行動経済学　第 12 巻 (2019) 37-50

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1 12 (2019) * a b 要旨キーワード : JEL Classification Numbers: B29, D90, D91 1. はじめに (Kahneman, D.) (Tversky, A.) (Thaler, R.) (prospect theory) * A , 16H02050 a kazupsy@waseda.jp b 2. 心理学と行動経済学とのかかわりの歴史 (Skinner, B. F.) 1930 (Pavlov, I. P.) (Thorndike, E.) (Hursh 1980) 2001 (Ed-

2 38 12 wards, W.) (behavioral decision theory) (Edwards 1961) (2017) 1952 (Allais, M.) (Savage, L. J.) (Coombs, C. H.) (Thrall, R. M.) (Thaler, R.) (CGSTDM) (Scott, W. D.) (Starch, D.) (Münsterburg, M.) 20 (Watson, J. B.) 1982 (International Association for Research in Economic Psychology: IAREP) (Journal of Economic Psychology) (Bettman, J. R.) (Psychonomic Society) (1988) (1997) (Slovic, P.) (Fischhoff. B.)

3 39 3. 古典的心理学と行動経済学の接点としての確率荷重関数の問題 19 (Fechner, G. T.) 1860 (psychophysical method) 18 (Bernoulli, D.) (2009) 19 (Kahneman and Tversky 1979, Tversky and Kahneman 1992) (Luce 2001, Prelec 1998, Prelec and Loewenstein 1991, Rachlin et al. 1991, Takahashi 2011, Tversky and Kahneman 1992) Prelec (1998) (Luce 2001) Compound invariance W(p) exp[ β( ln p) α ], (1) Luce (2001) Prelec (1998) Compound invariance Reduction invariance Reduction invariance Compound invariance Prelec and Loewenstein (1991) (Allais 1953) Rachlin et al. (1991) Rachlin et al. (1991) Takahashi (2011) Rachlin et al. (1991) (2017)

4 非線形期待効用理論と確率荷重関数 20 (von Neumann, J.) (Morgenstern, O.) (Allais, M.) (Ellsberg, D.) 2009, Takemura 2014 (Choquet integral) (Tversky and Kahneman 1992) (Schmeidler 1989) 1979 (Kahneman and Tversky 1979), 1992 (Tversky and Kahneman 1992) (e.g., Quiggin 1993, Starmer 2000, 1997) X Θ f: Θ X θ Θ x X f(θ) x x X f (θ 1 ) 1000 (x 1 ) (θ 2 ) 2000 (x 2 ) (Choquet 1955) (Fishburn 1988) θ i Θ θ i x i f (x i, θ i ) f 1 2 f f f(θ) 0 f (θ) f(θ) f(θ) 0 f (θ) 0 f(θ) 0 f (θ) f(θ) f(θ) 0 f (θ) 0 f (θ 1 ) 1000 f (θ 1 ) 2000 f (θ 1 ) 0 f (θ 2 ) 0 f g V(f) V(g) V(f) V(f ) V(f ), V(g) V(g ) V(g ), (2) (Savage 1954) Θ [0,1] W: 2 Θ [0,1] (W(φ) 0, W(Θ) 1) Θ A i A j A i A j W(A i ) W(A j ) 1, 3, v: X R e v(x 0 ) v(0) 0 v(x) 2x 0.8 V(f) V(f ) V(f ) V(f ) V(f ) V(f) V(f ) V(f ), (3)

