a) b) A Truthful Interdependent Value Auction Based on Contingent Bids Takayuki ITO a) and David C. PARKES b) Dasgupta and Maskin [1] Contingent Bids 1 A x y 1 2 Efficient Truthful 3 1. [2] Graduate School of Engineering, Nagoya Institute of Technology, Gokiso, Showa-ku, Nagoya-shi, 466 8555 Japan Division of Engineering and Applied Science, Faculty of Arts and Sciences, Harvard University, 209 Maxwell Dworkin, 33 Oxford University, Cambridge 02138, USA a) E-mail: ito.takayuki@nitech.ac.jp b) E-mail: parkes@eecs.harvard.edu Optimal Efficient [3] [1], [4] Dasgupta and Maskin [1] Contingent Bids Direct Revelation Framework D Vol. J90 D No. 5 pp. 1209 1218 c 2007 1209
2007/5 Vol. J90 D No. 5 1 2 y x Dasgupta and Maskin [1] Dash [5] Ito [6], [7] AI 1 2 Efficient Truthful 3 Likhodedov and Sandholm [8], [9] interim Individual Rationality: IR 1 [10] Ex post Nash Equilibrium i α i α Pareto Efficient Efficient Truthful Strategy-proof 1 2 1 2 2. 3. Dasgupta and Maskin [1] 4. 5. 6. 2. 1 Cremer and McLean [10] 1210
3. v i0 α ij Dash [5] Krishna [3] Dasgupta and Maskin [1] Dash [5] [6], [7], [11] Dash [5] [6], [7], [11] [6], [7], [11] [6], [7], [11] [6], [7], [11] strategy-proof 2 [12] Krishna [3] Dasgupta and Maskin [1] Dasgupta and Maskin [1] 3. [1] [3] Dasgupta and Maskin [1] Krishna [3] 10 10.1 VCG Generalized Vickrey-Clarke-Groves VCG VCG 2 i {1,...,N} S i s i S i 1211
2007/5 Vol. J90 D No. 5 i z i(s) 0 s =(s 1,...,s N) 2 p z i(s) p s i (s 1,...,s i 1,s i+1,...,s N ) (z 1,...,z N ) 2 ŝ i z i(ŝ) i max i{z i(ŝ)} p i p i =min s i s.t. z i(s i, ŝ i) (1) z i(s i, ŝ i) max{z j(s i, ŝ i)} z i(s) =max {z j(s)} s monotonicitysinglecrossing condition: SCC monotonicity z i(s) s i 0, s, i single-crossing condition: SCC z i(s) s i > zj(s) s i 1 VCG [3] 10 10.1 (1) i p i p i i ŝ i = s i SCC i ŝ i = s i p i s i SCC p i s i i s i i i Vickrey Dasgupta and Maskin [1] z i Dasgupta and Maskin [1] Contingent bids v i = z i(s) 0 i b i : R N 1 0 b i(v i) v i =(v 1,...,v i 1,v i+1,...,v N ) (v 1,...,v N ) (b 1(v 1),...,b N(v N )) v v z(s) i b i(z i(s i,s i)) = z i(s i,s i), s i, s i i i s i 2 z i : S 1...S N R 0 b i : R N 1 R 0 0 v i R 0 i z i (s) 1212
(v1,...,v N )=(z 1(s),...,z N(s)) Dasgupta and Maskin [1] well-defined 1 (v1,...,v N ) 2 i 3 i p i p i v i max {vj } min v i j i vj = b j(v i,v ij) i v i v ij i j v i =(v1,...,v i 1,v i+1,...,v N ) v i vj 2 SCC [1] 1 4. 4. 1 (2) SCC b i(v i) =v i0 + α ijv j, (2) v i0 0 0 i α ij [0, 1) i v i0 α z =(z 1,...,z N) 1 i αij < 1 {1, 2, 3} 1 2 3 b 1(v 2,v 3)=50+0.3v 2 +0.5v 3 (3) (3) 1 2 0.3 α 3 0.5 α 2 3 b 2(v 1,v 3)=60+0.4v 1 +0.4v 3 (4) b 3(v 1,v 2)=70 (5) 1 2 3 α 1.0 4. 2 (3) (4) (5) (v1,v 2,v 3) v 1 =50+0.3v 2 +0.5v 3 v 2 =60+0.4v 1 +0.4v 3 v 3 =70 (v1,v 2,v 3) = (126.6, 138.6, 70.0) 2 2 1213
