Implementation and Detection

Transcription

1 1 December Implementaton and Detecton Htosh Matsushma Department of Economcs Unversty of Tokyo

2 2 Ths paper consders mplementaton of scf: Mechansm Desgn wth Unqueness CP attempts to mplement scf f : ( A) allocaton f ( ) desrable to the general publc. However CP does not know whch state actually occurs. CP knows at least three players (partcpants) know : Majorty rule functons. However CP does not know who know. Hence CP needs to desgn ncentve mechansm for partcpants (players experts). Important Assumptons.e. acheve state-contngent CP accesses ex-post verfable nformaton channel. Players (partcpant experts) are not representatves of the general publc. State does not necessarly nclude detaled nformaton about players (cf. Unversal type spaces).

3 3 What knd of procedure ths paper consders? T K Round procedure At the begnnng CP desgns mechansm G ( M M0 g x) m ( m1... m T K) M m ( m ) N M Allocaton rule g: M ( A) Sde payment rule x : M M0 R At each round t from 1 to T each player N {1... n} accesses some nformaton channel (partton) C t and observes Ct ( ). He (or she) then announces message mt Mt. At each round t from T 1 to T K each player does not access any nformaton channel and contnues announce message mt Mt. At the end of round T K CP selects an allocaton accordng to gm ( ). Durng procedure CP can access some nformaton channel c 0 and observes C ( ) 0. After ths allocaton selecton CP s nformaton C ( ) 0 becomes publc and verfable. CP announces message m ( ) 0 C 0 truthfully. CP then makes sde payment for each player x ( mm ) R. by 0 Durng procedure we assume Imperfect Informaton!.

4 4 Ths paper consders severe uncertanty and severe ncentve requrements Very weak knowledge assumpton: Almost no common knowledge Very severe unqueness requrement: Strct Iteratve Domnance

5 5 Very weak knowledge assumptons (1) Fxed state space s common knowledge. But each player s pror dstrbuton on ( p P( 1) 1 0) s unknown to CP and others. (2) Sequence of nformaton channels c ( c) (( c t)) Fxed set of sequences of nformaton channels Ĉ s common knowledge. But each player s pror dstrbuton on Ĉ ( 2 CP and others. q Q( ) 2 0 Whch sequence n Ĉ s actually accessble s unknown. ) s unknown to (3) Profle of payoff functons v ( v ) V( ) ( v : A R ) are not common knowledge.

6 6 Very severe unqueness requrement: Strct teratve domnance Strategy for player : ( ) T K t s s t t 1 st : C Mt t t t t [ c( ) c ( ) ] [ st ( c) st ( c )] Player announces mt ( c s) at each round t {1... T K}. CP makes commtment: s0 : C0 M0 [ c 0 ( ) c 0 ( ) ] [ s 0 ( c 0 ) s 0 ( c 0 )] Strategy s s strctly domnated by s f and only f strct domnaton holds rrespectve of ( v p q ) E[ v( a ) t G pqs ] Ev [ ( a ) t G pqs s ] for all s S p P( 1) q Q( 2 ) and v V( ). (Ths s an extremely strong requrement) A strategy profle s strctly teratvely undomnated f and only f t survves through teratve removal.

7 7 Unque Implementaton n strct teratve domnance wth almost no sde payments Mechansm G unquely mplements scf f n strct teratve domnance f strctly teratvely undomnated strategy profle s unque and t acheves f. A scf f s unquely mplementable n strct teratve domnance f there s such a mechansm. A scf f s unquely mplementable n strct teratve domnance wth almost no sde payments f there s such a mechansm G and t permts just a tny sde payment: x ( m) 0 for all N and m M

8 8 Queston of Ths Paper Under very weak knowledge assumptons very severe unqueness requrements and almost no sde payments Can we mplement a socal choce functon? Answer: Yes we can! Why? Key Condton: People expect CP accesses verfable nformaton channels. Prevous works (Maskn (1999) Abreu and Matsushma (1992a 1992b 1994) Abreu and Sen (1991) Cheny and Kunmoto (2014) Duggan (1997) Dutta and Sen (2012) Elaz (2002) and Matsushma ( a 2008b 2008c 2013) ) commonly assume CP accesses no nformaton channels.

