Mechanism Design in Hidden Action and Hidden Information: Richness and Pure Groves

Transcription

1 1 December 13, 2016, Unversty of Tokyo Mechansm Desgn n Hdden Acton and Hdden Informaton: Rchness and Pure Groves Htosh Matsushma (Unversty of Tokyo) Shunya Noda (Stanford Unversty) May 30, 2016

2 2 1. Introducton Mechansm Desgn wth Sde Payments: Hdden acton (Ex-Ante Investment) hdden nformaton (Revelaton) ex. Aucton, Publc Goods, Prncpal-agent, Partnershps, Tmelne (General) n Stage 1 CP desgns and commts to mechansm ( gx, ), g: A, x ( x ): R. A denotes the set of alternatves, denotes the state space. Stage 2 (Hdden acton) Agents make acton choces 1 Stage 3 (Hdden nformaton) A state b( b,..., b ) B B at the expense of c ( b ). (,,,..., n ) occurs. Acton choces nfluences state dstrbuton f( b): [0,1]. Each agent N prvately observes as hs (or her) type.. After ther reports, CP and agents observe publc sgnal 0 0. n CP determnes g( 0, 1,..., n) A and x( 0, 1,..., n ) R. Stage 4 Each agent reports n N

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4 4 Rchness (Key Assumpton of Ths Paper) Each agent has varous aspects of actvtes. nformaton acquston, R&D nvestment, patent control, standardzaton, M&A, rent-seekng, postve/negatve campagns, envronmental concern, product dfferentaton, entry/ext decsons, preparaton of nfrastructure, headhuntng Each agent s acton choce has sgnfcant externalty effects. Each agent can change the state dstrbuton, ncludng the other agents types, n varous drectons.

5 5 Queston: Can CP solve both ncentves n hdden acton and n hdden nformaton? How? To what degree?

6 6 Example: Sngle-Unt Aucton Falure of Second-Prce Aucton (SPA) Each bdder N He (or she) obtans payoff maxm announces prce bd m 0. f he wns,.e., gm ( ) j j 0 f he loses,.e., gm ( ) (or (or m m maxm ). j maxm ). j j j Truth-tellng m s a domnant strategy n SPA. SPA solves ncentve n hdden nformaton, and acheves allocatve effcency. What s wrong wth SPA?

7 7 SPA fals to acheve effcency n hdden acton. Each bdder makes ex-ante nvestment that nfluences the other bdders valuatons. In order to save the wnner s payment maxm j j max, each bdder makes under-nvestment that decreases the others valuatons SPA fals to acheve ex-ante effcency n hdden acton. j j. We need another protocol desgn!

8 8 Each bdder has varous technologes: rchness n ths paper s termnology

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11 11 CP must take nto account technologes 1, 2, and 3 altogether. How should CP desgn mechansm? Answer: Let s desgn Pure Groves mechansm (PGM)!

12 12 What s Pure Groves Mechansm (PGM)? A Varant of Posted Prce Scheme CP fxes a prce z for each bdder N n advance. Each bdder N reports m 0. The wnner ( m m, j ) pays z j to CP. Each loser j receves loser s gan maxm z j from CP. N Truth-tellng s a domnant strategy n PGM, because PGM s Groves.

13 13 Pure Groves mechansm solves ncentve not only n hdden nformaton but also n hdden acton Each bdder s wllng to make ex-ante nvestment n PGM, Because t ncreases loser s gan maxm z, N whle keepng the wnner s payment z unchanged.

14 14 More Practce: Symmetrc Pure Groves Mechansm ( z 1 z z for all N) s equvalent to Descendng Aucton for Determnng Loser s Gan CP fxes (common) prce z n advance. CP conducts Descendng Aucton for the determnaton of losers gan. Frst bdder who drops hs hand becomes wnner, gettng commodty at prce z. The prce level at whch the wnner drops hs hand,.e., k R, s regarded as loser s gan. Each loser receves k from CP. Droppng hands at the prce level z s a domnant strategy, achevng effcency and the same payments as symmetrc PG, hence solvng both hdden acton and hdden nformaton.

