x78x3c von Neumann–Morgensternx48x79x7ax7bx7cx7d
x57x58x7fx59x20 x47x20 von Neumann–Morgenstern x55x13x6fx4ax7ax32x20x4x5cx15x50xdx29x52
x2bx7dx43x5ax5bx5cx20x78x79xcx30x25x73x11x5cx20x29 x47x20Allaisx20x4fx34Kahneman–Tverskyx20x30
x31x29x21x52Arrow–Prattx3dx30x22x23x58x3bx20x43x46x39x35x58x59x25x46x29x52x25x5ex20x55x13x6fx4ax7ax32
x56x4f Arrow–Prattx3dx30x22x23x58x3bx29x5cx59x21x52x58x53x1ax23x20xexfx29
x5ax2ex51x5bx5cx5dx20 x37x3ax11x32x29x5cx33xex41x4fx29x28x77x33x29
x66x5ex20
? 6.1 x10x6ex24x30x25x20x4fx13x20x4ex4fx29
? 6.2 von Neumann–Morgensternx55x13x6fx4ax7ax32x20x4x5cx15x50xdx29
? 6.3 Allaisx20x4fx34Kahneman–Tverskyx20x30x31x29
? 6.4 Arrow–Prattx3dx30x22x23x58x3bx29
? 6.5x0x1x1cx18x55x13x6fx4ax7ax32x29
? 6.6 x58x53x1ax23x20xexfx29
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x4fx54 Markowitz x2cx68x14x25x7x8x69x63x3x1ex6fx1ex37x62x4ax55x56x5bx11x12x70x1x1c
x1dx62x4ax55xex54x74x64x2ex10x54x60x1ax78 Ross x54 APT x69x55x5ax2ex1fx2x49x25x70x69
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x24 Hilbert x72x76x68x4fx31x4dx48x51x6cx0x1x10x50x62xbx21x54x62x4ax51x1cx1dx62x4ax55
xex41x7dx48x33x34x33x63x2 Markowitz x69x63x58x2x1ex6fx1ex37x62x4ax55x56x5bx73x2ex1f
x2x28x29x13x1ax7bx14x54x2ex10x54x60x1ax40x26x78 Markowitz x69x63x6ax27x67x2x5dx2cx68
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134
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x51x3ax67x4fx78
x5x75x48x5x3ex54x3dx3ex57x58x4fx21x68x19x54x21x51x31x63x5bx57x3dx3ex42x43x2cx54x57
x58x2fx30x31x54x4fx1x33x3x78x11x12x54x76x77x0x1x45x46x54x62x4ax2x26x66x57x58x2fx30x31
x38x39x74x36x54x15x4ex3cx21x48x7ax13x49x42x43x44x4bx10x4bx1ex65x4ax58x4ax48x6fx2x77x10x77
x1ex4ax4bx21x1ax57x15x13x48x25xbx57x58x2fx30x31x54x21x1fx4fx1x49x42x43x44x54x19x72x73x58
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x12x54x57x58x4fx59x3ex7fx19x2ex28x38x11x78x56x4bx3ax3ex6fx7cx39x72x73x62x4ax63x54x3ax3bx27
x65x63x51x3x5fx60x48x17xax54x2x75x5dx8x48xbx4bx68x19x54x70xbx31x63x39x72x73x62x4ax54
x3ax3bx27x3cx1cx7ax41x7dx78xfx2x33xbx6ex26x4x55x56x51x50x51x2ex5fx71x60x17x2ax48x39x58x48
x57x17x70x49x8x61x70x64x20x54x63x2cx69x55x48x2ex31x70x25x2x1x61x4ex70x57x19x68x17x10x49
x54x57x58x79x7ax48x6fx31x70x32x33x1ex1ex4dx61x1cx1dx62x4ax55xexdx40x2x26x37x34x27x44x62
x54x78x1x7ex48x3ax3ex6cx79x75x65x63x57x58x2fx30x31x49x2ex3cx62x70x71x56x54x44xex69x63x78
6.1 x49x1cx25x27x54x2ex3cx62x70x71x56x54x44xex5fx60x57x25x27x63x25x23x4fx4ex54x62
x63x70x26x6x31x78x17x18x48x25x72x5fx60x4ex2ex2x3bxdx4bx54x40x26x20x8x54x78x47x7ax2xd
x4bx5fx60x48x67x2ex47x75x3ax3bx65x68x75x72x54x25x22x48x58x73x47x75x47x48x1bx2cx7dx54x22x3ax78
xdx4bx7ex7fx2x23x24x2ex73x45x1x54x4bx23x1ax1bx78x74x64x2bx3dx7ax1x28x29x8x1ax53x7dx3e
x41x7dx40x41x28x29x2ax6cx58x17x33x6cx3x72x6(x68x4fx3ex2b)x3x69x7dx9x25x22x78x58x3ex2bx20
x21xfx36x7ax1ex67x41x1x23x34xdx4bx22x3ax54x5x1fx5dx38x78x7dx3ex0x1x48x2x35x6x68x46
x4bx70x43xdx4bx44xex48x49x23x25x54x74x75x56x48x23x7ex1fx6bx12x10x10x1bx2cx7dx27x70x54x3e
x2bx20x21x27x72x73x78x31x32x3ex2bx20x21x1x3bx48x33x34x10x66x46x4bx31x27x70x67x2x51x3cx54x48
x7bx1x25x79x7ax51x2ex79x41x42x3ax59xdxcx3ax48xfx2ex73xdx56x12x48x17x36x72x6x53x1x3bx48
x4x51x3bx54x20x21x78x3ex2bx20x21x1x50x54x74x75x68x28x7bx7ax78x58x3ex2bx20x21x1x7dx48xe
x2x79x41x42x70x43xdx6ex36xdx4bxfx78x25x26x48x3ex2bx20x21x54x4ex16x2xdx69x1dxdx24xfx3b
x27x10xdx4bx4ex4fx44xex54x74x75x49x5ex48x6fx31x70x48x3ex2bx20x21x23x24x3bx27x49xdx4bxdx62
x4axfx78
x7ax58x48x25x26x54x44xex72x73xfx77x67x11x60x34x78x25x70x60x34x2x26x4ax4bx68x4fx59
Nicolaus Bernoulli (1687–1759) x49x1ex69x62x54x25x26x3x25x27x63x54x6bx40x7dx75x70
Jacob Bernoulli (1654–1705)x54x28x52x18x661713x46x24x62x54x27x28x3x55x56x68x4fx59
Pierre R′emond de Monmort (1678–1719)x20x8x54x78x10x46x48x25x70x5fx60x26Mon-
mortx49x62x54x52x2ex2cx3bx26x5fx60x78 25x46x24x13x48x5fx60x11Nicolaus Bernoullix54
x29x2ax3x17x18x54x3dx17x6bx2bx4ex4fx56x56x4bDaniel Bernoulli (1700–1782)x661738
x46x1ex44x78x58x25x2bx5fx60x13x27x5ax24xdx3dx17x6bx2bx67x63xfx58x52xcx78
x21x22x3dx17x68x2bx67x63x44x58x54x2x70x43x2cx59x51x3bx7ax4bx54xdx4bx78x61xdx65 1
x6dx2cx10x48xdx24x23x682x38; x651x6dx36x2cx10x48x652x6dx2cx10x48xdx24x23x684x38x57
135
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n ? 1 x6dx5bx36x2cx10x48x65 n x6dx2cx10x48xdx24x23x68 2n x38x78x1ax49x75x5fx48x1x53x70
x2bxdx24x51xbx46x0x25x26x54xdx4bx48x62x7x8x79x66x2ex5fx5bx78x25x53x25x43xdx4bx41x1
xdx6ex36xdx4bxf?
x3ax3ex6cx79x61xdx59x51x6fx50x2x33x5fx60x54x48x4x0x1x46xdx31x59x6dx2cx2cx54x25
x27x21/2. x25x26x48x62x651x6dx2cx2cx54x25x27x21/2x48x651x6dx36x2cx2cx3x652x6dx78
x2cx2cx54x25x27x11/4x48x31x7ex9x9x48x70x71x1x2axbn?1x6dx36x2cx2c,x65nx6dx78x2c
x2cx54x25x27x21=2n. x7bx7ex48x46xdx31x23x25x3ax54x5bx54x68x4fx3ex2bx7x8x1
2￠ 12 +22 ￠ 122 +23 ￠ 123 +￠￠￠+2n ￠ 12n +￠￠￠ = 1:
x25x67x2x70x48x46xdx31x33x63x66x2ex5fx5bx48x25x43xdx4bx5bx2x10x62x51x3cx54x78
x7ax58x48x31x32x51x7dx3bx75x5dx25x43xdx4bx2bx2dx71x1ax48x4x53x46xdx31x2x2bx2ex53
x15x1fx69x6ax54x1ex2ax54xdx24, x17x46x4bx25x43xdx4bx54xbx3cx54x38x4ax43x7fx71x18x48x62
x10x26x4dx2fx30x2exbx48x2excx46x0x78x66x2x67x37x36x61x70x2bxdx67x63xf: x1x37x34x70x43
x69x63x44x54xdx6ex36xdx4bxfx48x71x72x44x71x73x51x31x14x78x4dxcx79x8x15x79x54x71x4ax12x4f
x1?
x17x71x25x2bxdx67x63xfx51x43x2x70x2b(x5cx7ax54)x4ex4fx5fx60x48x58x2x70x2bx10x7dx54
x4fx1x30x29x54x0x4ax5fx60x78x7bx7ex48 D. Bernoulli x49 1738 x46x2ex8x54x1bx1cx5ax2
x43x4ex4fx54x78x62x0x1x48x7dx2ex2x16x5ex17x23x68x54x5bx54x68x4fx3ex2bx27x4fx30x54x48x58
x2x16x5ex17xdx2ax6bx3ex2bxfx27x4fx30x54x78x62x54x70x55x43x7fx38x5dx66x17x72x3ax3bx4fx2x54
xdx17x72x3ax3bx49x3fxfx61xd,x4xdx2ax6bx3ex2bxfx51x2ex57x68x3cx2ex5fx41x3bx77x48x58x57x6a
x27x51x2ex5fx5bx51x3x78x6ax27xfx18x48x49x70x77x5bx67xfx9xax48x58x6ax27x19xfx51x5bx48x75
x49xfx2ex5bx78x4dx54x0x9x79x6ax3bx78x3bx68x4fx54x11x1x27x70, xdx2ax6bx3ex2bxfx7x8x2
x3cx21x54x1fx68x49x3fx54x7dx65x68x78 D. Bernoullix21x7x62x54xdx2ax6bx3ex2bxfx7dx65x68x2
x5bx68x54x10x68x65x68x78x25x26x48x7x8x6bx12x54x2ex7x2xx54x68x4fx3ex2bx48x58x2alogx
x54x68x4fx3ex2bx48x25x18x54a > 0x24x17x3cx62x78x0x61x10x68x5x12x13x48x6ax27x54x2ax1x33
x2ax4ex54x68x4fx3ex2bx6x11x4ex43x1
1
2alog2+
1
22alog2
2 +￠￠￠+ 1
2nalog2
n +￠￠￠ = alog2X
i=1
i
2i
= 2alog2 … 1:39a:
x25x2x70x2bx51x2ax6x48 D. Bernoullix0x1x10x53x2x7dx3ex10x25x43xdx4bxcx79x3fx2bx54
x6x48x17x2cax23x3bx71x72x50x64x27x3cx621x78x13x27x58x3bx51x7dx2ex1fx25x72x50x64x78x79x7a
x10x2x7bx7dx58x13x54x48xfx3cx71x5ax51x70x62x54x12x6bx4x62x1d.
1xfx11x20x2dx52x47x3bxax3ex15x20x2dx52x5x7cx70x2dx2ax62x57x53Karl Mengerx20x2dx52x29 D. Bernoulli
x4x4dx20x2dx52x37x38x50x6cx29x25x3fx40x20x1fx70x48x42x62x7cx7a1954x2x21x21x26x6dx24x29x20D. Bernoulli
x201738x2x20x4fx2e[3]x29
136
D. Bernoullix54x1cx6cx79x7ax23x13x2ex4ex70x57x65x63x48xfx2xfx23x25x51x7dx4dx0
x1, x1fxcx5dx25x2bxdx67x63xfx70x68x25x26x27x4ex48x25x26x4fxdx32x78x73x75x5dx25x2bx5fx60
x3bx57x71x72x61x4cx14x27x48x67x2ex4dx8x1axdx67x63xfx78x62x3ex4dx70x49x1ax71x2cx48x25x43xd
x4bx2x2ex23x25x33x2ax10x4ex4fx56x12x54x48x5x75x49x51x2ax6dx2cx31x18x78x58x25x2bx51x2ax6d
x68x2ex23x25xfx4ex78x45x31x48xdxdx4bx54x6dx68x110x6dx48x25x18x48x46xdx31x2cx10x70x6d
x23x25x3ax54x5bx54x68x4fx3ex2bx1
2￠ 12 +22 ￠ 122 +23 ￠ 123 +￠￠￠+210 ￠ 1210 = 10
x25x18x61xd10x6dx24x13x13x2cx2cx48x6fxdx31x67x2ex13x2bx5bx49x46xdx31x78x25x2bx68x79x2e
xfx4ex48x6fx2dx49x2ex47x48x2ax6bx5fx60x18x48xfx2ex7dx5bx4dxcx79x70x76x57x7bx1x46xdx31
x15x2ex2210x38x48x58x31x32x2cx68x28x48x15x2exex23x25x3a 210 = 1024x38x48x21x24x25x2
x70x2bx69x6ax2ex4ex54x3ax5bx29x4dx78x13x4ex70x57x70x48x26x66x6x4dx44x54x5xaxbx2x70x2b
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x61x3dx17x68x2bx67x63x54x71x60x78x2bx71x44x48x10x51x36x51x1bx1cx46xdx31x1x37x34xcx79x3fx2b
10x38x6f43x6bx38x27x46xdx48x73x2x7bx110x38x6f43x6bx38x10xfx2ex7dx27x70x2ex12x2
x2bx4ex68xax78x31x32x4cx5bx44x54x5xaxbx4x6bx53x68x25x43xdx4bx51x23x25x2cx441000000
x6dx48x33x34x58x51x2ex5fx7dxcx79x2bx441000000x6bx38x27x46x0x25x43xdx4bx75?
