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

Transcription

1 CIRJE-F-1015 Mechansm Desgn n Hdden Acton and Hdden Informaton: Rchness and Pure Groves Htosh Matsushma Unversty of Tokyo Shunya Noda Stanford Unversty May 2016 CIRJE Dscusson Papers can be downloaded wthout charge from: Dscusson Papers are a seres of manuscrpts n ther draft form They are not ntended for crculaton or dstrbuton except as ndcated by the author For that reason Dscusson Papers may not be reproduced or dstrbuted wthout the wrtten consent of the author

2 1 Mechansm Desgn n Hdden Acton and Hdden Informaton: Rchness and Pure Groves 1 Htosh Matsushma 2 Unversty of Tokyo Shunya Noda 3 Stanford Unversty May 30, 2016 Abstract We nvestgate general collectve decson problems related to hdden acton and hdden nformaton We assume that each agent has a wde avalablty of acton choces at an early stage, whch provdes sgnfcant externalty effects on the other agent s valuatons n all drectons We characterze the class of all mechansms that solve the hdden acton problem, and demonstrate equvalence propertes n the ex-post term Importantly, we fnd that pure Groves mechansms, defned as the smplest form of canoncal Groves mechansms, are the only effcent mechansms that solve such hdden acton problems We argue that the resoluton of the hdden acton problem automatcally resolves the hdden nformaton problem JEL Classfcaton Numbers: C72, D44, D82, D86 Keywords: Hdden Acton, Hdden Informaton, Externaltes, Rchness, Pure Groves, Equvalence 1 Ths study s supported by a grant-n-ad for scentfc research (KAKENHI , 26J10076, 16H02009) from the Japan Socety for the Promoton of Scence (JSPS) and the Mnstry of Educaton, Culture, Sports, Scence and Technology (MEXT) of the Japanese government Ths paper s also supported by the Funa Overseas Scholarshp, the E K Potter Fellowshp, the JSPS research fellowshp (DC1), and the CARF research fund at Unversty of Tokyo We thank Gabrel Carroll, Fuhto Kojma, Takuo Sugaya, Yuch Yamamoto, and all the semnar partcpants n Htotsubash Unversty, Unversty of Tokyo, Osaka Unversty, Kyoto Unversty, and Stanford Unversty All errors are ours 2 Department of Economcs, Unversty of Tokyo, Hongo, Bunkyo-ku, Tokyo , Japan (htosh[at]eu-tokyoacjp) 3 Department of Economcs, Stanford Unversty, 579 Serra Mall, Stanford, CA , USA (shunyanoda[at]gmalcom)

3 2 1 Introducton Ths study nvestgates a general class of collectve decson problems that ncludes ssues of mperfect montorng (hdden acton) as well as ncomplete nformaton (hdden nformaton), such as prncpal-agent relatonshps, partnershps, and general resource allocatons (ncludng auctons and publc good provson), wth the assumptons of quas-lnearty and rsk-neutralty Multple agents ndependently make ther acton choces at an early stage before a state occurs, nfluencng ther valuaton functons through stochastc state determnaton The central planner then determnes an allocaton that s relevant to the welfare of all agents as well as the central planner n a state-contngent manner We assume hdden acton n that the central planner cannot observe the agents acton choces The central planner, therefore, desgns a state-contngent mechansm, accordng to whch the central planner makes sde payments to agents, as well as an allocaton decson, ncentvzng the agents to select the acton profle that the central planner desres We frst demonstrate a benchmark model that addresses only the hdden acton problem We then ncorporate hdden nformaton by assumng that the central planner can observe nether the state nor the agents acton choces, and therefore requres agents to announce ther prvate nformaton regardng the state (e, ther types) The purpose of ths study s to clarfy whether, and how, the central planner overcomes the ncentve problem n cases of both hdden acton and hdden nformaton Ths study substantally dffers from prevous research on mechansm desgn and contract theory n that each agent potentally has varous aspects of actvtes such as 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, and headhuntng The central planner generally lacks nformaton about the breadth of these potental aspects, because of, for example, the separaton of ownershp and control Accordngly, the

4 3 central planner cannot know whch aspects of agents actvtes are actually relevant to the current problem In such cases, the central planner must, nevtably, account for all of these aspects n mechansm desgn One example s a dynamc aspect of auctons for allocatng busness resources such as spectrum lcenses, where each partcpant undertakes varous proft-seekng actvtes to gan an advantage over rval companes n the upcomng aucton event 4 Whether, and to what degree, such actvtes dstort the welfare crucally depends on the aucton format desgn Specfcally, the central planner worres that each agent s hdden acton has sgnfcant externalty effects on the other agents valuaton functons (e, each agent can smoothly change the dstrbuton of the state n all drectons va a unlateral devaton from the desred acton profle) Ths broad potental for externaltes, whch ths study terms rchness, dramatcally restrcts the range of possble mechansms that can ncentvze agents to make the desred acton choces (e, n ths study s termnology, t restrcts the range of possble mechansms that can nduce the desred acton profle) The man contrbuton of ths study s to characterze mechansms that can nduce the desred acton profle, even n the presence of such rchness Based on ths characterzaton, we present equvalence propertes n the ex-post term under the constrants of nducblty That s, the ex-post payments, the ex-post revenue, and the ex-post payments are unque up to constants We further clarfy the possblty that the central planner acheves an effcent acton profle and effcent allocatons wthout any trouble n lablty Wth the assumpton of prvate values, we defne pure Groves mechansms as the smplest form of canoncal Groves mechansms, n whch the central planner gves each agent the welfare of the other agents and the central planner and mposes on her a fxed monetary fee We show that a mechansm nduces an effcent acton profle f and only f t s pure Groves In other words, pure Groves mechansms are the only effcent mechansms that solve the hdden acton problem 4 See Klemperer (2004) and Mlgrom (2004), for example

