Dynamic Analysis of an Electoral Campaign
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Lambertini, Luca Working Paper Dynamic Analysis of an Electoral Campaign Quaderni - Working Paper DSE, No. 415 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca (2001) : Dynamic Analysis of an Electoral Campaign, Quaderni - Working Paper DSE, No. 415, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4884 This Version is available at: https://hdl.handle.net/10419/159256 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/
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iurp zklfk= Y&E| Y| 2YbE| Y| '2SE4nB t &E|Er rq+6:, Wkhuhiruh/ 43 Y&E| Y| 'fdw & 7 'q 2 E r eS 2 E4nB 2 +6;, dqg Y%E| Y| 'qE r 2S E 4nBB% E |'fdw % 7 'qE r 2S E 4nBB+6<, Wklv frqfoxghv wkh surri1 Wkh vwdelolw| surshuwlhv ri wkh v|vwhp dw wkh vwhdg| vwdwh surgxfhg e| vrfldo sodqqlqj duh dv iroorzv= Sursrvlwlrq ; Wkh vrfldo sodqqlqj htxloleulxp q % 7 c& 7 r lv d vdggoh srlqw1 Wkh surri lv rplwwhg iru euhylw|/ dv lw lv dqdorjrxv wr wkh surri ri Sursr0 vlwlrq 91 816 Sulydwh yv vrfldo rswlpxp Wkh frpsdudwlyh hydoxdwlrq ri wkh wzr uhjlphv fdq eh txlfo| ghdow zlwk/ dv lw lv txdolwdwlyho| dqdorjrxv wr vhfwlrq 6161 Hydoxdwlqj Sursrvlwlrqv 8 dqg :/ lw lv lpphgldwh wr yhuli| wkdw & 7 & W (% 7 % W +73, Dffruglqjo|/ dv lq prgho L/ dovr khuh wkh vrfldo rswlpxp lv dfklhyhg e| lq0 yhvlqj ohvv wkdw lq wkh sulydwh rswlpxp1 Dv d frqvhtxdqfh/ wkh vrfldoo| h!flhqw qxpehu ri yrwhuv lv orzhu xqghu vrfldo sodqqlqj wkdq dw wkh sulydwh htxloleulxp1 43 Djdlq/ wkh vroxwlrq n + w , @3 fdq eh glvplvvhg1 46
9 H{whqvlrqriprghoL=rswlpdoiudjphqwd0 wlrq Lq wkh iruhjrlqj dqdo|vlv/ L kdyh dgrswhg wkh dvvxpswlrq wkdw wkhuh h{lvw rqo| wzr sduwlhv/ dqg L kdyh hydoxdwhg wkh h!flhqf| ri vxfk d v|vwhp1 Krzhyhu/ pxowlsduw| v|vwhpv duh udwkhu frpprq/ dqg wr wklv uhjdug vhyhudo lqwhuhvwlqj txhvwlrqv fdq eh dgguhvvhg lq d jhqhudolvdwlrq ri dq| ri wkh deryh vhwwlqjv1 Rqh vxfk txhvwlrq lv zkhwkhu wkhuh vkrxog eh d olplw wr wkh qxpehu ri sduwlhv/ dqg/ li vr/ krz wr vhw wklv olplw1 Wzr uhodwhg txhvwlrqv duh +l, zkhwkhu wkh rswlpdo qxpehu ri sduwlhv rewdlqv e| pd{lplvlqj wkh ryhudoo qxpehu ri yrwhuv ru froohfwlyh zhoiduh/ dqg +ll, zkhwkhu rswlpdo iudjphqwdwlrq lv kljkhu lq wkh vrfldo ru lq wkh sulydwh rswlpxp1 Lq sduwlfxodu/ txhvwlrq +ll, lqyroyhv wkh frpsdulvrq ri wzr vlwxdwlrqv1 Rqh lv wkh uvw ehvw zkhuh d ehqhyrohqw vrfldo sodqqhu frqwurov erwk wkh qxpehu ri sduwlhv dqg wkhlu lqglylgxdo lqyhvwphqw lq wkh hohfwrudo fdpsdljq1 Wkh rwkhu lv d vhfrqg ehvw zkhuh wkh lqyhvwphqw lv qrqfrrshudwlyho| ghflghg e| sduwlhv/ zkloh wkh