12 (2019) 37 50 37 * a b 要旨 2018 6 22 2018 9 25 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 24243061, 16H02050 a e-mail: 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.) 1970 1980 (Hursh 1980) 2001 (Ed-
38 12 wards, W.) 1948 1961 (behavioral decision theory) (Edwards 1961) (2017) 1952 (Allais, M.) (Savage, L. J.) (Coombs, C. H.) (Thrall, R. M.) 1950 1950 (Thaler, R.) 1988 1980 1992 (CGSTDM) 20 20 (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) 1950 1960 1960 1970 1970 1975 1980 (Slovic, P.) (Fischhoff. B.) 1970 2018
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)
40 12 2016 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 ) 1000 2000 4000 (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, 5 0.1 0.4 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)
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 ( 5000 3000 1000 2000 4000 6000 ) 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 1/2 2000 4000 6000 1/6 f 5000 3000 1000 1/6 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 ) 1 2000 π 2000 w 4000 w π π v
42 12 1 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) 25 150 25 50 75 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)
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)
44 12 5 (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 1 1 6 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 1 2016
45 Takemura and Murakami (2016) 6.1.1. 刺激 174 (certainty equivalent; CE) 174 165 11 15 165 165 9 6.1.2. 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, 8 6.1.3. 実験参加者 50 35 19 24 0.70 4 46 6.1.4. 推定の結果 2 AIC (19) Prelec (25) 9 11 12 13 11 k 3 46 7 46 8 46 2 AIC AIC 1 2 3 4 5 6 7 16 15 13 2 0 0 0 6 3 0 11 8 18 0 8 12 11 7 7 0 1 0 1 1 0 11 4 29 Prelec type1 4 1 3 9 9 13 7 Prelec type2 8 11 16 11 0 0 0 Tversky and Kahneman 4 3 2 6 11 11 9
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 46.82 15.89 41.44 44.24 40.85 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) exp[ k(ln(d)) a ] 5 Rachlin et al. (1991) 40 1 6 1 5 10 25 50 $1,000 Green et al. (1997) Green et al. (1997) Green et al. (1997) Green et al. (1997) 24 3 6 1 3 5 10 20 $100 $2,000 $25,000 $100,000 Takahashi et al. (2007) 31 1,000 1 2 1 6 1 5 25 Takahashi et al. (2008) 26 100,000
47 6 AIC Prelec Rachlin et al. (1991) 13.39-17.47 15.93 1.87 0.58 0.20 13.59 Green et al. (1997) 4.77 11.62-16.78 11.37 7.12 9.17 15.49 Green et al. (1997) 19.23 28.75-28.97 3.16 1.32 1.08 23.25 Green et al. (1997) 22.98-25.14 24.31 4.62 3.32 2.57 21.20 Green et al. (1997) 20.50-33.00 31.23 4.30 2.78 2.23 31.03 Takahashi et al. (2007) 4.96 10.95-21.62 5.24 3.73 3.15 23.47 Takahashi et al. (2008) 3.36 8.35-48.39 6.52 4.94 4.43 36.26 12 α 14 Rachlin et al. (1991) 13 k 7. 結論と今後の課題 15 Green et al. (1997) $100
48 12 16 Green et al. (1997) $2,000 18 Green et al. (1997) $100,000 17 Green et al. (1997) $25,000 19 Takahashi et al. (2007) Prelec PET D1 D2 D1 D2 (Takahashi et al. 2010) Prelec α 0.5 0.6 PET D1 D2 D1 α D1 (18) Takahashi (2011)
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