5 4 Ths theoretcal fndng has mportant mplcatons not only n cases of hdden acton but also n cases of hdden nformaton Suppose that the central planner cannot observe the state, and, therefore, requres agents to announce ther prvate nformaton regardng the state The above-mentoned dynamc aucton s an example of ths case Importantly, once the central planner desgns any mechansm that nduces the effcent acton profle (e, solves the hdden acton problem), then ths mechansm automatcally solves the hdden nformaton problem, because of ts nternalzaton feature The generally accepted vew n mechansm desgn s that n some envronments, Groves mechansms 5 are the only mechansms that solve the hdden nformaton problem In contrast to ths vew, ths study shows that pure Groves mechansms (e, the specal form of Groves mechansms), are the only mechansms that solve the hdden acton problem Moreover, the resoluton of the hdden acton problem n ths manner generally and automatcally can solve the hdden nformaton problem Because the class of pure Groves mechansms s a proper subclass of Groves mechansms, t mght be more dffcult for the central planner to earn non-negatve revenues wth hdden acton than wthout hdden acton In fact, the popular Vckrey Clarke Groves (VCG) mechansm, whch algns each agent s payoff wth her margnal contrbuton, generally guarantees non-negatve revenues, but t s not pure Groves (e, t fals to satsfy nducblty) To be more precse, we show an mpossblty result that wth rchness, there exsts no pure Groves mechansm (e, no well-behaved mechansm) that satsfes the requrements of non-negatve revenues and ex-post ndvdual ratonalty, whle the VCG mechansm satsfes both Accordngly, our result suggests that the central planner, who wants to defeat rchness n order to overcome the hdden acton problem, should collect nformaton n advance about whch aspects of agents actvtes are actually relevant An example wthout rchness s a case of partnershps wth fnte acton spaces, where the central planner can successfully talor the sde-payment rule to the detal of specfcatons, makng nducblty compatble wth non-negatve revenues, or even 5 See Vckrey (1961), Clarke (1971), and Groves (1973)

6 5 budget-balancng (Legros and Matsushma (1991), for example) In contrast, wth rchness, the central planner cannot remove agents evasons when she replaces a pure Groves mechansm wth any complcated mechansm The dffculty n lablty depends on the presence of each agent s externalty effect on the other agents valuaton functons If any agent s acton choce has no externalty and the central planner recognzes ths n advance, then we would obtan the equvalence result for nterm payoffs, rather than ex-post payoffs Specfcally, to acheve effcency, a wder class of mechansms than the class of Groves mechansms (expectaton-groves mechansms) can solve the ncentve problems of both hdden acton and hdden nformaton, wthout defcts, or even wth budget-balancng The remander of ths paper s organzed as follows Secton 2 revews the lterature Secton 3 presents the benchmark model, where we account for only hdden actons Secton 4 ncorporates hdden nformaton nto the model Secton 5 focuses on the achevement of effcency Secton 6 examnes the central planner s revenues Secton 7 studes the case of no externalty Secton 8 presents an alternatve defnton of rchness from the vewpont of full dmensonalty Secton 9 concludes 2 Related Lterature Ths study makes mportant contrbutons to the lterature regardng the hdden acton problem as follows When an agent has a wde varety of acton choces, a complcated contract desgn mght motvate the agent to devate from desred behavor In ths case, a smply desgned contract could functon better than a complcated one For example, Holmström and Mlgrom (1987) studed prncpal-agent relatonshps n a dynamc context, where randomly determned outputs are accumulated through tme, and the agent flexbly adjusts the effort level dependng on output-hstores They showed that the optmal ncentve contract that maxmzes the prncpal s revenue must be lnear wth respect to the output accumulated at the endng tme Carroll (2014) nvestgated optmal contract desgn n a statc prncpal-agent relatonshp, where the prncpal experences

7 6 ambguty about the range of actvtes that the agent can undertake Carroll showed that the optmal contract must be lnear wth respect to the resultant output, provded that the prncpal follows the maxmn expected utlty hypothess Ths study examnes the hdden acton problem by ntroducng multple agents, general state spaces, general allocaton rules, and varous crtera such as effcency, nstead of revenue optmzaton We account delberately for the wde range of externalty effects of each agent s acton choce on other agents valuaton functons by assumng rchness However, we do not assume ambguty or behavoral modes such as maxmn utlty We then demonstrate the characterzaton result for well-behaved ncentve mechansms, mplyng that only a smple form of mechansm desgn (e, pure Groves) functons, and we further show equvalence propertes n the ex-post term Any well-behaved effcent mechansm must be pure Groves In ths regard, Athey and Segal (2013) showed that wth prvate values, but regardless of whether rchness s present, pure Groves mechansms nduce effcency n hdden acton We extend ther study to show that, wth rchness, pure Groves mechansms are the only mechansms that can acheve effcency n hdden acton The lterature regardng the hdden acton problem has shown that wthout rchness (e, when the scope of acton spaces s suffcently lmted), we can desgn effcent ncentve mechansms by talorng the payment rule to detaled specfcatons We can even make the ncentve constrants compatble wth ether full surplus extracton or budget-balancng See Matsushma (1989), Legros and Matsushma (1991), and Wllams and Radner (1995), for example See also Obara (2008), whose model s close to ths study s model, but wthout rchness Wth rchness, however, a mechansm s dependence on detal mght encourage each agent to devate, because such dependence nevtably results n a loophole that puts the agent n a more advantageous poston The poor functonng that results from complcaton n mechansm desgn makes t dffcult to cope wth ncentves n hdden acton as well as non-negatvty of revenues

8 7 Ths study makes mportant contrbutons to the lterature on the hdden nformaton problem Green and Laffont (1977, 1979) and Holmström (1979) showed that n hdden nformaton envronments wth dfferentable valuaton functons, smooth path-connectedness, and prvate values, Groves mechansms are the only effcent mechansms that satsfy ncentve compatblty n the domnant strategy Ths study reconsders Groves mechansms from the vewpont of hdden acton, and shows that only pure Groves mechansms resolve the hdden acton problem Whle the resoluton of the hdden acton problem automatcally resolves the hdden nformaton problem, the reverse s not true 6 Ths study also nvestgates the case n whch externalty effects are absent, showng that expectaton-groves mechansms, whch are more general than Groves and requre each agent to pay the same amount as Groves n expectaton, are the only mechansms that can acheve effcency Hatfeld, Kojma, and Komners (2015) s relevant to ths result, because they showed that when we confne our attenton to effcent mechansms that are detal-free (e, ndependent of detaled knowledge about the set of actons that the agents can take), the mechansms must be Groves We permt a mechansm to not be detal-free, and then show that expectaton-groves mechansms successfully acheve effcency Importantly, the class of expectaton-groves mechansms ncludes the AGV mechansm (see Arrow (1979) and D Aspremont and Gerard-Varet (1979)), whch satsfes budget-balancng Ths result contrasts wth that of Hatfeld et al because Groves mechansms generally fal to satsfy budget-balancng It s worth notng that our results are rrelevant to the fne detal of the state space and valuaton functons Consequently, ths study artculates the desrablty of pure Groves mechansms and expectaton-groves mechansms, even f the prerequstes of Green Laffont Holmström fals to hold (eg, even f type spaces are fnte) 6 Hausch and L (1993) and Persco (2000) are related They demonstrate that frst-prce and second-prce auctons provde dfferent ncentves n nformaton acquston that make the other agents valuaton more accurate We explan that ths dfference comes from the dfference n ex-post payoffs between these aucton formats

