8 OR (a) A A 3 1 B 7 B (game theory) (a) (b) 8.1: 8.1(a) (b) strategic form game extensive form game 1
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1 8 OR (a) A A 3 1 B 7 B (game theory) (a) (b) 8.1: 8.1(a) (b) strategic form game extensive form game 1
2 2 [5] player 2 1 noncooperative game 2 cooperative game ( ). A B A B B A 1, 1 8,, 8 4, 4 2 A B A B A B prisoners dilemma B A ( 1, 1) ( 8, 0) (0, 8) ( 4, 4) bimatrix 2 A B payoff payoff matrix A B 2 (a) A (b) B B B 1 8 A 1 < 0 A Λ Λ 8 < 4 0 4
3 Excel OR 8 3 A (a)b A A B A B A (b)a A B ( 4, 4) ( 1, 1) A B A, B =, Nash equilibrium A B A B B A B A, B =, A B 3 A, B =, () B F A I M B D5:E6 A K5:L6 B B5:B6 A B5 B6 =1-B5 D3:E3 B F5:F6 B A F5 =SUMPRODUCT(D5:E5,K$3:L$3) F6 D3 E3 B =K3 =L3 B3 A =SUMPRODUCT(B5:B6,F5:F6) B I5 I6 A =B5 =B6 K7:L7 A B K7 =SUMPRODUCT($I5:$I6,K5:K6) L7 I3 B =SUMPRODUCT(K3:L3,K7:L7) B5 0 K3 L3 0 1 A B J.F.Nash
4 4 8.2: A Excel B3 B5 GRG 2 B3 0-1 LP LP GRG 3. A B B 4. B5 1 K3, L A 5. 50%B5 K3 L best response strategy pure strategy 50% mixed strategy
5 Excel OR willingness to pay; WTPA WTP B 2 8.1(a) (a) A B B A A A B B 0 A (a) A A B A 1 A,B, 8.1: (a) (b) WTP B WTP (1, 1) (8, 0) A ( 1, 1) ( 1, 0) WTP B WTP (3, 1) (10 0) A ( 1, 3) ( 1, 0)
6 6 A B (b) (a) A A B (b) A B 8.1(a) (b) A B (a) A B (b) A B A B A B 8 4 W.S.Vickrey mechanism design Yahoo! second price auction max-min Excel 2
7 Excel OR () %, 40%, 20% Nash (0,0) (3,-3) (-6,6) (-3,3) (0,0) (6,-6) (6,-6) (-6,6) (0,0) zero-sum two-person game (p g, p c, p p ) (q g, q c, q p ) g c p p g q c ) p g + p c + p p = 1, p g, p c, p p 0; q g + q c + q p = 1, q g, q c, q p 0 2 (p g, p c, p p ) = ( 1 3, 1 3, 1 3 ) 1 3 (p g, p c, p p ) = (0, 1, 0) Π Π (p g, p p, p c q g, q p, q c ) = ( 0 q g +3 q c 6 q p )p g +( 3 q g +0 q c +6 q p )p c +( 6 q g 6 q c +0 q p )p c (q g, q c, q p ) (p g, p c, p p ) LP P (q g, q c, q p ) max p g,p c,p p Π (p g, p c, p p q g, q c, q p ) s.t. p g + p c + p p = 1, p g, p c, p p 0
8 8 (q g, q c, q p ) Π (p g, p c, p p ) (q g, q c, q p ) Π (p g, p c, p p ) LP P (p g, p c, p p ) max q g,q c,q p Π (q g, q c, q p p g, p c, p p ) s.t. q g + q c + q p = 1, q g, q c, q p 0 ((p g, p c, p p), (qg, qc, qp)) (p g, p c, p p) LP P (qg, qc, qp) (qg, qc, qp) LP P (p g, p c, p p) 1 ((p g, p c, p p), (q g, q c, q p)) (q g, q c, q p) ((p g, p c, p p), (q g, q c, q p)) Excel B G J N D5:F7 L5:N7 B5:B B8 =SUM(B5:B7) D3:F3 G5:G7 G5 =SUMPRODUCT(D5:F5,L$3:N$3) G6:G7 D3;F3 L3:N3 D3 =L3 M3:N3 B3 =SUMPRODUCT(B5:B6,F5:F6) J5:J7 A J5 =B5 J6:J7 L8:N8 L8 =SUMPRODUCT($J5:$J7,L5:L7) M9:N8 J3 =SUMPRODUCT(L3:N3,L8:N8) 2. (p g, p c, p p) = (q g, q c, q p) = (0, 1, 0) B5:B7 L3:N3 0,1,0 B3 J B3 B5:B7
9 Excel OR : Excel B8 = 1 GRG LP LP GRG (p g, p c, p p) = (q g, q c, q p) = ( 1 3, 1 3, 1 3 ) B5:B7 L3:N3 =1/ %,40%,20%(p g, p c, p p) = (q g, q c, q p) = (0.4, 0.4, 0.2)? B5:B7 L3:N3 0.4,0.4,0.2 40%,40%,20% %,40%,20% max-min max-min J.C.Harsanyi R.Selten Nash
