Inequalities and zones. New mathematical results for behavioral and social sciences

Harin, Alexander (2018): Inequalities and zones. New mathematical results for behavioral and social sciences.

Abstract

A theorem, mathematical method and model are introduced in the present article. Inequalities, allowed and forbidden zones, their relations, consequences, and applications are considered for the expectations of random variables. The method and model are based on the inequalities and zones of the theorem. The article is motivated by the need for theoretical support for the practical analysis performed for the purposes of behavioral economics.

• Item Type: MPRA Paper
• Language: English
• Keywords: probability; variance; noise; bias; measurement; utility theory; 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] Abramovich S., and L-E. Persson, Extensions and refinements of Fejer and Hermite–Hadamard type inequalities, Math. Inequal. Appl. 21, 3 (2018), 759–772.

[2] 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.

[3] 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.

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

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

[6] Awan, M., M. Noor, M. Mihai and K. Noor, Inequalities via harmonic convex functions: conformable fractional calculus approach, J. Math. Inequal. 12, 1 (2018), 143–153.

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

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

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

[10] Burqan, A. (2016). Improved Cauchy-Schwarz norm inequality for operators, J. Math. Inequal. 10, 1, 205–211.

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

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

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

[14] Chen, F. (2016). Extensions of the Hermite–Hadamard inequality for convex functions via fractional integrals, J. Math. Inequal. 10, 1, 75–81.

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

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

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

[18] Daryaei, M., and A. Doagooei (2018). Topical functions: Hermite–Hadamard type inequalities and Kantorovich duality, Math. Inequal. Appl. 21, 3, 779–793.

[19] 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.

[20] Dragomir, S. (2017). Inequalities of Hermite–Hadamard type for functions of selfadjoint operators and matrices, J. Math. Inequal. 11, 1, 241–259.

[21] 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.

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

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

[24] Gordji, M., M. Delavar, and D. Sen (2016). On j – convex functions, J. Math. Inequal. 10, 1 (2016), 173–183.

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

[26] 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.

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

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

[29] 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.

[30] 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.

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

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

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

[34] 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.

[35] Pavi´c, Z. (2017). The Jensen and Hermite–Hadamard inequality on the triangle, J. Math. Inequal. 11, 4, 1099–1112.

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

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

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

[39] Qaisar, S., F. Ahmad, S. Dragomir and A. Doagooei (2018). New Hermite–Hadamard inequalities via fractional integrals, whose absolute values of second derivatives is p-convex, J. Math. Inequal. 12, 3, 655–664.

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

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

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

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

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

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

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

[47] 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.

[48] Sun, W., and Q. Liu (2017). New Hermite–Hadamard type inequalities for (α, m) convex functions and applications to special means, J. Math. Inequal. 11, 2, 383–397.

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

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

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

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

[53] Yildiz, Ç. (2016). New inequalities of the Hermite–Hadamard type for n-time differentiable functions which are quasiconvex, J. Math. Inequal. 10, 3, 703–711.