D. Bernoullix54x1cx6cxex2x22x23x54x48x7bx1x3bx2x62x6cx79x68x3cx5dx8x48x7dx3e
x49x2ex3cx62x54x70x71x56x70x71x51x2ex24x38x39x18x38x3cx21x54x15x4ex68x4fx3ex2bx2ex1xax38,
x58x2x4dx51xdx21x71x54xf xdx2ax6bx3ex2bxfx78xfx2x48 D. Bernoulli x54xdx2ax6bx3ex2bxf
x3dx3ex27x51x2ex1x7dx3ex21x69x1ex78x70x18x19x17x72x3ax3bx4fx2x54x6bx40x7dCarl Menger
(1840–1921) x54x37x14x48x68x4fx59 Karl Menger (1902–1985) x5dx10x57x3ax3bx65x68
x61x4cx14x27x48x16x17x2J. von Neumann (1903–1957)x4O. Morgenstern (1902–
1976) x49x62x3ex54 1944 x46x5fx4cx54x38x52x65x10xex63x4x57x58x4fx1x6b([24]) x2cx10x10
x4ex4fx61x67x4bx54x6ex69x5cx63x29x48x78x6fx7cx11x19x68x4fx59x4x57x58x4fx59x54x45x15x78
x58x51x70x77x75x70x68x54x2x48 D. Bernoulli x54x1cx6cx51x2ex79x41x42x51x61xdx2ax6b
x3ex2bxfx24x13x48x67x34x35x61x21x51x68x4fx3ex2bx1x33x2ax4ex54x74x40x78x2bx71x44x48x4x53x0
x4ex61xdx2ax6bx3ex2bxfx48x6fx31x56x4bx75x70x54xdx3ex2bx3ax3bx65x68xfx48x72xdxdx3dx17x6bx2b
x67x63xfx54x74x40x58x4dx8x1a(x46x3cx7ax605)x78x62x54x1cx6cx73x2x70x68x48x3ax3ex2excx10
x25x64x68x4fx3ex2bx8x1ax33x2ax4ex54x74x40x7fx19x39x3ax78 1959x46Duranx4bx57x5fx60x61
x70x66x60x1x65x41x3dx56x6ax73x4x3dx17x6bx2bx67x63x6bx54x63x64[5]x48x5dx8x41x3dx56x6ax73x54
137
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x6ax73x54x4x6x75x4bx19x16x21x78
6.2 x3ax3bx65x68x10x26x2x49x62x63x44 (x2fx5ex5ex11) x0x14x3dx3ex1ax63x54x25x22x78
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x75x6ax7bx78x66x2x13x27x3ax3bx65x68x54x25x22x67x41x7dx11x7x28x25x22x21x4ex43x78xfx2von
Neumann–Morgensternx49x62x3ex54xex2cx70x48x31x32x2x49x1cx25x27x54x2ex3cx62x1dx54
x70x71x2cx48x51x0x1x7dx3ex38x39x54x2x3ax3bx65x68x54x68x4fx3ex2bx15x4ex48x33x34x49x70x62
x54x25x69x61x62x56x48x7x28x21x44x62x54x3ax3bx65x68x49x23x7bx7dx70x2bx36x77x4x7bxd(x4x48
x7bx7dx70x2bx74x7ax2ax67)x54x79x7ax56x2x1dx70x54x78x67x9x3fx70x48x49x1cx25x27x54x2ex3cx62
x1dx54x70x71x56x48x3ax3bx65x68x71x72x44x2x23x51x6cx54x48x73x2ex1fx25x2bx65x68x54x7dx77x4
x6dx6ex23x15x79x3cx62x78x67x44x51x6cx3cx3bx33x26x48x2ex10x54x7dx77x4x6dx6ex4dx41x2ex10x54
x1x55x3x3dx55x3cx3bx6bx48x58x26x7ex68x19x54x2ex10x3cx38x23x7ex1fx28x6dx54x67x12x25x7bx1f
x68x78x62x3ex54xex2cx2ex6dx20x19 D. Bernoulli x54xdx3dx17x68x2bx67x63xfx78x49x62x3ex3c
x27x48 D. Bernoullix54xdx2ax6bx3ex2bxfx17x71x67x2x3ax3bx65x68x54x3ex2bx6x78x49x2ex1cx2e
x3cx62x1dx54x70x71x56x48x65x68xx4alogxx21x44x62x54x7x28x2x70x26x54x57x39x58x15x47
x70x2bx27x2ex1x3ax3bx65x68x48x51x39x17x15x4ex6x48x21x68x19x54x57x58x2fx30x31x54x4fx1x5a
x2x70x26x54x78xfx2x49x1cx2ex3cx62x1dx54x70x71x56x48x31x32x0x1x57x58x2fx30x31x38x39x54
x2x3ax3bx65x68x54x68x4fx3ex2bx6x15x4ex48x33x34x25x78x2bx65x68x67x2ex25x5bx2ex1x3ax3bx65
x68x27x3cx7dx78x62x3ex54x25xbx74x0x48x49x17x46x4bx11x7dxcx1x2xdx20 Bernoulli x74x7a
x31xfx78x4fx56x54xdx20Bernoullix74x7ax31xfx54x63x52x23x46x3cFriedman–Savage [7]x48
x17x2cx3ax3ex23x24x3cx19x48x62x3ex2x32x26x27x7ax6ax70xbx7ax10x79x4fx54x78
x49x2ex1cx2ex3cx62x1dx54x70x71x57x58x72x73x54x65x63x2cx48x57x58x2fx30x31x54x4fx1x2
x7ex1fx10x62x54x3ax3bx65x68x54x15x4ex5cx27x44xex54x48x25x18x10x66x34x35x31x54x3ax3bx65x68
x2x62x21x34x35x54x45x46x6cx54x65x68x48x58x10x66x36x37x31x54x3ax3bx65x68x2x62x54x36x37x6b
x34(x10x26x3bx45x46x27x60x27)x54x65x68x78x49x1cx2ex3cx62x1dx54x70x71x57x58x72x73x54x65x63
x2cx48x70x64x73x69x2ex3cx62x1dx54x43x55x2x61x62x45x46x6cx5bx2x28x29x2ax6cx48x4x10x3ex54
x4ex16x53x49x45x66x2ex3cx62x54x70x71x78x31x32x1cx7ax3bx6ax27x54x3ax3bx65x68x48x33x34x3ax3b
x65x68x54x6x5ax53x49x45x66x70x71x54x28x29x2ax6cx48x53x68x7dx3ex33x55x18x38x7ex1fx3ax3bx65
x68x54x6x27x44xex78x49x25x64x74x75x56x48x7dx3ex23x24x64x2bx48x73x75x10x25x26x54x3ax3bx65
x68x39x72x6(x68x4fx3ex2b)x24x13x48x67x25x77x77x3ax3bx54x4ex16x78x5ax67x2x70x48x49x21x44
x58x54x28x29x45x46x6cxx69x25x44x18x38x62x7ax3ax3bx65x68u,x10x7x8x9xax56x5cx9x26x2a
E[u1(x)] = u(x);
x17x2cu1 x2x5dux1bx2ax19x43x28x29x6cx18x54x65x68x48x25x18x48 u1(x)x53x2x28x29x2ax6cx78
x45x31x48x37x1fx19x70x2bx24x25x27px47ax48x24x25x27(1?p)x47bx54x28x29x2ax6cxx48x25
x64x3ax3bx65x68x7x8x9xa
pu(a)+(1?p)u(b) = u(x):
138
x17x6ex7ax2x70x64xdx50x62x45x46xfx54x3ax3bx67x9x66x8xdx50x62x45x46xfx21x44x58x54xdx3c
x62x45x46xfx54x3ax3bx54x72x6x78x35xex4dx53x69x63xfx2ex25x69x78x9xax25x26x38x3dx54x3ax3b
x65x68xcx1x72x73x14xax49x42x6fvon Neumann–Morgensternx14xax49x42x78x25
x67x2von Neumann–Morgensternx49x62x3exex2cx6cx79x20x8x54x78x66x2x7dx4x67xc
x19x25x26x54x5fx60x2ax25x26x54x65x68x2x35x23x25x25x69x1ex62x7ax4x66x49x78
von Neumann–Morgenstern x3bx68x4fx6ex69x5cx54x69x55x73x69x61x25x70x5fx60x78
x62x3ex20x8x54x6ex69x1fx4cx51x2exfx1x2x78x16x17x2x3ax3ex56x4bx75x76x29x54x57x1fx13x7d
x17x4ex54x1fx4cx21x2x28x6dx78x2ex1fx1x60x54x25xbx6ex69x48x3ax3ex58x47x70xbx40x6fx25x22x78
x61xdx3ax3ex47x48x54x57x58x2fx30x31x2x34x35x31x48x25x70x34x35x31x10x62x54x34x35x2fx30x51
x17x36x57x54x7x28x78x62x54x34x35x2fx30x23x24x3bx62x34x35x54x45x46x6cx27x23x34x48xfx2x26
x66x1ax49x4bx2fx54x2x1cx2ex3cx62x1dx54x70x71x48x62x4bx2fx7x8x54x2bx3dx2ex13x2x3cx62x54
x45x46x24x6cx48x58x2x10x3ex54x23x25x1dx14x25x78x45x31x48 xdx51 30% x54x23x25 (x25x27) x68
x19x70x3ex36x4fx3fx48x5170%x54x23x25(x25x27)x68x19x70x6dx12x40x78xf2x9x9x78x34x35x31
x75x10x21x51x23x25x54x25x64x14x25x2ex8x17x6dx55x72x73x78xfx2x3ax3ex23x39x73x44x58x78x2b
x45x46x24x6cx4x70x2bx25x27x54x14x8x5fx27x4dx48x10x3ex78x45x31x48xd L = (x;y;p)x48x25
x18xx4y x2x78x2bx34x35x24x6cx48 p 2 [0;1]x2x25x27x57x1ex2bx14x54x6ex7ax1x2ax34x35
x31x24x25x27px68x19x45x46x24x6cxx48x24x25x27(1?p)x68x19x45x46x24x6cy. x7ax13x48x13
x10x78x2bx25x26x54x14 L1 = (x1;y1;p1) x4 L2 = (x2;y2;p2)x48x4dx48x71xex6dx54x14
L1 = (L1;L2;q)x48x17x6ex7ax2x72xdx54x78x10x25x64x71xex6dx54x14x48x58x25x40x41x21x71
xex6dx54x14x78x31x7ex9x9x78x15x52x68x19x54x71x72x44x2x70x2bx45x46x24x6cx14x48x17x2cx59
x70x2bx45x46x24x6cx5bx24x70x62x54x25x27x11x34x35x31x21x68x19x78x21x51x25x26x54x14x49 von
Neumann–Morgensternx54x6ax52x2cx51x50x49x8xcx5ax54x53xcx48xfx2x49x13x27x54x70
xbx64x67x2cx7dx3ex10x10x14x61x74x64x74x26x54x53x79x57x51xcx10x1xdx3fx2b (prospect)xf
x54 (x25x70x53xcx13x27x11 Kahneman–Tversky [13] x7fx27x2ex1x62x3ex54x69x63x54x22
x23x53x1e)x48x51xcx10x1xdx62x73(lottery)xfx54x48x51xcx10x1xdx50x62x45x46(contingent
commodity)xfx54x9x9x78x5ax2dx15x13x70x2bx53xcx77x77x25x24x78
x25x26x48 von Neumann–Morgensternx6ex69x54x8x5fx77x2xdx50x62x45x46x72x76xf
Lx48x10x2x26xdx3cx62x45x46x72x76xfx2cx54x24x6cx;yx9x40x41x40x1L = f(x;y;p)gx54
x14x4x21x71xex6dx54x14x21x14x41x78x2ex1fx24x13x3ax3ex67x24x;y;z x9x60x27Lx2cx54x38
x46x48x10x23x25x2xdx3cx62x45x46xfx48x5ax23x25x2xdx50x62x45x46xfx78x49x25x2bx72x76Lx44
x62x7ax61x70x2bx7x28x3x4c%. x21x22x7x28x3x4cx2x70x64x2bx41x3x4cx78x10x26x24x56x6e
x69x27x23x34x2a
x44x3f0
i) (x51x77x76) x % x:
2x27x44x59x3exexbx17x20x2dx52x2bx4dx5cx20x5x7ex66x19x1ax4ex48x15xbxax48x35x3ax25x26x5cx20x37xbx17x20
x10x77x75x56x57x13x20 x4a30%x20x2bx2dx65x55x3bx27x45x7cx18x5x4a70%x20x2bx2dx65x55x3bx47x74x28x29x6x32x33
xfxbx2bx6x41x65x6ex48x62x34x29 x1ex56x5x6x11x12x58x59x33x7x5x19x1ax2ax26xex8x2cx4axex41x4dx4ax5cx59
x4dx20x2bx7dx43x3ax29
139
ii) (x42x4ax76)x57x58x % yx6b y % zx6bx5dx5ex % z.
