The *q*-Exponential Probability Discounting of Gain and Loss

Affiliation(s)

Department of Behavioral Science, Center for Experimental Research in Social Science, Faculty of Letters, Hokkaido University, Sapporo, Japan.

Department of Forensic Psychiatry, National Institute of Mental Health National Center of Neurology and Psychiatry, Tokyo, Japan.

Department of Behavioral Science, Center for Experimental Research in Social Science, Faculty of Letters, Hokkaido University, Sapporo, Japan.

Department of Forensic Psychiatry, National Institute of Mental Health National Center of Neurology and Psychiatry, Tokyo, Japan.

ABSTRACT

Probability discounting is defined as the devaluation of outcomes as the probability of receiving or paying those decreases. A*q*-exponential probability discounting model based on Tsallis’ statistics has been proposed in econophysics (Takahashi, 2007, Physica A). We examined (a) fitness of the models to behavioral data of probability discounting of both gain and loss; and (b) relationships between parameters in the *q*-exponential probability discounting model across gain and loss. Our results demonstrated that, for both gain and loss, the *q*-exponential model better fits the behavioral data than exponential and hyperbolic functions, and there is the sign effect in *q*-exponential probability discounting. Relationships between Kahneman-Tversky’s prospect theory in behavioral economics and the *q*-exponential probability discounting are high-lightened.

Probability discounting is defined as the devaluation of outcomes as the probability of receiving or paying those decreases. A

Cite this paper

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. doi: 10.4236/am.2013.46120.

T. Takahashi, R. Han, H. Nishinaka, T. Makino and H. Fukui, "The

References

[1] J. von Neumann and O. Morgenstern, “Theory of Games and Economic Behavior,” 2nd Edition, Princeton University Press, Princeton, 1947.

[2] D. Kahneman and A. Tversky, “Prospect Theory: An Analysis of Decisions under Risk,” Econometrica, Vol. 47, No. 2, 1979, pp. 263-291. doi:10.2307/1914185

[3] A. Tversky and D. Kahneman, “Advances in Prospect theory: Cumulative Representation of Uncertainty,” Journal of Risk and Uncertainty, Vol. 5, No. 4, 1992, pp. 297-323. doi:10.1007/BF00122574

[4] M. P. Paulus and L. R. Frank, “Anterior Cingulate Activity Modulates Nonlinear Decision Weight Function of Uncertain Prospects,” Neuroimage, Vol. 30, No. 2, 2006, pp. 668-677. doi:10.1016/j.neuroimage.2005.09.061

[5] J. Peters and C. Büchel, “Overlapping and Distinct Neural Systems Code for Subjective Value during Intertemporal and Risky Decision Making,” The Journal of Neuroscience, Vol. 29, No. 50, 2009, pp. 15727-15734. doi:10.1523/JNEUROSCI.3489-09.2009

[6] T. Takahashi, H. Oono and M. H. B. Radford, “Comparison of Probabilistic Choice Models in Humans,” Behavioral and Brain Functions, Vol. 3, No. 20, 2007.

[7] T. Takahashi, K. Ikeda and T. Hasegawa, “A Hyperbolic Decay of Subjective Probability of Obtaining Delayed Rewards,” Behavioral and Brain Functions, Vol. 3, No. 52, 2007.

[8] C. Anteneodo, C. Tsallis and A. S. Martinez, “Risk Aversion in Economic Transactions,” Europhysics Letters, Vol. 59, No. 5, 2002, pp. 635-641. doi:10.1209/epl/i2002-00172-5

[9] T. Takahashi, “A Probabilistic Choice Model Based on Tsallis’ Statistics,” Physica A: Statistical Mechanics and its Applications, Vol. 386, No. 1, 2007, pp. 335-338.

[10] T. Takahashi, “Tsallis’ Non-Extensive Free Energy as a Subjective Value of an Uncertain Reward,” Physica A: Statistical Mechanics and Its Applications, Vol. 388, No. 5, 2009, pp. 715-719.

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

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

[13] 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.

[14] B. J. Weber and S. A. Huettel, “The Neural Substrates of Probabilistic and Intertemporal Decision Making,” Brain Research, Vol. 1234, 2008, pp. 104-115. doi:10.1016/j.brainres.2008.07.105

[15] P. Samuelson, “A Note on Measurement of Utility,” The Review of Economic Studies,” Vol. 4, No. 2, 1937, pp. 155-161. doi:10.2307/2967612

[16] J. E. Mazur, “An Adjustment Procedure for Studying Delayed Reinforcement,” In: J. E. Mazur, J. A. Nevin and H. Rachlin, Eds., The Effect of Delay and Intervening Events on Reinforcement Value, Michael L. Commons, Hillsdale, 1987.

