Vol. 31 No. 3 Aug VCG [16][12] yes/no VCG VCG 1 ( ) 1. 1 (VCG-equvalent n expectaton, VCG-EE) VCG VCG VCG-EE VCG VCG

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1 156 VCG-equvalent n Expectaton VCG-equvalent n expectaton VCG-equvalent n expectaton Vckrey-Clarke-Groves (VCG) VCG VCG-equvalent n expectaton VCG-equvalent n expectaton In ths paper, we develop a new class of teratve mechansms called a VCG-equvalent n expectaton mechansm. To guarantee that sncere strateges are an ex post equlbrum, t nevtably asks an rrelevant query, n whch a partcpant has no ncentve to answer the query sncerely. Such an rrelevant query causes unnecessary leakage of prvate nformaton and a dfferent ncentve ssue. In a VCG-equvalent n expectaton mechansm, the mechansm acheves the same allocaton as VCG, but the transfers are the same as VCG only n expectaton. We show that n a VCG-equvalent n expectaton mechansm, sncere strateges consttute a sequental equlbrum. Also, we develop a general procedure for constructng a VCG-equvalent n expectaton mechansm that does not ask any rrelevant query. To demonstrate the applcablty of ths dea n a practcal applcaton, we develop a VCG-equvalent n expectaton mechansm that can be appled to the Japanese 4G spectrum aucton. 1 Vckrey [13] () VCG-equvalent n Expectaton Mechansm. Etsush Fujta, Tak Todo, Makoto Yokoo,, Graduate School of ISEE, Kyushu Unversty. Atsush Iwasak,, Graduate School of Informaton Systems, Unversty of Electro- Communcatons.,Vol.31, No.3 (2014),pp [ ] ( ) ( ) [17] Vckrey-Clarke-Groves (VCG) [6][10][15] ( )

To guarantee that sncere strateges are an ex post equlbrum, t nevtably asks an rrelevant query, n whch a partcpant has no ncentve to answer the query sncerely.

2 Vol. 31 No. 3 Aug VCG [16][12] yes/no VCG VCG 1 ( ) 1. 1 (VCG-equvalent n expectaton, VCG-EE) VCG VCG VCG-EE VCG VCG

3 158 VCG-EE VCG VCG VCG-EE [11] VCG-EE (BDT) (BDT-based ) BDT-based 1 yes/no : ( ) 2 (A, B ) 1 9 A A 9 B 5 B A 5 Vckrey B B B A 9 VCG VCG-EE A B A (B 4.5 )

4 Vol. 31 No. 3 Aug [1][2] [3] [16] [12] [4] [5] [7] [14]. [7] [14] BDT-based BDT-based -externalty [8] VCG-EE expected-externalty 3 M = {1,, m} N = {1,, n} Θ Θ θ θ B M v(θ, B) v v(θ, ) = 0 B B v(θ, B) v(θ, B ) θ B t v(θ, B) + t ( t t B ). θ p(θ ) > 0 θ Θ p(θ ) = 1 p Θ Θ p(θ ) θ Θ p(θ ) Θ N Θ Θ j Θj θ = (θ 1,, θ n ) Θ θ Θ θ θ (θ, θ ) p(θ) N p(θ ) p(θ ) j p(θj) X X = (X 1,, X n ) X X M X X X X j = ( j) X X θ Θ N v(θ, X ) > N v(θ, X ) X X X (θ, M) a : Θ X t : Θ R n a a(θ) = (a 1 (θ),, a n (θ)) t t(θ) = (t 1(θ),, t n(θ)) VCG a t 1 (VCG ). a (θ) = X. X X θ M t (θ) = j v(θ j, X j ) j v(θ j, X j). X X j v(θj, X j) VCG 1. M = {1, 2} N = {1, 2, 3} 1 1