9 8 3 Hdden Acton Let us consder a settng wth one central planner and n agents ndexed by N {1,2,, n} As a benchmark model of ths study, we nvestgate an allocaton problem that conssts of the followng four stages Stage 1: The central planner commts to a mechansm defned as ( g, x ), where n g : A and x x ) : R Here, denotes the fnte set of all states and ( N A denotes the set of all allocatons We call g and x the allocaton rule and the payment rule, respectvely 7 Stage 2: Each agent N selects a hdden acton b B, where B denotes the set of all actons for agent The cost functon for the agent s acton choce s gven by c : B R We assume that there s a no-effort opton B B and b( b1,, bn ) B N b 0 B such that c b Let 0 ( ) 0 Stage 3: The state s randomly drawn accordng to a condtonal probablty functon f ( b) ( ), where b B s the acton profle selected at stage 2, ( ) denotes the set of all dstrbutons (e, lotteres) over states, and f ( b) denotes the probablty that the state occurs provded that the agents select the acton profle b 7 Ths study assumes that the central planner commts to the mechansm before agents take acton Wthout ths assumpton, the desred outcomes, such as effcency, may not be achevable Consder the central planner who commts to a mechansm after the agents acton choces In ths case, the central planner prefers the VCG mechansm because t yelds greater revenue than the mechansm that ths study dscusses The VCG mechansm, however, fals to nduce any effcent acton profle; by antcpatng that the central planner wll set a VCG mechansm, agents are wllng to select neffcent actons

10 9 Stage 4: The central planner determnes the allocaton g( ) A and the sde payment n vector pad to the central planner x( ) ( x( )) N R, where s the state that occurs at stage 3 The resultant payoff of each agent N s gven by (1) v ( g( ), ) x ( ) c ( b), where we assume that each agent's payoff functon s quas-lnear and rsk-neutral, and the cost of the agent s acton choce s addtvely separable Fgure 1 descrbes the tmelne of the benchmark model Ths secton ntensvely studes the ncentves n hdden acton at stage 2 Fgure 1: Tmelne wth Hdden Acton Defnton 1 (Inducblty): A mechansm ( gx, ) s sad to nduce an acton profle b B f b s a Nash equlbrum n the game mpled by the mechansm ( gx,, ) e, for every N, (2) Ev [ ( g( ), ) x( ) b] c( b) Ev [ ( g( ), ) x( ) b, b ] c( b ) for all b B, where E[ b] denotes the expectaton operator condtonal on b, e, for every functon : R, E[ ( ) b] ( ) f ( b)

11 10 Lemma 1: Consder an arbtrary combnaton of an acton profle and an allocaton rule ( bg, ) There exsts a payment rule x such that ( g, x ) nduces b f and only f there n exsts a functon w( w ) : R such that N (3) E[ w( ) b] c( b) E[ w( ) b, b ] c( b ) for all N and b B Proof: Consder an arbtrary ( b,( g, x )) We specfy w by w( ) v( g( ), ) x( ) for all N and It s clear from Defnton 1 that ( g, x ) nduces b f and only f (3) holds QED Let nt ( ) ( ) denote the set of all full-support dstrbutons over states We ntroduce a condton on an acton profle b B, namely rchness, as follows Defnton 2 (Rchness): 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 8 Rchness mples that each agent N has a varety of acton choces around b that can smoothly and locally change the dstrbuton over states n all drectons from f ( b) at a dfferentable cost 9 8 Ths study assumes that the state space s fnte We, however, can elmnate ths assumpton wthout substantal changes Instead of the fnte state space, we can assume that the state space s a subset of a mult-dmensonal Eucldan space and the dstrbuton over states s contnuous In fact, we can obtan the unqueness results such as Lemmas 2 and 3 almost everywhere 9 Secton 8 wll replace Defnton 2 wth an alternatve that concerns only fntely many drectons

12 11 Under the rchness, f ( g, x ) nduces b, e, b s a Nash equlbrum n the game mpled by ( g, x ), the followng frst order condton holds for every N and ( ): Ev [ ( g( ), ) x( ) (, ), b ] c( (, )) 0 Furthermore, the rchness assures that each agent can change the expected value of any non-constant payment rules by makng a unlateral devaton Consder an arbtrary non-constant functon : R Note that there exsts such that 0 ( ) E[ ( ) b] Let denote the degenerate dstrbuton where ( ) 1 Takng (, ):[, ] B as defned n Defnton 2, we have E[ ( ) (, ), b ] 0 0 f ( (, ), b ) f ( b) ( ) lm ( ){ ( ) f ( b)} ( ) E[ ( ) b] 0 Hence, each agent can make a unlateral devaton to take advantage of non-constant In the sngle-agent case, where we regard the set of all actons of agent 1 as the set of all full-support dstrbutons over states, e, B 1 nt ( ), the well-known drectonal dfferentablty for arbtrary drectons wll be a tractable suffcent condton for the rchness 1011 Proposton 1: In the sngle-agent case, an acton B1 nt ( ) s rch f for every ( ), the agent s cost functon c 1 :n t ( ) R has the drectonal dervatve along : 10 The exstence of drectonal dervatves of c 1 at for all drectons s mpled by the total dfferentablty of c 1 at 11 When s a contnuum, the drectonal dfferentablty s generalzed to the Gâteaux dfferentablty