15 15 However, n PGM, Each bdder (low valuaton) may have negatve payoff n ex-post term. Ex-Post Indvdual Ratonalty (EPIR) may be questonable. We need commtment devce such as depost requrement at the nterm stage (stage 3 ). Fortunately, we can show Interm Indvdual Ratonally (IIR) s generally harmless. End of SPA Example

16 16 Man Results of Ths Paper Result I: Inducblty (Incentve n hdden acton) Assumpton of Rchness dramatcally restrcts the range of mechansms that can nduce the desred acton profle as a NE outcome. Ex-post Equvalence: Payments, revenue, and payoffs are unque up to constants. Pure Groves: A mechansm nduces an effcent acton profle f and only f t s pure Groves. Defcts and IR: There may exst no mechansm that satsfes non-negatve expected revenue and ex-post ndvdual ratonalty (EPIR). Commtment devces guarantees nterm ndvdual ratonalty (IIR). Result II. Incentve Compatblty (Incentve n hdden nformaton, EPIC) Any mechansm that solves hdden acton automatcally solves hdden nformaton (EPIC). Result III. No Externalty (Prvate Rchness) Wthout externalty, a much wder class of mechansms, namely, expectaton-groves, solves both hdden acton and hdden nformaton wthout defcts, or wth budget-balancng.

17 17 2. Related Lteratures Green and Laffont (1977, 1979), Holmström (1979): Characterzaton of Groves from hdden nformaton cf. Characterzaton of pure Groves from hdden acton Bergemann and Valmak (2002): Prvate Values vs Interdependent Values Hatfeld, Kojma, and Komners (2015): No externalty, detal-freeness, Groves cf. Wth and/or wthout rchness, expectaton-groves Obara (2008): Mxed actons, unbounded sde payments, approxmate full surplus extracton cf. Bounded sde payments, pure actons, defcts Athey and Segal (2013): Suffcency of pure Groves cf. Necessty of pure Groves under rchness

18 18 3. Benchmark (wth hdden Acton, but wthout hdden nformaton) Tmelne (wthout hdden nformaton) Stage 1: CP desgns a mechansm ( gx, ). Stage 2 (Hdden acton) Agents make acton choces b B at the expense of c ( b ). Stage 3 (No-hdden-nformaton) A state occurs. CP and all agents observe. n Stage 4: CP determnes g( ) A and t( ) R.

19 19 Inducblty: Defnton A mechansm ( gx, ) s sad to nduce an acton profle b B f b s a NE,.e., (2) E[ v( g( ), ) x( ) b] c( b) E[ v( g( ), ) x( ) b, b ] c( b ) for all N b B. and

20 20 Rchness: Defnton (1) Each agent N can smoothly and locally change the dstrbuton of state n all drectons from f( b) through pure acton devaton. An acton profle b B s sad to be rch f for every N and ( ), there exst 0 and a path on B, (, ):[, ] B, such that (,0) b, f( (, ), b ) f( b) (4) lm ( ) f( b), 0 and c ( (, )) s dfferentable n at 0. *Rchness (1) sounds very restrctve, but we can replace t wth much weaker requrements wthout loss of substances.

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24 24 Rchness: Defnton (2) (Weaker than (1)) Each agent changes state dstrbuton n only fnte drectons through pure acton devaton. However, each agent can change state dstrbuton n all drectons through mxed acton devaton. For each N, there exst 0, { k } K k K { :[, ] B } k 1 k k K K dm( ( )) 1 vectors { ( ) f ( b) } k1 k for every k {1,..., K}, (0) b, k f( ( ), b ) f( b) k lm ( ) f( b), 0 k and c ( ( )) s dfferentable n 0. Alternatvely: An acton profle b B s sad to be rch f for every N a path on B, (, ):[, ] ( B) f( (, ), b ) f( b) (4) lm ( ) f( b),, and such that 1 are lnearly ndependent, and ( ), there exst 0 and, such that (,0) b, 0 and c ( (, )) s dfferentable n at 0.

25 * We can further replace (2) wth a weaker requrements wthout loss of substances (explan later). 25

26 26 Frst-Order Condton: Wth rchness (1) or (2), we can show: If ( gx, ) nduces b, for every N and ( ), Ev [ ( g( ), ) x( ) (, ), b ] c( (, )) 0 0. Non-Constant Devaton: For every N and every non-constant functon : R, there exsts b b such that E[ ( ) b, b ] E[ ( ) b]. By addng any non-constant fee, each agent has ncentve to devate.