5 Θ 1 = Θ 2 = {2, 4, 6, 8}, Θ 3 = {9} θ = (2, 2, 9) a (θ) = (,, {1, 2}), t (θ) = (0, 0, 4) θ = (8, 8, 9) a (θ) = ({1}, {2}, ), t (θ) = ( 1, 1, 0) 4 θ 1 2? θ 2 2? θ 2 2? θ 2 4? θ 1 4? θ 2 6? θ 1 6? θ 2 4? θ 1 4? ( ) (BDT-based ) BDT-based yes/no ( ) 2 (BDT-based ). N, M, Θ BDT-based d r, D n, D l, nt, q, ā, t D n D l d r D n d D n D l p(d) D n d D n lc(d), rc(d) (lc(d), rc(d) D n D l ) d D n nt(d) N q(d) Θ nt(d) ā t d nt(d) N q(d) Θ nt(d) d n () Θ d = N Θd Θ d Θ d Θ j Θd j 3 (). d d = d r Θ d d r p(d) = d, nt(d ) =, q(d ) = Θ d Θ d, d = lc(d ) Θ Θ d d = rc(d ) (Θ d \Θ ) Θ d 1 d r d q(d) yes lc(d) no rc(d) Θ d 2 d nt(d) = q(d) Θ d ( ). d D l ā(θ d ), t(θ d ) 2 ā t Θ Θ 1 1 BDTbased 3 1, 2 wn lose 3 BDT-based 1 2 {2, 4, 6, 8} {2, 4, 6, 8} (θ 1 2?) {2} {2, 4, 6, 8} {4, 6, 8} {2, 4, 6, 8}

$Θ d Θ j Θd j 3 (). d d = d r Θ d d r p(d) = d, nt(d ) =, q(d ) = Θ d Θ d, d = lc(d ) Θ Θ d d = rc(d ) (Θ d \Θ ) Θ d 1 d r d q(d) yes lc(d) no rc(d) Θ d 2 d nt(d) = q(d) Θ d ( ).$

6 Vol. 31 No. 3 Aug (θ 2 2?) {2} {2} {2} {4, 6, 8} 1 {6, 8} 2 {4, 6, 8} BDT-based VCG-equvalence 4 (VCG-equvalence). d θ, θ Θ d, a (θ) = a (θ ) = ā(θ d ) t (θ) = t (θ ) = t(θ d ) BDT-based VCG-equvalent 1 ( 2 ) VCG-equvalent VCGequvalent n expectaton (VCG-EE) 5 (VCG-equvalence n expectaton (VCG-EE)). d BDTbased VCG-EE θ, θ Θ d, a (θ) = a (θ ) = ā(θ d ) t(θ d ) t (Θ d ) = θ Θ d t ((θ d, θ )) p(θ ) p(θ d ) θ d Θ d VCG-EE d N, θ, θ Θ d, θ Θ d, t ((θ, θ )) = t ((θ, θ )) t (Θ d ) Θ d VCG 1 VCG-EE {6, 8} {4, 6, 8} VCG , 8 1 VCG 3, 1 2 p(4) = p(6) = p(8) = 1/4, p({4, 6, 8}) = 3/4 1 (4 9)/3 + (6 9)/3 + (8 9)/3 = BDT-based BDT BDT-based VCG-EE VCG-equvalent VCG-EE d, nt(d) = d θ d θ Θ d d (, θ ) yes/no s (, θ ) d

d BDTbased VCG-EE θ, θ Θ d, a (θ) = a (θ ) = ā(θ d ) t(θ d ) t (Θ d ) = θ Θ d t ((θ d, θ )) p(θ ) p(θ d ) θ d Θ d VCG-EE d N, θ, θ Θ d, θ Θ d, t ((θ, θ )) = t ((θ, θ )) t (Θ d ) Θ d VCG 1