5 41 V(f ) n i 0π i v(x i ), V(f ) 0 i mπ i v(x i ). (4) f (x 0, A 0 ; x 1, A 1 ;...; x n, A n ) f (x m, A m ; x m 1, A m 1 ;...; x 0, A 0 ) π 0,..., π n π m,..., π 0 π n W (A n ), π m W (A m ), (5) π i W (A i... A n ) W (A i 1... A n ), 0 i n 1, (6) π i W (A m... A i ) W (A m... A i 1 ), 1 m i 0. (7) π i x i x i π i x i x i W W π i A i i 0 π i π i i 0 π i π i n V(f) i m π i v(x i ), (8) f (x i, A i ) p(a i ) p i f (x i, p i ) π n W (p n ), π m W (p m ), (9) π i W (p i... p n ) W (p i 1... p n ), 0 i n 1, (10) π i W (p m... p i ) W (p m... p i 1 ), 1 m i 0, (11) W, W W (0) W (0) 0, W (1) W (1) 1 i 0 π i π i i 0 π i π i n V(f) i m π i v(x i ), (12) (Tversky and Kahneman 1992) x x 1,...,6 x 1000 x 1000 x f ( ) 1/6 f (0, 1/2; 2000, 1/6; 4000, 1/6; 6000, 1/6), f ( 5000, 1/6; 3000, 1/6; 1000, 1/6; 0, 1/2) f 0 1/ /6 f /6 0 1/2 V(f) V(f ) V(f ) v(2000 )[W (1/6 1/6 1/6) W (1/6 1/6)] v(4000 )[W (1/6 1/6) W (1/6)] v(6000 )[W (1/6) W (0)] v( 5000 )[W (1/6) W (0)] v( 3000 )[W (1/6 1/6) W (1/6)] v( 1000 )[W (1/6 1/6 1/6) W (1/6 1/6)] v(2000 )[W (1/2) W (1/3)] v(4000 )[W (1/3) W (1/6)] v(6000 )[W (1/6) W (0)] v( 5000 )[W (1/6) W (0)] v( 3000 )[W (1/3) W (1/6)] v( 1000 )[W (1/2) W (1/3)], (13) 2 V(f ) 1 V(f ) π 2000 w 4000 w π π v

6 V(f) W, W 2 + p W ( p) =, 1/ γ γ γ p +( 1-p) p W ( p) =, 1/ δ δ δ p +( 1-p) γ δ (15) γ 0.61 δ 0.69 δ γ 2 2 W (W ) Tversky and Kahneman (1992) α ( ) x, ( x 0 の場合 ) v x = β - λ ( - x ), and ( x < 0 の場合 ) (14) α β 0.88, λ 2.25 α β 1 λ 2 5. 遅延価値割引と確率荷重関数 (Takemura and Murakami 2016) Rachlin et al. (1986)

7 43 A V = 1 kd, (16) + V A D k Rachlin et al. (1986) W(D) ( ) A W D = 1 kd, (17) + (17) D 1 1 p 1 / p 1 (1 / p) 1 D (1 / p) 1 W(p) 1 W ( p) =, (18) 1 + k[ ( 1/ p) - 1] 2011 Takahashi (2011) 3, 4 1 W ( p) =, (19) 1 + k ( 1/ p) - 1 α { [ ]} Takahashi (2011) Takahashi (2011) Prelec (1998) Takemura and Murakami (2016) Rachlin et al. (1986) Takahashi (2011) 1 (17) D p 1 / p log p (17) (20) 1 W ( p) =, (20) 1 + klog ( 1 / p) (20) (21) 1 W ( p) =, (21) 1 - klog ( p) 3 (19) α (k 10) 4 (19) k (α 3)

8 (21) 6 (24) 1 F (D) ln(d) f (D) [1 k D] 1 W(p) {1 k [(1/p) 1]} α W(p) [1 k ln(p)] 1 W(p) [1 k ln(p)] β W(p) exp[k ln(p)] Prelec type1 F (D) [ln(d)] a f (D) exp( k D) W(p) exp[ ( ln(p)) a ] Prelec type2 W(p) exp[ k( ln(p)) a ] Tversky and Kahneman W(p) p γ /[p γ (1 p) γ ] 1/γ k 0 Tversky and Kahneman (1992) 5 A V = exp( kd ), (22) V A D k W(D) W(D) exp{ kd}, (23) W(p) exp{ [ k log(p)]} exp{k log p}, (24) k 0 k W(p) exp{ k{ log p} α }, (25) Prelec (1998) 6. 確率荷重関数と遅延時間割引関数の推定 6.1. 確率荷重関数の推定