10 ( ). A B A 2 B 5 A B 1 A 4 B 5 A B 2 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q p 1 1 (0, 0) ( 2, 2) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) p 2 2 (2, 2) (0, 0) ( 2, 2) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) p 3 3 ( 1, 1) (2, 2) (0, 0) ( 2, 2) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) p 4 4 ( 1, 1) ( 1, 1) (2, 2) (0, 0) ( 2, 2) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) p 5 5 ( 1, 1) ( 1, 1) ( 1, 1) (2, 2) (0, 0) ( 2, 2) (1, 1) (1, 1) (1, 1) (1, 1) p 6 6 ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) (2, 2) (0, 0) ( 2, 2) (1, 1) (1, 1) (1, 1) p 7 7 ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) (2, 2) (0, 0) ( 2, 2) (1, 1) (1, 1) p 8 8 ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) (2, 2) (0, 0) ( 2, 2) (1, 1) p 9 9 ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) (2, 2) (0, 0) ( 2, 2) p ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) (2, 2) (0, 0) A k p k A (p 1,..., p 10 ) p p 10 = 1, p 1,..., p 10 0 p 2 = 1, p 1 = p 3 = = p 10 = 0 A 2?? B (q 1,..., q 10 ) q q 10 = 1, q 1,..., q 10 0 A Π A Π A (p 1,..., p 10, q 1,..., q 10 ) := (2p 2 p 3 p 4 p 10 )q 1 +( 2p 1 + 2p 3 p 4 p 10 )q 2 + +(p 1 + p p 8 2p 9 )q 10 B (q 1,..., q 10 ) A Π A (q 1,..., q 10 ) B A
11 Excel OR 8 11 (q 1,..., q 10 ) A max-min max-min strategy { { } } q q 10 = 1, p p 10 = 1, max min Π A (p 1,..., p 10, q 1,..., q 10 ) : : (p i) (q j) q j 0, j = 1,..., 10 p j 0, j = 1,..., 10 A min (qj) Π A (p 1,..., p 10, q 1,..., q 10 ) A Π A A B B A (p 1,..., p 10 ) Π A (p 1,..., p 10, q 1,..., q 10 ) (q 1,..., q 10 ) a 1 := 2p 2 p 3 p 4 p 10, a 2 := 2p 1 +2p 3 p 4 p 10,..., a 10 := p 1 +p 2 + +p 8 2p 9 min (q j) Π A (p 1,..., p 10, q 1,..., q 10 ) = min (q j) a 1 q a 10 q 10 s.t. q q 10 = 1 q 1,..., q 10 0 LP LP min{a 1,..., a 10 } (8.1) max (p i ) min{2p 2 p 3 p 4 p 10, 2p 1 + 2p 3 p 4 p 10,..., p 1 + p p 8 2p 9 } s LP (8.1) max s,(p i) s s.t. s 2p 2 p 3 p 4 p 10 s 2p 1 + 2p 3 p 4 p 10. s p 1 + p p 8 2p 9 p 1 + p p 10 = 1 p 1, p 2,..., p 10 0 Excel s 2 s +, s s = s + s, s +, s 0 max-min LP max s +,s,(p i ) s + s s.t. s + s 2p 2 p 3 p 4 p 8 p 9 p 10 s + s 2p 1 +2p 3 p 4 p 8 p 9 p s + s p 1 +p 2 +p 3 +p 4 + 2p 8 +2p 10 s + s p 1 +p 2 +p 3 +p 4 + +p 8 2p 9 p 1 +p 2 +p 3 +p 4 + +p 8 +p 9 +p 10 = 1 s +, s, p 1, p 2,..., p Excel LP(8.2) A max-min (8.2)
12 12 1. LP(8.2) 8.4 Excel B2:M2 s +, s, p 1, p 2,..., p 10 B3 =B2-C2 C3:C12 (8.2) C3 =SUMPRODUCT(D$2:M$2,D3:M3) C4:C12 N2 =SUM(D2:M2) p 1 + p p B3 B2:M2 C3:C12 >= B3 N2 = 1 LP (a) Excel (b) 8.4: max-min [1] 1 LP A maxmin B B max-min 2 max-min (p 1, p 2, p 3, p 4, p 5, p 6,..., p 10, s +, s ) = (0.0625, , 0.25, , , 0,..., 0, 0, 0) 0 max-min max-min ( ) maxmin ((p g, p c, p p), (q g, q c, q p)) = ((0.4, 0.4, 0.2), (0.4, 0.4, 0.2)) (p g, p c, p p ), (q g, q c, q p ) max-min { { } } q g + q c + q p = 1 p g + p c + p p = 1 max min Π (p i) (q j) (p g, p c, p p, q g, q c, q p ) : : q g, q c, q p 0 p g, p c, p p 0