7 162 θ q(d ) s (d ) yes s (d ) no s d (, θ ) s (, θ ) (, θ ) s (, θ ) θ (, θ ) j θ j p(θ j ) 6 ( ). j(j ) s j d j(j ) θ j d p(θ j d) = p(θ j )/p(θ d j ) 7 (). θ s (, θ ) d j(j ) s j s d ( ) (, θ ) d s (d) [11]. 8 ( ). (a) (b) [9] [11] 1. VCG-equvalent d θ s E(s, θ d) 2 1. VCG-EE (, θ ) d E(s, θ d) VCG θ Θ d u (θ, (θ, θ )) p(θ ) p(θ d ). θ θ θ VCG u (θ, (θ, θ )) VCG VCG-EE 2. VCG-EE. (a) (b) (b)

8 Vol. 31 No. 3 Aug d 6 d (, θ ) d (, θ ) s s d E(s, θ d) E(s, θ d) d θ d θ 1 E(s, θ d) θ Θ d u (θ, (θ, θ )) p(θ ) p(θ d ). (1) E(s, θ d) lv(d) d s d lv(d) D lv(d) E(s, θ d) E(s, θ d) = (v(θ, ā (Θ d )) + t (Θ d )) d D p(θ d ) d D p(θ d ) v(θ, ā (Θ d )) + t (Θ d ) θ Θ d u (θ, (θ d, θ )) p(θ ) p(θ d ). θ Θ d θ Θ d d D d D Θ d = Θ d E(s, θ d) d D θ Θ d u (θ, (θ d, θ )) p(θ ) p(θ d ). (2) E(s, θ d) > E(s, θ d) θ Θ d 1 2 θ 1 u (θ, (θ d, θ )) > u (θ, (θ, θ )). VCG θ E(s, θ d) E(s, θ d) VCG 3. VCG-EE BDTbased 9 ( ). nt(d) = d θ q(d), θ Θ d \ q(d), θ Θ d, ā(θ d l ) = ā(θ d l ) t (Θ d l ) = t (Θ d l ) d d l, d l Θ d l (θ, θ ), Θ d l (θ, θ ) d d d ( ) 1 Θ 1 = {8} VCG 2 4. VCG-equvalent N, M, Θ VCG-EE

θ Θ d θ Θ d d D d D Θ d = Θ d E(s, θ d) d D θ Θ d u (θ, (θ d, θ )) p(θ ) p(θ d ). (2) E(s, θ d) > E(s, θ d) θ Θ d 1 2 θ 1 u (θ, (θ d, θ )) > u (θ, (θ, θ )).

9 ( ). Θ Θ ) n Θ = N Θ θ, θ Θ nd, θ j Θ j, a ((θ, θ )) = a ((θ, θ )) Θ nd Θ nd Θ nd Θ Θ Θ 5. VCG-EE = nt(d) d d Θ d Θ nd Θ d Θ nd q(d) Θ nd Θ d \ q(d). = nt(d) d Θ d Θ nd Θ nd q(d) Θ nd Θ d Θ d \ q(d) d θ q(d), θ Θ d \ q(d) θ θ a ((θ, θ )) a ((θ, θ )) θ Θ d θ d l (, θ ) s d l (, θ ) s VCG-EE (θ, θ ) Θ d l ā(θ d l ) = a ((θ, θ )) (θ, θ ) Θ d l ā(θ d l ) = a ((θ, θ )) a ((θ, θ )) a ((θ, θ )) ā(θ d l ) ā(θ d l ) d d 6. N, M, Θ, VCG-EE. Θ Θ d d N Θ d = 1 Θ d d θ Θ d Θ d > 1 Θ nd nt(d) =, q(d) = Θ nd 1 Θ d Θ nt(d) =, q(d) = Θ 5 Θ d 6 VCG-EE [18] VCG-EE GHz ( 200 MHz) 200 MHz 10 ( 20 MHz ) (Tme Dvson Duplex, TDD) (Frequency Dvson Duplex, FDD) 2 FDD 1

a ((θ, θ )) (θ, θ ) Θ d l ā(θ d l ) = a ((θ, θ )) a ((θ, θ )) a ((θ, θ )) ā(θ d l ) ā(θ d l ) d d 6. N, M, Θ, VCG-EE.

