Promoting Information Revelation in an Ascending-Bid Auction

Transcription

1 Technical Papers Promoting Information Revelation in an Ascending-Bid Auction 473 Shigeo Matsubara NTT Communication Science Laboratories, NTT Corporation keywords: auction, last minute bidding, game theory, electronic commerce, multiagent Summary An ascending-bid auction protocol with a fixed end time has been widely used in many Internet auction sites. In such auctions, we can observe bidders behavior called last minute bidding, namely, a large fraction of bids are submitted in the closing seconds of the auction. This may cause a problem of information revelation failure as well as a problem of the server s overload and network congestion. If almost all bids are submitted only during the last minute, each bidder cannot obtain information about the good through others bidding behavior, which will spoil the advantage of open-bid auctions. This results in having an inefficient allocation of the good. To solve this problem, we propose a new protocol that gives bidders an incentive to fix the maximum bid of a proxy agent. We examine the property of the protocol by using game theory and clarify which situations our protocol outperforms the existing protocol by using a computer simulation. 1. AI [Wurman 98] [Sandholm 99, Yokoo 00] [Boutilier 99] ebay Yahoo! [Roth 00, Bajari 00, Wilcox 00]1 2

2 C r i r 1 r end r k r k (1) (2) 2 ɛ (3) [Rasmusen 89]

3 [v 1,v 1 ] f F [v 2,v 2 ] g G v 2 <v 1 6 x q i i y q j j q 1 q n m [Milgrom 82] A i x 2 i r end 1 x 1 i j yj 2 v 1 yj 2 yj 1 1 (n + m) A A A 3 2 r end 1 r i <r end 1 v 1 v 1 A v 1 v 1 v 1 n + m r end 1 v 1 v 1 1 n n + m n 4 r end 1

4 C r end A yj 2 yj 2 2 A v 1 v 1 3 A A (1) v t (2) p c (3) (4) S fix S fix (5) Q (6) (7) r end 2 +ɛ (8) Q = 1 S fix p c 3 3 S fix

5 477 2 S fix 1 v t 3 v t = v Q =1 (Q =0) 1 B B i x 2 i x 1 i j yj 2 Q =1 yj 2 yj 1 Q =0 6 n 3 (n + m) B 3 7 n 3 B B i x 1 i i i 2 S fix i i i x 1 i 8 B B 3 9 B Q v t v 1 Q =1 Q 12 Q =1 4 6

6 C2001 x 1 E[u e (x1 i )] = i E[u e (x 1 i )] = (n 1)(F (z)) n 2 (x 1 i z)f(z)dz (1) v 1 x 1 i v 1 (n + m 1)((F (z)) n+m 2 (x 1 i z)f(z)dz +(F (x 1 i )) n 1 p c x 1 i v 1 (n 1)((F (z)) n+m 2 p c f(z)dz (2) B B 10 1 E[u e (x1 i )] E[u e (x1 i )] p c x 1 i E[u e(x 1 i )] E[u e (x1 i )] 1 r end 1 x 1 i n 1 (1) (1) 2 n 1 2 z n 2 z (F (z)) n 2 i z + ɛ ɛ i (x 1 i z) 4 2 z f(z) (1) 4 (x 1 i z) (1) (x 1 i z) (x 1 i z) 7 i 1 (2) n 11 x 1 i E[u e (x1 i )] E[u e (x1 i )] (n + m) B 10 p c p c p c B n n + m n + m p c 1 F (z)=(z v 1 )/(v 1 v 1 ) E[u e (x1 i )] = 1 (x 1 i v1 ) n n (v 1 v 1 ) n 1 E[u e (x 1 i )] E[u e(x 1 i )] E[u e (x1 i )] p c p c 1 n (x1 i v1 ) 1 (x 1 i v1 ) m+1 n+m (v 1 v 1 ) m 1 n 1 (x 1 i v1 ) m n+m 1 (v 1 v 1 ) m p c p c p c x1 i = v 1 p c [v1,v 1 ] x 1 i p c