27 27 Ex-post Equvalence Theorem Theorem 1: Consder an arbtrary ( b,( g, x )). Suppose that b s rch ((1) or (2)) and ( gx, ) nduces b. For every payment rule x, the assocated mechansm ( gx, ) nduces b f and only f there exsts n a vector z R such that for every, x( ) x ( ) z. Consder an arbtrary ( bg,, ) x, and x. Assume both ( gx, ) and ( g, x ) nduce b. Let U E[ v ( g( ), ) x( ) b] c( b) and U E[ v ( g( ), ) x ( ) b] c ( b). From Theorem 1, we have: Ex-post payment equvalence: x ( ) x ( ) U U Ex-post revenue equvalence: x ( ) x ( ) ( U U ) N N N Ex-post payoff equvalence: v ( g( ), ) x ( ) c( b) { v ( g( ), ) x ( ) c ( b)} U U Theorem 1 mples that the class of all well-behaved mechansms s qute lmted. Cf. Green and Laffont (1977, 1979), Holmström (1979): Hdden Informaton

28 28 * Even wthout rchness, nducblty may stll be a severe requrement.

29 29 * Even wthout rchness, nducblty may stll be a severe requrement.

30 30 * Even wthout rchness, nducblty may stll be a severe requrement.

31 31 * Even wthout rchness, nducblty may stll be a severe requrement.

32 32 Even wthout rchness, Pvot (or VCG) fals to satsfy nducblty. Let z ( ) mn v ( a, ) aa j N {0} j.

33 33 4. General Model: Hdden Acton and Hdden Informaton Tmelne (general)

34 34 Defnton 3 (Ex-Post Incentve Compatblty, EPIC): A mechansm ( gx, ) s sad to be ex-post ncentve compatble (EPIC) f truth-tellng s an ex-post equlbrum; for every N,, and, v ( g( ), ) x( ) v ( g(, ), ) x(, ). Defnton 4 (Bayesan Implementablty, BI): A combnaton ( b,( g, x )) s sad to be Bayesan mplementable (BI) f the selecton of the acton profle b at stage 2 and the truthful revelaton at stage 4 results n a Nash equlbrum; for every N, every b B, and every functon :, E[ v ( g( ), ) x( ) b] c( b) E[ v ( g( ( ), ), ) x( ( ), ) b, b ] c( b).

35 35 Theorem 2: Consder an arbtrary ( bg., ) Assume b s rch. 1. Suppose that there exsts a payment rule x such that ( gx, ) nduces b and satsfes EPIC. For every payment rule x, whenever ( g, x ) nduces b, t satsfes EPIC. 2. Suppose that there exsts a payment rule x such that ( b,( g, x )) satsfes BI. For every payment rule x, whenever ( g, x ) nduces b, ( b,( g, x )) satsfes BI. [We fnd a mechansm that satsfes nducblty but does not satsfy IC] [We can never fnd a mechansm that satsfes both]

36 36 5. Effcency: Pure Groves Mechansm An allocaton rule g s sad to be effcent f v ( g( ), ) v ( a, ) N{0} N{0} for all a A and. A combnaton ( bg, ) s sad to be effcent f g s effcent and the selecton of b maxmzes the expected welfare: E[ v ( g( ), ) b] c ( b ) N{0} N{0} E[ v ( g( ), ) b] c ( b ) for all b B N{0} N{0}. A payment rule x s sad to be Groves f there exsts y : R for each N x ( ) ( ( ), ) ( ) v g y j. jn{0}\{ } such that

37 37 Pure Groves Mechansm: Defnton Groves mechansm wth constant fees n A payment rule x s sad to be pure Groves f there exsts a vector z ( z ) R such N that x ( ) v ( g( ), ) z. j jn{0}\{ } Note Pvot (VCG) mechansm (SPA) s not pure Groves. In sngle-unt allocatons (wth symmetry), PGM (symmetrc) s equvalent to Posted Prce wth Descendng Aucton determnng loser s gan

38 38 Wth the constrants of effcency, nducblty, and rchness, we can safely focus on pure Groves. Theorem 3: Suppose that ( bg, ) s effcent. For every payment rule x, ( gx, ) nduces b f x s pure Groves. Suppose that b s rch and ( bg, ) s effcent. For every payment rule x, ( gx, ) nduces b f and only f x s pure Groves. (Proof s straghtforward from ex-post equvalence.)

