The proof is a straightforward probabilistic extension of Palfrey and Rosenthal (1983). Note that in our case we assume N

Transcription

1 PROOFS OF APPDIX A Proo o PROPOSITIO A pure strtey Byesn-s equlbr n te PUprtcpton me wtout lled oters: Te proo s strtorwrd blstc etenson o Plrey nd Rosentl 983 ote tt n our cse we ssume A B : It s esy to see [c condton 7 ] tt c > prtcpton s too costly or ny oter nd ull bstenton s te only equlbrum to : I c < n order or turnout to be pure strtey Byesn-s equlbrum outcome no non-prtcpnt my recee strctly er epected pyo by detn to bstenton prtcpton For een eery decson s potl wen In ll oter cses nobody s potl becuse Ts mples tt or cnn one s decson ects reenues nd or t does not Usn ts we cn dere necessry nd sucent condtons or pure strtey Byesn-s equlbr wt een turnout to est Frst te epected ncrese n reenue non-prtcpnt decdes to ote must be equl to or smller tn te costs: c A were s te blty tt te ote wll ect te outcome wc becuse s een only occurs tere s te: mn[ ] m[ ] Second te epected decrese n reenue prtcpnt decdes to bstn must be equl to or lrer tn te costs sed: c A were denotes te blty tt te swtc wll ect te outcome wc becuse s een only occurs tere s te: mn[ ] m[ ]

2 For odd we rst estbls te cses were n bstner s potl Ts occurs wen mplyn tt cn orce te I oweer s one o te prtcpnts se s potl becuse swtc to bstenton would reduce ctory to te In ll oter cses 3 nd cnn one s decson s no eect on reenues Usn ts we cn dere necessry nd sucent condtons or pure strtey Byesn-s equlbr wt odd turnout to est Frst or non-prtcpnt A must old ow s en by te blty tt s n roup wt one-ote deet to te oter roup: mn[ ] m[ ] Second or prtcpnt A must old Here s te blty tt s n roup wt one-ote ctory oer te oter roup: mn[ ] m[ ] et we nestte weter nd wc pure strtey Byesn-s equlbr est tt ulll A nd A We consder ll possble cses Full bstenton : Full bstenton cnnot be n equlbrum We only need to consder A becuse A B Ts reduces to c A3 snce wc contrdcts our ssumpton tt c < Full bstenton n one roup nd poste prtcpton n te oter roup ; > : Full bstenton n nd poste prtcpton > n cnnot be n equlbrum Ts s esy to see Gen te pure strtey o bstenton ollowed by eeryone n or ny > eery prtcpnt n s n ncente to bstn untl wc suces to wn te electon But t s dnteous or eery bstner n to prtcpte becuse te lue rom turnout s wc eceeds c <

3 Full prtcpton : For some c < equlbr wt ull prtcpton est We only need to consder A becuse A B For een {odd} ts reduces to c A4 { c } Hence or c { c } A s stsed wc proes o te proposton Full prtcpton n one roup nd possbly some n te oter roup < : cnnot be n equlbrum becuse oters n e n ncente to swtc to otn For < < tere est equlbr or some c < wt ull prtcpton n nd possbly some prtcpton < n For een {odd} A ppled to es c { c nd A ppled to es nd to } c { c } c { c } A5 Hence c { c c } re te only cses were A nd A cn be ontly ullled wc proes o te proposton Oter equlbr wt < < : ote tt by ssumn symmetrclly dstrbuted roup szes we cn restrct our nlyss to oters n For suc equlbr to est we need to sow tt tere s some c tt ontly ullls A nd A : 3

4 c A6 We e emples or beore prodn enerl proo tt c est tt ulll A6 mple : For en ] te blty tt te only ote cst s n te oter roup s [ perceed by n bstner n s Ten < Furtermore or te only prtcpnt te blty tt te own roup s one more ote tn te oter s Hence Tereore we e > nd or ny c A6 olds nd pure strtey Byesn-s equlbr est wt ectly one oter turnn out to ote mple : From A6 t ollows tt s n equlbrum outcome or ny c [ ] Wt needs to be sown s tt ts set s non-empty For en ] te blty tt te two otes re dded eqully cross te two [ roups en tt bstns s { ote n ote n } Smlrly Te set s non-empty Ten ssumn or te moment we e Snce te blty dstrbuton o s symmetrc round ts es or ter some rerrnements nd becuse te lst two terms o te sum re equl to zero et note tt te rcton s poste or < Hence > It s esy to see tt ts stll olds wen we e up our ssumpton tt > sows tt te rne 4