13 Excel OR 8 13 LP max s +,s,(p i ) s.t. s + s s + s 3p c + 6p p s + s 3p g 6p p s + s 6p g + 6p c p g + p c + p p = 1 s +, s, p g, p c, p p 0 (p g, p c, p p ) = (0.4, 0.4, 0.2) max-min (q g, q c, q p ) = (0.4, 0.4, 0.2) min (PK ). PK or or or 6 or or 3 PK max-min (0,1 ) (1,0 ) (1,0 ) (0.3,0.7) (0.9,0.1) (0.9,0.1) (0.8,0.2) (0,1 ) (0.9,0.1) (0.8,0.2) (0.1,0.9) (0.9,0.1) (1,0 ) (1,0 ) (0,1 ) (0.8,0.2) (0.7,0.3) (0.6,0.4) max-min max-min max-min max-min PK
14 A B 2 B (1) (2) A (1) (4, 4) (0, 8) (2) (8, 0) (2, 2) (1) 2. (2) 3. - (1) 4.- (2) 5. (1) % 50%
15 Excel OR : (a) F3:G4 A K3:L4 B 6. B11:C210 U(0, 1) 8.5(b) (a) E C vs. 6 E 1 tt E11 1 F 6 F11 =IF($B11<0.5,1,2) 4. A G11 =INDEX($F$3:$G$4,E11,F11) B H11 =INDEX($K$3:$L$4,E11,F11) E11:H11 E12:H vs. 6 J M J 2 1 K M (a) (b) 2 8.5: vs O11 1 O12 =IF(P11=1,1,2) O13:O210 O12 P R 2 K N 11
16 vs T vs Y Y12 =IF(PRODUCT(Z$11:Z11)=1,1,2) Y12:210 PRODUCT( ) PRODUCT(Z$11:Z11) : AVERAGE 8.7:
17 Excel OR !? : 8.4 cooperative game
18 ε ( ) coalition characteristic function 3 L J I {L}{L,J} 3 x L, x J, x I 3 10 x L + x J + x I = 10 (8.3) 1 x J 0.8 x J < x L x L + x J x L 1.0 x J 0.8 x I 0.7 x L + x J x L + x I x J + x I S {L, J, I} (x L, x J, x I ) x j v(s) j S v S v(s) (x L, x J, x I ) x L + x J + x I = v({l, J, I}) (= 10) S {L, J, I} x j v(s) j S
19 Excel OR 8 19 (x L, x J, x I ) core : 3 v 8.8 ε- ε-core ε x L + x J + x I = v({l, J, I}) (= 10) C(ε) := (x L, x J, x I ) : S {L, J, I} ε + x j v(s) j S ε = 0 C(0) ε- ε C(ε) ε C(ε) ε C(ε) ε- least ε-core LP min. x,ε ε s.t. ε + j S x j v(s) S N j N x j = v(n) (8.4) N := {L, J, I} 3 LP min x max S v(s) x j : S N : x j = v(n) j S j N e(s) := v(s) j S x j S excess demand, regret x S S N LP 8.4 x N ε-
20 (a) Excel A S B v(s) D8:F8 x L, x J, x I G8 ε D2:G7 LP 2. C2:C8 LP(8.4) C2 =SUMPRODUCT(D$8:G$8,D2:G2) C3:C7 ε + j S x j C8 =SUM(D8:F8) j N x j 3. H2:H7 e(s) H2 =B2-C2+G$8 H3:H7 4. G8 D8:G8 B2:B7 <= C2:C7 B8 = C8 LP (a) Excel (b) 8.9: 5. (a), (b) S e(s)
21 Excel OR 8 21 (a) ε- L J I (b) S L J I LJ JI IL e(s) ε (4 ). T Y K A Jay-G J 4 S v(s) S v(s) S v(s) S v(s) T 0 Y 3 A 1 J 0.5 T,Y 4 T,A 1.5 T,J 5 Y,A 2 Y,J 2.5 A,J 3 T,Y,A 7 T,Y,J 8 T,A,J 8.5 Y,A,J 6 T,Y,A,J 10 T, Y, A, J x T, x Y, x A, x J ε- LP min. x,ε ε s.t. ε + x T 0.0 (= v({t})) ε + x Y 3.0 (= v({y})) ε + x A 1.0 (= v({a})) ε + x J 0.5 (= v({j})) ε + x T + x Y 4.0 (= v({t, Y})) ε + x T + x A 1.5 (= v({t, A})) ε + x T + x J 5.0 (= v({t, J})) ε + x Y + x A 2.0 (= v({y, A})) ε + x Y + x J 2.5 (= v({y, J})) ε + x A + x J 3.0 (= v({a, J})) ε + x T + x Y + x A 7.0 (= v({t, Y, A})) ε + x T + x Y + x J 8.0 (= v({t, Y, J})) ε + x T + x A + x J 8.5 (= v({t, A, J})) ε + x Y + x A + x J 6.0 (= v({y, A, J})) x T + x Y + x A + x J = 10.0 (= v({t, Y, A, J})) ε- (x T, x Y, x A, x J ) = (4.75, 2.25, 0.25, 2.75) 0.75 H ε-