iii) (x7dxbx76) x % yx71y % xx7fx38x39x73x7axcxaxbx0
x25xbx75x39x2xdx2bx41x3x4cxfx54x62x7ax78x68x4fx44xcx1%x49Lx44x62x7ax61x70x64
xdx7fx28x3x4cxfx78x2ex1fx49x25x64x2bx41x3x4cx2cx51x43x15x1fx78x2bx38x46x76x5bx51x79x13x48
x4x49x44x29x54 iii) x2cx48x51x23x25x78x31x5bx41x7dx48xf x 6= y. x21x51x33x79x13x54x38x46
x7fx1fx48x49x57x58x4fx44x7fx7fxcx10x3ex10x34x35x31x27x70xdx33x7dx3fxfx78x3ax3ex23x10x10x53
x3bx44x49x2a ?. x31x32x % yx48x58x 6? yx48x33x34x3ax3ex44x2ex ′ y.
x44x3f1
i) x = (x;y;1):
ii) (x;y;p) = (y;x;1?p):
iii) ((x;y;p);y;q) = (x;y;pq):
x25x18x54i)x4ii)x2xfx36x7ax54x57 i)x2x70x5dx3cx62x45x46x24x6cx4x24x25x271x68
x19x54x45x46x24x6cx3cx41x70x1bx2bx48x10x71x72x44x23x24x3cx2ex70x2bx62x7ax48x39x58xdx50x62x45
x46x72x76xf Lx45x6exdx3cx62x45x46x72x76xfx2ex1x14x69x78xfx25x18x54x;yx5ax23x69x1ex1
x2x50x62x45x46x78 ii)x2x70x48x25x18x54x;yx75x76x36x51x3fx28x54x7dx3fx78 iii)xex2x70x48
x34x35x31x73x1fx79x10x23x25x68x19x54x45x46x24x6cx54x15x52x25x27x48x58x10x68x19x10x54x1fx6a
x2ex3x21x78x25x38x6ex69x2cx543x38x5bx77x36x7ax48x7dx3exfxfx20x8x13x36x78
x44x3f2 (x43x44x59x44x3f) x6dx6ex6fx70x;y;z 2L,
fp 2 [0;1]j(x;y;p) % zg x71 fp 2 [0;1]jz % (x;y;p)g
x24x67x45x5bx0
x25x38x7fx0x1dx6ex69x70x69x4bx7ax7bx2bx5x33x4ex14x4ex65x2ax5cx48x4dx70x69x4bx5ax2
x68x4fx44x54x47x75x78x35xex68x19x54x65x68x33x7fx0x1dx48x73x69x14x27xfx2ex69x66x78x2ex1f
x29x2fx76x2dx70x56x25x38x6ex69x54x6ex7ax48x51x23x53x7dx13x34x75x73x78x3ax3ex65x63x31x56x78
x6cx79x3ax3ex5dx8x56x5cx4x60x78
x59x23 6.1 xdx;y;z 2 L,x9xax ′ z ′ y. x31x32x6ex690–2x41x7dx48x33x34
x66x49q 2 (0;1),x53x68(x;y;q) ? z.
xdx27 x12A = fp 2 [0;1]j(x;y;p) % zg, B = fp 2 [0;1]jz % (x;y;p)g.
x33x34x26x6ex69 1x48 A x4 B x5bx2ex2x72x69x48x7bx1 1 2 A, 0 2 B. x26x6ex69 0,
A[B = [0;1]. x26x6ex692x48x10x3ex5bx2x67x69x78x12qA = infp2A p. qB = supp2B p.
140
x33x34x26Ax4Bx5bx2x67x69x48 qA 2 A, qB 2 B. xfx2x7fx26x66[0;1] = A[B,xc
x2cx51qA = qB, x35xe(qB;qA)x75x76x54x68x47x2ex66x66A, x5ax2ex66x66B. x25x2x2e
x23x25x54x78 q = qA = qB x67x2x4x60x2cx21x39x54q 2 (0;1). 2
x25x70x4x60x49x68x4fx44x3cx27x70x73x5bxfx36x7ax78xfx2x31x32x10 x;y;z x40x24x71
x72x79x7ax48x67x51x43x21x51x7dx5bx4dx10x79x78x45x31x48xdxx2x68x3c101x38x48 yx2x7x8
x3bx37x48 z x2x68x3c100x38x78x33x34x17x7ax4dx0x1x ′ z ′ y. xfx2x24x4ex2ex68x7d
x5bx2ex4dx0x1x1ax62x68x3c100x38x57x23x25x53x3bx37x54x15x6ax12x1ax47x68101x38x2xdx33
x7dx3fxfx54x78x22x23x31x7ex48x10x7fx0x1dx6ex69x58x2x36x51x37x34x7dx1dx13x36x48x7bx1x44x29
x45x14x2x2bx24x14x54x45x14x78x47x7ax49x65x63x1cx2ex3cx62x1dx54x5fx60x48x16x25x27x54x17x45
x7dx46x5x2x0x1x66x49x54x48x52x10x23x2bx54x2bx5cx2x2ex49x47x48x75x45x78x7bx7ex48x49x69
x63x44x0x1x44x29x78x3dx2bxdx33x7dx3fxfx51x33x2ex5fx2ex47x75x73x78
x44x3f 3 (x48x31x59x44x3f) x57x58 x ? yx6bx5dx5ex6dx6ex6fx70 p 2 [0;1] x71
z 2Lx48 (x;z;p) ? (y;z;p).3
x25x38x7cx7dx1dx6ex69 (x10x5ax57x7fx11xcx1x2xdx43x4ex1dx6ex69xf) x2x70 x x57 y
x54x33x7dx3fx2ex11x17x10x23x25x1dx54x13x14x78x60x4bx44x3cx27x10xdx15xfx36x7ax48xfx2x10
x11x19 1988 x46x54x7x59x5bx57x58x4fx5ax68x74 M. Allais (1911–) x54x25xax6x7x78x3
x66x25x70x77x48x3ax3ex32x19x13x4bx27x28x78
x0x4ex61x6ex693x24x13x48x3ax3ex23x24x51x56x5cx4x60x2a
x59x23 6.2 x20 x;y 2 L, x ′ y. x57x58x76x9 0–3 xaxbx6bx5dx5ex6dx6ex6fx70
p;q 2 (0;1), (x;y;p) ′ (x;y;q)x3x2x68x3p > q.
xdx27 x3ax3ex79x2cx68x48x10x66x15x1f p 2 (0;1), x ′ (x;y;p) ′ y. x35xex51
(x;y;p) % x ′ y x6fx31 x ′ y % (x;y;p). x31x32 (x;y;p) % x ′ y, x33x34x26
x4x60 6.1, x66x49 fi 2 (0;1], x53x68 ((x;y;p);y;fi) ? x. x4 (x;y;pfi) ? x. x12
T = ffl 2 [0;1]j(x;y;fl) ? xg. x33x34 pfi 2 T, x51x6cx26x6ex69 2, x10x2x2bx67x69
(x78x2bx67x69x54x66x69)x78x12 qT = minfi2T fi. xe qT 2 (0;pfi] (x7bx1 x ′ y)x48x51
x6c (x;y;qT) ? x. xfx26x6ex69 3x48x7fx51 ((x;y;qT);y;pfi) ? (x;y;pfi) ? x, x4
(x;y;qTpfi) ? x. x7bx1qTpfi < qT,x25x57qT x54x62x7ax5dx23x78x7bx7ex48 (x;y;p) %
x ′ yx2ex23x25x78x72xdx1ex23x2cx68x ′ y % (x;y;p)x5ax2ex23x25x78x25x26x48x3ax3ex67
x2cx68x61x10x66x15x1fp 2 (0;1), x ′ (x;y;p) ′ y.
x4dx70x69x4bx48x31x32p > q,x33x34x66x49fi 2 (0;1),x53x68fip = q. x39x58x48x49
x44x4bx2cx68x2cx54x31x32x3b(x;y;p)x47x4ex,x3ax3ex68x19
(x;y;p) ′ ((x;y;p);y;fi) = (x;y;pfi) = (x;y;q)(′ y):
x7ax75x48x31x32(x;y;p) ′ (x;y;q),x33x34x26(x;y;p) ′ (x;y;q) ′ y x4x4x606.1,
3xbx36x4x5cx48x2ax63x37x46x2cx2cx52p = 1=2xax2ax5x7cx7aHerstein–Milnor [9] Theorem 2.
141
x66x49 fi 2 (0;1), x53x68 ((x;y;p);y;fi) = (x;y;pfi) ? (x;y;q). x26xbx4bx54x2c
x68x48x25x4cx17q = pfix18x78x51x23x25x78 2
x1ax49x3ax3ex5dx8x48x49x44x29x61xdx56x51x24x56x62x69:
x36x3f 6.1 (von Neumann–Morgensternx36x3f) x57x58(L;%)xaxb
x7cx17x5cx5fx76x9x6bx5dx5ex61Lx7cx60x61(x61x38x21x25x51x7axcx7ex65x3bx49x16x59x21)x3
x7ax79x8x9x64x6cx64x65ux6bxaxb
x % y x3x2x68x3 u(x) > u(y);
u((x;y;p)) = pu(x)+(1?p)u(y): (1)
xdx27 x3ax3ex79x61x62x2a x61Lx7fx60x61bx71wx6bx4x5Lx7fx79x6fx70xx6bx24
xaxbb % x % w. x25x18x54bx4wx33x3fx2Lx2cx54xdx15x28x54(x50x62)x45x46xfx4
xdx15x70x54(x50x62)x45x46xfx78x51x61x25x2bx61x62x24x13x48x3ax3ex71x72x44x73x47x47x48x51x5b
x54x3ax3bx65x68x78
x12u(b) = 1, u(w) = 0. x26x44x29x61x62x4x4x60 6.1, x3ax3ex10x66x15x1fLx2c
x54xx5bx23x39x68x25x27px,x53x68
x ? (b;w;px) = pxb+(1?px)w:
x10x18x48x26x4x606.2,x3ax3ex58x23x5dx8x25x2bpx x2x1dx70x54x78
x1ax49x3ax3ex62x7au(x) = px. xex26x6ex691 iii)x48x75x16x52x2cx48
(x;y;p) ? ((b;w;px);(b;w;py);p)
? (b;w;ppx +(1?p)py)
? (b;w;pu(x)+(1?p)u(y)):
x7bx7ex48 (2)x41x7dx78
x58x47x52x2cx25x26x62x7ax54ux2x3ax3bx65x68x48x4x % y x9x4ax66u(x) > u(y).
x25x2x7bx1x26x4x606.2x48x3ax3ex74x7ax51x % yx9x4ax66px > py.
