167 p (n) im p(n+1) im p (n+1) im p(n) im < ε (3.264) ε p (n+1) im 1 4 [1],, :, Vol.43, pp.14-21, 2001. [2] Rust, J., Optiml Replcement of GMC Bus Engines: An Empiricl Model of Hrold Zurcher, Econometric, Vol.55,pp.999-1033, 1987. [3] Hotz, V. J., nd R. A. Miller, Conditionl Choice Probbilities nd the Estimtion of Dynmic Models, Review of Economic Studies, Vol.60, pp.497-529, 1993 [4] V,Aguirregbiri., P,Mir., Swpping the Nested Fixed Point Algorithm: A Clss of Estimto for Discrete Mrov Decision Models 3.3 3.3.1 ().. TDM ITS.
168. W rdrop. W rdrop ( ).. (b).. 1.. 2. ( ) OD W rdrrop. 3.4. r s x t f c q δ, 3.4 ( ) OD OD OD { δ, = 1 if OD 0
3.3 169 W rdrop (3.65). ( c u ) 0 c u = 0 (3.265) u OD. (3.65). OD q x. (3.66). min z (x) = x 0 t (ω) dω (3.266) x x = (..., x...). (3.70) (3.71). f = q (3.267) f 0 (3.268) (3.72) (3.73). x = r f δ, (3.269) s c = t δ, (3.270) (3.72) OD (3.73).
170 (c) (3.66) (3.70) (3.73). L (f, u) = z [x (f)] + u ( q f ) (3.271) f f = (..., f,...). u OD u = (..., u,...). (3.71). (3.73) (3.272). f L (f, u) = 0 nd f L (f, u) f 0 (3.272) (3.273). L (f, u) u = 0 (3.273) (3.273) (3.70) f (3.274). f mn l L (f, u) = f mn l z [x (f)] + f mn l (3.274) (3.275). ( u q ) f (3.274) f mn l z [x (f)] = z (x) x b x b f mn l (3.275) (3.275). (3.276). z (x) x b = x b x 0 t (ω) dω = t b (3.276)
3.3 171 (refue5) (3.275) (3.277). x b f mn l = δ mn b,l (3.277) (3.276) (3.277) (3.275) (3.73) (3.278). z [x (f)] f mn l b t b δ mn b,l = c mn l (3.278) OD mn l. (3.274). (3.279). f f mn l = { 1 if r = m, s = n nd = l 0 (3.279) q u fl mn. (3.274) (3.280) f mn l ( u q f ) = u mn (3.280) (3.278) (3.280) (3.274) (3.281). f mn l L (f, u) = c mn l u mn (3.281) (3.281) (3.272) (3.273) (3.71) (3.282). ( f c u ) = 0 c u 0 f = q f 0 (3.282) (3.282) (3.66) (3.70) (3.71).
172 3.3.2 () W rdrop. W rdrop. W rdrop. W rdrop... (3.283) (3.285). min z (x) = x t (x ) (3.283) (b) f = q (3.284) f 0 (3.285). ( ).. 3.65 6. q = 6. (3.286).
3.3 173 t 1 [x 1 ] = 50 + x 1 t 2 [x 2 ] = 50 + x 2 t 3 [x 3 ] = 10x 3 (3.286) t 4 [x 4 ] = 10x 4 3.65. O 1 4 D O 3 2 D 50 + 3 + 10 3 = 83 498. 3.66 5. 5 (3.287). t 5 = 10 + x 5 (3.287) 3.66. O 1 4 D O 3 2 D O 3 5 4 D 92 552 5.. 3.3.3 ()...