9 45 Takemura and Murakami (2016) 刺激 174 (certainty equivalent; CE) 推定方法 Gonzalez and Wu (1999) 165 Gonzalez and Wu (1999) v(ce) w(p)v(x) [1 w(p)]v(y) 8 v(2,500), v(10,000) 11 w(0.01), w(0.50) Gonzalez and Wu (1999) p p w(p) w(p ), x x v(x) v(x ) 11 7, 実験参加者推定の結果 2 AIC (19) Prelec (25) k AIC AIC Prelec type Prelec type Tversky and Kahneman

46 12 9 AIC 1 19 AIC AIC Prelec 6.2. 遅延時間割引関数の推定 4 5 AIC 6 14 20 3 7 AIC AIC 10 AIC 4 25 Prelec type1 Prelec type2 Tversky and Kahneman -55.41 38.28 46.

85 4 f(d) exp( k D) F(D) ln(d) f(d) (1 k D) 1 f(d) (1 k D) a f(d) exp( k ln(d)) f(d) [1 k ln(d)] 1 f(d) [1 k ln(d)] β 11 k 7 Prelec F(D) [ln(d)] a f(d)

10 AIC 1 19 AIC AIC Prelec 6.2. 遅延時間割引関数の推定 4 5 AIC AIC AIC 10 AIC 4 25 Prelec type1 Prelec type2 Tversky and Kahneman f(d) exp( k D) F(D) ln(d) f(d) (1 k D) 1 f(d) (1 k D) a f(d) exp( k ln(d)) f(d) [1 k ln(d)] 1 f(d) [1 k ln(d)] β 11 k 7 Prelec F(D) [ln(d)] a f(d) exp[ k(ln(d)) a ] 5 Rachlin et al. (1991) $1,000 Green et al. (1997) Green et al. (1997) Green et al. (1997) Green et al. (1997) $100 $2,000 $25,000 $100,000 Takahashi et al. (2007) 31 1, Takahashi et al. (2008) ,000

11 47 6 AIC Prelec Rachlin et al. (1991) Green et al. (1997) Green et al. (1997) Green et al. (1997) Green et al. (1997) Takahashi et al. (2007) Takahashi et al. (2008) α 14 Rachlin et al. (1991) 13 k 7. 結論と今後の課題 15 Green et al. (1997) $100

12 Green et al. (1997) $2, Green et al. (1997) $100, Green et al. (1997) $25, Takahashi et al. (2007) Prelec PET D1 D2 D1 D2 (Takahashi et al. 2010) Prelec α PET D1 D2 D1 α D1 (18) Takahashi (2011)

13 49 引用文献 20 Takahashi et al. (2008) Prelec (Green et al. 1999, 2004) (Takemura, and Murakami 2018) (2011) Allais, M., Le comportement de l homme rationnel devant le risque: critique des postulats et axiomes de l école Américaine. Econometrica 21, Choquet, G., Theory of capacities. Annales de l Institute Fourier 5, Edwards, W., Behavioral decision theory. Annual Review of Psychology 12, Fechner, G. T., Elemente der Psychophysik. Breitkopf and Hartel, Leipzig. Fishburn, P. C., Nonlinear Preference and Utility Theory. Johns Hopkins University Press, Baltimore. Gonzalez, R. and G. Wu, On the shape of the probability weighting function. Cognitive Psychology 38, Green, L. and J. Myerson, A discounting framework for choice with delayed and probabilistic rewards. Psychological Bulletin 30, Green, L., J. Myerson, and E. McFadden, Rate of temporal discounting decreases with amount of reward. Memory and Cognition 25, Green, L., J. Myerson, and P. Ostaszewski, Amount of reward has opposite effects on the discounting of delayed and probabilistic outcomes. Journal of Experimental Psychology: Learning, Memory, and Cognition 25, Hursh, S. R., Economic concepts for the analysis of behavior. Journal of the Experimental Analysis of Behavior 34, , Kahneman, D. and A. Tversky, Prospect theory: An analysis of decision under risk. Econometrica 47, Luce, R. D., Reduction invariance and Prelec s weighting functions. Journal of Mathematical Psychology 45, Prelec, D., The probability weighting function. Econometrica 66, Prelec, D. and G. Loewenstein, Decision making over time and under uncertainty: A common approach. Management Science 37, Quiggin, J., Generalized Expected Utility Theory: The Rank Dependent Model. Kluwer Academic Publishers, Boston. Rachlin, H., A.W. Logue, J. Gibbon, and M. Frankel, Cognition and behavior in studies of choice. Psychological Review 93, Rachlin, H., A. Raineri, and D. Cross, Subjective probability and delay. Journal of the Experimental Analysis of Behavior 55,