x15x13x48x58x75x52x2cx1dx70x1dx78xd u1 x2x4dx70x2bx9xa (2) x54x3ax3bx65x68x78x33
x34x10xcx62x9xa
u1(x) = pxu1(b)+(1?px)u1(w)
= (u1(b)?u1(w))u(x)+u1(w):
x4u1 x57ux73x7bx7dx70x2bx74x7ax2ax67x78
142
x1ax49x3ax3ex27x32x24x66x49xdx15x28x54 (x50x62) x45x46xfx4xdx15x70x54 (x50x62) x45
x46xfx54x61x62x78x61x62x3ax3ex19x57x10xdx7bx10x15x28x54 (x50x62) x45x46xfb x4xdx7bx10x15
x70x54(x50x62)x45x46xfwx24x58x21x51x9xab % x % wx54x 2Lx2cx68x61x3ex2bx3ax3b
x65x68ux54x66x49x78xfx2x49Lx2cx58x51xdx21x28x54(x50x62)x45x46xfb0 ′ bx4xdx21x70
x54(x50x62)x45x46xfw0 (w ′ w0) (x6fx31x73x66x49x17x2cx75x70x48x10x73x66x49x17x2cx75x70
x54x74x40x54x2cx68x7ex7fx70x26)x78x66x2x3ax3ex7fx23x3bx10x26x54x69x55x68x19x24x3bx66x21x51
x9xab0 % x % w0 x54x 2 Lx54x3ex2bx3ax3bx65x68u0. x25x2bu0 x17x7ax57x44x4bx39x68
x54ux2ex70x26x48x7bx1x10x9xa u0(b0) = 1 x4 u0(w0) = 0, x58u0(b) < u0(b0) = 1,
u0(w) > u0(w0) = 0. xfx2x25x18x3ax3ex23x24x10x21x51x9xab0 % x % w0 x54x 2 L,
x62x7a
?u(x) = u
0(x)?u0(w)
u0(b)?u0(w);
x33x34x10x1cx7ax2x21x51x9xab0 % x % w0 x54x 2 Lx54x3ex2bx3ax3bx65x68x48x51x6cx10
x58x9xa ?u(b) = 1, ?u(w) = 0. x26x44x4bx54x2cx68x23x24x7dx4x5fx1ax48x25x2b ?ux10x21x51
x9xab % x % w x54x 2 L, x57x44x4bx68x19x54ux45x25x78x25x26x48x3ax3ex71x72x44x5d
x44x4bx68x19x54ux4bx1dx19x21x51x9xab0 % x % w0 x54x 2L. x7bx1x25x26x54x4bx1dx5
x2x23x25x54x48x21x24x3ax3ex15x13x23x24x5dux4bx1dx19x1ex2bLx444x78 2
x25x26x48x3ax3ex67x49xdx50x62x45x46Lxfx54x68x4fxex4cx56x48x2cx68x61vonNeumann–
Morgensternx3ax3bx65x68x54x66x49x62x69x78x25x70xex4cx10x11xbx77x1x2x54x57x58x5fx60
x54x55x56x27x70x48x23x25x58x2ex63x78xfx2x26x7ex8x5fx48x74x7ax23x24x68x19x21x70x71x54x31
x32x78x45x31x48x3ax3ex23x24xdx55x20x8x10x66x3ax3exbx4bx57x7fx3bx19x54x26x69x7dx51x2ax54
x28x29x2ax6cx21x4dx41x54xdx50x62xbx21 Hilbertx72x76xfx20x8x72xdx54x31x32x78x2dx2ex57
x58x4fx59x4x68x4fx59x5bx4bx10x7ex4ex4fx22x1xcxdx78x25xbxcxdx5bx42xax66x28x5cx62x69
x54x76x29x4x2cx68x48x20x8x70xbx20x54x6ex69x60x54x40x26x48x76x77x19x21x1x70x71x54x40x26x78
x5x75x48x25x70x62x69x53x3ax3ex10x2ex3cx62x1dx38x3dx56x54x44xex5fx60x54xcxdx51x69x26x49
x70x62x38x3dx56x38x11xdx3ex2bx3ax3bx65x68x61xdxfx48x4x7dx3ex49x4bx2fx1cx2ex3cx62x1dx54x4a
x8x5fx60x18x48x23x24x7ex1fx10x70x2bx3ax3bx65x68x39x17x68x4fx3ex2bx15x4ex27x2ax7dx55x56x78
Friedman–Savage [7]x5dx8x48x7dx3ex7fx39x78x69x4bx27x7ax10x25x70x61xdx2ax70x64x79x4f
x2x0x1x25x2ex2x10x71x72x1ax44x54x70x64x51x3bx6fx25x69x54x1ex4dx78x10x7ex33x55x1ax4cx48
x73x25x3bx71x72x54xcxdx41x32x27x2cx71x78x4dx70x64x79x4fx2x16x6fx7ax10x3bx70x2bx65x68x27
x51x6cx7dx3ex49x2ex3cx62x1dx70x71x56x54x4ax8x4fx1x78x10x7ex48x62x3ex0x1x2x2ex25x38x11
x54x48x7bx1x25x31x10x7ax10x1ex2bx57x58x4fx60x54x69x63x54x69x26x78
6.3 x31x44x21x29x48 von Neumannx4Morgensternx20x8x62x3ex54x3ax3bx65x68
x6ex69x1fx4cx54x6ax79x2xdxex5dx8x48x49x1cx2ex3cx62x1dx54x70x71x56x48x7dx3ex2x51x23x25
x27xdx51x6cxfx3ax3bx65x68x54x78x26x66x62x3ex21x3bx54x68x4fx6ex69x5cx69x55x48x25x49x36x37x44
x2x4bx67x2ex4cx54x78x31x32x75x51x37x34x5fx60x48x33x73x25x19x62x3ex20x8x54x6ex69x1fx4cx2c
4x32x33xfx5dx17xex2dx5xbx53x4axex47x10x3ex7ex7cxfx52x13x29
143
x12x60x78x581952x46Allaisx49x70x6dx3x66von Neumannx4Morgensternx54xex54
x65x63x4dx2cx20x8x54x42xf ([1])x48x3bx2x25x26x54x5fx60x78x25x4dx53x7dx3ex2ex68x2ex4ex70
x57x22x5ex78x1ax49x3ax3ex49x64x67x2cx23x24x3cx19x48x7dx3ex5bxfx1fx79x5dvon Neumann–
Morgensternx3ax3bx65x68x54x25x22x4x7ex7fx54x3ax3bx65x68x25x22x48x33x6fx27x78xbx31x7f
x3bx54x53xcx2x72x73x14xax78x25x67x2x70x48x3ex2bx3ax3bx51x43x2x34x35x31x3bx3bx54x3ax3bx48
x10x58x45x6ex42x34x35x31x10x2ex3cx62x70x71x54x47x48x48x77x7ex7fx54x3ax3bx51x21x2ex54x74x76
x7bx46x78x25x64x4ex4fx54x56x55x49xfx4ex6ax3bx44x2Allaisx54x20x74x78
Allaisx54x23x67x74x75x49x66x5dx8x2ax50xbx76x76x9xax51x10x72x64x78 Allaisx4dx48
x61x70x2bx45x14x48x51x3bx25x2bx45x14x77xbx39x39x61x79x4fx48x17x2cx45x46x2dx2ex25x69x4bx54
x22x59x78x31x32x52x4ex2ex68x7dx5bx2ex8x61x57x7cx7dx1dx6ex69x7bx7ax54x72x73x78x7fx5dx7cx7d
x1dx6ex69x3bx4dx70x64x40x26x1ex4ex61x62x54x53x52x2a x65x12x6bx4fx40x41x6b(1954) ([20]x48x49x33
x18x2ex31x7fx20x8x4dx70x6dx47x29x74x76x25x27x54x72xdx66von Neumann–Morgenstern
x3ex2bx3ax3bx65x68x54x6ex69x1fx4cx48x26x7ex23x1fx8x74x76x25x27x54x66x49x1d)x54x57x58x4fx59
x4x68x4fx59L. J. Savege (1917–1971)x5bx2ex68x2ex6bx0Allaisx2x51x69x54x78x39x7ex48
Allaisx54x25x2bx45x14x67x24xdAllaisx52x40xfx52xcx78
Allaisx54x45x14x29x33x28x6dx78x59x6ex26x31x5ax5bx23x24x2ex1cx7x4fx1ex2ex8x36x57x54
x72x73x48x3cx2x35x57Allaisx21xdx55x54x70x2ax78x3ax3ex49xbx4bx65x63x7fx0x1dx18x71x72x44
x5ax19x20x8x61x70x2bx0x1c Allais x67x63x51x62x54x45x14x78xfx2x49x33x18x58x25x53x25x36
x28x17x70x48x58x49x25x18x10x66x7cx7dx1dx6ex69x27x70x48xex2x2ax4x54x78
Allaisx67x63x54x45x14x31x56:x61xdx51x24x56x78x14x2bx3dx48x47x75x2ex8x72x73x78
8
>><
>>:
A1 = x54x62x5x36 100x55x8x56
A2 =
8
<
:
x4810%x79x3dx13x5x36 500x55x8x56
x4889%x79x3dx13x5x36 100x55x8x56
x481%x79x3dx13xax5x5f
8>
><
>>:
A3 =
‰ x4810%x79x3dx13x5x36 500x55x8x56
x4890%x79x3dx13xax5x5f
A4 =
‰ x4811%x79x3dx13x5x36 100x55x8x56
x4889%x79x3dx13xax5x5f
x10x66x70x71x7dx27x70x48x49 A1 x4 A2 x75x76x5x2x4dx7x8 A1x48x58x49 A3 x4 A4 x75
x76x5x2x4dx7x8A3. x25x26x54x7x8x36x7ax23x24x70x8xfx2ex69x26x27x48x17x2cx15x74x75
x54x33x43x2x75x49x68x3cx4x53x6ax75x76x2ex70x36xbx73x78x7bx7ex48x49 Allais x10x44x61x2b
x61x1ex25x27x63x54x7dx21x2ex54x50x64x2cx48x52x4ex2ex68x54x7dx5bx2ex61x25x26x54x7x8x48x7f
Savegex5bx36x51x45x19x78
xfx2x25x26x54x7x8x68x28x2x48x77x7cx7dx1dx6ex69x54x78x2bx71x44x48x3ax3ex12
L1 = (1;1;0:11); L2 = (5;0;10=11);
144
x17x2cx6dx6ex47x2ex61x6x55x57x78x33x34x74x7ax51
L1 = A1;
x58
(L2;1;0:11) = ((5;0;10=11);1;0:11) = A2:
x4dx70x69x4bx48x3ax3ex7fx51
A4 = (1;0;0:11);
(L2;0;0:11) = ((5;0;10=11);0;0:11) = A3:
x25x26x48x26x7cx7dx1dx6ex69x48 A1;A2 x75x76x4A3;A4 x75x76x54x7x28x7x8x5fx60x48x53
x7ex7fx47x44x66x101x57L2 x54x7x8x78x31x321 ? L2x48x33x34x26x7cx7dx1dx6ex69x7x8x5a
x51 A1 ? A2x48 A3 ? A4. x58x31x32x0x1 A1 ′ A2x48xex7x8x51 1 ′ L2x48x39x58
x1fx2aA4 ′ A3:x5ax67x2x70x48 A1 ′ A2 x73x25x57A4 ′ A3 x7bx75x78x25x67x2x21x22
Allaisx67x63x78
x7dx3ex10x66Allaisx67x63x51x74x64x71x3bx78x70x64x0x1Allaisx67x63x54x45x14x7exc
x5ax48x7cx7dx1dx6ex69x70x71x27x70x58x2x41x7dx54x78x66x2x62x3ex58x2x4x0x3bvonNeumann–
Morgensternx3ax3bx65x68x56x74x64xcxdx78x4dx70x64x71x3bx2x0x1Allaisx67x63x54x8
x1ax2x7bx1x70x71x54x7dx3ex5bx2ex7ex68x2bx69x48x21x24x33x79x2cx2ex61x6ex6fx72x73x78x66x2
x7cx7dx1dx6ex69x2ax41x61x57x58x44xex5dx1fx48x58x2ex2x77x7ex7dx54x57x58x44xex54x7ax7bx78
x58x51x70x64x71x3bxex2x0x1xcx2cx32x24x7cx7dx1dx6ex69x78x33x6cx79x2 Allais x6fx7dx54
x71x3bx78x62x0x1x48x1fx12x3x29x46x27x2dx2ex2ax7dx49x7cx7dx1dx6ex69x40x41x44x54xcxdx48x16
x17x2x2ax7dx49x38x39x3ex2bx3ax3bx15x4ex40x41x44x54xcxdx48x5bx2x2ex4ex25x71x72x54x48x7b
x1x10x3ex5bx58x0x61x7dx54x21x69x7bx46x10x25x27x33x6cx54x13x14x78
Allaisx67x63x24x13x48x7fx51x70xbx4fx31x20x8x74x64x74x26x54x67x63x48x17x2cx77x77x52
x53x54x511961x46x20x8x54Ellsbergx67x63(x46x3cx45x31x48 [6]x48 [15]x48x10x17x71x21x2ex1e
x2x45x10Savagex54x6ex69x4cx12)x9x78 1970x46x4ex14x48x78x6ex21x69x4fx59Kahneman
(1934–)x4Tversky (1937–1996)x59x4ex61x25x70xcxdx1dx74x78x62x3ex24x21x69x4fx59
xcx51x54x5ex7ex48x27x64x65x7dx3ex49x2ex3cx62x70x71x56x54x44xex5fx60x48x20x8x61x2dx2ex20
x54x76x22x78x62x3ex6cx79x49xfx4ex6ax3bx44x38x11x61 Raiffa [17] x54x76x77x48x0x1x10x2e
x3cx62x70x71x56x4ex4fx44xex54x33x14x75x48x3fx60x64x2ex10x54x69x55x78x70x64x2x5ax41x27x5b
(normative analysis)x48x10x3x1fx54x2x44xex5fx60x54x69x1dx1ex1cx48x25x64x1ex1cx2x71
x72x44xex43x39x4dx5dx54x78 von Neumann–Morgensternx3ex2bx3ax3bx69x63x67x2x25x64
x33x14x78x70x64x2x5cx5dx27x5b(descriptive analysis)x48x10x3x1fx54x2x1ax71x54x7dx3ex71
x72x44x2x32x26x44xex54x78x62x3ex0x1x62x3ex54x69x63x21x38x5dx66x25x64x33x14x48x22x23x49
x11x64x6ax3bx44x48x62x3ex54x69x63x5ax25x68x4fx6ex69x5cx48xfx33x44x2ex2x62x3ex38x39x54xa
x38x78x58x51x70x64x2x5ex5fx27x5b(prescriptive analysis)x48x10x3x1fx54x2x71x4x2ax36x48
x24x51x6x57x7dx3ex23x3bx27x2ex8x77x1x25x69x54x44xex78x25x64x69x55x67x5ex3ax3ex45 (x23
x69) x4fx56x54x6cx45x33x14x20x33x26x48x70x71x51x36x51x4cx12x54x69x63x23x49x5ex48x58x2ex44
145
x2xbx49x7dx24x76x27x54x57x52x75x5fx78x58xdx36x51x70x64x7x8x69x63x47x49x33x5ax44x2x25
x24x54x7fx49x47x29x44x2x46x3cx54 (no theory of choice can be both normatively