13 12 1((1 ) ) ( ) 1( ) lm c c c 0 Proof: For every ( ), specfy (, ):[, ] 1 B by (, ) (1 ) 1 for all [, ], where 0 s selected suffcently small so that 1(, ) nt ( ) B1 holds for all [, ] (snce s a full-support dstrbuton, such 0 always exsts) Then, t follows from c ( (, ) ) c ( ) c ((1 ) ) c ( ) lm lm c1( ) 0 0 that the drectonal dfferentablty of c 1 along assures the dfferentablty of c ( (, )) n at 0 Hence, s rch 1 1 QED Note that the converse of Proposton 1 does not hold, because the drectonal dfferentablty does not account for the dfferentals along curves whose tangent at b satsfes (4) Note also that t follows from Proposton 1 that n ths sngle-agent case, f c 1 s drectonally dfferentable everywhere on B 1, then every acton b1 B1 s rch We further consder the followng class of mult-agent problems For every N, specfy where we denote B nt ( ) B, 2 b ( b, b ) nt ( ) B Each agent N makes a recommendaton about the state dstrbuton as the frst component of her acton, b nt ( ), and lobbes her recommendaton as the second component of her acton, 1 b B Denote b ( b) N and b 2 ( b 2 ) N Defne f b b b, 2 1 ( ) ( ) () N

14 13 where and 2 ( b ) (0,1) N 2 ( b ) 1 The state dstrbuton s determned as the compromse among the agents recommendatons, e, the average of ther recommendatons 1 b weghted by each agent s nfluence 2 ( ( b )) N, whch s the consequent of the agents lobbyng actvtes We assume that for every N and b B, 2 c (, b ) s drectonally dfferentable In the same manner as Proposton 1, we can prove that every acton profle b B s rch n ths case Regardless of the acton profle b B, each agent N can smoothly and locally change the dstrbuton n all drectons from f( b) by manpulatng her recommendaton (the frst component) b 1 nt ( ) Note that the class of problems dscussed here ncludes substantally general mult-agent problems In fact, t does not requre any restrcton on the lobbyng acton spaces 2 ( ) N B, the N nfluence functons ( ) c N, and the cost functons ( ), besdes the drectonal dfferentablty of 2 ( c(, b )) N The followng lemma mples that wth rchness, the functon w that satsfes (3) s unque up to constants Lemma 2: Suppose that an acton profle b s rch, and a functon w satsfes (3) For n every functon w ( w ) : R, w satsfes the propertes mpled by (3), e, for N every N, Ew [ ( ) b] c( b) Ew [ ( ) b, b ] c( b ) for all b B, n f and only f there exsts a vector z ( z ) R such that N w ( ) w( ) z for all N and Proof: The proof of the suffcency s straghtforward We present the proof of the necessty as follows Take an arbtrary w whch satsfy (3) and consder an arbtrary agent N Take an arbtrary w such that w w s a non-constant functon

15 14 Then, there exsts such that ( ) E[ ( ) b] Let denote the degenerate dstrbuton where ( ) 1 Due to rchness, there exst 0 and (, ):[, ] B such that f( (, ), b ) f( b) lm ( ) f ( b) 0 Snce w satsfes (3), the frst order condton along (,) must hold, e, (5) Ew b c On the other hand, [ ( ) (, ), ] ( (, )) 0 Ew [ ( ) (, ), b ] c( (, )) 0 0 E[ ( ) w( ) (, ), b ] c( (, )) 0 0 E[ ( ) (, ), b ] ( ) E[ ( ) b] 0 Hence, f w w s non-constant, agent has ncentve to ncrease along (,) from 0 Accordngly, whenever w and w satsfy (3), w w s constant, e, there exsts z n R such that ( ) ( ) w w z for all QED Based on Lemmas 1 and 2, we show the followng theorem, whch states that the payment rule that guarantees nducblty s unque up to constants Theorem 1: Consder an arbtrary combnaton of an acton profle and a mechansm ( b,( g, x )) Suppose that b s rch and ( g, x ) nduces b Accordngly, for every payment rule x, the assocated mechansm ( g, x ) nduces b f and only f there exsts a vector z n R such that

16 15 x( ) x ( ) z for all Proof: The proof of the suffcency s straghtforward We present the proof of the necessty as follows Suppose that ( g, x ) nduces b Accordng to the proof of Lemma 1, we specfy w by (6) w( ) v( g( ), ) x( ) for all N and Suppose also that ( g, x ) nduces b Smlarly, we specfy w by (7) w ( ) v( g( ), ) x ( ) for all N and Lemma 2 mples that there exsts By subtractng (6) from (7), we have x( ) x ( ) z for all z n R such that ( ) ( ) w w z for all QED Theorem 1 mples the followng equvalence propertes n the ex-post term Consder an arbtrary combnaton of an acton profle and an allocaton rule (, bg ) Consder two arbtrary payment rules x and x such that both ( gx, ) and ( gx, ) nduce b Let U each agent N: and R and U R denote the respectve ex-ante expected payoff for U E[ v ( g( ), ) x ( ) b] c ( b), U E[ v ( g( ), ) x ( ) b] c ( b ) Accordngly, Theorem 1 exhbts that the ex-post payment for each agent s unque up to constants n that x ( ) x ( ) U U for all, the ex-post revenue for the central planner s unque up to constants n that x ( ) x ( ) ( U U ) for all, N N N

17 16 and the ex-post payoff for each agent s unque up to constants n that v ( g( ), ) x ( ) c ( b ) { v ( g( ), ) x ( ) c ( b )} U U for all 4 Hdden Informaton Let us specfy an nformaton structure where the state s decomposed as 0 1 n (,,, ) Here, we call 0 a publc sgnal and a type for each agent N Let 0 denote the set of all publc sgnals and N{0} denote the set of all types for each agent N Let We assume that the publc sgnal 0 0 becomes observable to all agents as well as the central planner just before the central planner determnes an allocaton and sde payments, and t s, therefore, contractble The central planner, however, cannot observe the profle of all agents types, whch s denoted by 0 ( ) N 0 Each agent N can observe her type, but cannot observe the profle of the other agents types, denoted by ( ) {0}\{ } j j N j jn{0}\{ } Because of the above-mentoned hdden nformaton structure, we replace stages 3 and 4 wth the followng stages (e, stages 3 and 4, respectvely) Importantly, the central planner requres each agent to reveal her type, whch the central planner cannot drectly observe 12 N 12 Because the central planner nduces a pure acton profle, we can safely focus on revelaton mechansms where each agent only reports her type If the central planner attempts to nduce a mxed acton profle, we need to consder mechansms where each agent reports not only her own type but also her selecton of pure acton See Obara (2008) For further dscussons, see Footnote 17