9 9 Why we need CP s accessblty? Proposton 1 (Impossblty Theorem) Suppose CP accesses no nformaton channel. If scf f s unquely mplementable n strct teratve domnance wth almost no sde payments then f must be strctly monotonc; for every ether f( ) f( ) or there exst N and a ( A) such that v ( f( ) ) v ( a ) and v ( f( ) ) v ( a ). Monotoncty mples dctatorshp by Gbbard (1973) and Satterthwate (1975)

10 10 Now let us permt CP to access verfable nformaton channels. We show a suffcent condton for unque mplementaton n strct teratve domnance wth almost no sde payments. Ths condton s a mld requrement: All we need s for CP to access just a margnal nformaton channel whch s unrelated to the socal choce functon. Key Words: Detecton Self-Detecton

11 11 Detecton Informaton functon : 2 detects channel aganst channel under channel f and only f for every and satsfyng ( ) ( ) there exsts ( ) such that ( ) ( ). A player has observed ( ) before. He addtonally observes ( ). He s requred to tell ether a cell of or I don t know. He tells a le.e. ( ). He expects the detector observes ( ) wth a postve probablty. he cannot exclude the possblty s real because ( ). If ( ) ( ) s empty he expects the detector to fnd out hs le. Ths prevents hm from lyng; he prefers tellng I don t know.

12 12

13 13 If ( ) ( ) s non-empty he expect the detector fals to fnd out hs le; hs le s not detected.

14 14 If ( ) ( ) then he s wllng to tell the truth by sayng ( )

15 15 Self-detecton Detectors ncentve matters: We need detector s detector detector s detector s detector. We need chan of detectors. Order of Detecton ( ) :{1... nt} N :{1... nt} {1... T} { l {1... nt} ( l) } T for all N [l l () l ( l ) ] [ () l ( l ) ] Sequence of nformaton channels c Cˆ s self-detectve n Cˆ Cˆ f there s a chan of detectors: For every l {1... nt} and c Cˆ ( c ) c detects c ( l) ( l) aganst l l ( l ) ( l) ( l ) ( l ) 0. ( l ) c ( l ) Detector 1 should be central planner because of verfablty.

16 16 Example: Fnancal Rescue at Crss The central planner wants to rescue Frm L but only knows Frm 0 s. Frm Moth-Eaten State Space l' s condton Frm ( l 1)' s condton

17 17 Example: Nuclear Plants at Sesmc Belt The central planner knows three experts know the truth but doesn t know who.

18 18 Self-detecton (General Verson) I know CP fals to access nformaton channels. But I stll prefer honesty. Informaton functon c ( c ): 2 : We defne t t ct ( c )( ) ct ( ) f c ( ) c ( ) c t( ) 1 ct ( c )( ) f t c ( ) c ( ) t 1 ( cc ) c 0 l Informaton functon ( cc ): 2 : l ( cc )( ) c ( ) { c ( c )( )}. 0 ( l ) ( l ) ( l ) l l ( l ) ( l) t

19 19 Sequence of nformaton channels c N {0} ˆ C s sad to be self-detectve n Cˆ Cˆ f there exsts ( ) such that for every l {1... nt} and c ˆ ( l ) C ( l ) there exsts ( c c ) Cˆ Cˆ satsfyng the followng propertes: 0 N\{ ( l)} 0 j j N\{ ( l)} c ( c c c ) belongs to Ĉ 0 ( l) N\{ ( l)} l ( cc ) detects c ( l ) ( l aganst ) ( l ) c ( l ).