adequate and descriptively acurate)xfx78x25x26x54x76x77x57x3ax3ex49x65x70x51x2cx20
x19x54xdx4ex4fx64x5cxfx57xdx64x4fx64x5cxfx54x48x3fx51x77x72xdx78x6fx2dx10x23x24x69x1ex1x2a
x49xdx4ex4fx64x5cxfx54x10x32x58x51xdx4ex4fxfx57xdx64x4fxfx75x33x48x4xdx33x5axfx57xdx47
x29xfx75x33x78x25x67x2x70x48x3ax3ex2ex7x8x5dxaxbx10x4fx1x3dx3ex4fx13x14x24x22x23x54
Kahneman–Tversky x69x63x3cx2e von Neumann–Morgenstern x3ex2bx3ax3bx65x68
x69x63x54x60x10x4bx48x58x21x7x8x3cx2ex2x70x64x25x14x4bx78
Kahneman–Tversky x35x4x6bx61 Allais x67x63x56x54x45x14x19x48x58x4dx48x61x2d
x2ex51x70x12x43x54x45x14x27x5dx8von Neumann–Morgensternx69x63x54x2exax78x25x18
x3ax3ex5cx12x70xbx25x26x54x45x14x78
x451x2ax47x48x11x7dx61x5ex7dx12x62x56x3cx68x8x48x62x23x25xcx19x25x26x54x78x64x72xd
x54x74x75x2ax70x64x74x75x2x62x19x61x62x56x18x48x5fx1ax19x70x54x78x0x73x63x61x78x25x18x62
xdx40x2x13x70x73(x61x62x58x25x70x19x3x15x10x26x54x73)x3cx68x8x48x58x2x1bx59x7ex4dx70
x64x74x75x2x62x2cx74x70x73x54x5bx63x61x78x25x18x62xdx40x2x13x64x5bx70x73(x61x62x62x58x51
x5b)x3cx68x8x48x58x2x1bx59x7e Kahneman–Tverskyx5fx1ax48x4ex2ex68x7dx49x65x70x64x74
x75x56x4dx7x8x1bx59x57x58x49x65x8x64x74x75x56x4dx7x8x13x70x73x78x10x25x2bx45x14x54x31
x32x3ax3ex23x24x70x8x2dx2ex21x69x44x54x6ax7bx48xfx2x3axfxfx3bx3ex2bx3ax3bx65x68x27x23
x34x48x7bx1x3bx5bx27x73x6cx18x48x63x73x57x63x5bx54x31x32x2x70x26x54x78
x45 2 (x4f [22])x2ax61x62x51x11x2bxbx11x65x74x10x66x54x6dx14x78x25x2bx6dx14x51 600
x7dx78x7fx61x62x51x78x2bx6bx67x69x6cx2ax65x70x2bx69x6cx2x53 400 x7dx73x48 200 x7dx5bx67x57
x4dx70x2bx69x6cx2600x7dx7fx73x54x25x27x22/3, 600x7dx7fx5bx67x54x25x27x21/3. x25
x78x2bx69x6cx54x5bx67x7dx68x54x3ex2bx6x2x70x26x54x78x15x3ex2bx3ax3bx65x68x54x69x63x48x26
x66x70x71x7dx5bx68x50x69x6ax48x7ex7fx5bx4dx7x8x65x70x64x69x6cx78xfx2x26x66x25x2bx5fx60
x44x58x7dx3ex54x36x73x48x7dx3ex54x71x72x7x8xfx4ex6ax3bx44x47x44x66x32x26x27x76x29x25x78
x2bx69x6cx78x49x70x6dx50x64x2cx48x25x78x2bx69x6cx11x76x29x1x2ax65x70x69x6cx2 200 x7dx5b
x67x57x65x8x69x6cx2600x7dx5bx67x54x25x27x21/3,x36x51x7dx5bx67x54x25x27x22/3. x31
x32x5dx41x65x70x69x6cx54x7dx68x57x5dx41x65x8x69x6cx54x7dx68x54x77x45x172:28 (x11x50x64
x5x7dx68152x48x71x72x7dx68x54x77x45x1109:43)x78x49x4dx70x6dx50x64x2cx48x7fx5dx6bx67x69
x6cx76x29x1x2ax65x70x69x6cx2 400 x7dx73x12x48x65x8x69x6cx2x53x7fx1fx5bx67x54x25x27x2
1/3, 600x7dx73x12x54x25x27x22/3. x31x32x5dx41x65x70x69x6cx54x7dx68x57x5dx41x65x8x69
x6cx54x7dx68x54x77x45x40x7ax1fx27x2ax4122:78 (x11x50x64x5x7dx68155x48x71x72x7dx68x54x77
x45x134:121)x78
x25x78x2bx45x14xfx25x70x68Kahneman–Tverskyx54x5ex55x78x31x32x70x48 Allaisx67
x63x2cx73x44x58x4ax8x7dx54xdx3cx21xfx4x2ex3cx62x1dx48x53x68x4ax8x7dx2ex1xdx6x4dx7dxf
x54xcx39x58x2exfx68x74x48x33x34Kahneman–Tverskyx54x45x14xex7ex7fx23x8x61x4ax8
x7dx54xdx6x4dx7dxfxcx39x78x3b Kahneman–Tversky x54x7bx11x27x70x48x62x3ex0x1x7d
146
x3ex49x2ex3cx62x70x71x56x54x4ax8x48xfx4ex6ax3bx44x47x44x66xdx44xex54xex4c(framing of
decision)xfx78x65x70x2bx45x14x2cx79x7ax5bx73x44x58x78x0x62x56x73x54x4ax6x48xfx2x78x0
x73x6fx50x57x78x0x73x54x4ax6x10x4ax8x7dx27x70x48x1cx7ax2x2ex70x26x54x7bx46x78x65x8x2b
x45x14x2cx7ex7fx70x26x54x69x6cx48x67x7bx1x39x5bx67x58x2x39x73x69x54x2ex10x7fx11x27x47x48x48
x7dx3ex54x4ax8x5ax4dx2ex10x78x25x18x44x58x54x5bx3x62x3x36x3x73x9x5bx57xdx6x4dx7dxfxc
x39x51x3x78x21x22x2ex10x54xdx44xex54xex4cxfx73x2ex1fx2x49xex4cx2cx75x25x50xbxbxc
x39x78x58x39x3ax3ex49x65x70x51x2cx21x20x19x54x36x37x3ax3bx27x47x48x48x33x34x6fx2dx3ax3ex23
x24x70x48 Kahneman–Tverskyx54x45x14x71x72x44x5dx8x48x49x44x58x74x7x69x6cx54x10x29
x18x48x7dx3ex57x7fx53x3bx54x2x10x29x36x37x48x58x2ex2x19x2ex36x37x78x19x2ex44x70x26x54x10
x44x48x26x66x7dx3ex69x1ex54x10x29x2ex10x48x67x4dx51x2ex10x54x36x37x76x69x31x32x78x21x22x2e
x10x54xdx44xex54xex4cxfx48x71x72x44x2x49x10x10x70x2b (x19x2e) x10x44x40x24x2ex10x54x10
x29x78x39x25x2bx3ax3bx27x3cx48x3ax3ex5ax25x69x1evon Neumann–Morgensternx3ex2bx3a
x3bx65x68x69x63x54x1bx2ax78x25x64x69x63x2x39x41x1x1x54x6ex69x1fx4cx8x5fx54x48x25xbx6e
x69x19x57x2ex6ax7dx76x6bx6cx48x58x2ax1x48x37x54x68x4fx4ex49x78xfx2x2fx36x36x54x7dx49x2e
x4ax8x18x44x2ex4dx3bx48x37x54x68x79x57x4ex49x27x5ex47x78x62x3ex4dx55x19x5bx3x62x3x36x3x73
x9x9x10x36x50x54x2dx2ex33x55x1x60x54x6ex7ax48x58x25xbx6ex7ax2x2ex23x25x7ex7fx1fx1ax49
x33xbx6ex69x2cx12x54x78
x6fx2dx5ax67x7bx1x25x26x48 Kahneman–Tverskyx54xdx3fx2bx69x63(prospect the-
ory)xfx21x2ex1ex11x25x50x1x2x70x64x47x29x33x14x48x53x7dx3ex49x7x3bx25x64x69x63x18x75x2e
x47x48x2fx36x36x54x1ax71x4cx5bx48x58x2ex75x73x78x6dx49x41x1x1x54x68x4fx4ex49x2cx78x7bx7ex48
x49x62x3ex54x33x14x2cx48x62x3ex57x7fx5x3bx2dx2ex71x52x6cx45x27x70x68x5fx60x48x17x2cx16x17
x2x3bx71x52x6cx45x27x20x8x70x4cx5cx57x3ex2bx3ax3bx65x68x69x63x7bx67x54xdx3ax7xfx78x25xb
xdx3ax7xfx51x58x36x59x14x43 (certainty effect) (x57x1ax62x31x32x7bx77x6fx4x23x25x5fx36
x54x31x32)x3x6ex78x14x43(reflection effect) (x10x54x21x57x2dx2ex54x7ax7x6fx31x70x5ex48x4
x6fx4x54x21x48x71x4x2dx2e)x3x71x31x14x43(isolation effect) (x10x10x70x2bx3dx54x2ex10x72
x4bx25x50x48x4dx1fx2ax52x7ax7bx7ax54x7x28)x9x9x78x10x3ex5bx2x7dx3ex49x5x3bx3ex2bx3ax3b
x65x68x69x63x18x47x75x1fx79x54x78
x3fx2bx69x63x15x13x1fx1ax1x3ex2bx3ax3bx65x68x54x70x64x27x3bx78x1cx7ax5x3bx44x4bx65
x63x54x3ex2bx3ax3bx65x68x54x69x63xex4cx48x3x66x50x62x45x46 (x;y;p), x10x3ex2bx3ax3bx65
x68ux27x70x48
u((x;y;p)) = pu(x)+(1?p)u(y);
xfx2x10xdKahneman–Tverskyx29x49x42(value function)xf vx27x70x48
v((x;y;p)) = …(p)v(x)+…(1?p)v(y);
x17x2cx2ex1x6dx2ax6cx65x68x54x6x65x68v,x2ex13x5ex3ax3bx65x68x70x71x2x7dx65x68x48x58x2
x10x36x2ax6cx54x3bx6 (xdx54x21 (gain)xf) x2x7dx65x68x48x10x36x2ax6cx54x50x6 (xdx2dx2e
(loss)xf) xex2x26x65x68x57 … x2x31x73x49x42 (weighting function)x48x10x2x10x25x27
x54x70x64x4x6bx48x7ex7fx5dx6fx25x27x71x4x48x5dx71x25x27x6fx4x78x78x31x54xex44x31xe1x78
147
x661: Kahneman–Tverskyx1cx64x65(x74)x14x43x27x64x65(x75)x0xax51 [13]
Kahneman–Tverskyx54xcxdx0x14x61x57x58x4fx5bx54x24x4ex45x15x48x16x17x2x3d
x3ex57x58x4fxcxdx54x24x4ex45x15x78x3dx3ex57x58x4fx54x70x2bx20xcxdx69x24x2ax4fx1x3dx3e
x4fx7bx7ex58x7x5x58x36x78x2dx2ex25x69x4bx54x4fx31x3bx62x3ex54x69x63x27x1ex4dx3dx3ex42x43
x54x7ax7fx1ax44 (x4x2ex4ex25x57x4fx3dx3ex57x58x4fx54x1ax44x48x45x31x6ax42x54x1fx3bx7ax7)
x9x48x49x70x62x6ax3bx44x5bx68x41x20x78 Kahneman x5ax7bx7ex57x71x52x57x58x4fx54xbx46
x31V. L. Smith (1927–)x70x14x18x5b2002x46x54x7x59x5bx57x58x4fx5ax78x58x5x1cx54
Tverskyxex41x61x4Blackx24x13x54x7fx70x6ex49x7x59x5bx57x58x4fx5ax6ex58x44x11x68x3c
x60x72x54x19x46x4fx31x78
x22x23x31x7ex48x18x19x1ax49x1x1bx48x49x1cx2ex3cx62x1dx54x57x58x4fx5fx60x54xcxdx2cx48
x6bx0von Neumann–Morgensternx3ax3bx65x68x54x58x2x74x59x78 Allaisx54x13x4x31
x79x7ax49x32x24x7cx7dx1dx6ex69x9x69x4bx2ex61xfx2exfx22x1x54xcxdx48xfx2x26x66x17x69
x63x44x1fx66x1x2x48x7cx52x36x51x54x19x47x4evon Neumann–Morgensternx3ax3bx65x68
x54x1ex57x78x3ax3ex49Fishburn [6]x25x26x54x22x52x2cx23x24x60x19x2dx2ex39x74x69x4bx27x76
x77x3ex2bx3ax3bx65x68x69x63x54x6ex69x1fx4cx54xcxd(x74x64x43x1cx1dx3x43x23x0x3x43x11x49
x3ax3b)x78xfx40x26x76x77x73x2x70x64x41x1x1x54x68x4fx48x58x5fx60xex49x66x2fx36x36x54x7dx78
x7bx7ex48x2ax7dx49x4ex6cx21x69x71x52x40x41x44x54Kahneman–Tverskyx69x63x79x7ax49x68
x4fx44x51x2ex71x22x48x2fx5ex4cx4cx2x3ex2bx3ax3bx65x68x54x70x64x28x6dxdx27x3bxfx48x7ax58x41
x61x7dx3ex3x1fx54x76x77x78x10x49x11x64x79x7ax56x48x2ax41x61x10x3ex2bx3ax3bx65x68x69x63x54
x70x64x20x54x1ax62x78x25x26x48x3ax3ex1cx7ax7x8x5dx3ex2bx3ax3bx65x68x69x63x2ex1x3dx3ex57
x58x4fx54x70x49x4ax77xaxbx48x4ex4fx1ex5dx10x7cx66x56x27x78
6.4 x1ax49x3ax3ex4x0x65x63x3ex2bx3ax3bx65x68x69x63x78xdx11x57x58x2fx30x31x54x3e
x2bx3ax3bx65x68x1x6dx2ax6cx65x68 u(x). x3ax3ex2ex40xdx25x18x36x2ax6cx54x6ex7ax67x2x20
x1x78x61xdx;y > 0x1x78x64x23x25x54x20x1x57x68x19xx54x25x27x1px48x58x68x19y x54
x25x27x1(1?p). x33x34x26von Neumann–Morgensternx3ax3bx65x68x54x62x7ax48x51
148
x1cx3bx6ax27x54x44x49x48x23x68x25x70x2bx3dx54x3ax3bx1
u((x;y;p)) = pu(x)+(1?p)u(y):
x7ex7dx10 (x;y;p) x25x70x2bx3dx2cx21x45x6ex54x69x6ax54x71x3bx23x26x25x2bx6x57 u(px +
(1?p)y)x54x77x77x27x23x34x78x31x32
u(px+(1?p)y) = u((x;y;p));
x33x34xcx8x57x58x2fx30x31x1x28x29x44x59x34x78x31x32
u(px+(1?p)y) > u((x;y;p));
x33x34xcx8x57x58x2fx30x31x1x28x29x78x79x34x78x31x32
u(px+(1?p)y) < u((x;y;p));
x33x34xcx8x57x58x2fx30x31x1x28x29x7ax7bx34x78x75x1fx79x54x2x48x25xbx20x55x5bx49x45x66
x;yx4p.