18 17 Stage 3 : The state ( 0, 1,, n ) s randomly drawn accordng to the condtonal probablty functon f( b) ( ), where b B s the acton profle selected at stage 2 Each agent N observes hs type, but cannot observe at ths stage Stage 4 : Each agent N announces about her type Afterward, all agents, as well as the central planner, observe the publc sgnal 0 0 Accordng to the profle of the agents announcements 0 ( ) N 0 and the observed publc sgnal 0 0, the central planner determnes the allocaton g 0 0 payment vector x( 0, 0) R n (, ) A and the sde The resultant payoff of each agent s gven by v( g(, ), ) x (, ) c( b) Fgure 2: Tmelne wth Hdden Acton and Hdden Informaton Fgure 2 descrbes the tmelne of the model wth hdden acton and hdden nformaton We consder the agents ncentves n hdden nformaton by ntroducng the concept of ex-post ncentve compatblty

19 18 Defnton 3 (Ex-Post Incentve Compatblty): A mechansm ( gx, ) s sad to be ex-post ncentve compatble (hereafter EPIC) f truth-tellng s an ex-post equlbrum; for every N and, v( g( ), ) x( ) v( g(, ), ) x(, ) for all 13 EPIC s ndependent of the acton profle, because of the addtve separablty of the cost of actons We further ntroduce a weaker noton, namely Bayesan Implementablty, as follows Defnton 4 (Bayesan Implementablty): A combnaton of an acton profle and a mechansm (,( b g, x )) s sad to be Bayesan mplementable (hereafter BI) f the selecton of the acton profle b at stage 2 and the truthful revelaton at stage 4 results n a perfect Bayesan 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) BI ncludes the nducblty of b n order to account for the possblty of contngent devatons; BI requres (,( b g, x )) to exclude the possblty that each agent benefts by devatng from both the acton choce b at stage 2 and the truthful revelaton at stage 4 The followng theorem states that f there s a mechansm that nduces an acton profle b but fals to be ncentve compatble, then t s generally mpossble to dscover mechansms that satsfy both nducblty and ncentve compatblty 13 Because of addtve separablty, the ncentve compatblty s rrelevant to the shape of the cost functons

20 19 Theorem 2: Consder an arbtrary combnaton of an allocaton rule and an acton profle (, bg ), where we assume that 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 ( gx, ) 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 ( gx, ) nduces b,( b,(, g x )) satsfes BI Proof: Suppose that both ( gx, ) and ( gx, ) nduce b From Theorem 1, there exsts z n R such that x( ) x ( ) z for all, whch mples that ( gx, ) satsfes EPIC f and only f ( gx, ) satsfes EPIC We can smlarly prove that ( b,( g, x )) satsfes BI f and only f ( b,( g, x )) satsfes BI QED () () Theorem 2 mples that the followng two statements are equvalent There exsts a mechansm that satsfes nducblty but fals to satsfy ncentve compatblty There exsts no mechansm that satsfes both nducblty and ncentve compatblty 5 Effcency We denote by v : A R 0 the valuaton functon for the central planner Ths secton and the next ntensvely study allocaton rules and acton profles that are effcent (e, maxmze the welfare that ncludes the central planner s welfare as well as the agents welfare); an allocaton rule g s sad to be effcent f

21 20 v ( g( ), ) v ( a, ) N{0} N{0} for all a A and A combnaton of an acton profle and an allocaton rule (, bg ) s sad to be effcent f g s effcent and the selecton of b maxmzes the expected welfare: such that 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}\{ } for all N and Accordng to a Groves payment rule, the central planner gves each agent N the monetary amount that s equvalent to the other agents welfare plus the central planner s welfare (e, jn{0}\{ } y( ) that s ndependent of v ( g( ), ) ), and mposes on the agent the monetary payment j A payment rule x s sad to be pure Groves f t s Groves and y () s constant n for each N; there exsts a vector z ( z ) R such that N x ( ) v ( g( ), ) z j jn{0}\{ } for all N and Pure Grove payment rules are specal cases of Grove payment rules where the central planner mposes a fxed amount z on each agent as a non-ncentve term The followng theorem states that an effcent mechansm nduces an effcent acton profle f and only f the payment rule s pure Groves Accordngly, under effcency and nducblty, wthout loss of generalty, we can focus on pure Groves mechansms (e, combnatons of an effcent allocaton rule and a pure Groves payment rule) Ths study makes a slght extenson of the canoncal defnton of a Groves mechansm, by accountng for the valuatons of the central planner

22 21 Theorem 3: Suppose that ( b, g ) s effcent For every payment rule x, ( g, x ) nduces b f x s pure Groves Suppose that b s rch and ( bg, ) s effcent For every payment rule x, ( g, x ) nduces b f and only f x s pure Groves Proof: If (, bg ) s effcent, for every N and b B, E[ v ( g( ), ) x ( ) b] E[ v ( g( ), ) x ( ) b, b ] E[ v ( g( ), ) b] E[ v ( g( ), ) b, b ] j j jn{0} jn{0} 0, whch mples the nducblty of b If b s rch, t s clear from Theorem 2 and the defnton of the pure Groves payment rule that any payment rule that guarantees nducblty must be pure Groves QED Note that we can construct any pure Groves mechansm wthout detaled knowledge of ( f, Bc, ) Even f the central planner s allowed to utlze such knowledge, the central planner s best choce would be pure Groves The followng theorem demonstrates a necessary and suffcent condton for the satsfacton of both nducblty and ncentve compatblty on the assumpton of effcency and rchness 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 :,