qxpehu ri sduwlhv lv frqwuroohg e| d ehqhyrohqw sodqqhu1 KhuhLuhfrqvlghuprghoLzlwk?sduwlhv +dqg ?fdqglgdwhv,/ zkhuh/ iru wkh vdnh ri vlpsolflw|/ L dgrsw wkh v|pphwu| dvvxpswlrq q 'qiru doo Wklv dprxqwv wr h{foxglqj wkh xvh ri d txruxp vr dv wr ghwhuplqh wkh rswlpdo qxpehu ri sduwlhv +ru fdqglgdwhv,/ vlqfh wkh htxloleulxp vl}h +l1h1/ wkh yroxph ri yrwhv lq vwhdg| vwdwh, lv wkh vdph dfurvv sduwlhv1 Wkh glhuhqwldo htxdwlrq ri wkh vwdwh yduldeoh lv= Y% E | Y| '& E|r [ ' & E|B% +74, Dffruglqjo|/ wkh forvhg0orrs irupxodwlrq ri fdqglgdwh v Kdplowrqldq qrz uhzulwhv dv= K E |'e 3 4| ; ? = q% E |S 2 d & E | o 2 nb E | 5 7 & E |r [ ' & E |B% 6 8 n [ ' b E | 5 7 & E |r [ 6 ' & 6 E |B% 6 8 < @ > +75, Rqfh djdlq/ lw fdq eh vkrzq wkdw wkh rshq0orrs vroxwlrq lv d ghjhqhudwh forvhg0orrs vroxwlrq/ dqg wkhuhiruh +75, fdq eh uhirupxodwhg e| vhwwlqj b E |' b E |dqg b E |'firu doo 47
Sursrvlwlrq < Lq wkh sulydwh rswlpxp zlwk ?fdqglgdwhv/ wkh ryhudoo qxp0 ehuriyrwhuvlvpd{lplvhgdw? W % 'nr 2rczkloh wkh htxloleulxp froohfwlyh xwlolw| lv pd{lplvhg dw ? W 'BE n 2rn24E n r erE4nBczlwk ? W % :? W lq wkh dgplvvleoh udqjh ri sdudphwhuv1 Surri1 Iluvw rughu frqglwlrqv iru wkh sulydwh rswlpxp duh= YK E| Y& E |'S& E|nb E|'f +76, YK E | Y% E |'Yb E | Y| 4b E |,+77, Yb E | Y| ' E 4nBb E |q soxv wkh vdph lqlwldo dqg wudqvyhuvdolw| frqglwlrqv dv lq vhfwlrq 6141 Iurp +74,/ +76, dqg +77, L rewdlq= 44 & W 'q S E 4nB(% W 'q d r E ? o S E 4nBB:f;??'nr r+78, Khqfh/ wkh wrwdo qxpehu ri yrwhuv dw htxloleulxp lv f W '?% W czkloh wkh froohfwlyh sd|r dffuxlqj wr sduwlhv lv W '? q% W S 2E& W 2 'q 2 ?dBE n 2rE ? n 24E n rE ?o 2S E 4nB 2 B+79, Lw lv wkhq lpphgldwh wr yhuli| wkdw f W lv pd{lplvhg dw ? W % 'nr 2r ? +7:, zkloh W lv pd{lplvhg dw ? W 'B E n2rn24 E nr er E 4nB ? +7;, 44 L rplw frpsxwdwlrqdo ghwdlov iru euhylw|1 Lw fdq eh dovr hdvlo| vkrzq wkdw + 78 , lv d vdggoh srlqw1 48
zlwk ? W % ? W 'B erE4nB:firu doo rc Bc 4 +7<, Wklv frqfoxghv wkh surri1 Wkh deryh uhvxow whoov wkdw/ dw wkh sulydwho| rswlpdo vwhdg| vwdwh/ wkh iudjphqwdwlrq wkdw hqvxuhv wkh kljkhvw froohfwlyh xwlolw| surgxfhv wrr olwwoh sduwlflsdwlrq wr hohfwlrqv rq wkh sduw ri yrwhuv1 Wklv/ lq wxuq/ fdoov iru wkh hydoxdwlrq ri wkh vdph lvvxh xqghu vrfldo sodqqlqj1 Sursrvlwlrq 43 Xqghu vrfldo sodqqlqj zlwk ?fdqglgdwhv/ erwk wkh ryhudoo qxpehu ri yrwhuv dqg wkh froohfwlyh xwlolw| duh pd{lplvhg dw ? 7 'nr r Surri1 Wkh ehqhyrohqw vrfldo sodqqhu pd{lplvhv K 7 E |'e 3 4| q% E |S 2 d & E | o 2 nb E | d & E | E r E ? B% o +83, z1u1w1 lqglylgxdo dgyhuwlvlqj hruw & E |Wkh xvxdo surfhgxuh |lhogv wkh iro0 orzlqj vdggoh srlqw htxloleulxp= & 7 'q d r E ? o S E 4nB:f;??'nr r(% 7 'q d r E ? o 2 S E 4nBB +84, Wkhvh ydoxhv doorzv ph wr fdofxodwh f 7 '?q d rE?o 2 SE4nBB+85, zklfk lv pd{lplvhg dw ? 7 % 'nr r ? +86, Ilqdoo|/ urxwlqh fdofxodwlrqv duh qhhghg wr yhuli| wkdw ? 7 % dovr pd{lplvhv 7 '? q% 7 S 2 & 7 2 +87, dqg wkhuhiruh ? 7 '? 7 % '? 7 D uhohydqw frqvhtxhqfh ri Sursrvlwlrqv <043 lv= Fruroodu| 6 Iru doo r5 fc 2 c 49