x44x29x25xbx20x55x23x24x25x26x27x69x1ex2ax70x71x5x0x1ux2x20x1x54x7fx0x49x54
x65x68x78x66x2x10x66x69x6ax68x50x31x27x70x48x53 u(x0) = u((x;y;p)) x54 x0 x7x8x77
px+(1?p)yx16(x31xe1)x48x58x10x66x69x6ax2cx28x31x27x70x48x7x8x77px+(1?p)y
x4ex57x58x10x66x69x6ax2cx1dx31x27x70x48x78x31xex5x7bx9x78
-
6
0 x
u
r
r
r
r
r
x0
rr
y
ru(y)
x
r
u(x) r
px+(1?p)y
r
ru(px+(1?p)y)
rpu(x)+(1?p)u(y)=u((x;y;p))
x66 2: x69x3bx3cx64x6cx64x65x79x66x73
x25x26x70x48x58x2x77x77x77x44x78x3ax3ex13x4ex70x57x5d x x1ex4dx1x3bx7fx20x1x48x4
x23x68x19x10x54x25x27px77x38x5dx661. x31x32y x2x70x43x45xfx48x4y x7fx16x66xx48x33
149
x34x?x0 x23x24x3cx2ex57x58x2fx30x31x1x45xfy xcx79x3fx2bx54x7cx6ax35x3bx78x25x18x3a
x3ex23x24x3cx8x2ax69x6ax68x50x31xcx79x3fx2bx77x71x54x35x3bx48x58x69x6ax2cx28x31xex2exc
x79x78x7ax75x48x31x32y x2x70x47x38x4ex54x62x73x20x1x48 y x7fx4ex66xx48x33x34x0 ?xx23
x24x3cx2ex57x58x2fx30x31xcx79x17x70x54x62x73x4ax4bx78x25x18x51x2ax69x6ax2cx28x31xcx79x17
x70x4ax4bx77x71x54x62x73x48x58x69x6ax68x50x31xex2excx79x78x25x78x2bx45x14xax24x70x68x44
x4bx25xbx53xcx54x6ex7a.
x33x34x32x26x27x23x34x68x50x69x6ax54x6ax3bx75? x31x32x39x65x68x54xex44x27x3c(x13x46
x3cxe 2)x48x36x7ax2x39x1cx24x44x7cx68x4bx7dx7ex48x10x69x6ax67x4bx68x50x78x39x1cx54x7cx39
x6ax3bx23x24x3bx65x68x54x8x7bx1fx68x27x23x34x48xfx7ex2x8x7bx1fx68x58x2ex4fx48x7bx1x10
x2ex2x10x74x7ax2ax67x2ex2ax54x6cx78x79x7ax8x7bx1fx68x2ex11x3ax3bx65x68x67x7dx77x54x13x14x48
xfx2x75x11x3ax3bx65x68x67x6dx6ex54x13x14x78x39x58x2ex10x54x7dx75x76x54x69x6ax68x50x6ax3b
x67x33x55x77x77x78x1x61x53x25x2bx6cx2ex11x67x6dx6ex54x13x14x48 Arrow [2]x4Pratt [16]
x2ax36x69x6ax68x50x6ax3bx3b
Au(x) = ?u
00(x)
u0(x)
x27x73x6cx78x25x67x2x21x22Arrow–Prattx28x29x78x79x37x2dx48x6fx31Arrow–Prattx7f
x14x28x29x78x79x49x42x78 Tu(x) = 1=Au(x)xexcx1x28x29x0x1x49x42x48x58Ru(x) =
xAu(x)xexcx1x76x14x28x29x78x79x49x42x78
x1x61x69x1ex25x70x3bx6cx54x79x7ax48x3ax3ex47x75x0x4ex49x42x76x3dx54x25x22x78xda1;a2;
:::;an > 0x1nx2bx3bx68x78x7ex7fx54x12x7bx36x72x62x7ax1
a1 +a2 +￠￠￠+an
n :
x36x69x36x72x62x7ax1
a21 +a22 +￠￠￠+ann
n
?1=2
:
x3x1fx36x72x62x7ax1
npa1a2￠￠￠an = exp
loga
1 +loga2 +￠￠￠+logan
n
?
:
x10x66x70x71x54x62x7ax74x1(0;+1)x54x67x4bx6dx50x65x68f x62x7ax54f-x76x3dx1x2a
Mf(a) = f?1
f(a
1)+f(a2)+￠￠￠+f(an)
n
?
;
x25x18f?1 x60x27f x54x7ax65x68x78
x1bx2x6dx2ax6cx26x65x68x54x62x7ax48x3ax3ex23x24x3cx19x48x10x66x26x65x68x27x70x48x51
f
a
1 +a2 +￠￠￠+an
n
?
6 f(a1)+f(a2)+￠￠￠+f(an)n :
150
x31x32 f x2x67x4bx6dx50x65x68x48x33x34x10x44x26x78x14x2ex3b f?1x48x10x5ax23x60x54x1x2a
(x67x4bx6dx50)x26x65x68f x67x2f-x36x72x2ex16x66x12x7bx36x72x54x65x68x78x10x66x7dx65x68
uxex23x24x70x48 (x67x4bx6dx50)x7dx65x68ux67x2u-x36x72x2ex4ex66x12x7bx36x72x54x65x68x78
x31x32x10x66x15x1fx3bx68x24x6c a = (a1;a2;:::;an), a x54 g-x36x72x5x2ex16x66
f-x36x72x48x33x34x17f x54x6x74x2(0;+1)x18x48x2exfx52x2cgx57f?1 x54x1x25x65x68
h = g–f?1 x54h-x36x72x5x2ex16x66x12x7bx36x72x78x7bx7ex48 h = g–f?1 x2x26x65x68x78
x31x32x10x8x7bx23x1fx48x33x34h00 > 0. x26x1x25x65x68x4x7ax65x68x39x1fx55xex48x23x24x39
x68
h0(x) = (g(f?1(x))0 = g0(f?1(x))(f?1(x))0 = g
0(f?1(x))
f0(f?1(x)):
h00(x) =
g00(f?1(x))(f0(f?1(x))?1f0(f?1(x))?g0(f?1(x))f00(f?1(x))(f0(f?1(x))?1
(f0(f?1(x))2
> 0:
x7bx7ex48
f00(f?1(x))
f0(f?1(x)) 6
g00(f?1(x))
g0(f?1(x)):
x26xx54x15x79x1dx48x4x68
f00
f0 6
g00
g0 :
x25x26f00=f0 x23x24x3cx2e f x54x26x54x6ax3bx54x3bx6cx48 ?u00=u0 x67x23x3cx2e ux54x7dx54
x6ax3bx54x3bx6cx78x25x67x2Arrow–Prattx52x10x69x6ax68x50x3bx6cx54x68x4fx79x7ax78
x5dx44x29x31x32x1ex41x68x4fx40x26x48x4x3ax3ex51x56x5cx4x60x2a
x59x23 6.3 x20 f;g x1dx62x59x61 (0;1) x7cx71x60x1cx61 (0;1) x7cx79x3x4x49
x7x69x6ax64x65x0x5dx5e,
8x;y 2 (0;1); 8p 2 [0;1];
f?1(pf(x)+(1?p)f(y)) 6 g?1(pg(x)+(1?p)g(y)) (2)
x26x63x6e
8x 2 (a;b); f
00(x)
f0(x) 6
g00(x)
g0(x) (3)
x61x4cx3x66x1ex37x62x4ax40x6fx62x69x54x65x63x48x3ax3ex23x24x3cx19x48x31x32x5dx1ex37
x54x4ax4bx3cx2ex70x2bx3ex2bx3ax3bx65x68x48x33x34x62x69 5.4’ x71x72x44x2x70x48x66x49x70x2b
151
x9x4ax25x27x51x3bx48x53x68x25x2bx3ex2bx3ax3bx65x68x2x69x6ax2cx1dx54x78x7bx7ex48x64x67x2c
x5ax57x7fx5dx6dx3cx62x4axcx1x28x29x44x59x36x28x78
6.5 x24x56x3ax3ex27x27x28x70xbx7fx3bx54x3ex2bx3ax3bx65x68x78x49x44x70x51x65x63x1e
x37x62x4ax65x8x40x6fx62x695.5x18x21x44x58x54x3ax3bx65x68x57x25x27x33x3x48x7bx58x5ax36x51
x44x58x3ex2bx3ax3bx65x68x78xfx2x31x32Sx2bx50x27x70x71x54x25x27x33x3fx1q1;:::;qS > 0,
(q1 +￠￠￠+qS = 1),x33x34x70x64x7fx3bx54x3ax3bx65x68x1
u(z0;z1;:::;zS) = v(z0)+–
SX
s=1
qsv(zs);
x17x2cvx1x62x7ax49R+ x44x54x67x4bx49x54x7fx0x7dx65x68x48 – 2 (0;1)x4ex60xdx13x1ax7b
x14xfx78x25x18x48 ux67x23x3cx2ex3ex2bx3ax3bx65x68x78
x40x26x1
u(x) = 1?
ax
1? +b
?