14 50 12 Savage, L. J., The Foundations of Statistics. Wiley, New York. Schmeidler, D., Subjective probability and expected utility without additivity. Econometrica 57, WIN- PEC Working paper Series No. J1701, Starmer, C., Developments in nonexpected-utility theory: The hunt for descriptive theory of choice under risk. Journal of Economic Literature 38, Takahashi, H., H. Matsui, C. Camerer, H. Takano, F. Kodaka, T. Ideno, S. Okubo, K. Takemura, R. Arakawa, Y. Eguchi, T. Murai, Y. Okubo, M. Kato, H. Ito, and T. Suhara, Dopamine D1 receptors and nonlinear probability weighting in risky choice. Journal of Neuroscience 30, Takahashi, T., K. Ikeda, and T. Hasegawa, A hyperbolic decay of subjective probability of obtaining delayed rewards. Behavioral and Brain Functions 3. doi: / Takahashi, T., Oono, H., & Radford, M. H. B Psychophysics of time perception and intertemporal choice models. Physica A: Statistical Mechanics and its Applications 387, Takahashi, T., Psychophysics of the probability weighting function. Physica A: Statistical Mechanics and Its Applications 390, Takemura, K., Behavioral Decision Theory: Psychological and Mathematical Representations of Human Choice Behavior. Springer, New York CD Takemura, K. and H. Murakami, Probability weighting functions derived from hyperbolic time discounting: Psychophysical models and their individual level testing. Frontiers in Psychology 7. doi: /fpsyg Takemura, K., & Murakami, H A testing method of probability weighting functions from an axiomatic perspective. Frontier in Applied Mathematics and Statistics, 4, 48. doi: /fams T 1997 Thaler, R. H. and W.T. Ziemba, Parimutuel betting markets: Racetracks and lotteries. Journal of Economic Perspectives 2, Tversky, A. and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty 5, Psychology and Behavioral Economics: Focus on the Relationship between Classical Psychophysics and Probability Weighting Functions in Decision Making Kazuhisa Takemura a and Hajime Murakami b Abstract We firstly explain the relationship between psychology and behavioral economics from historical perspective. We also theroretically and emprically demonstrated that classical psychophysics which was considered to be an origin of psychology is closely related to probability weigting function in prospect theory which is typical behavioral economic theory. (Received: June 22, 2018, Accepted: September 25, 2018) Key words: Psychology, Behavioral economics, Probability weighting function, Prospect theory JEL Classification Numbers: B29, D90, D91 a Department of Psycholgy, Waseda University/Institute for Decision Research, Waseda University/Waseda Research Institute of Science and Engineering kazupsy@waseda.jp b Department of Psychology, Waseda University

時間割引：双曲割引と弱加法性

時間割引：双曲割引と弱加法性 Discussion aper No. 666 June 2006 The Institute of Social and Economic Research Osaka University 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan Time Discounting: Declining Impatience and Interval Effect

行動経済学 第 12 巻 (2019) 37-50

行動経済学　第 12 巻 (2019) 37-50