23 22 (9) E[ v ( g( ), ) b] c ( b) Ev [ ( g ( ( ), ), ) jn{0} jn{0}\{ } j v ( g( ( ), ), ( ), ) b, b ] c ( b) j Proof: Consder the proof for the case of EPIC From Theorem 3, ths proof only requres us to clarfy a necessary and suffcent condton for a pure Grove payment rule to guarantee EPIC Let x denote the pure Groves payment rule gven by x ( ) v ( g( ), ) jn{0}\{ } j for all N and Accordngly, ( gx, ) satsfes EPIC f and only f for every N,, and, v ( g( ), ) x ( ) v ( g( ), ) j j jn{0} v ( g(, ), ) v ( g(, ),, ) j jn{0}\{ } v ( g(, ), ) x (, ), whch s equvalent to (8) We can prove the case of BI n a smlar manner QED Note that (8) s a necessary and suffcent condton for a Groves mechansm to satsfy EPIC However, f each agent s payoff functon has nterdependent values, any Groves mechansm does not satsfy EPIC, and, therefore, (8) generally fals 15 Based on ths outcome, we shall focus on the case of prvate values, where for every N, the valuaton v ( a, ) s ndependent of the profle of the other agents types, and v ( a, ) s ndependent of 0 0 Wth prvate 0 ( j ) jn \{ } 0 jn \{ } j values, any Groves mechansm satsfes (8) (e, EPIC) Wth prvate values, we wrte 15 Maskn (1992), Dasgupta and Maskn (2000), and Bergemann and Välmäk (2002) proposed the generalzed VCG mechansm and showed that t satsfes EPIC for some envronments, even wth nterdependent values The generalzed VCG mechansm, however, fals to nduce an effcent acton profle, because t s not pure Groves Mezzett (2004), however, showed that f the realzed valuaton v ( g( ), ) s observable as an ex-post publc sgnal and s contractble, we can acheve effcency wth EPIC even wth nterdependent values See Noda (2016) for more general ex-post sgnals

24 23 v ( a, 0, ) nstead of v( a, ) for each N, and v 0 ( a, 0 ) nstead of v 0 ( a, ) 16 Wth prvate values, EPIC s equvalent to ncentve compatblty n a domnant strategy (thereafter DIC), where for every N,, and, v ( g( ),, ) x ( ) v ( g(, ),, ) x (, ) 0 0 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 Proof: From effcency, on the assumpton of prvate values, nequaltes (8) always hold Because, wth rchness, nequaltes (8) are necessary and suffcent for a Groves mechansm to satsfy DIC, Theorems 3 and 4 mply the latter part of Theorem 5 QED Wth prvate values, any DIC mechansm that satsfes nducblty and effcency must be pure Groves, although any Groves mechansm generally satsfes DIC and effcency For example, consder the VCG mechansm ( gx,, ) whch s defned as an effcent (Groves) mechansm specfed by x ( ) v ( g( ),, ) mn v ( g(, ),, ) 0 j 0 j jn{0} \{ } jn{0} for all N and Clearly, the VCG mechansm s not pure Groves; thus, t fals to satsfy the nducblty of effcent acton profle We permt the valuatons to depend on the publc sgnal 0 For convenence, we sometmes wrte v0( a, 0, 0) nstead of v 0 ( a, 0 ) 17 Obara (2008) showed that, when the acton space s fnte, full surplus extracton s achevable n approxmaton, under a weaker condton than exact full surplus extracton By carefully mxng

25 24 6 Revenues and Defcts By assumng prvate values, ths secton determnes whether the central planner can acheve effcency even wthout defcts We defne the central planner s ex-post revenue by (10) v ( ( ), ) ( ) 0 g 0 x, 18 N and the expected revenue n the ex-ante term by E[ v ( g( ), ) x ( ) b] 0 0 N We ntroduce three concepts of ndvdual ratonalty as follows The tmngs of ext opportuntes are depcted n Fgure 2 Defnton 5 (Ex-Ante Indvdual Ratonalty): A combnaton of an acton profle and a mechansm (,( b g, x )) s sad to satsfy ex-ante ndvdual ratonalty (hereafter EAIR) f Ev [ ( g( ),, ) x( ) b] c( b) 0 for all N 0 Defnton 6 (Interm Indvdual Ratonalty): A combnaton of an acton profle and a mechansm (,( b g, x )) s sad to satsfy nterm ndvdual ratonalty (hereafter IIR) f E [ v ( g(, ),, ) x (, ) b, ] 0 0 for all N and, where E [ (, ) b, ] (, ) f ( b, ) agents actons, Obara generated a correlaton between ( b, ) and ( b, ), and then appled the technque of Cremer and McLean (1985, 1988) Obara s argument, however, requred extremely large sde payments and very detaled knowledge about model specfcatons 18 Note that the central planner s revenue ncludes not only payments from agents, but also the valuaton of the central planner In the sngle agent case, where the allocaton space s degenerate, the value of (10) corresponds to the prncpal s payoff n a standard prncpal-agent model wth rsk-neutralty

26 25 denotes the expectaton of a functon (, ): condtonal on (, b ) R Defnton 7 (Ex-Post Indvdual Ratonalty): A mechansm ( gx, ) s sad to satsfy ex-post ndvdual ratonalty (hereafter EPIR) f v( g( ),, ) x( ) 0 for all N and 0 EPIR automatcally mples IIR IIR, however, does not necessarly mply EAIR, because the cost for the acton choce at stage 2 s a sunk cost The followng proposton shows that EPIR mples EAIR Proposton 2: Suppose that ( gx, ) nduces b Whenever ( gx, ) satsfes EPIR, (,( b g, x )) satsfes EAIR Proof: Because c b, t follows from EPIR and nducblty that 0 ( ) 0 E[ v ( g( ),, ) x( ) b] c( b) E[ v( g( ), 0, ) x( ) b, b ] c( b ) 0, whch mples EAIR QED EPIR s the strongest requrement for voluntary partcpaton among the above three concepts The purpose of ths secton s to show that EPIR s not compatble wth the non-negatvty of revenue n expectaton The followng proposton calculates the maxmal expected revenues (e, the least upper bounds of the central planner s expected revenues)