? W % :? 7 :? W l B 4:2Enr 2r ? W % :? W :? 7 l B 45 % fc2Enr 2r $ Iru doo r5 2c c? W % :? 7 :? W iru doo Bc4 Surri1 Iurp Sursrvlwlrq < zh douhdg| nqrz wkdw ? W % :? W Pruhryhu/ ? W % :? 7 dozd|v1 Wr frpsohwh wkh surri/ revhuyh wkdw ? 7 ? W 'B E 2r24 E nr 2r E 4nB:fl B 4:2 E nr 2r+88, zkhuh 2 E nr 2r2dqg lqfuhdvlqj lq rc iru doo r5 fc 2 c+89, zkloh lw lv qhjdwlyh iru doo r5 2c Wkhuhiruh/ lq wkh odwwhu lqwhuydo/ ? 7 ? W :firu doo Bc4 Qrz h{dplqh wkh wzr dowhuqdwlyh vfhqdulrv1 Wkh uvw lv vrfldo sodqqlqj/ zkhuh d ehqhyrohqw glfwdwru frqwurov wkh dgyhuwlvlqj lqyhvwphqw ri hdfk sduw|/ dqg pd{lplvhv froohfwlyh zhoiduh lq vwhdg| vwdwh z1u1w1 ? Wkh vhfrqg vfhqdulr lv d sulydwh htxloleulxp gulyhq e| lqglylgxdo lqfhqwlyhv/ zlwk wkh sodqqhu frqwuroolqj rqo| wkh qxpehu ri sduwlhv1 Qhhgohvv wr vd|/ wkh zhoiduh ohyho dvvrfldwhg wr wkh uvw htxloleulxp lv vwulfwo| odujhu wkdq wkdw dvvrfldwhg zlwk wkh vhfrqg1 Krzhyhu/ wkh vrfldoo| rswlpdo iudjphqwdwlrq ghshqgv xsrq wkh vl}h ri sdudphwhuv1 Lq sduwlfxodu/ lw lv kljkhu xqghu sodqqlqj wkdq lq wkh sulydwh rswlpxp +l, iru doo ydoxhv ri sdudphwhuv ghvfulelqj wkh lqwhuwhpsrudo surshuwlhv ri wkh prgho/ li wkh qhjdwlyh h{whuqdolw| lv odujh> dqg +ll, li wkh ghsuhfldwlrq udwh Blv frqvlghudeo| odujhu wkdq wkh glvfrxqw udwh 4c zkhq wkh qhjdwlyh h{whuqdolw| lv orz1 Uhvxow +l, lqglfdwhv wkdw kdylqj d odujh qxpehu ri fdqglgdwhv +ru sduwlhv, lv vrfldoo| suhihudeoh zkhq hdfk sduw|*v dgyhuwlvlqj h{shqglwxuh lv kljko| hhfwlyh lq vwhdolqj yrwhuv iurp wkh ulydov/ zkloh uhvxow +ll, vwdwhv wkdw d odujh qxpehu ri fdqglgdwhv lv vrfldoo| h!flhqw zkhq wkh rssrvlwh krogv/ surylghg wkdw wkh shufhlyhg gxudwlrq ri wkh jdph +dv phdvxuhg e| 4,lvorzhqrxjk frpsduhg wr wkh ghfd| udwh ri wkh vwrfn ri hdfk sduw|*v frqvhqvxv1 4:
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dffxpxodwlrq lv olqhdu/ wkhq wkh qxpehu ri sduwlhv dw wkh uvw ehvw lv kljkhu +orzhu, wkdq wkh qxpehu ri sduwlhv dw wkh vhfrqg ehvw/ li wkh vslooryhu hhfw lv odujh +orz, hqrxjk1 Li lqvwhdg wkh frvw ri lqyhvwphqw lv olqhdu/ wkh uvw ehvw qxpehu ri sduwlhv lv dozd|v orzhu wkdq wkh vhfrqg ehvw qxpehu ri sduwlhv/ luuhvshfwlyho| ri wkh vslooryhu ohyho1 53
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