39 39 Wth effcency, nducblty, and rchness, t s dffcult to satsfy IC (EPIC or BI) n nterdependent values, whle IC automatcally holds n prvate values. Theorem 4: Suppose that b s rch and ( bg, ) s effcent. There exsts a payment rule x such that ( gx, ) nduces b and satsfes EPIC f and only f for every N,, and, (8) v ( g( ), ) v ( g(, ), ) v ( g(, ),, ) j. j jn{0} jn{0}\{ } There exsts a payment rule x such that ( b,( g, x )) satsfes BI f and only f for every N, b B, and :, (9) E[ v ( g( ), ) b] c ( b ) j jn{0} E[ v ( g( ( ), ), ) v ( g( ( ), ), ( ), ) b, b ] c ( b) j jn{0}\{ } (8) and (9) mples Groves satsfes EPIC. Theorem 5: Suppose that ( bg, ) s effcent. Wth prvate values, for every payment rule x, ( gx, ) nduces b and satsfes DIC f x s pure Groves. Suppose that b s rch and ( bg, ) s effcent. Wth prvate values, for every payment rule x, ( gx, ) nduces b and satsfes DIC f and only f x s pure Groves.

40 40 6. Revenues and Defcts: Effcency n Prvate Values CP s ex-post revenue: 0 CP s expected revenue: 0 0 v ( g ( ), ) x ( ) 0 N E[ v ( g( ), ) x( ) b] N Defnton 5 (Ex-Ante Indvdual Ratonalty): A combnaton ( b,( g, x )) s sad to satsfy ex-ante ndvdual ratonalty (EAIR) f Ev [ ( g( ),, ) x( ) b] c( b) 0 for all N 0. Defnton 6 (Interm Indvdual Ratonalty): A combnaton ( b,( g, x )) s sad to satsfy nterm ndvdual ratonalty (IIR) f E [ v ( g(, ),, ) x (, ) b, ] 0 for all N 0. and Defnton 7 (Ex-Post Indvdual Ratonalty): A mechansm ( gx, ) s sad to satsfy ex-post ndvdual ratonalty (EPIR) f v ( g( ),, ) x( ) 0 for all N and. 0 [EPIR] [IIR and EAIR]

41 41 Proposton 3: Suppose that ( bg, ) s effcent, b s rch and nducble. Wth prvate values, the maxmal expected revenues are gven by EPIR (11) R nmn v ( g( ),, ) ( n1) E v ( g( ),, ) b j 0 j j 0 j jn {0}. jn{0} (12) R IIR mn E v ( g (, ),, ) b, ( n 1) E v ( g ( ),, ) b j 0 j j 0 j N jn {0}. jn{0} EAIR (13) R E v ( g( ),, ) b c ( b ) j 0 j j j jn {0}. jn EAIR, IIR (14) R Ev ( g( ), ) b mn 0 0 E v ( g( ),, ) bc ( b ), 0 N mn E v ( g(, ),, ) b, E v ( g( ),, ) b. j 0 j j 0 j jn{0} jn{0}\{ }, R EPIR R IIR EAIR mn[ R IIR, R EAIR ]

42 42 Wth EPIR, CP fals to acheve effcency wthout defcts when a null state exsts. Proposton 4: Assume prvate values. Suppose that ( bg, ) s effcent and b s rch. Suppose also c( b) 0 N and there exsts a null state (,..., ) 0 n where v ( a, ) 0 for all a A, 0 0 and for every N, v ( a,, ) 0 for all a A and Then, wth EPIR, CP has a defct n expectaton: EPIR R 0. cf. Pvot mechansm satsfes EPIR and DIC, but does not satsfy nducblty.

43 43 Wth IIR and EAIR, maxmal expected revenue s rrelevant to nducblty, and t s generally non-negatve. Proposton 5: Assume prvate values and Condtonal Independence: For every b B and, f ( b) f ( b ). N{0} Suppose that ( bg, ) s effcent and b s rch. Then, revenue acheved by Groves mechansms that satsfy IIR and EAIR. IIR, EAIR R s the maxmal expected Proposton 6: Assume the suppostons n Proposton 4, condtonal ndependence, and the followng condtons. Non-Negatve Valuaton: For every N {0} and, v ( g( ),, ) 0. 0 Non-Negatve Expected Payoff: For every N, (15) E[ v ( g( ),, ) b] c ( b ). 0 j Wth IIR and EAIR, the central planner has non-negatve expected revenue: IIR, EAIR R 0. Non-negatve valuaton excludes the case of blateral barganng (Myerson and Satterthwate (1983)). Non-negatve expected payoff excludes the case of opportunsm n hold-up problem (excludes large sunk cost c( b )).

