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1 玉田康成研究会 < オークションパート > 赤星仁田中さゆり若森直樹

3 1 1 1 First Price Auction 2

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9 3 First Price Auction Second Price Auction ,. 3 6 ( ) ( ) ( ) =

10 3.1.2 English Auction Dutch Auction First Price Auction Second Price Auction English Auction Dutch Auction First Price Auction (sealed bid auction) Second Price Auction Vickrey Vickrey Auction 6 6 Matthews[4] 9

11 n v i i =1,,n Assumption 1 Assumption 2 v 1,,v n Assumption 3 v i F ( ) Assumption 4 Prob[win](v i p w ) + Prob[lose]( p l ) 10

12 Prob[win] p w Prob[lose] p l 0 max Prob[win](v i p w ) First Price Auction V V A [0, 1] k V A =0.6,k = B max B (V B)Prob[B >kv A ] B>kV A B/k > V A 7 B/k max B (V B) B k B B = V 2 n v i v i [0,1] 7 11

13 F ( ) F (v) =Prob[ṽ v] F (v) =v F 0 )=f( ) f( ) f(v) =1 i v i B( ) B( ) b i i Prob[ b i >B(v j )] for all j 6= i b i > B(v j ) v j < B 1 (b i ) for all j 6= i Prob[v j <B 1 (b i )] for all j 6= i [F (B 1 (b i ))] n 1 i max π i = max b i (v i b i )[F (B 1 (b i ))] n 1 (1) i π b i =0 (2) b i dπ/dv i = π i / v i + ( π i / b i )( b i / v i ) dπ i dv i = π i v i =[F (B 1 (b i ))] n 1 (3) i B b i = B(v i ) B 1 (b i )=B 1 (B(v i )) = v i dπ i dvi =[F (v i)] n 1 (4) 8 B(v i )=v i Z vi r [F (u)] n 1 du F [v i ] n 1 (5) [F (v i )] n 1 =(v i ) n 1 v l =0) B(v i )= n 1 n 8 0 v l 1 B(v l )=v l v i 12

14 r B(v i )= n 1 n v r n i + n(v i ) n ( ) z 1. z>5000 z 5000 z z>5000 z 2. z =5000 z 5000 = z z 3. z<5000 z<

15 z z v b B(b, v) z h( ) B(b, v) B(b, v) = Z π(b, z, v)h(z)dz (b, z, v) π(b, z, v) π(v, z, v) Z π(b, z, v)h(z)dz Z π(v, z, v)h(z)dz B(b, v) B(v, v) b = v h( ) b = v 14

16 Strong Cartel Weak Cartel Side Payment Side Payment Side Payment Strong Cartel Side payment Weak Cartel Strong Cartel r 9 T (v) r/n 1 Strong Cartel 2000 Strong Cartel v A =2600 v B = 2400 v C = Weak Cartel v<r v r r v r n (v r)/n Weak Cartel Side Payment Strong Cartel Strong Cartel 9 T (v) T (v) =[F (v)] n 1 (u v)(n 1)[F (u)] n 1 f(u)du + r 15 Z v r

17 Weak Cartel Weak Cartel v i i v i i v i v i E i w i h(w i,v i ) w i T (w i,v i ) i π i i π i = E i [v i h(w i,v i ) T (w i,v i )] i π i w i dπ dv i = E i h(v i,v i ) v i E i h(v i,v i ) 0 Weak Cartel n k n k v [0, 1] v p = Prob[v] n k k n k πn k = p(1 r)(1 p) n k G N (b) =[( 1 r 1 b ) 1 1 p n k 1] p 10 Weak Cartel 16

18 k πc k =(1 r)(1 p) n k ( 1 k )[1 (1 p)k ] 1 r (1 p)( 1 b G C (b) = ) 1 n k (1 p) k 1 (1 p) k G C (r) > 0; r n k +1 (k +1)π k+1 C >kπ k C + π k N (6) k k +1 k +1 1/(k +1) r r 17

19 r v r b r v<r (v b)prob[win] < 0 r r r r (b = r) r r

20 19

22 5.2.2 H9 H10 H11 H12 H

24 6 23

25 [1] Gibbons, R. (1992), Game Theory for Applied Economists, Princeton University Press., (1995), [2] Graham, D.,and R.Marshall(1987), Collusive Bidder Behavior at Single-Object Second-Price and English Auctions, The Journal of Political Economy, 95(6), [3] Mailath, G.,and P.Zemsky(1991), Collusion in Second Price Auctions with Heterogeneous Bidders, Games and Economic Behavior, 3, [4] Matthews, S. (1995), A Thecnical Primeron Auction Theory 1: Independent Private Values, Discusion Paper No.1096, Northwestern Univ. [5] McAfee, P., J.McMillan(1992), Bidding Rings, The American Economic Review, 82(3), [6] McMillan, J.(1992) Games, Strategies, and Managers, Oxford Univeristy Press. (1995) [7] Riley, J.,and W.Samuelson (1981), Optimal Auctions, The American Economic Review, 71(3), [8] Wolfstetter, E. (1999), Topics in Microeconomics -Industrial organization, Auctions, and Incentives, Cambridge University Press. [9] Yeon-koo che, Ian Gale. (1998), Standard Auctions with Financially Constrained Bidders, The Review of Economic Studies, 65(1), [10] (2001) [11] (2000) [12] (2000), [13],, (2000), 24

2/50 Auction: Theory and Practice 3 / 50 (WTO) 10 SDR ,600 Auction: Theory and Practice 4 / 50 2

2/50 Auction: Theory and Practice 3 / 50 (WTO) 10 SDR ,600 Auction: Theory and Practice 4 / 50 2 stakagi@econ.hokudai.ac.jp June 24, 2011 2.... 3... 4... 7 8... 9.... 10... 11... 12 IPV 13 SPSB... 15 SPSB.... 17 SPSB.... 19 FPSB... 20 FPSB.... 22 FPSB.... 23... 24 Low Price Auction.... 27 APV 29...