., and J. Hershey (1992). Utility measurement: Signal, noise, and bias. Organ. Behav. Hum. Dec. 52, 397–424.

[37] Shannon, C. (1949). Communication in the presence of noise. Proc. Institute of Radio Engineers 37(1), 10–21.

[38] Sharma, R., and. R. Bhandari (2014). On Variance Upper Bounds for Unimodal Distributions. Commun. Stat. A-Theor. 43(21), 4503–4513.

[39] Sharma, R., M. Gupta, and G. Kapoor (2010). Some better bounds on the variance with applications, J. Math. Inequal. 4(3), 355–363.

[40] Smith, J. (1971). The information capacity of amplitude- and variance constrained scalar Gaussian channels. Inform. Control 18(3), 203–219.

[41] Starmer, C., and R. Sugden (1991). Does the Random-Lottery Incentive System Elicit True Preferences? An Experimental Investigation. Am. Econ. Rev. 81, 971–78.

[42] Steingrimsson, R., and R. D. Luce (2007). Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function. J. Math. Psychol. 51, 29–44.

[43] Thaler, R., (2016). Behavioral Economics: Past, Present, and Future. Am. Econ. Rev. 106(7), 1577–1600.

[44] Tversky, A. and D. Kahneman (1992). Advances in Prospect Theory: Cumulative Representation of Uncertainty. J. Risk Uncertainty 5, 297–323.

[45] Von Neumann, J., and O. Morgenstern (1944). Theory of games and economic behavior. Princeton: Princeton University Press.

[46] Vostrikova, L. (2017). Expected utility maximization for exponential Lévy models with option and information processes. Theory Probab. Appl. 61(1), 107–128.

[47] Wolfowitz, J. (1975). Signalling over a Gaussian channel with feedback and autoregressive noise. J. Appl. Probability 12(4), 713–723.