174 3.65 : 3.66 :. W rdrop... (b) ( )..OD V (3.288). V = c (3.288)
3.3 175. OD K P P (3.289). = exp ( ) θ c exp ( θ c ) (3.289) K θ. θ. θ θ. OD f f (3.291). = q P = q exp ( ) θ c exp ( θ c ) (3.290) K (3.291) (3.292). min z (x) = x 0 t (ω) dω Ω 1 θ q K ( f ln f ) q q (3.291) f K = q f 0 (3.292) Ω OD. (3.293). x = Ω f K δ, (3.293)
176 3.3.4.. F rn W olfe. () F rn W olfe F rn W olfe.. u n n OD. 0: x 0 = 0 t0. ll or nothing ( 0 ) ( )x 1. n = 1. 1: x n tn. OD u n. 2: t n OD y n.dn = yn xn d n. 3: (3.294) x n+1 (3.66) α n. (3.295) α n. α n x n+1
3.3 177. t n+1 u n+1. OD x n+1 = x n + αn d n = x n + αn ( y n xn ) (3.294) min 0 α 1 4: x n +αn (y n xn ) 0 t (ω) dω (3.295) (3.296) ( ) n = n + 1 1. x n+1. (3.296) κ. u n+1 u n u n+1 κ (3.296) 3.3.5 F rn W olfe. (MSA; MethodofSuccessiveAverges) SimplicilDecomposition. ().. (Dil )..
178 0: x 0 = 0 t0. ( )x 1. n = 1. 1: x n tn. 2: t n OD y n.dn = yn xn d n. 3: α n (3.297). (3.294) x n+1. t n+1 4: α n = 1 n + 1 (3.297). (3.298) ( ) n = n + 1 1. x n+1. (3.298) κ. ( x n+1 x n x n x n (3.299). ) 2 < κ (3.298) x n = xn + xn 1 + x n 2 + + x n m+1 m (3.299)
3.3 179 4 n n x n n + 1 xn+1. (3.299) m. (b) (=).. (3.291) 2 ( ) (3.300). ) q ( f q ln f q Ω K = { x r ln xr + ( ) ( ln r R j N I j x r I j x r )} (3.300) x r ( r ) R N I j j. (3.300).. 0: x 0 = 0 t0. OD ( )x r,1 n = 1.. 1: x n = tn. r R x r,n
180 2: t n OD y r,n.d r,n = y r,n x r,n d r,n. 3: α n (3.302). x r,n+1 = x r,n + α n d r,n (3.301) x r,n+1 α. (3.303) α. min 0 α 1 r R 4: (c) x n+1 t 0 (ω) dω ( x r,n + α n d r,n 1 + { ( x r,n θ j N I j { ( ln x r,n I j ) ln ( x r,n ) } + α n d r,n ) } + α n d r,n ) + α n d r,n (3.302) (3.296) ( ) n = n + 1 1. x n+1 SimplicilDecomposition. SimplicilDecomposition... 0: OD ˆK.
3.3 181 x 0 = 0 t0. ˆK. ll or nothing f,1 x 1. n = 1. 1: x n tn. ˆK. 2: f n+1 = (f,n+1 ) (3.303). min f n+1 Ω A x n+1 0 t (w) dw 1 θ q ˆK ( ) f,n+1 q ln f,n+1 q (3.303) (3.304). f,n+1 0 f,n+1 ˆK = q x n+1 = Ω f ˆK δ, (3.304). 2-0: g,1 = f,1 y 1 = x1. m = 1. 2-1: (3.305) t m t m = t. ( ) y m (3.305)
182 t m OD h, z m (3.306) (3.307). ( ) h,m = q z m exp ˆK exp = Ω θ ( A θ e,m d m. δ, tm A h,m ˆK δ, tm ) (3.306) δ, (3.307) (3.308) (3.309) e,m = h,m 2-2: g,m (3.308) d,m = z m ym (3.309) η m (3.310) (3.311). g,m+1 = g,m + η m e,m (3.310) y m+1 = y m + ηm d m (3.311) g,m+1 y m+1 η. (3.312). min 0 η m 1 Ω A 1 θ q y m +ηm d m 0 t (w) dw ˆK ( g,m,m +η m e ln g,mq,m +η m e q ) (3.312)
3.3 183 2-3: g m = h m f,n+1 = g,m+1 x n+1 = y m+1 3. m = m + 1 2-1. 3: (d) n = n + 1 1. x n+1 f,n+1. (Genetic Algorithm) ( GA ) DNA 1 ( ) GA 3.67 ( ) ( ) GA 3 0 : 10 3 (2, 3, 5) 3.68 10 2 (3, 7) STEP2 ( )
