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1 CIRJE-J

2 New Progress n Repeated Games: Implct Colluson wth Prvate Montorng Htosh Matsushma Faculty of Economcs, Unversty of Tokyo October 5, 001 Abstract The present paper provdes a survey on the recent progress n the theory of repeated games. Many recent works nvestgated nfntely repeated games wth dscountng, and newly assumed that montorng s prvate. These works provded ther respectve folk theorems or effcency theorems on ths assumpton. In partcular, t was shown n Matsushma (001a) that n repeated prsoner dlemma games, approxmate effcency can be attaned by a perfect equlbrum when prvate sgnals are condtonally ndependent and players are patent enough, rrespectve of the accuracy of prvate montorng technology. It was also shown n Matsushma (001b) that n general two player games, approxmate effcency can be attaned even though prvate sgnals are mperfectly correlated. The effcency theorem can be appled to the study of cartel behavor wth secret prce cuts.

3 (Matsushma (001a, 001b)) Repeated Games wth Prvate Montorng E-mal htosh@e.u-tokyo.ac.jp

4 4 5 6 (Sekguch (1997)) (Ely-Valmak (1999))

5 3 (code of behavor) 1 G = ( N, ( A, u ) N ) N = { 1,..., n} A A = A 1 A n u : A R N δ [0,1) a ( 1), a(),... v = (1 δ ) t= 0 δ u ( a( t)) t (Pearce (199)) (1994)

6 4 s : U H ( t) ( A ) H (t) A ) t= 1 t 1 mn max u ( a), mn max u ( a)) ( 1 a a1 a1 a δ (

7 5 G c d c 1 d 1 1 {1, } c ( c, c 1 ) ( c, d 1 ) (, c ) 1 ( 1,1) ( 0,0) ( 3,), 3) d d = d, d 1 c = ( ) d (

8 6 ( 1,1 ) ( 1,1) (1 δ ){ u ( c) + δu ( c) + δ u ( c) + } = (1 δ )(1 + δ + δ + ) = 1 u ( c / d ) = u ( d) = 0 (1 δ ){ u ( c / d ) + δu ( d) + δ u ( d) + } = (1 δ )( + δ + δ = (1 δ ) + ) (1 δ ) 1 δ δ

9 7

10 8 (Fudenberg and Maskn (1986)) G B 1 ε G ε B G ε B 1 ε ε 0. 5 G B G G B

11 9 v v = 1 δ ){ u ( c) + δ (1 ε) v } ( 1 εδ = 0 1 δ + εδ > ε ( 1,1 ) ε δ 1 3ε 1 ε ε v = 1 δ ){ u ( c / d ) + δεv } ( v = (1 δ ) 1 εδ 1 v v δ 3ε ( 0,0) ( 1,1 )

12 10 (Radner-Myerson-Maskn (1986)) (Matsushma (1989)) p( ω a) G B B ε (Kandor-Obara (000)) (Ely-Valmak (1999))

13 11 (Fudenberg-Levne-Maskn (1994)) (Abreu-Pearce-Stacchett (1990)) B

14 1 (coordnate) B G p( ω 1, ω a) = p1( ω1 a) p ( ω a) (Matsushma (1991))

15 13 (Sekguch (1997)) 1 1 G 1 B 1 G B B G G B G

16 14 3 (Ely-Valmak (1999)) (nterchangeablty) G B ( c, c 1 ) ( c, d ) ( d, c ) 1 1 ( d, d 1 ) 4 3 (Bhaskar-Obara (000))

17 15

18 16 (Matsushma (001a)) ε 0.5 ε 0.5 T T G M T T+1 T G M T T+1 T G T T G G M T M ( 1 ε)t ε T ( 1,1 ) 1 T G ( 1 ε) T M T+1

19 17 M ( 1 ε) T M ( 1 ε) T G G (Radner (1985, 1986))

