Quaderni di Dipartimento. A note on the Exclusion Principle. Paolo Bertoletti (University of Pavia) # 179 (12-05)
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1 Quaderi di Diartimeto A ote o te xclusio Pricile Paolo Bertoletti Uiersity of Paia # Diartimeto di ecoomia olitica e metodi quatitatii Uiersità degli studi di Paia Via Sa Felice 5 I-700 Paia Dicembre 005
2 A ote o te xclusio Pricile Paolo Bertoletti Diartimeto di ecoomia olitica e metodi quatitatii Uiersity of Paia Abstract Accordig to te so-called xclusio Pricile itroduced by Baye et alii 993 it migt be rofitable for te seller to reduce te umber of fullyiformed otetial bidders i a all-ay auctio. We sow tat te xclusio Pricile does ot aly if te seller regards te bidders riate aluatios as belogig to te class of idetical ad ideedet distributios wit a mootoic azard rate. Keywords: all-ay auctios xclusio Pricile mootoic azard rate ecoomic teory of lobbyig. JL Classificatio: D44 D7 D8. Faculty of coomics Uiersity of Paia Via Sa Felice 7 - I Paia ITALY; mail: aolo.bertoletti@ui.it; Tel ; Fax I am grateful to Carolia Castagetti ad Pietro Rigo for useful suggestios ad i articular to Domeico Meicucci wo gracefully kees correctig my mistakes so e is fully resosible for tose remaiig ad roidig aluable commets.
3 . Itroductio Baye et alii 993 demostrate te followig somewat surrisig result te so-called xclusio Pricile. I a all-ay auctio wit comlete iformatio it migt be i te best iterest of te seller if se is able to to exclude some otetial bidders from te sort list of te auctio articiats. Ad i tis case se sould exclude tose wit te largest riate aluatios willigess to ay for te uique object to be sold. Te result ca be alied to seeral social games suc as atet races ad sorts ad otably to lobbyig games: see e.g illma ad Riley 989. It is due to te fact tat te reeue exected te biddig equilibrium is i mixed strategies by te seller is decreasig i te largest aluatio amog bidders call it wile icreasig wit resect to te secod-largest aluatio te oter bidders bid zero wit robability. xcludig te strogest bidders iduces some of te weakest oes to bid more ad may icrease te oerall exediture. I articular it turs out tat te exected total aymet to te seller is / / / < were te latter amouts are te exected aymets of bidders ad tose wit te largest ad te secod largest aluatios resectiely. Te object is assiged to bidders ad resectiely wit robabilities - / ad / ad te former bidder exects - i te equilibrium all te oter bidders exect zero. Te oerall exected welfare is te w - < were w > ad tus te outcome does ot belog to te Core of te corresodig excage game. As idicated aboe te quoted literature refers to te case of comlete iformatio 3 wic is a somewat uusual assumtio i auctio teory. Moreoer te role ad te iformatio aailable to te desiger if ay of te auctio are someow left uexlaied. I a comaio aer Bertoletti 005 we argue for examle tat te xclusio Pricile is affected by te imlicit assumtio tat te auctio resere rice is ull. 4 Ideed as far as te lobbyig models are cocered te oly cosistet justificatio for te adoted settig seems to be tat te oliticia te seller wo receies te lobbies'bidders cotributios as ery little bargaiig ower. oweer te assumtio tat a fully iformed seller ca credibly exclude some bidder from er sort list wile se is uable to ask im a rice ot iger ta is aluatio does ot aear geerally alatable as a bargaiig feature. More robust results sould te be based o te exlicit Ce ad Gale 998 sow a someow related result amely tat te imositio of a exogeous ca o idiidual lobbyig cotributios may ae te aderse effect of icreasig total exediture. Te ossibility of ties i te aluatios is igored ere sice we assume tat te aluatios are ex ate cotiuously distributed ties may imly te existece of multile Nas equilibria wic are ot ecessarily reeue equialet: see Baye et alii illma ad Riley 989: also deal wit te case of icomlete iformatio amog coteders ad Ce ad Gale 998:. 648 claim tat teir result would old ee uder icomlete iformatio if tere were asymmetry amog bidders. 4 I additio tere migt also be oter ossibly more efficiet ways to motiate te less faourite coteders for examle offerig if ossible multile diided rizes: see Moldoau ad Sela 00.