; b > 0; > 0;
x54x3ax3bx65x68xcx1x5x6x7fx14x28x29x78x79 (hyperbolic absolute risk aversion,
HARA)x72x3ax3bx65x68x78x17xcx77x49x66x17x69x6ax68x50x3bx6cru(x) = ?u00(x)=u0(x)
x9xa
Tu(x) = 1A
u(x)
= x1? + ba:
x25x2x70x2bx1cx1dx65x68x48x7bx58 HARA x72x65x68x5ax57x7fx11xcx1x74x59x28x29x0x1
(linear risk tolerance, LRT)x72x65x68x78x25x72x65x68x54xcx77x1x17x69x6ax68x50x6ax3b
x54x52x10x6x28x42xx54x52x10x6x54x4ex58x3fx5f(xdx5bx4bx2ex67x4bx10x69x6ax2ex49x15xf)x78
xfx2x17x4ex49x58x47x44x66 x2< 1x58x2> 1;xbx31x10x7xdx5bx4bx2ex67x4bx2ex7x69
x6axfx57x13x31x10x7xdx5bx4bx2ex67x4bx2ex55x53x6axfx78x26x66x25x18x1x7cx2cx34x65x68x51x62
x7ax48x58x51x70x2bx65x18x38x3dx2a b+ax=(1? ) > 0,x25x53x68xx10 < 1x51x56x5bx48
x10 > 1x51x44x5bx78x7bx7ex48x31x32x3bx2dxx47x50x6x48x13x70x9x3fx10x24x2ax74x75x27x70
x7x8x2xdx8x68x4bx2ex67x4bx2ex49x15x69x6axfx78
= 1x10x7u(x) = ax. x25x2x69x6ax2cx1dx74x40x78 = 2x2x8x6dx65x68x74x40x78
x17b = 1,x12 ! ?1, u(x) ! ?e?ax. x12b = 0, < 1, ux2x34x65x68x48x25x18
x7bx10x69x6ax68x50x65x68Ru x2x7fx681? . u(x) = logxx5ax23x24x3cx2e HARAx65
x68x54x70x64x48x25x2x7bx1logx = lim !0(x ?1)= . x7bx7ex48x10x49x11x64x79x7ax56x48
x23x3cx2e = 0x54HARAx65x68x78x17x7bx10x69x6ax68x50x65x68Ru = 1. x26x7ex23x4fx48
HARAx3ax3bx65x68x72x2x7bx17x77x54x65x68x72x48x10x24x3bx66x2dx2ex3dx3ex5fx60x54x65x63x78
6.6 x3bx3ex2bx3ax3bx65x68x27x72x73x78x2bx28x29x2bx3d (x50x62x45x46) x75x76x54x6d
x55x2x4cx4cx45x10x6dx2bx57x58x2fx30x31x58x1x54x78x66x2xfx36x7ax1ex23x20x8x25x26x54x5f
152
x60x48x25x35x10x57x58x2fx30x31x54x21x1fx27x20x8x10x26x54x5fx60x7ex31x32x1cx7ax3bx3ex2bx3a
x3bx65x68x27x65x63x48x25x79x41x42x10x70x72x3ex2bx3ax3bx65x68x27x20x8x10x26x54x5fx60x78x49
x68x4fx44x48x10x23x24x40x41x25x26x54x70x2bx5fx60x2axd X x2x70x2bx28x29x2ax6cx54x69x25x78
U x2x70x2bx6dx2ax6cx65x68x69x25x78x49X x2cx62x7ax61x70x64x44x28x3x4c<: x10x66x15x1f
x;y 2 X,
x < y () 8u 2 U; E[u(x)] > E[u(y)]:
x25x18x;y x54x6ex7ax23x24x2x78x64x69x6ax2cx68x54x20x21x27x48x58ux54x6ex7ax23x24x2x11
x2bx4fx1ex31x54x3ex2bx3ax3bx65x68x48x58U x54x6ex7ax23x24x2x11x2bx4fx1ex31x21x1fx54x3ex2b
x3ax3bx65x68x7fx1fx78x39x58x48x25x26x62x7ax54x44x28x3x4cx79x41x42x25x21x4fx1ex31x5bx0x1
x13x70x64x2cx68x2ex77xbx70x64x2cx68x7dx78x25x26x54x25x22x10x3dx3ex57x58x4fx54x79x7ax2x74
x58x16x4fx54x78x25x18x47x75x1fx79x54x2x2ax70x71x27x70x48x25x64x3x4cx2x70x64x44x28x3x4cx48
x58x2ex2x7fx28x3x4cx57x5ax67x2x70x48x51x2ex2x15x1fx78x64x69x6ax2cx68x5bx23x77x77x6dx55
x54x78x10x18x48x25x2ex1fx2x3bx3ex2bx3ax3bx65x68x69x63x27x23x34x4fx1ex31x21x54x7x28x48x25x26
x54x65x63xcx62x11x19x3ex2bx3ax3bx65x68x69x63x54x1bx2ax78x49Markowitzx69x63x2cx48 xdx20
x21x4ex48x69x6ax16xfx5ax2x10x2cx68x54x20x21x27x62x7ax61x70x64x44x28x3x4cx78x28x16x54x2x48
x25x64x44x28x3x4cx51x2ex25x28x6dx1ex30x31x1x44x29x74x40x78x10x7ex48x3ax3ex24x13x13x4ex4f
x65x63x78
x10x66x3dx3ex4fx5fx60x27x70x48x44x4bx54x28x29x2ax6cx69x25X xdx15x7x8x47x50x62xb
x21 Hilbertx72x76M. xfx2x26x66xaxbx54x5fx60x2ex44x58x62x4ax75x72x54x10x75x48x7dx3e
x57x7fx65x63x54X x67x2(x71x68x6ax44)x54x28x29x2ax6cx7fx1fx78x44x29x54x44x28x3x4cx67x11
xcx1x2ax2bx9x21 (stochastic dominance)x78x3ax3bx65x68x69x25 U xex23x47x25x26x78
x64x2a 1. x6dx50x2ex3fx65x68x72x57 2. x6dx50x2ex3fx7dx65x68x72x78x58x7bx7x54x28x29xfx6dx5a
x11xcx1x18x67x41x1fx67x2ax2bx9x21x78x31x32x5dx28x29x2ax6cx69x1ex1x2cx68(x6fx21x70x71x54
x50x62xbx21) x54x20x21x27x48x33x34x25x78x72x65x68x23x33x3fx69x1exdx20x21x4bx2ex4bx28xfx4
xdx69x6ax68x50xfx78x51x18x58x47x75x65x63xax67x2ax2bx9x21x48x10x2x5dU x1x51x43x50x60x7b
x1fx68x54x49x54x7dx65x68x72x78x52x10x69x6ax68x50x49x3fx54x3ax3bx65x68x66x66x25x70x72x48xf
x10x58x45x6ex21x2ex54x65x68x78
x3x66x28x29xfx6dx54x74x75x31x32x2x24x56x78x38x62x69x78xfx2x1x61xdx1ex70xbx68
x4fx44x54xexfx48x3ax3ex61x62Xx1x21x51x73x47x51x2ax48x76x2cx54x6x54x28x29x2ax6cx25x26x48
x10x54x68x4fx3ex2bx3x69x7dx24x58x10x70x71x54x3ax3bx65x68x54x3ex2bx3ax3bx5x66x49x78x25x70
x61x62x10x66x71x72x7x3bxax63x48x51x6cx5ax2exfx3bx70xbx7fx33x54x68x4fx26x27x5dx25x70x61
x62x47x34x78x10x18x48x3ax3ex58x75x1fx79x19
1. x10x66x70x7bx28x29xfx6dx54x3ax3bx65x68x72x1x43x50x23x51x65x68x54x2x33x48x4u(t) =
u(0)+Rt0 u0(s)ds,x6cu0 > 0.
2. x10x66x8x7bx28x29xfx6dx54x3ax3bx65x68x72x1x44x29x65x68x72x2cx54x70x2bx14x72x48x17
x2cx54x65x68 u x54x1fx68 u0 x2x43x3bx23x51x65x68x54x2x33x48x4 u0(t) = u0(0) +R
t
0 u
00(s)ds,x6cu00 6 0.
153
x36x3f 6.2 (x18x67x2ax2bx9x21x36x3f) x20 x;y x1dx0xcxbx60x73x52x78x45x7fx79
x1cx79x39x3ax3bx3cx6b F(s) = Pfx 6 sgx71G(s) = Pfy 6 sgx49xfx1dx15x5ax79xc
x5cx49x6dx64x65x0x5dx5exx7axdx39x3axex72x6eyx79x62x49x63x4ex5fx60x1d
8s 2R; F(s) 6 G(s): (4)
xdx27 x21x22xx70x7bx28x29xfx6dx66y x2x5dx10x66x44x29x65x68x72x2cx54x15x1fu
x51
E[u(x)] > E[u(y)]; x4
Z 1
?1
u(s)dF(s) >
Z 1
?1
u(s)dG(s):
xfx26x33x32x2x33x55
Z 1
?1
u(t)d(F(t)?G(t)) = u(t)(F(t)?G(t))
flfl
fl
1
?1
?
Z 1
?1
(F(s)?G(s))du(s)
= ?
Z 1
?1
(F(s)?G(s))du(s) = ?
Z 1
?1
(F(s)?G(s))u0(s)ds:
x17x2cx3ax3ex75x1fx79x19x48x26x66 F ? G x71x72x44x73x49x70x2bx51x2ax48x76x2cx2ex1x7dx48
x44x29x54x2x33x17x71x5bx2x73x49x51x2ax48x76x2cx4ex4fx54x78x25x70x9x26x10x66x15x1fx43x50
x23x51x65x68u0 x41x7dx78x47x48x19x28x29x2ax6cx54x6fx2x33x6cx65x68x5bx2x1fx7fx0x49x54x65
x68x48x75x16x2cx68x48x15x13x70x2bx60x54x26x43x50x54x1dx75x38x3dx1(4)x78 2
x26x66u(s) = sx2x9xax44x29x75x39x54x6dx50x2ex3fx65x68x48x46xx70x7bx28x29xf
x6dx66 y x54x70x2bxcx75x38x3dx1x2a E[x] > E[y]: x39x58x48x10x66x50x62xbx21x54x20x21
x27x54x77x77x48x70x7bx28x29xfx6dx6cx79x79x41x42x3ex2bx20x21x27x44x54xdxfx6dxfx78x4dx70x69
x4bx48x3ax3ex58x75x1fx79x19x48x6fx2x33x6cx65x68x54xcx77x5bx2x39?1x73x540x2ax191
x73x541. x7bx7ex48x70x2bx6fx2x33x6cx65x68x73x73x5bx2ex4ex66x4dx70x2bx6fx2x33x6cx65x68x48
x79x41x42x10x3ex54xex44x2x30x49x62xax38x540x571x75x76x54x1cx40x2cx54x39?1x73x54
0x6dx50x2ax191x73x541x54x78x38x39x1cx48x17x2cx70x38x7cx52x2ex4bx1fx4dx70x38x78
x36x3f 6.3 (x1fx67x2ax2bx9x21x36x3f) x20 x;y x1dx0xcxbx60x73x52x78x45x7fx79
x1cx79x39x3ax3bx3cx6b F(s) = Pfx 6 sgx71G(s) = Pfy 6 sgx49xfx1dx15x5ax79xc
x5cx49x6dx64x65x0x5dx5exxfxdx39x3axex72x6eyx79x62x49x63x4ex5fx60x1d
8t 2R;
Z t
?1
F(s)ds 6
Z t
?1
G(s)ds: (5)
xdx27 x12
S(s) =
Z s
?1
(F(s)?G(s))ds:
154
xeS x73x49x51x2ax48x76x2cx2ex1x7dx48x51x6cdS(s) = (F(s)?G(s))ds. x26x44x70x62
x69x54x2cx68x48x13x5x3bx33x32x2x33x55x48x23x68
Z 1
?1
u(t)d(F(t)?G(t)) = ?
Z 1
?1
(F(s)?G(s))u0(s)ds
= ?S(s)u0(s)
flfl
fl
1
?1
+
Z 1
?1
S(s)u00(s)ds =
Z 1
?1
S(s)u00(s)ds:
x26x66x25x18x54u00x23x47x15x1fx43x3bx23x51x65x68x48x7bx58x44x29x9x26x54x1fx14x43x50x54x1d
x75x38x3dx1(5)x78 2
x10x66x8x7bx28x29xfx6dx27x70x48x10x26x26x66u(s) = sx2x9xax75x39x54x6dx50x2e
x54x7dx65x68x48 E[x] > E[y] x1cx7ax2 x x8x7bx28x29xfx6dx66 y x54xcx75x38x3dx78xf
x2x26 x x8x7bx28x29xfx6dx66 y, x71x2ex25x1fx8 x x54x69x7dx77 y x54x69x7dx16x78x5ax67
x2x70x8x7bx28x29xfx6dx51x43x2 Markowitz x72x6x64x69x7dx2cxex54x1dx33x38x3dx78x2b
x71x44x48x3ax3exfx75x16x4dx48x7ax45x27x70x68x25x70x77x78x56x5cx7ax45x47x36Hanoch–Levy
[9]x2axdx28x29x2ax6cxx4yx73x470;1;2x60x2bx6x48xfx21x47x54x25x27x2ex10x78 xx47x25
x60x2bx6x54x25x27x1x2a 0.65, 0.25, 0.10;x58yx47x25x60x2bx6x54x25x27x1x2a 0.86, 0.89,
0.11. x66x2x3ax3ex51
F(0) = 0:65 G(0) = 0:86 F(0)?G(0) = ?0:21 R0?1(F ?G)ds = 0
F(1) = 0:90 G(1) = 0:89 F(1)?G(1) = 0:01 R1?1(F ?G)ds = ?0:21
F(2) = 1:00 G(2) = 1:00 F(2)?G(2) = 0 R2?1(F ?G)ds = ?0:20
x7bx1x10x15x1ft,x51S(t) = Rt?1(F ?G)ds 6 0,x46x26x62x696.3, xx8x7bx28x29xf
x6dx66y. xfx2E[x] = 0:45 > E[y] = 0:25, Var[x] = 0:4475 > Var[y] = 0:4075:
5
x4dx70x69x4bx48x3ax3ex5ax23x4dx48x7ax45x70x68x48x8x7bx28x29xfx6dx5ax2ex2Markowitz
x72x6x64x69x7dx2cxex54xcx75x38x3dx78x4dx70x2bx7ax45x31x56x2axdx28x29x2ax6c x x24 0.80
x54x25x27x47 1, x24 0.20 x54x25x27x47 100; y x24 0.99 x54x25x27x47 10, x24 0.01 x54x25
x27x47 1000. xe E[x] = 20:8 > E[y] = 19:9, Var[x] = 1568:16 < Var[y] =
9702:99:x4xx72x6x64x69x7dx6dx66y,xfx10x66x49x54x7dx65x68u(s) = log10 sx27x70x48
E[u(x)] = 0:4 < E[u(y)] = 1:02. x4xx51x50x8x7bx28x29xfx6dx66y.