10 Vol. 31 No. 3 Aug FDD 1 (2 ) 2 Ausubel [2] VCG-EE m TDD FDD p, q TDD 1 p FDD 1 p + q p = q = 0 p, q TDD m p, q TDD m p TDD q = p q p VCG-EE 7 1 VCG-EE VCG BDT-based VCG-equvalent VCG-EE VCG-EE VCG-EE VCG-EE VCG-EE (S) ( ) 3

11 166 [ 1 ] Ausubel, L. M. and Mlgrom, P. R.: Ascendng Auctons wth Package Bddng, Fronters of Theoretcal Economcs, Vol. 1, No. 1 (2002), pp [ 2 ] Ausubel, L. M.: An Effcent Ascendng-Bd Aucton for Multple Objects, Amercan Economc Revew, Vol. 94, No. 5 (2004), pp [ 3 ] Ausubel, L. M.: An effcent dynamc aucton for heterogeneous commodtes, Amercan Economcs Revew, Vol. 96, No. 3 (2006), pp [ 4 ] Blumrosen, L. and Nsan, N.: On the Computatonal Power of Demand Queres, Sam Journal on Computng, Vol. 39, No. 4 (2009), pp [ 5 ] Blumrosen, L. and Nsan, N.: Informatonal Lmtatons of Ascendng Combnatoral Auctons, Journal of Economc Theory, Vol. 145, No. 3 (2010), pp [ 6 ] Clarke, E. H.: Multpart Prcng of Publc Goods, Publc Choce, Vol. 2 (1971), pp [ 7 ] Conen, W. and Sandholm, T.: Preference Elctaton n Combnatoral Auctons (Extended Abstract), n Proceedngs of the ACM conference on electronc commerce, MIT Press, 2001, pp [ 8 ] d Aspremont, C. and Gerard-Varet, L.-A.: Incentves and ncomplete nformaton, Journal of Publc Economcs, Vol. 11, No. 1 (1979), pp [ 9 ] Fudenberg, D. and Trole, J.: Perfect Bayesan equlbrum and sequental equlbrum, Journal of Economc Theory, Vol. 53, No. 2 (1991),pp [10] Groves, T.: Incentves n teams, Econometrca, Vol. 41 (1973), pp [11] Kreps, D. M. and Wlson, R. B.: Sequental Equlbra, Econometrca, Vol. 50, No. 4 (1982), pp [12] Mshra, D. and Parkes, D. C.: Ascendng prce Vckrey auctons for general valuatons, Journal of Economc Theory, Vol. 132, No. 1 (2007), pp [13] Parkes, D. C.: Iteratve Combnatoral Auctons, Combnatoral auctons, MIT Press, 2006, pp [14] Sandholm, T. and Boutler, C.: Preference Elctaton n Combnatoral Auctons, Combnatoral auctons, MIT Press, 2006, pp [15] Vckrey, W.: Counter Speculaton, Auctons, and Compettve Sealed Tenders, Journal of Fnance, Vol. 16 (1961), pp [16] Vres, de S., Schummer, J. and Vohra, R. V.: On ascendng Vckrey auctons for heterogeneous objects, Journal of Economc Theory, Vol. 132, No. 1 (2007), pp [17],, [18] 4G : Japanese Package Aucton(JPA), handle.net/2261/51497 (2012) NTT 2013 ( ) IEEE ( ) NTT ( ) 1992, 2002

and Nsan, N.: On the Computatonal Power of Demand Queres, Sam Journal on Computng, Vol. 39, No. 4 (2009), pp. 1372 1391. [ 5 ] Blumrosen, L. and Nsan, N.

12 Vol. 31 No. 3 Aug ACM SIGART Autonomous Agent Research Award AAAI

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