27 26 Proposton 3: Suppose that ( bg, ) s effcent, b s rch, and the prvate value assumpton s satsfed Then, the maxmal expected revenue from the mechansm ( gx, ) that nduces b and satsfes EPIR s gven by EPIR (11) R nmn vj( g( ), 0, j ) j ( n1) E vj( g( ), 0, j) b N{0} jn{0} The maxmal expected revenue from the mechansm ( gx, ) that nduces b and satsfes IIR s gven by IIR (12) R mn E v ( (, ), 0, ), j g j b N jn{0} ( n1) E vj( g( ), 0, j) b jn{0} The maxmal expected revenue from the mechansm ( gx, ) that nduces b and satsfes EAIR s gven by EAIR (13) R E vj( g( ), 0, j) bcj( bj ) jn{0} jn The maxmal expected revenue from the mechansm ( gx, ) that nduces b and satsfes IIR and EAIR s gven by (14) EAIR, IIR R Ev ( g( ), ) b mn Ev ( g( ),, ) bc ( b), N mn E v ( (, ), 0, ), j g j b E vj( g( ), 0, j) b jn{0} jn{0}\{ } Proof: See the Appendx EAIR Clearly, R s equal to the maxmzed expected socal welfare From the relatve strength of ncentve compatblty constrants, t s also clear that EPIR IIR, EAIR IIR R R R and R IIR, EAIR EAIR R

28 27 IIR However, whch s greater between R and R EAIR depends on specfcatons The followng proposton ndcates that wth the constrants of EPIR, t s generally dffcult for the central planner to acheve effcency wthout defcts Proposton 4: Suppose that ( bg, ) s effcent, b s rch, and the prvate value assumpton s satsfed Suppose also that c ( b ) 0 N and there exsts a null state 0 n (,, ) n the sense that and for every N, v0( a, 0) 0 for all a A, v ( a,, ) 0 for all a A and EPIR Wth EPIR, the central planner has a defct n expectaton: R 0 Proof: See the Appendx EAIR If R 0, the concluson s mmedate from R R EPIR EAIR EAIR R Suppose 0 It follows from c ( b ) 0 N that the second term of to the presence of the null state, the frst term of EPIR R n (11) s negatve Due R EPIR n (11) s non-postve EPIR Accordngly, R s negatve Wth prvate values, the VCG mechansm generally satsfes EPIR (and DIC) and guarantees that the central planner wll earn a non-negatve ex-post revenue at all tmes In contrast, once we requre nducblty, the VCG mechansm does not functon, and the central planner fals to earn non-negatve revenue, even wth an ex-ante expectaton By replacng EPIR wth weaker constrants, such as IIR and EAIR, and addng some restrctons, t becomes much easer for the central planner to acheve effcency wthout defcts The followng proposton states that the maxmal expected revenue of the central planner s rrelevant to whether nducblty s requred

29 28 Proposton 5: Suppose that ( bg, ) s effcent, b s rch, and the prvate value assumpton s satsfed Suppose also that we have: Condtonal Independence: For every b B and, f ( b) f( b ), N{0} where f ( b ) denotes the probablty of occurrng when the agents select the acton profle b The expected revenue acheved by any Groves mechansm that satsfes IIR and EAIR s less than or equal to R IIR, EAIR Furthermore, there exsts a pure Groves mechansm that satsfes IIR and EAIR and acheves IIR, EAIR R Proof: See the Appendx We mght expect that the central planner can receve larger expected revenue than IIR, EAIR R once we remove the requrement of nducblty However, wth condtonal ndependence, no Grove mechansm s able to make the expected revenue greater than IIR, EAIR IIR, EAIR R Ths mples that R s the upper bound of expected revenue regardless of whether we requre nducblty Proposton 5 s related to the observaton from rsk-sharng n a classcal prncpal-agent model When the prncpal s rsk-averse but the agent s rsk-neutral, the prncpal s best choce s to sell the company to the agent by gvng the entre outcome (externaltes to the prncpal) n exchange for a fxed constant fee 19 By regardng pure Groves mechansms as an extenson of such sellng-out contracts, Proposton 5 ndcates that wth rchness, sellng the company s the best choce even f the prncpal s rsk-neutral It s well-accepted that effcency s achevable by a Groves mechansm wthout runnng expected defcts f we do not requre nducblty Wth condtonal 19 See Harrs and Ravv (1979), for example

30 29 ndependence and some moderate restrctons, we can guarantee the non-negatvty of IIR, EAIR R as follows 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) Ev [ ( g( ), 0, j) b] c( b) 0 Wth IIR and EAIR, the central planner has non-negatve expected revenue: IIR, EAIR R 0 Proof: See the Appendx Non-negatve valuaton excludes the case of blateral barganng addressed by Myerson and Satterthwate (1983), where t s mpossble for the central planner to acheve effcency wthout defcts Non-negatve expected payoff excludes the case of opportunsm n the hold-up problem, where the sunk cost c ( b ) s so sgnfcant that t volates nequalty (15) By elmnatng these cases, and replacng EPIR wth IIR and EAIR, we can derve the possblty result n lablty mpled by Proposton 6 7 No Externalty So far, we have assumed rchness, e, that each agent s acton choce provdes a sgnfcant externalty effect on the other agents types and the publc sgnal Ths secton assumes that no such externalty exsts Specfcally, ths secton assumes ndependence of the nformaton structure, requrng condtonal ndependence and requrng that each agent s acton choce

31 30 b B only nfluences the margnal dstrbuton of the agent s type and b B, f( ( ) ( ) b) f0 0 f b N ; for every Here, f( b) denotes the dstrbuton of each agent s type that s assumed to depend only on b, and f () denotes the dstrbuton of the publc sgnal 0 0 that s assumed to be ndependent of the acton profle b B When we have such ndependence, agent s acton space s equvalent to the set of avalable margnal dstrbutons on agent s type Accordngly, ths restrcton expresses no externalty Wth ndependence, for every : R and :, we can smply wrte E R [ (, ) b ] and E[ ( ) b ] nstead of E [ (, ) b, ] and E[ ( ) b], respectvely On the acton profle b, we ntroduce a condton of prvate rchness, the no-externalty verson of rchness, as follows Defnton 8 (Prvate Rchness): An acton profle b B s sad to be prvately rch f we have the ndependence of the nformaton structure, and for every N and ( ), there exst 0 and a path on B, (, ):[, ] B, such that (16) (,0) b, 0 f( (, )) f( b) lm ( ) f( b), and c( (, )) s dfferentable n at 0 Prvate rchness mples that each agent N has a wde varety of acton choces that can smoothly and locally change the dstrbuton of the agent s type (not of the entre state ) n all drectons from f ( b ) Snce the acton choce problem of each agent s separated by the ndependence of the nformaton structure, Proposton 1