5 5 ] [ s non-empty so ll combntons o strtees yeldn consttute pure strtey Byesn-s equlbr or c n ts rne We now turn to te enerl cse o turnout ben n equlbrum outcome < < nd een: Smlr to te rument n our emples A6 s used to determne rne ] [ n wc c sould le to mke n equlbrum outcome We ten proceed to sow ts rne s non-empty Once n consder roup szes nd [ ] ] ]mn[ m[ For outsde o ts rne te blty o te t s nd drops out o te clculton o For en n ts rne te blty tt te otes re splt eqully en tt bstns s } otes n otes n { A7 Smlrly } otes n otes n { A8 Denn es nd -equlbr est or some c Ten ssumn < or te moment we e

6 6 A9 were we use Obously te second term dsppers snce Ts sows tt ssumn < so r s nnocent ow usn ] m[ nsted nd becuse te dstrbuton o s symmetrc round we e: ] m[ ] m[ [ ] ] m[ > A becuse > or < Hence or eery PU-prtcpton me wt symmetrclly dstrbuted roup szes we cn nd some c suc tt pure strtey Byesn-s equlbrum ests n wc n een number o oters rom eter roup prtcptes nd ll oters bstn wt < < < < nd odd: Smlrly we use A6 to determne ] [ s rne n wc c sould le to mke n equlbrum outcome nd proceed to sow tt ts rne s nonempty Consder roup szes nd [ ] ] ]mn[ m[ Once n s outsde o te rne te blty o te s For en te blty tt te otes re splt suc tt tere s one ote less n en tt bstns s

7 7 } otes n otes n { A And en nd tt prtcptes te blty tt te otes re splt suc tt tere s one ote more n s } otes n otes n { A Denn ϕ es ϕ nd ϕ -equlbr est or some c Ten ssumn < or te moment we e ϕ ϕ ϕ ϕ

8 ϕ ϕ A3 were we use ote tt te second term o A3 s poste nd dsppers we drop te ssumpton < Becuse te dstrbuton o s symmetrc round we e: ϕ ϕ ] m[ ϕ ϕ [ ϕ ϕ ] m[ ] ϕ A4 ote tt te rst term n A4 s poste snce ϕ > ϕ or < But te second term s poste or zero only I ts s te cse ten > Howeer > bot terms n A4 e to be eluted ncluse te possble second term n A3 Weter s poste or zero depends on te symmetrc blty dstrbuton t nd wc we e not speced urter Hence or eery PU-prtcpton me wt ny symmetrclly dstrbuted roup szes nd < tere s some rne or c suc tt pure strtey Byesn-s equlbrum ests n wc n odd number o oters rom eter roup prtcptes nd ll oters bstn For > > tese equlbr my est dependn on te speccton o te symmetrc roup sze dstrbuton nd proe o te proposton An nterestn property o te -equlbr ust descrbed s tt te rnes or dcent turnouts re dcent too nd c s non-ncresn n To see ts look t c mn nd c m Obously c < c m 8

9 es te upper lue It s redly ered tt c mn c m olds or een or wc we e c ϕ c mn m nd or odd or wc we e A5 c c ϕ mn m Usn ts t s esy to see tt te derence or een we e c s lso equl to zero snce mn c m c mn c m nd or odd we e A6 c c m mn Hence we estblsed tt te rnes o possble equlbrum costs c or dcent re dcent s well nd tt te costs re non-ncresn n To : Condtons A to A re necessry nd sucent or te estence o pure strtey Byesn-s equlbr QD Proo o PROPOSITIO A pure strtey Byesn-s equlbr n te PUprtcpton me wt lled oters: Te proo s strtorwrd etenson o te proo o proposton A Becuse lotn oter knows er preerence roup se cn updte te blty dstrbuton o roup szes so enerlly: ecept or te cse een wt Due to symmetry nd ence ll lled lotn oters e te sme posteror blty dstrbuton o roup szes Contrry to tt o lotn oters te preerences roup membersps o lled oters re dentble Dene totl rete prtcpton o lled lotn oters by nd te derence n prtcpton between bot lled roups by 9