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

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

[19] T. Takahashi, “Theoretical Frameworks for Neuro-Economics of Intertemporal Choice,” Journal of Neuroscience, Psychology, and Economics, Vol. 2, No. 2, 2009, pp. 75-90. doi:10.1037/a0015463

[20] Y. Ohmura, T. Takahashi and N. Kitamura, “Discounting Delayed and Probabilistic Monetary Gains and Losses by Smokers of Cigarettes,” Psychopharmacology (Berlin), Vol. 182, No. 4, 2005, pp. 508-515.

[21] T. Takahashi, “Loss of Self-Control in Intertemporal Choice May Be Attributable to Logarithmic Time-Perception,” Medical Hypotheses, Vol. 65, 2005, pp. 691-693. doi:10.1016/j.mehy.2005.04.040

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

[1] J. von Neumann and O. Morgenstern, “Theory of Games and Economic Behavior,” 2nd Edition, Princeton University Press, Princeton, 1947.

[2] D. Kahneman and A. Tversky, “Prospect Theory: An Analysis of Decisions under Risk,” Econometrica, Vol. 47, No. 2, 1979, pp. 263-291. doi:10.2307/1914185

[3] A. Tversky and D. Kahneman, “Advances in Prospect theory: Cumulative Representation of Uncertainty,” Journal of Risk and Uncertainty, Vol. 5, No. 4, 1992, pp. 297-323. doi:10.1007/BF00122574

[4] M. P. Paulus and L. R. Frank, “Anterior Cingulate Activity Modulates Nonlinear Decision Weight Function of Uncertain Prospects,” Neuroimage, Vol. 30, No. 2, 2006, pp. 668-677. doi:10.1016/j.neuroimage.2005.09.061

[5] J. Peters and C. Büchel, “Overlapping and Distinct Neural Systems Code for Subjective Value during Intertemporal and Risky Decision Making,” The Journal of Neuroscience, Vol. 29, No. 50, 2009, pp. 15727-15734. doi:10.1523/JNEUROSCI.3489-09.2009

[6] T. Takahashi, H. Oono and M. H. B. Radford, “Comparison of Probabilistic Choice Models in Humans,” Behavioral and Brain Functions, Vol. 3, No. 20, 2007.

[7] T. Takahashi, K. Ikeda and T. Hasegawa, “A Hyperbolic Decay of Subjective Probability of Obtaining Delayed Rewards,” Behavioral and Brain Functions, Vol. 3, No. 52, 2007.

[8] C. Anteneodo, C. Tsallis and A. S. Martinez, “Risk Aversion in Economic Transactions,” Europhysics Letters, Vol. 59, No. 5, 2002, pp. 635-641. doi:10.1209/epl/i2002-00172-5

[9] T. Takahashi, “A Probabilistic Choice Model Based on Tsallis’ Statistics,” Physica A: Statistical Mechanics and its Applications, Vol. 386, No. 1, 2007, pp. 335-338.

[10] T. Takahashi, “Tsallis’ Non-Extensive Free Energy as a Subjective Value of an Uncertain Reward,” Physica A: Statistical Mechanics and Its Applications, Vol. 388, No. 5, 2009, pp. 715-719.

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

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

[13] 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.

[14] B. J. Weber and S. A. Huettel, “The Neural Substrates of Probabilistic and Intertemporal Decision Making,” Brain Research, Vol. 1234, 2008, pp. 104-115. doi:10.1016/j.brainres.2008.07.105

[15] P. Samuelson, “A Note on Measurement of Utility,” The Review of Economic Studies,” Vol. 4, No. 2, 1937, pp. 155-161. doi:10.2307/2967612

[16] J. E. Mazur, “An Adjustment Procedure for Studying Delayed Reinforcement,” In: J. E. Mazur, J. A. Nevin and H. Rachlin, Eds., The Effect of Delay and Intervening Events on Reinforcement Value, Michael L. Commons, Hillsdale, 1987.

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

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

[19] T. Takahashi, “Theoretical Frameworks for Neuro-Economics of Intertemporal Choice,” Journal of Neuroscience, Psychology, and Economics, Vol. 2, No. 2, 2009, pp. 75-90. doi:10.1037/a0015463

[20] Y. Ohmura, T. Takahashi and N. Kitamura, “Discounting Delayed and Probabilistic Monetary Gains and Losses by Smokers of Cigarettes,” Psychopharmacology (Berlin), Vol. 182, No. 4, 2005, pp. 508-515.

[21] T. Takahashi, “Loss of Self-Control in Intertemporal Choice May Be Attributable to Logarithmic Time-Perception,” Medical Hypotheses, Vol. 65, 2005, pp. 691-693. doi:10.1016/j.mehy.2005.04.040

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