32 31 guarantees that an acton profle a A satsfes prvate rchness f the cost functon for each agent N, c, has drectonal dervatves at b Prvate rchness s a much weaker restrcton than the orgnal verson of rchness Accordngly, we can expect a much wder class of payment rules than pure Groves to satsfy nducblty The man purpose of ths secton s to characterze payment rules that guarantee nducblty under prvate rchness Lemma 3: Suppose that an acton profle b s prvately rch, and a functon w n satsfes (3) For every functon w ( w ) : R, w satsfes the propertes mpled by (3), e, for every N, E[ w( ) b] c( b) E[ w ( ) b, b ] c( b ) for all b B, f and only f E [ w (, ) w (, ) b ] s ndependent of N Proof: The proof of ths lemma parallels that of Lemma 2 See the Appendx Proposton 7: Consder an arbtrary combnaton of an acton profle and a mechansm ( 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 Proof: The proof of ths proposton parallels that of Theorem 1 See the Appendx Fx an arbtrary combnaton of an acton profle and an allocaton rule (, bg ) Consder two arbtrary payment rules x and x such that both ( gx, ) and ( gx, ) nduce b Let U each agent N: R and U R denote the respectve ex-ante expected payoff for U E[ v ( g( ), ) x ( ) b] c ( b),

33 32 and U E[ v ( g( ), ) x ( ) b] c ( b ) Proposton 7 mples that the nterm (rather than ex-post) payment for each agent s unque up to constants, n that E [ x (, ) b ] E [ x (, ) b ] U U for all Proposton 8: Consder an arbtrary combnaton of an allocaton rule and an acton profle (, bg ), where b s prvately rch If there exsts a payment rule x such that (,( b g, x )) satsfes BI, then, for every payment rule x, whenever ( gx, ) nduces b, (,(, b g x )) satsfes BI Proof: The proof of ths proposton parallels that of Theorem 2 See the Appendx Let us consder an arbtrary combnaton of acton profle and allocaton rule (, bg ) that are effcent A payment rule x s sad to be expectaton-groves f for each N, there exst r : R such that for every N, x ( ) v ( g( ), ) r( ) j jn{0}\{ } for all, and E [ r(, ) ] b s ndependent of Note that, wth effcency, any expectaton-groves payment rule guarantees nducblty Note that any Groves payment rule s expectaton-groves Whenever x s expectaton-groves, then 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, ( gx, ) nduces b f x s expectaton-groves Suppose that b s prvately rch and (, bg ) s effcent Wth ndependence, ( gx, ) nduces b f and only f x s expectaton-groves

34 33 Proof: See the Appendx The replacement of rchness wth prvate rchness dramatcally reduces the dffculty of achevng effcency wthout defcts In fact, the VCG payment rule s not pure Groves, but s expectaton-groves and satsfes EPIC Wth prvate values, t generally guarantees the non-negatvty of the ex-post payment from each agent to the central planner Wth prvate values, the mechansm ( gx, ) explored by Arrow (1979) and D Aspremont and Gerard-Varet (1979) (e, the AGV mechansm) s expectaton-groves, where r ( ) ( (17) x ( ) v ( g ( ), ) ) s specfed as jn {0}\{ } j r( ) v ( g( ), ) E [ v ( g(, ), ) b ] j j j j jn{0}\{ } jn{0}\{ } 1 n 1 E [ v ( g(, ), ) b ] j h j j h j jn\{ } hn{0}\{ j} Ths mechansm clearly satsfes budget-balancng: x ( ) 0 for all N Accordngly, wth ndependence, the central planner can acheve effcency under the constrants of BI and budget-balancng Hatfeld, Kojma, and Komners (2015) studed a relevant problem by focusng on mechansms that are detal-free (e, ndependent of specfcatons such as acton spaces and cost functons) They showed that effcent and nducble mechansms must be Groves In contrast, we permt mechansms to not be detal-free, and then show that expectaton-groves mechansms, whch ncludes Groves as a proper subclass, are necessary and suffcent 20 8 Alternatve Defnton of Rchness: Full Dmensonalty 20 Note that the constructon of an expectaton-groves mechansm does not depend much on fne detals of the specfcaton of ( f, Bc, ) In fact, the only knowledge needed for ths constructon s the shape of f( b) at the effcent acton profle b

35 34 So far, we have defned rchness as a condton mplyng that each agent can change the dstrbuton n all drectons by selectng pure actons Ths secton ntroduces an alternatve defnton of rchness, n whch each agent can change the dstrbuton n fntely many drectons by selectng pure actons Denote K dm( ( )) 1 Takng advantage of the fnteness of the state space, we assume the followng condton as an alternatve to rchness: For each N, there exst 0, { k }, and k K 1 { :[, ] B } 1 such that the K vectors K k 1 K k k k { ( ) f ( b) } are lnearly ndependent, and for every k {1,, K}, k (0), b k f( ( ), b ) f( b) k lm ( ) f ( b), 0 k and c ( ( )) s dfferentable n at 0 In contrast to the orgnal defnton of rchness, we only assume that each agent can change the dstrbuton n fntely many k K drectons, e, { ( ) f ( b) } k1, by selectng pure actons However, accordng to the k K lnear ndependence of { ( ) f ( b) } 1, t follows that for every ( ), there k exsts a mxed acton for agent N, by selectng whch ths agent can change the dstrbuton n the drecton of ( ) f( b) at a dfferentable cost Hence, wth the alternatve rchness, each agent can change the dstrbuton n all drectons by selectng not pure but mxed actons Ths fndng has mportant mplcatons Even f we replace the orgnal defnton of rchness wth the above-mentoned alternatve, we can prove all results of ths study due to rchness For the satsfacton of the alternatve rchness, we do not requre each agent to have a huge breadth of acton space All we have to do for the satsfacton of the alternatve rchness s to clarfy whether each agent can change the dstrbuton by selectng pure actons n fntely many drectons In the same manner as the above argument, we can also replace the defnton of prvate rchness wth an alternatve concernng only fntely many drectons