20 20 ˆ n 2 { } ˆ ˆ c j C for all { n 2 n 1 n} ˆ C { } for all {0... n 3} c C C : For every {0... n 3} whenever c and for all j { 1 n 2 n 1 n}. 1 c 1 then Player l 2 expects player l 1 knows frm l 1 s condton wth a postve probablty. He however knows CP fals to know Frm 0 s.

21 21 Theorem 2 (Man Theorem) * Suppose that there s a self-detectve sequence c n Ĉ. * Suppose that scf s measurable w.r.t.. Suppose that for every possble sequence c n Ĉ there are at least three players who receve full nformaton about f ( ) * at round T.e. c. Then scf f s unquely mplementable n strct teratve domnance wth almost no sde payments. Ths s very permssve: Self-detecton s a weak requrement for CP s accessblty. We can permt many sequences are not self-detectve. Probablty of self-detectve sequences to be accessble s as close to zero as possble. We even permt self-detectve sequence does not belong to Ĉ. A player may sure expect CP never access any channels. Despte of ths he prefers beng honest. T

22 ( T K) 22 Sketch of Proof: How to desgn mechansm? -round Sequental Revelaton mechansm wth mperfect nformaton Round 1~T : Revelaton Phase * Each player announces whether he observes a cell of c t or even better nformaton. If he answers yes he also announces whch sell. Hence * Mt {{ ct ( )} }. All we need s just tny fnes. Round T 1: Recommendaton Phase Each player announces whether he knows * ( ) (or f ( )). If he answers yes he also announces * ( ). From round 1 through 1 T any detected player s fned a small amount. Any player who answers No s fned a smaller amount. These fnes ncentvze players because lyng gves no nfluence on allocaton. All we need s just tny fnes.

23 23 Round T 2~ T K: Abreu-Matsushma Method T : * ( ) or (I don t know). T. Each player repeatedly answer the same queston as round 1 Any player s fned a small amount f he s the frst devant from round 1 CP randomly selects the message profle from T 2 to T K. CP then selects the allocaton accordng to a varant of majorty rule. All we need s just tny fnes.

24 24 Abreu-Matsushma Revsted: Non-Expected Utlty We can replace expected utlty wth contnuty and ncreasngness: We defne payoff functon as u : R R. We assume: u( t ) s contnuous wth respect to and t u( t ) s ncreasng n t R. R The paper shows the same permssve result as the man theorem n a smplfed mplementaton problem.

25 25 Concluson Ths paper nvestgates unque mplementaton n strct teratve domnance. We make very weak knowledge assumptons; pror dstrbutons on state space and nformaton accessblty and even payoff functons are not common knowledge. We desgn a detal-free mechansm and apples a detal-free verson of strct teratve domnance. We permt CP to access verfable nformaton channels; otherwse t s mpossble to mplement scfs. Ths s the frst paper to nvestgate such very weak assumptons and verfable nformaton accessblty. Ths paper shows a suffcent condton for unque mplementaton n strct teratve domnance; self-detecton s suffcent. Ths s a permssve result; the central planner s channels s margnal and even unrelated to the socal choce functon. The probablty of each player as well as the central planner accessng non-degenerate channels s as close to zero as possble. I cannot gnore the possblty that the others detect my le even f t could be a small probablty. I am convnced CP fals to access nformaton ths tme. But I stll prefer honesty.

26 26 Q & A Q1: Are there other mechansm desgns? What are advantages of AM? A1: Modulo game and nteger game (Maskn (1999)). Modulo game has unque pure NE but multple mxed NE. Integer game acheves unqueness of mxed NE but s not robust w.r.t. announcement cost and s dscontnuous w.r.t. sze of message spaces. AM acheves unqueness of mxed NE as well as contnuty. Q2: I worry about f AM has dsadvantage of Level-K (Sefton and Yabas (1996)) and focal pont (Glazer and Rosenthal (1992)). A2: We need practcal devces to encourage players to follow strct teratve domnance. Learnng sessons teachng regularty of removal and framng encourage players ratonalty.