7 479 x 1 i E[u e(x 1 i )] E[u e (x 1 i )] p c p c = n + m 1 n(n + m) (v1 v 1 ) (3) F (z)=(z v 1 )/(v 1 v 1 ) (3) R R 2 (4) (5) (4) z 2 n z (1 F (z)) 2 n 1 f(z) n 2 z (F (z)) n 2 (5) 1 2 (n(n 1))/((n + m)(n + m 1)) R R 2 (6) R >R ((n 1)m 2 +(n 2 3n)m 3n 2 +1)m n =2m 5 n 3 m [1000, 2000] F (z)=(z 1000)/( ) [1000, 2000] F (z)= k 1 k 2 exp( (z 1000)/100) k 1,k 2 F (1000) = 0,F(2000) = 1 a b v t 1000 (3) p c 95%1000 x 1 i E[u e (x1 i )] E[u e (x1 i )] p c (a) 4 (a) 10

8 C2001 R = R = v 1 v 1 n(1 F (z))(n 1)(F (z)) n 2 zf(z)dz (4) v 1 v 1 (n + m)(1 F (z))(n + m 1)((F (z)) n+m 2 zf(z)dz (1 n(n 1) (n + m)(n + m 1) )p c (5) R R = m((n 1)m2 +(n 2 3n)m 3n 2 +1) (n + 1)(n + m + 1)(n + m) 2 (v 1 v 1 ) (6) (a) 10 (a) (b) 100 (b) (b) 4 (b)

9 481 E[u bi e (x1 i )] = m(f (z)) m 1 (F (B 1 (z))) n 1 (x 1 i z)f(z)dz v 1 bi + v 1 (n 1)(F (B 1 (z))) n 2 (F (z)) m (x 1 i z)f(b 1 (z))dz +(F (x 1 i )) n 1 p c (7) 5 B v 2 B B v 1 v 2 2 (1) v ɛ (2) S fix p c 1 B vj 2 v1 2 x 1 i b i b i = B(x 1 i ) B (2) 5 (7) 1 2 S fix 2 2 S fix B 1 (B(z)) = z (7) b i b i = x 1 i E[u e (x 1 i )] p c E[u e(x 1 i )] E[u e (x 1 i )] x 1 i Fujishima [Fujishima 99]

10 C from ebay Auctions, Working Paper (2000). [Boutilier 99] Boutilier, C., Goldszmidt, M., and Sabata, B.: Sequential Auctions for the Allocation of Resources with Complementarities, in Proceeding of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI- 99), pp (1999). [Fujishima 99] Fujishima, Y., McAdams, D., and Shoham, Y.: Speeding Up Ascending-Bid Auctions, in Proceeding of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI-99), pp (1999). [Milgrom 82] Milgrom, P. R. and Weber, R. J.: A Theory of Auctions and Competitive Bidding, Econometrica, Vol. 50, No. 5, pp (1982). [Rasmusen 89] Rasmusen, E.: Games and Information, Basil Blackwell (1989). [Roth 00] Roth, A. E. and Ockenfels, A.: Last Minute Bidding and the Rules for Ending Second-Price Auctions: Theory and Evidence from a Natural Experiment on the Internet, Working Paper (2000). [Sandholm 99] Sandholm, T.: An Algorithm for Optimal Winner Determination in Combinatorial Auctions, in Proceeding of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI-99), pp (1999). [Wilcox 00] Wilcox, R. T.: Experts and Amateurs: The Role of Experience in Internet Auctions, Marketing Letters, Vol. 11, No. 4, pp (2000). [Wurman 98] Wurman, P. R., Wellman, M. P., and Walsh, W. E.: The Michigan Internet AuctionBot: A configurable auction server for human and software agents, in Proceeding of the Second International Conference on Autonomous Agents (Agents-98), pp (1998). [Yokoo 00] Yokoo, M., Sakurai, Y., and Matsubara, S.: Robust Combinatorial Auction Protocol against False-Name Bids, in Proceeding of the Seventeenth National Conference on Artificial Intelligence (AAAI-2000), pp (2000) NTT [Bajari 00] Bajari, P. and Hortaçsu, A.: Winner s Curse, Reserve Prices and Endogenous Entry: Empirical Insights