20 18

21 19 (Matsushma (001b)) θ θ f ( θ a) p ( ω1, ω a, θ ) p ( ω 1, ω a) = p( ω1, ω a, θ ) f ( θ a) θ p ω, ω a, ) ( 1 θ p ( ω, ω a, θ ) = p1( ω1 a, θ ) p( ω a, 1 θ {1, } a { c, d} p ( a, θ ) : Ω [0,1] ( a j, θ ) A j Ξ Ω Ξ θ j )

22 0 Ω A j Ξ a A ω Ω µ ( a, ω ) T M θ µ ( a j, ω ) p ( ω a, θ ) θ ω µ ( a, ω ) a j = c j a j = d j µ ( a, ω )

23 1 (Matsushma (001b)) a 1 A1 = { 0,1,..., a1} a A = { 0,1,..., a} θ Ξ h { 1,..., n} a A θ θ h Ξ h {1, } ω θ θ h θ ω > a + 1) Ξ ( (Stgler (1964))

24 (Green-Porter (1984)) (Kandor-Matsushma (1998)) (Fudenberg-Levne-Maskn (1994)) (Abreu-Pearce-Stacchett (1990))

25 3 (Matsushma (001a, 001b))

26 4 Abreu, D., D. Pearce, and E. Stacchett (1990): Towards a Theory of Dscounted Repeated Games wth Imperfect Montorng, Econometrca 58, Bhaskar, V. and I. Obara (000): Belef-Based Equlbra n the Repeated Prsoners Dlemma wth Prvate Montorng, mmeo. Ely, J. and J. Valmak (1999): A Robust Folk Theorem for the Prsoner s Dlemma, mmeo. Fudenberg, D. and E. Maskn (1986): The Folk theorem n Repeated Games wth Dscountng or wth Incomplete Informaton, Econometrca 54, Fudenberg, D., D. Levne, and E. Maskn (1994): The Folk Theorem wth Imperfect Publc Informaton, Econometrca 6, Kandor, M. and H. Matsushma (1998): Prvate Observaton, Communcaton and Colluson, Econometrca 66, Kandor, M. and I. Obara (000): Effcency n Repeated Games Revsted: The Role of Prvate Strateges, mmeo. Matsushma, H. (1989): Effcency n Repeated Games wth Imperfect Montorng, Journal of Economc Theory 48, Matsushma, H (1991): On the Theory of Repeated Games wth Prvate Informaton, Part I: Ant-Folk Theorem wthout Communcaton, Economcs Letters 35, (1994) Matsushma, H. (001a): The Folk Theorem wth Prvate Montorng, Dscusson Paper CIRJE-F-13, Unversty of Tokyo. Matsushma, H. (001b): Repeated Games wth Prvate Montorng and Secret Prce Cuts, mmeo. Pearce, D. (199): Repeated Games: Cooperaton and Ratonalty, n Advances n Economc Theory: Sxth World Congress, ed. by J.-J. Laffont, Cambrdge Unversty Press. Radner, R. (1985): Repeated Prncpal Agent Games wth Dscountng, Econometrca 53, Radner, R. (1986): Repeated Partnershp Games wth Imperfect Montorng and No Dscountng, Revew of Economc Studes 53, Radner, R., R. Myerson, and E. Maskn (1986): An Example of a Repeated Partnershp Game wth Dscountng and wth Unformly Ineffcent Equlbra, Revew of Economc Studes 53, Sekguch, T. (1997): Effcency n Repeated Prsoners Dlemma wth Prvate Montorng, Journal of Economc Theory 76, Stgler, G. (1964): A Theory of Olgopoly, Journal of Poltcal Economy 7,

Vol. 31 No. 3 Aug. 2014 157 VCG [16][12] yes/no VCG VCG 1 ( ) 1. 1 (VCG-equvalent n expectaton, VCG-EE) VCG VCG VCG-EE VCG VCG

Vol. 31 No. 3 Aug. 2014 157 VCG [16][12] yes/no VCG VCG 1 ( ) 1. 1 (VCG-equvalent n expectaton, VCG-EE) VCG VCG VCG-EE VCG VCG 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