Mathematical Neurolaw of Crime and Punishment: The *q*-Exponential Punishment Function

ABSTRACT

Whether people tend to punish criminals in a socially-optimal manner (i.e., hyperbolic punishment) or not is unknown. By adopting mathematical models of probabilistic punishment behavior (i.e., exponential, hyperbolic, and *q*-exponential probability discounting model based on Tsallis thermodynamics and neuroeconomics, Takahashi, 2007, Physica A; Takahashi et al., 2012, Applied Mathematics), we examined 1) fitness of the models to behavioral data of uncertain punishment, and 2) deviation from the socially optimal hyperbolic punishment function. Our results demonstrated that, the *q*-exponential punishment function best fits the behavioral data, and people overweigh the severity of punishment at small punishing probabilities and underweigh the severity of punishment at large punishing probabilities. In other words, people tend to punish crimes too severely and mildly with high and low arrest rate (e.g., homicide vs. excess of speed limit), respectively. Implications for neuroeconomics and neurolaw of crime and punishment (Takahashi, 2012, NeuroEndocrinology Letters) are discussed.

KEYWORDS

Neurolaw; Neuroeconomics; Econophysics; Tsallis’ Statistics; Hyperbolic Punishment Function

Neurolaw; Neuroeconomics; Econophysics; Tsallis’ Statistics; Hyperbolic Punishment Function

Cite this paper

Yokoyama, T. and Takahashi, T. (2013) Mathematical Neurolaw of Crime and Punishment: The*q*-Exponential Punishment Function. *Applied Mathematics*, **4**, 1371-1375. doi: 10.4236/am.2013.410185.

Yokoyama, T. and Takahashi, T. (2013) Mathematical Neurolaw of Crime and Punishment: The

References

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[16] T. Takahashi, T. Hadzibeganovic, S. A. Cannas, T. Ma kino, H. Fukui and S. Kitayama, “Cultural Neuroeco nomics of Intertemporal Choice,” Neuroendocrinology Letters, Vol. 30, No. 2, 2009, pp. 185-191.

[17] R. Han and T. Takahashi, “Psychophysics of Valuation and Time Perception in Temporal Discounting of Gain and Loss,” Physica A: Statistical Mechanics and Its Ap plications, Vol. 391, No. 24, 2012, pp. 6568-6576.

[18] C. Tsallis, “Nonadditive Entropy Sq and Nonextensive Statistical Mechanics: Applications in Geophysics and Elsewhere,” Acta Geophysica, Vol. 60, No. 3, 2012, pp. 502-525. http://dx.doi.org/10.2478/s11600-012-0005-0

[19] T. Takahashi, H. Oono and M. H. Radford, “Comparison of Probabilistic Choice Models in Humans,” Behavioral and Brain Functions, Vol. 3, No. 1, 2007, p. 20. http://dx.doi.org/10.1186/1744-9081-3-20

[20] T. Takahashi, K. Ikeda and T. Hasegawa, “A Hyperbolic Decay of Subjective Probability of Obtaining Delayed Rewards,” Behavior and Brain Functions, Vol. 3, 2007, p. 52. http://dx.doi.org/10.1186/1744-9081-3-52

[21] J. Bentham, “An Introduction to the Principles of Morals and Legislation,” In: The Utilitarians, Anchor Books, Garden City, 1973.

[22] T. Takahashi, “Loss of Self-Control in Intertemporal Choice May Be Attributable to Logarithmic Time-Per ception,” Medical Hypotheses, Vol. 65, No. 4, 2005, pp. 691-693. http://dx.doi.org/10.1016/j.mehy

.2005.04.040

[23] T. Takahashi, R. Han and F. Nakamura, “Time Discount ing: Psychophysics of Intertemporal and Probabilistic Choices,” Journal of Behavioral Economics and Finance, Vol. 5, 2012, pp. 10-14.

[1] G. S. Becker, “Crime and Punishment: An Economic Approach,” Journal of Political Economy, Vol. 76, No. 2, 1968, pp. 169-217. http://dx.doi.org/10.1086/259394

[2] N. Garoupa, “Optimal Magnitude and Probability of Fines,” European Economic Review, Vol. 45, No. 9, 2001, pp. 1765-1771. http://dx.doi.org/10.1016/S0014-2921(00)00084-2

[3] A. Al-Nowaihi and S. Dhami, “Hyperbolic Punishment Functions,” Review of Law and Economics, Berkeley Electronic Press Journals, Vol. 8, No. 3, 2012, pp. 759-787.

[4] K. Prehn, F. Schlagenhauf, L. Schulze, C. Berger, K. Vohs, M. Fleischer, K. Hauenstein, P. Keiper, G. Domes and S. C. Herpertz, “Neural Correlates of Risk Taking in Violent Criminal Offenders Characterized by Emotional Hypo and Hyper-Reactivity,” Social Neuroscience, Vol. 8, No. 2, 2013, pp. 136-147. http://dx.doi.org/10.1080/17470919.2012.686923

[5] T. Takahashi, “Molecular Neuroeconomics of Crime and Punishment: Implications for Neurolaw,” Neuro Endo crinology Letters, Vol. 33, No. 7, 2012, pp. 667-673.

