早稲田大学現代政治経済研究所 ダブルトラック オークションの実験研究 宇都伸之早稲田大学上條良夫高知工科大学船木由喜彦早稲田大学 No.J1401 Working Paper Series Institute for Research in Contemporary Political and Ec

早稲田大学現代政治経済研究所 ダブルトラック オークションの実験研究 宇都伸之早稲田大学上條良夫高知工科大学船木由喜彦早稲田大学 No.J1401 Working Paper Series Institute for Research in Contemporary Political and Economic Affairs Waseda University 169-8050 Tokyo,Japan

60 Double-track auction is a new auction mechanism for allocating multiple indivisible goods to buyers. Buyers do not only view items as substitutes but also as complements. This is a remarkable auction mechanism since existing auction mechanisms with multiple goods can only allocate substitutes. One experimental study with two goods and two buyers, however, suspects that doubletrack auction does not work as the theory indicates due to an existence of theoretically unnoticed price, which is called the pitfall price. This study investigates performances of double-track auction with controlling the pitfall price in experiment. In the experiment, two goods are for sale. Two buyers view these goods as complement and bid for them. The experiment consists of two conditions. One is with pitfall condition, and the other is without n-uto@akane.waseda.jp 1

pitfall condition. The main result is that the pitfall price negatively affects the performances. Achievement rate of competitive equilibrium in with pitfall condition is lower than that in without pitfall condition by 60%. On the other side, without pitfall condition, the auction works as the theory indicates. Therefore double-track auction allocates only when there is no pitfall price. 1 Sun and Yang [5] Sun and Yang [5] Gul and Stacchetti [2] (Federal Communication Commission :FCC) [1, 3] FCC [4] Sun and Yang [5] S 1 S 2 2 S 1 (S 2 ) S 1 S 2 2

[6] (pitfall) 2 Sun and Yang [5] 3 [6] Sun and Yang [5] 4 3 5 6 7 2 3

2.1 1 b n I = {1, 2,, b} n N = {β 1, β 2,, β n } p = (p 1, p 2,, p n ) R n + β h N p h β h N S 1, S 2 S 1 S 2 = N S 1 S 2 = i I u i : 2 N R u i ( ) = 0 2 N N u i p i I D i (p) v i (A p) D i (p) = arg max A N {ui (A) p h } (1) β h A v i (A p) = u i (A) β h A p h (2) i I v i N π = (π(i), i I) i, j I, i j π(i) π(j) = i I π(i) = N π(i) i I π ρ i I ui (π(i)) i I ui (ρ(i)) 2.1. (p, π) p R n + π i I π(i) D i (p) p 2.2. i I u i (Gross Substitutes and Complements: GSC) p R n + S j (j = 1, 2) β k S j δ 0 i I A D i (p) B D i (p + δe(k)) [A S j ] \ {β k } B [A c S c j] B c k(1 k n) e(k) k 1 0 R n 4

i I A D i (p) S 1 β k S 1 [A S 1 ] \ {β k } B S 1 β k β k β k S 1 [A c S c 2] B c β k S 2 β k B D i (p + δe(k)) β k S 2 GSC S j (j = 1, 2) S 1 S 2 i I u i 2.1. i I u i 0 u i : 2 N Z + 2.2. i I u i GSC 2.2 R n g p R n A 2 N p(a) = β k A p ke(k) p, q R n p g q p(s 1 ) q(s 1 ) p(s 2 ) q(s 2 ) W R n p W p g p p W W R n p W p g q q W n = {δ R n 0 δ k 1, β k S 1 ; 1 δ l 0, β l S 2 } n δ p(t) t δ(t) t t + 1 p(t + 1) = p(t) + δ(t) n 5

S 1 S 2 δ i I D i (p) max { ( min δ i I C D i (p) β h C δ h ) δ h } (3) β h N = Z n (3) g δ Step1: p(0) g p p(0) Z n i I p g p t = 0 Step2 Step2: i I u i p(t) D i (p(t)) i I D i (p(t)) (3) δ(t) p(t + 1) := p(t) + δ(t) p(t + 1) = p(t) t := t + 1 Step2 p(0) g p S 1 S 2 2.1 (Sun and Yang [5]). 2.1 2.2 p 2.3 N = {A, B} S 1 = {A} S 2 = {B} 2 1 2 I = {1, 2} 6

A B AB 1 0 5 11 18 2 0 5 3 14 1: 2 1 1 0 A B (AB) A B A B 2 2 D i (p) N t p(t) = (p A, p B ) D 1 (p(t)) D 2 (p(t)) δ(t) 0 (3,15) A A (1,-1) 1 (4,14) A A (1,-1) 2 (5,13), A, AB, A (0,0) 2: 0 p(0) = (3, 15) 1 2 A D 1 (p(0)) = D 2 (p(0)) = {A} (3) 1 δ(0) = (1, 1) 1 p(1) = p(0) + δ(0) = (3, 15) + (1, 1) = (4, 14) 1 p(1) = (4, 14) 1 2 A D 1 (p(1)) = D 2 (p(1)) = {A} 0 (3) 2 δ(1) = (1, 1) 2 p(2) = p(1)+δ(1) = (4, 14) + (1, 1) = (5, 13) 1 1 7