184 STEP0: STEP1: STEP2: No STEP3: 3.67 Yes GA
3.3 185 1 2 2 3 3 5 1 2 3 4 5 6 7 3.68 8 9 10 11 12 / encode decode 1 1 : (t ) 1 STEP2: 3 ( )
186 2 ( ) 2 1 1 2 2 3.69 1 0 1 0 1 1 0 1 1 A B C D E 1 A 2 3 D 5 2 1 2 3 4 5 2 1 B C 4 E 3.69 2 2
3.3 187 3 : t t + 1. GA 2 GA GA ( ) GA GA
188 (e) 3.70. (3.313). [ t 1 = 10 1 + 0.15 ( x 1 ) ] 4 [ 2 t 2 = 20 1 + 0.15 ( x 2 ) ] 4 [ 4 t 3 = 25 1 + 0.15 ( x 3 ) ] 4 (3.313) 3 F rn W olfe 3.71 3.73. x 1 = 3.58 x 2 = 4.62 x 3 = 1.81. κ = 0.001. 3.70 3.71 F rn W olfe : 0 (f) Bell nd Cssir(2002)[5] n n
3.3 189 3.72 F rn W olfe : 3.73 F rn W olfe : 1 2 3, 4 1 OD n (i) Wrdrop h j = 0 g j (h) > min g (h)for ll pths j (3.314) h j > 0 g j (h) = min g (h) = g OD (h) (3.315) h g n h j = pj n h j = 0 p j = 0 p j j g j (h ) OD g OD (h) h p j = 0 (ii) n s π j π j j
190 s = j π j p j (3.316) p j j n s c = j p j c (s, π j ) (3.317) c (s, π j ) j n-1 s ( ) j p 1j = p 2j =... = p nj = p j (3.318) c 1 (s 1, π 1j ) = c 2 (s 2, π 2j ) =... = c n (s n, π nj ) (3.319) h p n j h j c (s, π j ) = g j (h) (3.320) j n h j n p j c (s, π j ) > min c (s, π ) = 0 (3.321) p j > 0 c (s, π j ) = min c (s, π ) (3.322) (Nsh, 1951) n
3.3 191 Networ demon n+1 demon demon 1 demon n+1 G 1 G 1 : solve simultnenously c (s, q) = p j q c (s, π j ) for ech networ user, (1,..., n) j c n+1 (s, q) = q c n+1, (s ) for the demon plyer n + 1 (3.323) q c (s, π j ) j n-1 s c n+1, (s) demon n s c n+1, (s) c n+1, (s) = p j c (s, π j ) = j c (s) (3.324) (Nsh(1951)) n B 1
192 B 1 : solve simultnenously U : mx c (s, q) = q g j (h) subject to q = 1, q 0 q j L : min c n+1 (s, q) = vu(h) q t u (x)dx (3.325) h u 0 subject to v u = uj h j, h j = n, h 0 j j g j (h) h j t u (v u ) u uj j u 1 0 n B 1 G 1 B 1 demon ( ) F or U : q = 0 j q > 0 j F or L : h j = 0 j h j > 0 j g j (h)h j < mx r g j (h)h j = mx r g j (h)q < min r g j (h)q = min r g jr (h)h j (3.326) j g jr (h)h j (3.327) j g r (h)q (3.328) g r (h)q (3.329) n g j (h)h j = cn+1, (s) (3.330) j
193 n q = 0 c n+1, (s) < mx c n+1,r (s) r (3.331) q > 0 c n+1, (s) = mx c n+1,r (s) r (3.332) n j n g j (h)q = c (s, π j )q = c (s, π j ) (3.333) h j = pj n = p j n p j = 0 c (s, π j ) < mx c (s, π r ) r (3.334) p j > 0 c (s, π j ) = mx c (s, π r ) r (3.335) n n+1 (Nsh, 1951)[6] n B 1 n+1 G 1 [1] : I,, 2003 [2] : 2,, 2006 [3] Yosef Sheffi: Urbn Trnsporttion Netwo: Equilibrium Anlysis With Mthemticl Progrmming Methods, Prentice Hll, 1985 [4] : D Vol.62 No.4 pp505-518 2006.10 [5] Michel G. H. Bell, Chris Cssir: Trnsporttion Reserch Prt B,Vol.36,pp.671-681,2002. [6] Nsh,J.:Non-coopertive gmes:annls of Mthemtics,Vol.54,pp.286?295,1951.