2007/5 Vol. J90 D No. 5 v 2 v 2 = max{v 1,v 3} v 1 = b 1(v 2,v 3) v3 = b 3(v 2,v 1) 2 b 2(v 1,v 3)=v 2 v 1 = b 1(v 2,v 3) = 50 + 0.3v 2 +0.5v3 v3 = b 3(v 2,v 1) = 70 v1 = 85 + 0.3max{v1,v 3} v1 > v3 v1 =85+0.3v1 v 2 = v1 v1 = 850/7 121.4 2 121.4 v1 1 2 (6) (2) b i(v i) =v i0 + α max v j. (6) α [0, 1) 4. 2 SCC (v 1,...,v N ) (b 1(v 1),...,b N (v N )) α α ij [13] 3 (Y,d) 3 f : Y Y Y x, y Y q<1 d(f(x),f(y)) q d(x, y) f x x Y x 0 f limit x y d(x, y) =d(f(x),f(y)) q d(x, y) q<1 d(x, y) q d(x, y) d(x, y) =0 d x = y Y N R N 0 f b(v) =(b 1(v 1),...,b N(v N )) b i (2)R N 0 d v, w R N 0 d(v, w) = max v i w i. i {1,...,N} α 12 = α 21 =0.5 v =(v 1,v 2) R 2 0 w =(w 1,w 2) R 2 0 d(v, w) =max( v 1 w 1, v 2 w 2 ) d(b(v),b(w)) = max( v 10 +v 2/2 (v 10 +w 2/2), v 20 + v 1/2 (v 20 + w 1/2) ) =max( v 1/2 w 1/2, v 2/2 w 2/2 ) b(v) =v 10 + v 2/2 (2) max( v 1/2 w 1/2, v 2/2 w 2/2 ) < max( v 1 w 1, v 2 w 2 ) 1 i αij < 1 b : R N 0 R N 0 v 0 R N 0 3 d Y x, y, z Y d(x, y) +d(y, z) d(x, z) d(x, y) = d(y, x) d(x, x) = 0 d(x, y) = 0 x = y lim m d(x m,x m+1 ) = 0 (0, 1) 1/2, 1/3,... [0, 1] R d R d 0 1214
v, w R N 0 max i{ b i(v) b i(w) } < max i{ v i w i } i b i(v) b i(w) < max { v j w j } v i0 + αijvj (vi0 + αijwj) < max { v j w j } (vj wj) < max { v j w j } αij vj w j < max { v j w j } αij < 1 αij [0, 1) αij vj wj < max { v j w j } i αij < 1 SCC 2 i αij < 1 i, j vj / v i0 < vi / v i0 N w i v i0 v1 v 2 v 1 vj = α 21 + α 2j w 1 w 1 w 1 w 1 j 1 j {1,2} α1j < 1 α1j 0 N>2 j {1, 2} v j / w 1 > v 2/ w 1 v3/ w 1 > v2/ w 1 v1/ w 1 3 α 31 v 1 w 1 + α 32 v 2 w 1 + j {1,2,3} α 3j v j w 1 > v 2 w 1 N>3 v4/ w 1 > v3/ w 1 vi / v i0 0 v b(v) 4 expressive expressive 1 2 2 well-defined j i b i i b i i b i(v i) =v i0 SCC i i v i0 α 5. Likhodedovand Sandholm [8] i =0 Myerson [14] β i [0, 1] β β [8], [14] α β α β Likhodedov and Sandholm [8] Myerson [14] x =(x 0,x 1,...,x N) 1215
2007/5 Vol. J90 D No. 5 x i {0, 1} N xi =1 x0 {0, 1} i=0 { N } max x β 0v 0(x)+ β ivi (x), i=1 x i =1 0 vi (x) =vi x 0 =1 0 v 0(x) =v 0 v 0 vi v 0 β β 1 2 3 4 1 v 0 β β [0, 1] N+1 2 (v1,...,v N ) v 0 3 4 i 0 v i β iv i =max, j=0 {β jvj } 4 vj v i i j j vj = b j(v i,v ij) i α β v 0 =0 Myerson [14] i 0 s i s i Uniform(0, 100) i L i =10+iδ δ = 300 10 N Uniform(0,L i) i α (1) j i 0.5 α ij =0 0.5 Uniform(0, 1) α ij (2) α m i (3.1) 0.5 0.25 [0, 1) (3.2) µ i =0.3 +iδ δ = 1.0 0.3 0.25 N [0, 1) α (4) α m i α ij α i (v 0,β 0,β 1,...,β N ) (v 0,β 0,β 1,...,β N ) 1 4 1 1 2 β β 0 =1/N n βi =1 i=1 n =(1+N)/N i=0 i>1 β 1 v 0 {0, 10,...,100} β i {0, 0.1,...,1} {1/N } 4j i β j v j j =0 j 0 j i i {1,...,N} j =0 1216
1 Fig. 1 Average efficiency in the optimal auction for the symmetric and asymmetric environments. 5 β β 3 β 1 0.5 2 0.3 3 0.2 β Myerson [14] β v 0 50 110 80 6. 2 Fig. 2 Average normalized revenue in the optimal auction for the symmetric and asymmetric environments. Likhodedovand Sandholm [9] 1 4000 1 5000 1 2 Dasgupta and Maskin [1] 2 [15] [1] P. Dasgupta and E. Maskin, Efficient auctions, The Quarterly Journal of Economics, vol.cxv, pp.341 388, 2000. [2] T. Ito and D.C. Parkes, Instantiating the contingent bids model of truthful interdependent value auctions, Proc. International Joint Conference on Autonomous Agents and Multi Agent Systems (AAMAS-06), pp.1151 1158, 2006. [3] V. Krishna, Auction Theory, Academic Press, 2002. [4] P. Jehiel and B. Moldovanu, Efficient design with 1217
