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2 1. 2. ( ) 3. ( ) 2

3 1. 3

4 (procurement auctions) etc. ( ) : : 4

5 5

6 : (sealed-bid auctions) : (1st-price auctions) (2nd-price auctions) 6

7 : ( ) (open auctions) (English auctions) (Dutch auctions) 7

8 : (private value models) : 8

9 : ( ) (interdependent value models) (i.e., ) : (common value models) 9

10 : 1: ( ) 2: u({x 1 }) + u({x 2 }) u({x 1, x 2 }). (package auctions) (combinatorial auctions) 10

11 : (double auctions) : (all-pay auctions) : (scoring auctions) : 11

12 1. :. 2. :. ( ) a 3. :. 4. : u i : A Θ i R. t(a): (1) 5. : θ i [0, 1] : f(θ i ) = 1 θ i [0, 1]. 12

13 : X X X : 1 X : 1/6. X : 1 X = a X = b X [a, b] 13

14 : ( ) 1. X x F : R [0, 1] : (2) : X a b (X [a, b]) P r.({a X b}) =. X x (X x). P r.({x x}) =. 14

15 : ( ) x P r.({x X x + ϵ}) =. F ( ) f( ), f = F, : F (x + ϵ) F (x) (3) X [x, x + ϵ] ϵ f(x) 2 X F ( ) F = f f : R R + 15

16 : ( ) x 0 0 x X [a, b] : F (b) F (a) = (4) F ( ) 16

17 : ( ) E[X] : (5) x f(x)dx X [0, 1] : (6) (7) 17

18 f( ) 18

19 2. 19

20 : 2 : (8) 20

21 ( ) θ 1 = 100, θ 2 = 200. a 1 = 70, a 2 = 150. : : u 1 (a, 100) =. u 2 (a, 200) =. S i : i (S i R [0,1] + ) S i S 1 S i 1 S i+1 S n. 21

22 (strategy-proofness) 2nd 1 : 3 in s D i : Θ i A i i : (9) (9) θ i i s i 22

23 (strategy-proofness) 2nd / BNE 2 s = (s 1, s 2,, s n) :. : θ 1st > θ 2nd > > θ nth. : θ 1st : x 2 (s ) = θ 2nd. 23

24 (strategy-proofness) 4 ( ) : (10) 24

25 BNE : Θ 1 = Θ 2 = {10, 20}. f i (10) = f i (20) = 1/2. : θ i Θ i s 1 (θ i ) = 100. (11) s 2 (θ i ) = 0. (12) BNE 25

26 b A i R + : i 26

27 ( ) ( ) a (a 1, a 2,, a n ): x(a) (x 1 (a), x 2 (a),, x n (a)):. a ( i ) :. ; s i : Θ i R +. 27

28 ( ) i θ i b 1. b > θ i θ i b < b < θ i b θ i b > 0. 1, 2 i. 28

29 θ ( ) θ : θ 1st > θ 2nd > > θ nth. s i ( ) : a θ (θ 1st ) (b = θ : 29

30 ( ) 3 s = (s 1, s 2,, s n ) :. ( ) 30

31 3. 31

32 : : (13) (s i = s ) s : 1. θ i > θ i s (θ i ) > s (θ i )

33 i s ( ) θ i i a i : (14) i i s ( ) j a i : s 1 ( ): s ( ). s ( ) 33

34 ( ) θ i i a i : ( ) i : j s 1 (a i ) :. s 1 (a i ) :. i :. a i i : (15) (15) a i (s ) : (15) a i ( ) 34

35 ( ) (15) a i (s ) ( ) : (16) : (f(x)g(x)) = f (x)g(x) + f(x)g (x). f 1 (x) = 1/(f (y)). y = f 1 (x). i s ( ) a i = s (θ i ) i : s 1 (a i ) = θ i. 35

36 ( ) s ( ) : (17) (17) : (18) (18) : (19) 36

37 (19) (20) (20) 37

38 (20) i (20) i a i = s (1) i a i s (1) θ i i z : Π(z, θ i ) = = (21) 38

39 s ( ) z : Π(θ i, θ i ) Π(z, θ i ) = θ i n 1 (θ i θ i ) + θ i n = z n 1 (z θ i ) 1 n (zn θ i n ) n zn 1 (θ i z) zn n = (22) (22) z θ i z θ i 39

40 Ý Ý Ý Ü Ò ½ Ý Ü Ò ½ Þ Ò ½ Þ Ò ½ ¼ ( ) z θ i ( ): (22) = z n 1 (z θ i ) z θ i ( ): Þ Ü z ¼ θ i x n 1 dx = > 0. Þ Ü (22) = θi z x n 1 dx z n 1 (θ i z) = > 0. 40

41 4 s ( ) : (23) 4 : : : (2nd ) (23) 41

42 b : 1. 42

43 ( ) : 2. 1st 2nd 43

44 1996 : Vickrey(1961): 2nd 44

45 : : : 2nd ( ) 1st BNE 45

REALV5_A4…p_Ł\1_4A_OCF

$REALV5_A4…p_Ł\1_4A_OCF$