x33x34 Markowitz x72x6x64x69x7dx2cxex25x35x5ax23x7ex1fx11x70x72x28x29x2ax6cx6f
x11x70x72x3ax3bx65x68x72x27x62x7ax75x7ex10x7ex48 Tobin [21]x4bx51x1fx78x2bx2ax36x78 1. x5
x3bx8x6dx3ax3bx65x68x57 2. x61x62x28x29x2ax6cx12x39x3bx71x33x6cx78x10x66x651x77x48x31x32
x58x75x39x3ax3bx65x68x2x6dx50x2ex3fx7dx65x68x48x33x34x25x2bx65x68x54x8x6dx66x4cx68xcx2c
5x16x12x5x19x1ax79x48x7axe F x23x2bx7fx2bx9x53G,x32x3ex43x5c6.2, xx23x2bx3bx12x58x53x1ax23x53y.
x2bx3bx5xbx14x19x5x4ex2dx42x22x12x58x53x1ax23x7cx32x33x55x13x38x9x23x2bx47x3bx12x58x53x1ax23x20x74x77x36
x6cx29
155
x2x50x54x48x51x6cx17x10x54x19x15x4ex6x13x48x15x4ex6x77x1fx17x54xdx56x10xdx5x1cxfx67x2e
x25x7ex3bx78x31x32x2ex47x48x25x32x33xdx56x10xdx5x1cxfx48x33x34x23x24x2cx68x48x72x6x64x69
x7dx2cxex2x10x25x72x3ax3bx65x68x28x29xfx6dx54x1dx33x38x3dx57x4x31x32 E[x] > E[y],
Var[x] 6 Var[y],x33x34x10x66x15x1fu(t) = ?at2 +bt+c, a > 0,x49ux1x49x54x7d
x65x68x54x48x76x10x48x51 E[u(x)] > E[u(y)] (x7ax60)x78x7ax58x48x7ax75x1cx7ax2ex41x7dx48
x4x72x6x64x69x7dx2cxex51x43x2x10x25x72x3ax3bx65x68x28x29xfx6dx54xcx75x38x3dx48x7bx1
xbx4bx54x65x70x2bx7ax45x49x25x64x74x40x1cx7ax51x3a(x3ax3ex23x24x47x48x49[0;2]x44x54x49
x54x7dx8x6dx3ax3bx65x68)x78x10x66x65 2 x77x48x74x75x1cx7ax72xdx78x25x18x48x3ax3ex47x48x51
Taylorx3fx6fx54x3ax3bx65x68x78x26x66x38x2cx3bx71x33x6cx28x29x2ax6cx54x10x7bx20x5bx1x7dx48
x14x7bx20x5bx2x8x7bx20x54x34x57x3bx7fx68x54x16x2x48x15x1fx6dx50x2ex3fx7dx65x68x54x3ex2b
x3ax3bx5bx72x10x66x70x2bx8x6dx3ax3bx65x68x54x3ex2bx3ax3bx78x7bx7ex48x72x6x64x69x7dx2cxe
x10x3bx71x33x6cx54x28x29x2ax6cx27x70x2x8x7bx28x29x15x6dx54x1dx33x38x3d (x11xbx68x4fx2f
x51x78x47x2cx68)x78xfx7ax75x58x2x2ex41x7dx78x25x73x47x47x48x70x2byx54x6fx2x33x6cx65x68
x73x73x4ex66xx54x6fx2x33x6cx65x68x54x45x14x4x23x78x26x62x69 6.2 x4 6.3, x25x18xx70
x7bx3x8x7bx28x29xfx6dx66y,xfx51x2ex25x1fx68xx72x6x64x69x7dx6dx66y,x7bx7ax48x3ax3e
x23x24x53xx54x69x7dx15x79x4ex78x5x75x48x22x23Markowitzx6fx7dx4bx57x22x43x55x53x62x54
x72x6x64x69x7dx2cxex55x1x3ex2bx3ax3bx65x68x54xex4cx48x31x44x21x29x48x72x6x64x69x7dx2c
xex71x72x44x2x2ex23x25x7ex7fx55x1x25x70x11x2ax54x78x3bx31x3ax3exbx3x51x2cx21x19x57x19
x68x54x48 Markowitzx69x63x17x71x73x2x1cx1dx62x4ax55xex54x70x64x60x54x40x26x48x58x1cx1d
x62x4ax55xex51x2ex47x75x3ex2bx3ax3bx65x68x69x63x78
x2fx6a x27x53x10x6ex24x30x25x20x4fx13x34 von Neumann–Morgenstern x6fx4ax7ax32x60x2exf
x21x50x47x36x70x3x4x20x4fxd [25]x29xbx18x3bx5bx60x47x52x1ax58x1cx20x5x71x17x5dx3bx2ex28x30x58x59x33x20
x7x64x2fx60x2cx4xdx28x1ax29x19x1ax4ex48x42x62x7cx77x58x1cx2ex4a Bernoulli [2] x34 von Neumann–
Morgenstern [24]x29 x37x38x41x2dx25 x73x11x20x3exdx48x6Kreps [15]x7x5dxex29xbx1x11x20x5x6bx4e
x2bx61x29 Fishburn [6]x4fx47x3bx2ex7fxex52x1ax73x11x25 x30x1x24x36x41x45x20x4dx2fx5 x35x7x4ex61x10x2bx61
x25x27 Kahneman–Tversky x5cx4fx20x3bx5bx29x7dx53 Kahneman–Tversky x20x41x42x10x60x13x5c
x4f (Prospect Theory)x13x5 x48x42x62x7cx7axex1ax20x58x1cx4fx2e [13]x5xbx3dx4fx2ex6ax2dx6ex58xax37
Econometrica (x56x3bx58x59x33x6cx6d)x25x49x3x4dx46x41x72x4cx32x20x26x20x4fx2ex29xax35x47x28x6exax37
x7cx37x11x12x33x20x60x1bx29x52x53x7bx48x33x4dx2dx5xex1ax20x5cx4fx20x25x5ex5ex2ax48x7cx7a Kahneman–
Riepex531998x2x24x29x20x3bx3dx3dx25x7fx45x20x4fx2ex27x7bx48x70x76x5cx33x20x71x14x1x11x28([14])x29
x27x53von Neumann–Morgensternx55x13x6fx4ax7ax32x5cx4fx5x19x1ax20x5bxdx69x7dx60x2exf
x41x45Varian [21]x5 x3cx6ex63x6ax4x10x2ex4aHerstein–Milnor [9]x7bx7cx47x5bx74x29 x35xex47x63x6ax3b
x18x49x65x56x3ex24x20x19x1ax45x46x20x5cx20x29 Huang–Linzenberger [10], Ingersoll [11], Jarrow
[12] x4ex60x25x25x27x4fxdx5 x35x7x3 [10] x20x5bxdx20x1cx5x5 [12] x20x3ex1bx29 x3cx47x1cx19x60x46x4ex4f
x3dx55x71x20x76x35xbx69x29x3bx56x20xex41x60x14xfx20 von Neumann–Morgenstern x6fx4ax7ax32x20
x2cx6x3axcx30x52x1ax50x6cx29 [6]x20x12x3bx6bx20x20x26x25xbx24x2cx6x43x5cx20x9x14x6fx2ex29
Arrow–Pratt x3dx30x22x23x58x3bx36x37xcxdx6fx4ax7ax32x20x29x2ax37x58x20x10x78x2bx2bx62x3bx13x6e
x58x46x1ax22x0x4ax29x6x306.3 x21x2cx47x28x18x7ax32x20x7ex3ax23x14x20x27x28x29xbx3bx41x5cx2cx2ax6Pratt
[16]x7x5x7ex53x79x25x3bx18xfx57x7ax32x7ex3ax20x35xex36x6cx29 Ingersoll [11]x20x121x6bx7x20x43x5c5
156
x4ex47x24x16x20xex44x29x6x5fx26x2ex4ax7x2cx2ax20 HARA x6fx4ax7ax32x24x4ex48x6 Ingersoll [11] x20
x121x6bx7x5dxex29
x58x53x1ax23xcx30x6Huang–Linzenberger [10]x7x4ax7ex7ex3bx6bx4dx7bx7cx4ex4fx29 Ingersoll
[11]x28x1x1x3ex3fx2ex39x29 Jarrow [12]x28x72x73x24x25x2exex29x19x1ax20x5bxdx5bx2cx63x6aHanoch–
Levy [8]x5x4ex37x38x62x57x53Ingersoll [11]x7x20x5bxdx29 x8x2cx3bx7fx20x47x5x19x1ax20x22x12x58x53x1a
x23x20x43x46x18[10]x7x2bx16x29x6[10]x7x5x22x12x58x53x1ax23x1cx51x2cx2cx9x14x58x53x62x3bx20x32x33x55
x13x52x16x29xbx17x20x43x46x47 Rothschild–Stiglitz [18], [19] x41x2ex2cx20x29x28x48x3cx50x22x12x58x53
x1ax23x47x5ex13x7cx1x32xcx28x20x74x77x36x6cx29x6x7ex53x5x36x70x4ex72x42x47x3bx14x58x53x62x3bx48x29x68x37
x3bx14x22x12x58x53x1ax23x53x28x20x58x53x62x3bx18x3bx14x10x2cx70x13x20x34x29 Rothschild–Stiglitzx20xbx9
x3dx21x2cx63x36x5x52x50x4bx58x59x33x20x24x60x25x3dx9x2ex2fx29
x44x30x23x41x45x23
1. x37x34x2xdx3dx17x6bx2bx67x63xfx7ex3ax10x10x51x37x34x3cx55x7e
2. x37x34x2 von Neumann–Morgenstern x3ax3bx65x68x7ex10x54x74x75x1dx60x2x37
x34x7ex10x2x16x5exbx3x38x6ex69x2ax7dx14x27x54x7e
3. x51x61von Neumann–Morgenstern x3ax3bx65x68x24x13x48 xdx3dx17x6bx2bx67x63xf
x2x35x67x2ex4dx5fx36x7ex76x47x48x70x2bx17 x ! +1 x18x48 u(x) ! +1 x54x3e
x2bx3ax3bx65x68x48x51x26x7ex4dx48x70x43x3bux27x73x6cx20x21x54xdx4bx48x10x1cx7ax51
xdx3dx17x6bx2bx67x63xfx54xcx39x78
4. x3ax10Allaisx67x63x54x45x14x32x26x72x73x7ex1x37x34x70x10x57xdx7cx7dx1dx6ex69xfx5d
x23x7e
5. xdx1ax62x68 800 x38xfx57xdx24 85% x54x25x27x5bx68 1000 x38xfx78x31x3ax7x8xb
x70x2bx7ex51x7dx7x8xbx31x48x25x70x68x62x54x3ex2bx3ax3bx65x68x51x37x34xcx77x7e
6. x39x56x5cx3ex2bx3ax3bx65x68x54Arrow–Prattx52x10x69x6ax68x50x65x68x2a
u(x) = x? b2x2; ?e?bx; blogx;
x17x2cb > 0x2x7fx68x78
7. xdux2x9xau0(x) > 0, u00(x) < 0x54x3ex2bx3ax3bx65x68x78x2cx68x48x31x32x10
x54x52x10x69x6ax68x50x65x68Au x9xaA0u(x) 6 0,x33x34u000(x) > 0.
8. x37x34x2x28x29xfx6dx7ex10x51x37x34x57x58x4fx6ex7ax7e
9. xdx28x29x2ax6c x x4 y x9xa E[x] > E[y], Var[x] 6 Var[y], x76x2cx48x10x66
x15x1fu(t) = ?at2 + bt + c, a > 0,x49ux1x49x54x7dx65x68x54x48x76x10x48x51
157
E[u(x)] > E[u(y)]. x70x71x74x75x56x48x23x2cx48x17a;b > 0x18x48 E[x] > E[y],
Var[x]+(E[x])2 6 Var[y]+(E[y])2,x23x1fx8E[u(x)] > E[u(y)]:
x2fx30x31x32
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2001x48x41x42x48 196–219)x78
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