2007/5 Vol. J90 D No. 5 interdependent valuations, Econometrica, vol.69, no.5, pp.1237 1259, 2001. [5] R.K. Dash, A. Rogers, and N.R. Jennings, A mechanism for multiple goods and interdependent valuations, Proc. 6th Int. Workshop on Agent-Mediated E-Commerce, pp.197 210, 2004. [6] T. Ito, M. Yokoo, and S. Matsubara, Designing an auction protocol under asymmetric information on nature s selection, Proc. 1st International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS02), pp.61 68, 2002. [7] T. Ito, M. Yokoo, and S. Matsubara, A combinatorial auction protocol among versatile experts and amateurs, Proc. 3rd International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS04), pp.378 385, 2004. [8] A. Likhodedov and T. Sandholm, Methods for boosting revenue in combinatorial auctions, Proc. National Conference on Artificial Intelligence (AAAI), pp.232 237, 2004. [9] A. Likhodedov and T. Sandholm, Approximating revenue-maximizing combinatorial auctions, Proc. National Conference on Artificial Intelligence, pp.267 274, 2005. [10] J. Cremer and R.P. McLean, Full extraction of surplus in Bayesian and dominant strategy aucitons, Econometrica, vol.56, no.6, pp.1247 1257, 1988. [11] T. Ito, M. Yokoo, and S. Matsubara, Towards a combinatorial auction protocol among experts and amateurs: The case of single-skilled experts, Proc. 2nd International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS03), pp.481 488, 2003. [12] M. Perry and P.J. Reny, An efficient auction, Econometrica, vol.70, no.3, pp.1199 1212, 2002. [13] R. Vohra, Advanced Mathematical Economics, Routledge, 2005. [14] R.B. Myerson, Optimal auction design, Mathematics of Operations Research, vol.6, pp.58 73, 1981. [15] D vol.j90-d, no.5, pp.1219 1228, May 2007. 18 6 20 10 4 12 11 13 12 13 13 15 17 18 18 AAMAS2006 Best Paper Award 2005 16 IPA 66 AAAI ACM David C. Parkes is the John L. Loeb Associate Professor of the Natural Sciences and Associate Professor of Computer Science at Harvard University. He received his Ph.D. degree in Computer and Information Science from the University of Pennsylvania in 2001, and an M.Eng. (First class) in Engineering and Computing Science from Oxford University in 1995. He was awarded the prestigious NSF CAREER Award in 2002, an IBM Faculty Partnership Award in 2002 and 2003, and the Alfred P. Sloan Fellowship in 2005. Parkes has published extensively on topics related to electronic markets, computational mechanism design, auction theory, and multi-agent systems. He serves on the editorial board of the Journal of Artificial Intelligence Research and the Electronic Commerce Research Journal, and has served on the Program Committee of a number of leading conferences in artificial intelligence, multiagent systems and electronic commerce, including ACM-EC, AAAI, IJCAI, UAI and AAMAS. He is the co-program chair of the ACM Conference on Electronic Commerce, 2007. 1218