4 assumtio tat te seller does ot kow te bidders refereces. oweer Meicucci 005 strikigly sows tat for some iformatio structures te xclusio Pricile also alies to te case i wic te seller regards te bidders riate aluatios as idetically ad ideedetly distributed iid ad uses o resere rice. Namely for te distributioal structures tat e cosiders excludig from te all-ay auctio wit comlete iformatio amog te bidders all but two of tem radomly selected icreases te seller s reeue. Meicucci s examle uses a discrete distributio wit small te seller is almost certai about te bidders aluatios ucertaity. I tis ote we sow tat te xclusio Pricile does ot aly to te class of iid cotiuous distributios wit a mootoic azard rate somewat more geerally we sow tat for te xclusio Pricile to aly te distributio of aluatios must be suc tat te coditioal exected alue of te differece betwee te igest ad te secodigest order statistics is somewere icreasig wit resect to te alue of te secod-igest order statistics.. Te settig ad te result Cosider te followig settig: m risk-eutral agets will ossibly bid for a uique rize i a all-ay auctio tere is o resale ossibility. Bidder i's aluatio of te rize is i i m ad is ad iterim before biddig takes lace commo kowledge amog bidders ad we order tem i suc a way tat > > > m- > m > 0. Te seller oly kows tat ex ate eac i is iid accordig to a commo strictly icreasig ad atomless cotiuous cumulatie distributio fuctio wit suort [ ] 0. 5 From er oit of iew te te reeue se exects ex ate by radomly selectig bidders m to articiate i te auctio is gie by { } were ad are resectiely te first igest ad te secod secod-igest order statistics of ideedet draws from see e.g. Krisa 00: Aedix C. Te followig Proositio olds. Proositio Cosider a all-ay auctio wit comlete iformatio amog bidders o resere rice o resale ossibility. Suose tat te bidders aluatios are ex-ate idetically ad ideedetly distributed accordig to a strictly icreasig atomless cotiuous distributio wit a mootoic azard rate. I tis case te seller maximizes er exected reeue by gettig te largest ossible set of actual articiats. Proof. Sice te desity fuctio of te joit distributio of te first ad secod order 5 Tese are of course te assumtios of te well-kow Reeue quialece Teorem: see e.g. Klemerer 004:. 7. 3
5 4 statistics see e.g. Krisa 00:. 67 is gie by: I g were is te desity fuctio wic corresods to ad I is te aroriate idicator fuctio te desity fuctio of coditioal o is gie by: g c o te suort [ ] ote tat it does ot deed o. Clearly { } { { }} wit obious otatio for te reious exectatios. Now comute te deriatie of: { } d 3 wit resect to : { } ] ] [ [ d. 4 Te by otig tat is a coex fuctio: { } 5 } { ] [ ] [ d d d d d } { } { were / is te so-called azard rate of. Tus { } is a icreasig fuctio of if te azard rate is mootoic. Fially recall tat te ucoditioal distributio fuctio of gie symmetric articiats to te auctio is:. 6 Sice 0 7
6 it follows tat first-order stocastically domiates - ad tus tat ay exclusio from te set of te otetial bidders strictly decreases te exected reeue of te seller if te azard rate of is mootoic. QD 3. Coclusio deriatie: Te ituitio for te reious result is straigtforward: te key is te sig of te followig { } -. 8 Te exected alue of te differece of te first ad te secod order statistics of te articiats aluatios would cage wit te umber of bidders accordig to te sig of 8 if tis were costat. Moreoer a somewere ositie alue for 8 is a ecessary coditio for te xclusio Pricile to aly to a ex-ate symmetric all-ay auctio wit comlete iformatio. Tat is te additio of aoter idetical bidder caot decrease te seller s exected reeue if { - } is owere icreasig wit resect to. Sice { - } {/ } ad it is well-kow tat first-order stocastically domiates - were is te ucoditioal distributio fuctio of i te case of ideedet draws from if te azard rate is mootoic te additio of aoter bidder does decrease { - } ad raises te seller s exected reeue. Note tat te exected welfare is gie by {w } { } { - }. So ay bidder exclusio rofitable for te seller would te raise te exected welfare by a triial reealedreferece argumet if it were also to icrease { - }. But tis ca eer be te case if te azard rate is mootoic ad te imact o te exected welfare of icreasig te umber of bidders set remais ambiguous ee i suc a case. oweer it is easy to see tat a sufficiet coditio for a exected welfare imroemet to follow ay bidder additio uder a mootoic azard rate is 6 > {w } icreases wit resect to te umber of bidders if / / } 0 for ay. {05 6 Peras iterestigly uder te same assumtios o bidder exclusio wic always decreases exected welfare troug a ositie resere rice would be otimal for te seller i a stadard see Klemerer 004: sectio.. auctio wit icomlete iformatio. 5
7 Refereces Baye M. R. Koeock D. ad de Vries C. G. 993 Riggig te lobbyig rocess: A alicatio of te all-ay auctio America coomic Reiew Baye M. R. Koeock D. ad de Vries C. G. 996 Te all-ay auctio wit comlete iformatio coomic Teory Bertoletti P. 005 O te resere rice i all-ay auctios wit comlete iformatio ad lobbyig games Paia: mimeo; aailable at te website <tt://ecoomia.ui.it/bertoletti/aers/lobby.df>. Ce Y. K. ad Gale I. 998 Cas o olitical lobbyig America coomic Reiew illma A. L. ad Riley J. G. 989 Politically cotestable rets ad trasfers coomics ad Politics Klemerer P. 004 Auctios: Teory ad Practice Priceto: Priceto Uiersity Press. Krisa V. 00 Auctio Teory Sa Diego: Academic Press. Meicucci D. 005 Baig bidders from all-ay auctios coomic Teory fortcomig. Moldoau B. ad Sela A. 00 Te otimal allocatio of rizes i cotests America coomic Reiew
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