.
, N N. , .
(0,1)- .
1. 2- . .
. , u, (0,1)- u1, :
u1(1) = 0, u1(2) = 0, u1(1,2) = 1
u1(2) = u1(1,2) u1(1) = 1 0 =1,
(*). , .
, .
2. 3- . u (0,1)- ,
u(1) = u(2) = u(3) = 0, u(1,2,3) = 1.
:
u(1,2) = u(1,2,3) u(3) = 1 0 =1,
u(1,3) = u(1,2,3) u(2) = 1 0 =1,
u(2,3) = u(1,2,3) u(1) = 1 0 =1,
, , . , : .
3. 4- . .
(0,1)- u
u(1) = u(2) = u(3) = u(4) = 0
u(1,2,3,4) = 1.
,
u(1,2,3) = u(1,2,3,4) u(4) = 1 0 =1;
u(1,2,4) = u(1,2,3,4) u(3) = 1 0 =1;
u(1,3,4) = u(1,2,3,4) u(2) = 1 0 =1;
u(2,3,4) = u(1,2,3,4) u(1) = 1 0 =1.
.
(1,2), (1,3), (1,4), (2,3), (2,4), (3,4).
:
u(1,4) = 1 u(2,3),
u(1,3) = 1 u(2,4),
u(1,2) = 1 u(3,4).
, , . x1, x2, x3, ..
u(1,4) = x1, u(2,4) = x2, u(3,4) = x3,
u(2,3) = 1 x1, u(1,3) = 1 x2, u(1,2) = 1 x3.
0 £ x1, x2, x3 £ 1,
, ( ), , ( 1 ). (x1, x2, x3) , .. .
|
|
, .
4. , 4- , .
, n , .. .
.
1. 2- N ={1,2},
u(Æ) = u(1) = u(2) = u(1,2) = 1.
, .
2. 3- N ={1,2,3},
u(Æ) = u(1) = u(2) = u(3) = 0; u(1,2,3) = 1.
( )
u(1,2) = C3, u(1,3) = C2, u(2,3) = C1,
0 £ C1, C2, C3 £ 1.
, , 4- .
3- .
, .
. x = (x1,..., xn) y = (y1,..., yn) G = {N,u}, K Ì N . x y K,
1) £ u(K) ( )
2) xi > yi iÎK ( )
, x , : . , , , , .
: xi > yi , .. K y , x, x y K, , xi > yi .
x y K
.
. x y, K, x y. :
|
|
x > y.
x > y , N , x y. , , , . . , , .
.
. u u1 , x y , x > y > .
, , , , ( ) (0,1)- , .
, .
.
. (0,1)- ( 1). 1,2,3, , , {1,2}, {1,3}, {2,3}.
. x1, x2, x3, 60, x3 x1 x2 (.1), x = (x1, x2, x3) x1, x2, x3, , .1. (, x .1. x1 < 0, x2 > 0, x3 > 0).
x1 + x2 + x3 = 1.
x1, x2, x3, (6). , . , (0, 1)- , x (. . 2). x1, x2, x3
x1 + x2 £ u(1, 2), x1 + x3 £ u(1, 3), x2 + x3 £ u(2, 3).
, ,
x1 + x2 = 1 - x3 £ 1 = u(1, 2), x1 + x3 £ 1, x2 + x3 £ 1.
x = (x1, x2, x3) y = (y1, y2, y3)
{1, 2}, x1 > y1, x2 > y2;
{1, 3}, x1 > y1, x3 > y3;
{2, 3}, x2 > y2, x3 > y3,
.. y ( , x) . 3, x y, , x.
x3 = - 1 x2 = - 1
x = (x1, x2, x3)
x3 = 1 - C3
x1 = 0
x1 = 1 - C1 x2 = 1 - C2
.3. 4
, x y - , , .
. (0, 1)- .
x = (x1, x2, x3) y = (y1, y2, y3) {1, 2}. :
, C3 < 1, (7) , - . , x
x1 + x2 = C3.
, (6),
|
|
x3 = 1 + C3.