2 p(2) = (5, 13) 1 A A B 0 B 0 0 D 1 (p(2)) = {, A, AB} 2 A B A B 1 0 D 2 (p(2)) = {, A} (3) 3 δ(2) = (0, 0) p(3) = p(2) π(1) = {A, B}, π(2) = { } π i I ui (π(i)) = 18 π 3 [6] D i (p) [6] D i (p) D i (p) 3.1 [6] 2 2 8

D i (p) [6], D i (p) { {A u(a) D i β (p) = h A p h > 0} if max{u(a) β h A p h} > 0 {A u(a) β h A p h = 0} otherwise (4) D i (p) i I D i (p) 3.1. ( p, π) p Z n + π i I π(i) D i ( p) p i I D i (p) 3.2 N = {A, B}, I = {1, 2} 3 t p(t) = (5, 10) 2 A B AB 1 0 12 4 17 2 0 3 7 14 3: 9

D i (p) D 1 ( p(t)) = {A}, D 2 ( p(t)) = { } (3) δ(t) = (0, 1) p(t) p(t + 1) = p(t) + δ(t) p(t) = (5, 10) p(t) = (5, 10) p(t) = (5, 10) 2 D i (p) D 1 ( p(t)) = {A, AB}, D 2 ( p(t)) = { } 1 A 7 A B 2 D i (p) 1 A AB D i (p) (3) δ(t) = (0, 0) p(t) = p(t+1) π π π(1) = {A, B}, π(2) = { } 17 π(1) D 1 ( p(t)) π(2) D 2 ( p(t)) p(t) = (5, 10) 19 π(1) = {A}, π(2) = {B} π p(t) = (5, 10) π 17 ( p, π) π D i (p) 2 3 4 4.1 10

i I D i (p) 1 2 4.2 2 N = {A, B} S 1 = {A} S 2 = {B} I = {1, 2} 2 1 1 2 (3) 4.3,A,B,AB 1,2 11

3,4 4.3.1 1. 1 ( ) A B A S 1, B S 2 A B 2 2 B 4.3.2 3 3 3 p D i (p) 4 3 3 4 p 12

1: 1 13 2: 2

3: 3,4 14

D i (p) 4.4 1 Step1 Step4 Step1 A B p(0) g p p(0) Z 2 + 1 2 p t = 0 Step2 Step2 1 2 A B A B AB 4 4 2 1 AB A B 2 (3) p(t + 1) p(t) p(t + 1) Step2 p(t) = p(t + 1) Step3 Step3 2 1 A 2 B AB 1 A 2 B 1 2 A B 1 2 1 15

2 3 2 1 Step4 Step4 Step3 Step4 5 4 16 4 A B 1 4, 8, 12, 16 2 3, 7, 11, 15 3 1, 5, 9, 13 4 2, 6, 10, 14 4: 4.5 2012 6 7 3 52 3 2 Step3 Step2 2 Step2 16

4: 17

1 2 16 16 3 16 800 50 90 1500 5 1 1 D i (p) 1 2 2 B A 2 18

3 3 4 4 3 6 6.1 2.1 p ρ 6.1.1 1,2 3,4 5 a 1,2 80% 3,4 20% 1,2 3,4 60% 19

60% 5 b 3,4 60% a b a+b 1 89.2% 89.2% 2 86.3% 0% 86.3% 3 18.4% 59.2% 77.6% 4 20.2% 63.5% 83.7% 5: a: b: 80% 60% 60% 6.1.2 6 c c 1 100% 2 100% 3 94.7% 4 95.2% 6: c: 6 D 3 4 20

6.1.3 1,2 80% 1,2 3,4 1,2 60% 60% 3 4 3,4 y i = α + β 1 D1 i + β 2 D2 i + u i (5) y i i 2 1 D1 i 1 0 2 D2 i 1 0 2 7 0.89 1% 90% D1 0.70 1% 70% D2 0 21

Coefficients: Estimate Std.Error t-value Pr(> t ) (Intercept) 0.89154 0.04458 19.999 3.80E-11 *** D1-0.72788 0.0546-13.332 5.86E-09 *** D2-0.02885 0.06304-0.458 0.655 Signif.codes: 0 *** 7: 6.2 6.2.1 1 6 1 12 12 92% 1 ( ) 7 1 A B 1 A B 1 B 6.2.2 2 6 2 7 7 92.3%. 2 22

6: 23

7: 2 A B 7 2 A B 4 A B 4 B 6.2.3 3 6 3 9 9 80% 9% 3 24

6 A B 7 3 A B 1,2 3 A 4 A B 1,2 4 B 6.3 4 6 4 14 3 14 11.5% 73.1% 4 7 4 A B 3 A 14 3 B 3 4 B 7 25

B 1,2 B A B 1,2 B A A 3,4 B A 14 A B A B 3,4 A B 2 2 26

[1] F. Gul and E. Stacchetti: Walrasian equilibrium with gross substitutes, Journal of Economic Theory, Vol. 87 (1999), 95 124. [2] F. Gul and E. Stacchetti: The English auction with differentiated commodities, Journal of Economic Theory, Vol. 92 (2000), 66 95. [3] P. Milgrom: Putting auction theory to work: The simultaneous ascending auction, Journal of Political Economy, Vol. 108 (2000), 245 272. [4] P. Milgrom: Putting Auction Theory to Work (Cambrige University Press, New York, 2004). [5] N. Sun and Z. Yang: A Double-Track Adjustment Process for Discrete Markets With Substitutes and Complements, Econometrica, Vol. 77 (2009), 933 952. [6],,,,, (, 2013), 89 110. 27