[6] L. G. A. Alves, H. V. Ribeiro and R. S. Mendes, “Scaling Laws in the Dynamics of Crime Growth Rate,” Physica A: Statistical Mechanics and Its Applications, Vol. 392, No. 11, 2013, pp. 2672-2679.

[7] T. Pachur, Y. Hanoch and M. Gummerum, “Prospects behind Bars: Analyzing Decisions under Risk in a Prison Population,” Psychonomic Bulletin and Review, Vol. 17, No. 5, 2010, pp. 630-636. http://dx.doi.org/10.3758/PBR.17.5.630

[8] T. Takahashi, “A Comparison of Intertemporal Choices for Oneself versus Someone Else Based on Tsallis’ Sta tistics,” Physica A: Statistical Mechanics and Its Applica tions, Vol. 385, No. 2, 2007, pp. 637-644.

[9] T. Takahashi, R. Han, H. Nishinaka, T. Makino and H. Fukui, “The q-Exponential Probability Discounting of Gain and Loss,” Applied Mathematics, Vol. 4, No. 6, 2013, pp. 876-881.

[10] C. Tsallis, C. Anteneodo, L. Borland and R. Osorio, “Nonextensive Statistical Mechanics and Economics,” Physica A: Statistical Mechanics and Its Applications, Vol. 324, No. 1-2, 2003, pp. 89-100.

[11] H. Rachlin, A. Raineri and D. Cross, “Subjective Prob ability and Delay,” Journal of Experimental: Analysis of Behavior, Vol. 55, No. 2, 1991, pp. 233-244. http://dx.doi.org/10.1901/jeab.1991.55-233

[12] T. Takahashi, “Psychophysics of the Probability Weight ing Function,” Physica A: Statistical Mechanics and Its Applications, Vol. 390, No. 5, 2011, pp. 902-905.

[13] D. O. Cajueiro, “A Note on the Relevance of the q-Ex ponential Function in the Context of Intertemporal Choices,” Physica A: Statistical Mechanics and Its Ap plications, Vol. 364, 2006, pp. 385-388.

[14] T. Takahashi, H. Oono and M. H. B. Radford, “Psycho physics of Time Perception and Intertemporal Choice Models,” Physica A: Statistical Mechanics and Its Appli cations, Vol. 387, No. 8-9, 2008, pp. 2066-2074.

[15] T. Takahashi, “Theoretical Frameworks for Neuroeco nomics of Intertemporal Choice,” Journal of Neurosci ence, Psychology, and Economics, Vol. 2, No. 2, 2009, pp. 75-90. http://dx.doi.org/10.1037/

a0015463

[16] T. Takahashi, T. Hadzibeganovic, S. A. Cannas, T. Ma kino, H. Fukui and S. Kitayama, “Cultural Neuroeco nomics of Intertemporal Choice,” Neuroendocrinology Letters, Vol. 30, No. 2, 2009, pp. 185-191.

[17] R. Han and T. Takahashi, “Psychophysics of Valuation and Time Perception in Temporal Discounting of Gain and Loss,” Physica A: Statistical Mechanics and Its Ap plications, Vol. 391, No. 24, 2012, pp. 6568-6576.

[18] C. Tsallis, “Nonadditive Entropy Sq and Nonextensive Statistical Mechanics: Applications in Geophysics and Elsewhere,” Acta Geophysica, Vol. 60, No. 3, 2012, pp. 502-525. http://dx.doi.org/10.2478/s11600-012-0005-0

[19] T. Takahashi, H. Oono and M. H. Radford, “Comparison of Probabilistic Choice Models in Humans,” Behavioral and Brain Functions, Vol. 3, No. 1, 2007, p. 20. http://dx.doi.org/10.1186/1744-9081-3-20

[20] T. Takahashi, K. Ikeda and T. Hasegawa, “A Hyperbolic Decay of Subjective Probability of Obtaining Delayed Rewards,” Behavior and Brain Functions, Vol. 3, 2007, p. 52. http://dx.doi.org/10.1186/1744-9081-3-52

[21] J. Bentham, “An Introduction to the Principles of Morals and Legislation,” In: The Utilitarians, Anchor Books, Garden City, 1973.

[22] T. Takahashi, “Loss of Self-Control in Intertemporal Choice May Be Attributable to Logarithmic Time-Per ception,” Medical Hypotheses, Vol. 65, No. 4, 2005, pp. 691-693. http://dx.doi.org/10.1016/j.mehy

.2005.04.040

[23] T. Takahashi, R. Han and F. Nakamura, “Time Discount ing: Psychophysics of Intertemporal and Probabilistic Choices,” Journal of Behavioral Economics and Finance, Vol. 5, 2012, pp. 10-14.