22 ( ). Akiko Bob Cecile A 3,500 B 5,000 C 10,000 A,B 8,000 B,C 12,000 C,A 12,000 A,B,C 14,000 ε- A, B 2 {A,B} v({a, B}) A B 3, 500+5, 000 = 8, , S v(s) S v(s) S v(s) A 0 B 0 C 0 A,B 500 B,C 3000 C,A 1500 A,B,C x A, x B, x C ε- LP min. x,ε ε s.t. ε + x A 0 ε + x B 0 ε + x C 0 ε + x A + x B 500 ε + x B + x C 3000 ε + x A + x C 1500 x A + x B + x C = (x A, x B, x C ) = (750, 750, 3000) A 2750 B 4250 C 7000
23 Excel OR 8 23 Shapley Shapley Shapley value Shapley L J I 3 L J I3 {L, I, J} {L} {L, I} {L, I, J} L = 1.0 J = 0.8 I = 8.2 I L J S {I} {L, I} {L, I, J} +I +L +J v(s) v(s) ! = = 6 n n! L J I L J I L I J J L I J I L I L J I J L Shapley
24 24 Shapley Shapley 3 99 A B C Shapley A, B A B C Shapley-Shubik A B C A B C A C B B A C B C A C A B C B A Shapley-Shubik
25 [1] M. I [2] [3], [4] [5] [6] Vol.54, pp ,
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2012 A, N, Z, Q, R, C 1 2009 9 2 2011 2 3 2012 9 1 2 2 5 3 11 4 16 5 22 6 25 7 29 8 32 1 1 1.1 3 1 1 1 1 1 1? 3 3 3 3 3 3 3 1 1, 1 1 + 1 1 1+1 2 2 1 2+1 3 2 N 1.2 N (i) 2 a b a 1 b a < b a b b a a b (ii)
(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/98/136512264.jpg "(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
1 (utility) 1.1 x u(x) x i x j u(x i ) u(x j ) u (x) 0, u (x) 0 u (x) x u(x) (Marginal Utility) 1.2 Cobb-Daglas 2 x 1, x 2 u(x 1, x 2 ) max x 1,x 2 u(
1 (utilit) 1.1 x u(x) x i x j u(x i ) u(x j ) u (x) 0, u (x) 0 u (x) x u(x) (Marginal Utilit) 1.2 Cobb-Daglas 2 x 1, x 2 u(x 1, x 2 ) x 1,x 2 u(x 1, x 2 ) s.t. P 1 x 1 + P 2 x 2 (1) (P i :, : ) u(x 1,
EP760取扱説明書
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06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
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3 m = [n, n1, n 2,..., n r, 2n] p q = [n, n 1, n 2,..., n r ] p 2 mq 2 = ±1 1 1 6 1.1................................. 6 1.2......................... 8 1.3......................... 13 2 15 2.1.............................
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空き容量一覧表（154kV以上）
1/3 A. 電気所 ( 発電所, 変電所, 配電塔 ) における変圧器の空き容量 覧 < 留意事項 > (1) 空容量は 安であり 系統接続の前には 接続検討のお申込みによる詳細検討が必要となります その結果 空容量が変更となる場合があります (2) 熱容量を考慮した空き容量を記載しております その他の要因 ( や系統安定度など ) で連系制約が発 する場合があります (3) 表 は 既に空容量がないため
Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.
![Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.](/thumbs/93/113887245.jpg "Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.")
Basic Mathematics 16 4 16 3-4 (10:40-12:10) 0 1 1 2 2 2 3 (mapping) 5 4 ε-δ (ε-δ Logic) 6 5 (Potency) 9 6 (Equivalence Relation and Order) 13 7 Zorn (Axiom of Choice, Zorn s Lemma) 14 8 (Set and Topology)