10 To : See proo o proposton A To nd : I c < n order or otes to be pure strtey Byesn-s equlbrum no prtcpnt non-prtcpnt my recee strctly er epected pyo by detn to bstenton prtcpton Ten t s necessry nd sucent or equlbr wt turnouts to est tt te ollown condtons old or ll lled nd lotn non-prtcpnts: o non-prtcpnt nd no prtcpnt nd A B wll cne er decson c c A7 nd A B wll cne er decson c c A8 were te bltes o ben potl or lled nd lotn oters re smlr to tose n A nd A ecept tt we now use updted bltes o roup szes only or lotn oters c proo o proposton A et we estbls weter nd wc pure strtey Byesn-s equlbr est tt ulll A7 nd A8 We consder ll possble cses Due to te bnoml dstrbuton o roup szes wt p 5 te blty o ben potl o lotn lled non-prtcpnt nd n lled prtcpnt or een s en by 5 oterwse A9 were corrects or te dentble prtcptons o lled oters Obously te strct nequltes n A7 nd A8 cnnot be ullled smultneously becuse lotn nd lled non-prtcpnts s well s lled prtcpnts e te sme blty o ben potl Hence or n equlbrum to est t must old tt c We nestte weter tese equltes cn be ullled ontly wt c or lotn prtcpnts A lotn prtcpnt s blty o ben potl s en by

11 5 oterwse A wc s en tt te blty o ll oter oters s equl to c lrer tn smller tn; equl to c or lotn prtcpnts n te roup wt < > ; It ollows tt or een A7 nd A8 ben ullled ontly cn only occur c nd ll lotn prtcpnts re members o roup wt For < oweer snce t s known tt only lotn oters n prtcpte tey re only potl But ten lled prtcpnts n would preer to bstn Hence tese cses cnnot be equlbr ote urter tt or te trl cses wt ll oters n bltes o ben potl equl to c occur Tese equlbr re not urter dscussed ere o oter pure strtey Byesn-s equlbr est or een n wc bstners nd prtcpnts coest For odd we cn wrte te blty or n lled non-prtcpnt s 5 oterwse A nd tt or n lled prtcpnt s 5 oterwse A It s redly ered tt A nd A re equl or lled non-prtcpnts nd lled prtcpnts But te blty o lled prtcpnts ben potl s lrer tn smller tn; equl to tt o lled non-prtcpnts > < It ollows tt lled prtcpnts nd lled bstners cnnot coest n te sme roup respectely A7 nd A8 cnnot be ullled ontly unless or eery lled oter n prtcptes nd none n Dscussn rst t s esy to see tt A A nd te blty o lotn non-prtcpnts ben potl re te sme or 5 A3

12 wc s smlr to A9 or een As beore we nestte net weter c nd c or lotn prtcpnts cn be ullled ontly or were te blty o ben potl o lotn prtcpnt s en by 5 A4 Te epresson n A4 s lwys lrer tn tt n A3 Hence or odd pure strtey Byesn-s equlbr wt bstners nd prtcpnts toeter ndeed est or > c Wt respect to te second cse t s redly ered tt t cnnot be n equlbrum wt > nd ll lled oters n prtcptn nd ll lled oters n bstnn Ts s becuse te blty o ben potl o requred lotn prtcpnts s en by 5 oterwse A5 wc s smller tn tt o lled bstners n Hence suc equlbr cnnot est o oter pure strtey Byesn-s equlbr est n wc bstners nd prtcpnts coest Ts leds us to te nlyss o two more possble equlbr let: Full bstenton : Full bstenton cnnot be n equlbrum becuse ny snle oter cn rse pyo by by turnn out wc s lrer tn te prtcpton costs c < Full prtcpton : qulbr wt ull prtcpton est or F een {odd} wt F F F F F 5 5 c F F F F F F { 5 5 c F > F s lon s } A6 To : Condtons A7 to A8 re necessry nd sucent or te estence o pure strtey Byesn-s equlbr QD