44 44 7. No Externalty: Prvate Rchness Prvate Rchness mples each agent can change the dstrbuton of (but not ) n all drectons. Independence of nformaton structure: for every and b B, f( b) f ( ) f ( b ) 0 0. N Defnton 8 (Prvate Rchness): An acton profle b B s sad to be prvately rch f we have ndependence of nformaton structure, and for every N and ( ), there exst 0 and (, ):[, ] ( B ) such that (,0) b, f ( (, )) f ( b ) (16) lm ( ) f ( b ), 0 and c ( (, )) s dfferentable n at 0.

45 45 Interm Equvalence Theorem Proposton 7: Consder an arbtrary ( b,( g, x )). Suppose that b s prvately rch and ( g, x ) nduces b. For every payment rule x, the assocated mechansm ( g, x ) nduces b f and only f E [ x (, ) x (, ) b ] s ndependent of. Fx an arbtrary ( bg, ) and two arbtrary payment rules x and x. Assume ( g, x ) and ( g, x ) nduce b. Let U E[ v ( g( ), ) x( ) b] c( b) and U E[ v ( g( ), ) x ( ) b] c ( b). We have Interm payment equvalence: E [ x (, ) b ] E [ x(, ) b ] U U

46 46 Wth prvate rchness, nducblty automatcally mples BI. Proposton 8: Consder an arbtrary ( bg., ) Assume prvate rchness. If there exsts a payment rule x such that ( b,( g, x )) satsfes BI, then, for every payment rule x, whenever ( g, x ) nduces b, ( b,( g, x )) satsfes BI.

47 47 Wth effcency, nducblty, and prvate rchness, we can safely focus on Expectaton-Groves nstead of PGM. Assume ( bg, ) s effcent. A payment rule x s sad to be expectaton-groves f for each N, there exst r : R such that for every N and, x ( ) ( ( ), ) ( ) v g r, and j jn{0}\{ } E [ r(, ) b ] s ndependent of. Expectaton-Groves guarantees nducblty under prvate rchness. Groves are expectaton-groves. Whenever x s expectaton-groves, any payment rule x s expectaton-groves f and only f E [ x (, ) x (, ) b ] s ndependent of. Proposton 9: Suppose that ( bg, ) s effcent. Wth ndependence, ( g, x ) nduces b f x s expectaton-groves. Suppose that b s prvately rch and ( bg, ) s effcent. Wth ndependence, ( g, x ) nduces b f and only f x s expectaton-groves.

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49 49 Wth prvate values, Expectaton-Groves guarantees non-negatve ex-post payments. Wth prvate values, AGV mechansm s expectaton-groves: (17) r( ) v ( g( ), ) E [ v ( g(, ), ) b ] j j j j jn{0}\{ } jn{0}\{ } 1 E [ v ( g(, ), ) b ]. j h j j h j n 1 jn\{ } hn{0}\{ j} AGV satsfes budget-balancng: x ( ) 0 for all. N Wth ndependence, CP can acheve effcency even wth the constrants of BI and budget-balancng!

50 50 9. Concluson We studed mechansm desgn wth sde payments that ncludes hdden acton and hdden nformaton. We assumed rchness n that each agent has a wde avalablty of ex-ante actvtes that have a sgnfcant externalty effect on other agents valuatons. We showed that the class of mechansms that nduce the desred acton profle s restrctve as follows. The payment rule that satsfes nducblty s unque up to constants. We have the ex-post equvalence theorem. Effcent mechansms that satsfy both nducblty and ncentve compatblty must be pure Groves, correspondng to a posted-prce scheme wth descendng aucton. It s dffcult to satsfy both nducblty and ncentve compatblty n nterdependent values, whle t s generally possble n prvate values. CP has to struggle to avod defcts. But we have possblty results once we permt nterm commtments (IIR). Wth no externalty, expectaton-groves, ncludng AGV, are only well-behaved mechansm. Wth no externalty and prvate values, we can acheve effcency wth BI and BB. End