, x ,
x1 ³ 1 - C1, x2 ³ 1 - C2, x3 ³ 1 - C3,
, . 4, , x y.
(8) , , , 2 - , . 5, x y.
x1 = 1 - C 1 x2 = 1 - C 2 x2 = 1 - C 2 x1 = 1 - C 1
x3 = 1 - C3
x
. 5. 6
, , , . . , .. , . - .
. x - u, , K
(9) x, , - .
- :
1) - ;
2) - .
(0; 1)- (. 7).
Ÿ :
u(Æ) = u(1) = u(2) = u(3) = 0;
u(1, 2, 3) = 1,
u(1, 2) = 3; u(1, 3) = 2; u(2, 3) = 1,
0 £ 1, 2, 3 £ 1.
x -
x1 + x2 ³ C3, x1 + x3 ³ C2, x2 + x3 ³ C1
, x1 + x2 + x3 = 1,
x3 £ 1 - C3, x2 £ 1 - C2, x3 £ 1 - C1.
1 2
. 7
, x i- (. . 7),
x i = 1 - i (i = 1,2,3)
(10)
x1 + x2 + x3 £ 3 - (1 + 2 + 3)
, , x1 + x2 + x3 = 1,
1 + 2 + 3 £ 2.
(12) -. , (12) , e1, e2, e3,
,
xi = 1 - Ci - ei (i = )
xi (10), .. x = (x1, x2, x3) - -.
- (. 7), - , (11)
3 3
1 2 1 2
. 8. 9
,
x1 + x2 + x3 = 1,
(11) . , , -
x1 = 1 - 1, x2 = 1 - 2.
0 £ 1 £ 1, 0 £ 2 £ 1, x1, x2 ³ 0.
x3 = 1 - x1 - x2 = 1 - (1 - 1) - (1 - 2) = 1 + 2 - 1.
, x3 ³ 0,
1 + 2 - 1 ³ 0
1 + 2 ³ 1.
- .7 . - . , 1 + 2 < 1, - - (.8). , ‑, (11) . , ,
|
|
1 + 2 < 1; 2 + 3 < 1; 1 + 3 < 1,
- , .9.
, , - . , , -, - , .. . - , , - , , . . , , .
. . , : - , , - , , , . .
. - (--) R , :
1) : R ;
2) : S R, r, R, S.
-- , - , --, ; , , .
. - C -- R, CÌ R.
--.
-- , .. .
--.
1. , --. , - , --. -- , - .
2. , --, , , . , --, : , , .
3. , . , - -- .
4. -- .
: , .
-- . , , , .
. u - T,
u(S) = u(S Ç T)
S.
T , , T, , .
. u n , p N . pu - , S = { i1, i2,..., iS }
u ({p(i1), p(i2),..., p(iS)}) = u(S).
pu , u p, - pu.
.
1. . S u,
|
|
= u(S)
, , , , .
2. . p iÎN
(pu) = j i (u),
.. , , .
3. . u¢ u¢¢,
j i (u¢ + u¢¢) = j i (u¢) + j i (u¢¢),
.. , .
. ( ) u n-
j (u) = (j1(u), j2(u),..., jn(u)),
.
. j, .
. wS(T), S, ,
wS(T) =
, S , S.
,
t T.
: , i - T,
u(T) - u(T \{ i })
i- ; g i (T) , i - T \{ i }; j i (u) i - . , u ,
,
T T, T \{ i } .
. ,
a1 = 10, a2 = 20, a3 = 30, a4 = 40.
, . , 1. , :
{2; 4}, {3; 4},
{1; 2; 3}, {1; 2; 4}, {2; 3; 4}, {1; 3; 4},
{1; 2; 3; 4}.
.
j1 , T = {1; 2; 3}, , T \{1} = {2; 3} . T t = 3 ,
.
, , 2- : {2; 4}, {1; 2; 3}, {2; 3; 4}.
.
, , .
, . , , ,
,
, , .
, 2- 3- , . , 2- 3- . 1- 4- , .
[DK1] [DK1]