A Truthful Interdependent Value Auction Based on Contingent Bids

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1 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, Japan Division of Engineering and Applied Science, Faculty of Arts and Sciences, Harvard University, 209 Maxwell Dworkin, 33 Oxford University, Cambridge 02138, USA a) ito.takayuki@nitech.ac.jp b) 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 c

2 2007/5 Vol. J90 D No 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 Dasgupta and Maskin [1] Cremer and McLean [10] 1210

3 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] VCG Generalized Vickrey-Clarke-Groves VCG VCG 2 i {1,...,N} S i s i S i 1211

4 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] (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

5 (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] (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} b 1(v 2,v 3)=50+0.3v v 3 (3) (3) α α 2 3 b 2(v 1,v 3)=60+0.4v v 3 (4) b 3(v 1,v 2)=70 (5) α (3) (4) (5) (v1,v 2,v 3) v 1 =50+0.3v v 3 v 2 =60+0.4v v 3 v 3 =70 (v1,v 2,v 3) = (126.6, 138.6, 70.0)

6 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) = v v3 v3 = b 3(v 2,v 1) = 70 v1 = max{v1,v 3} v1 > v3 v1 =85+0.3v1 v 2 = v1 v1 = 850/ 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

7 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 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

8 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 v 0 β β [0, 1] N+1 2 (v1,...,v N ) v 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 = 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, 1) (3.2) µ i =0.3 +iδ δ = 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 ) 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

9 1 Fig. 1 Average efficiency in the optimal auction for the symmetric and asymmetric environments. 5 β β 3 β β Myerson [14] β v Fig. 2 Average normalized revenue in the optimal auction for the symmetric and asymmetric environments. Likhodedovand Sandholm [9] Dasgupta and Maskin [1] 2 [15] [1] P. Dasgupta and E. Maskin, Efficient auctions, The Quarterly Journal of Economics, vol.cxv, pp , [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 , [3] V. Krishna, Auction Theory, Academic Press, [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.

1st International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS02), pp.61 68, 2002. [7] T. Ito, M. Yokoo, and S.

10 2007/5 Vol. J90 D No. 5 interdependent valuations, Econometrica, vol.69, no.5, pp , [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 , [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, [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 , [8] A. Likhodedov and T. Sandholm, Methods for boosting revenue in combinatorial auctions, Proc. National Conference on Artificial Intelligence (AAAI), pp , [9] A. Likhodedov and T. Sandholm, Approximating revenue-maximizing combinatorial auctions, Proc. National Conference on Artificial Intelligence, pp , [10] J. Cremer and R.P. McLean, Full extraction of surplus in Bayesian and dominant strategy aucitons, Econometrica, vol.56, no.6, pp , [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 , [12] M. Perry and P.J. Reny, An efficient auction, Econometrica, vol.70, no.3, pp , [13] R. Vohra, Advanced Mathematical Economics, Routledge, [14] R.B. Myerson, Optimal auction design, Mathematics of Operations Research, vol.6, pp.58 73, [15] D vol.j90-d, no.5, pp , May AAMAS2006 Best Paper Award 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 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 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,

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