Corruption and the choice of auction format

(1)

❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s

❊s❝♦❧❛ ❞❡

Pós✲●r❛❞✉❛çã♦

❡♠ ❊❝♦♥♦♠✐❛

❞❛ ❋✉♥❞❛çã♦

●❡t✉❧✐♦ ❱❛r❣❛s

◆◦ ✸✻✽ ■❙❙◆ ✵✶✵✹✲✽✾✶✵

❈♦rr✉♣t✐♦♥ ❆♥❞ ❚❤❡ ❈❤♦✐❝❡ ❖❢ ❆✉❝t✐♦♥ ❋♦r✲

♠❛t

❋❧❛✈✐♦ ▼❛rq✉❡s ▼❡♥❡③❡s✱ P❛✉❧♦ ❑❧✐♥❣❡r ▼♦♥t❡✐r♦

❋❡✈❡r❡✐r♦ ❞❡ ✷✵✵✵

(2)

❖s ❛rt✐❣♦s ♣✉❜❧✐❝❛❞♦s sã♦ ❞❡ ✐♥t❡✐r❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❞❡ s❡✉s ❛✉t♦r❡s✳ ❆s

♦♣✐♥✐õ❡s ♥❡❧❡s ❡♠✐t✐❞❛s ♥ã♦ ❡①♣r✐♠❡♠✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛

❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s✳

❊❙❈❖▲❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❈❖◆❖▼■❆ ❉✐r❡t♦r ●❡r❛❧✿ ❘❡♥❛t♦ ❋r❛❣❡❧❧✐ ❈❛r❞♦s♦

❉✐r❡t♦r ❞❡ ❊♥s✐♥♦✿ ▲✉✐s ❍❡♥r✐q✉❡ ❇❡rt♦❧✐♥♦ ❇r❛✐❞♦ ❉✐r❡t♦r ❞❡ P❡sq✉✐s❛✿ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r

❉✐r❡t♦r ❞❡ P✉❜❧✐❝❛çõ❡s ❈✐❡♥tí✜❝❛s✿ ❘✐❝❛r❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❈❛✈❛❧❝❛♥t✐

▼❛rq✉❡s ▼❡♥❡③❡s✱ ❋❧❛✈✐♦

❈♦rr✉♣t✐♦♥ ❆♥❞ ❚❤❡ ❈❤♦✐❝❡ ❖❢ ❆✉❝t✐♦♥ ❋♦r♠❛t✴

❋❧❛✈✐♦ ▼❛rq✉❡s ▼❡♥❡③❡s✱ P❛✉❧♦ ❑❧✐♥❣❡r ▼♦♥t❡✐r♦ ✕ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✿ ❋●❱✱❊P●❊✱ ✷✵✶✵

✭❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s❀ ✸✻✽✮

■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛❢✐❛✳

(3)

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Judgxdwh Vfkrro ri Hfrqrplfv

Jhwxolr Ydujdv Irxqgdwlrq

HSJH2IJY

Sudld gh Erwdirjr 4<3/ vdod 4458

Erwdirjr/ Ulr gh Mdqhlur/ UM/ 555860<33

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(4)

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(5)

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(6)

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(7)

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Wkh dxfwlrqhhu lv fruuxsw1 Zh frqvlghu wzr srvvleoh w|shv ri fruuxsw sudfwlfhv1 Lq wkh uvw fdvh/ wkh dxfwlrqhhu dssurdfkhv wkh zlqqhu dqg rhuv klp wr uhgxfh klv elg wr wkh vhfrqg kljkhvw elg lq h{fkdqjh iru d euleh 1 Li wklv elgghu djuhhv wr sd| wkh euleh/ wkhq kh zlqv wkh remhfw sd|lqj zkdw lv hhfwlyho| wkh vhfrqg kljkhvw elg +soxv wkh euleh wr wkh dxfwlrqhhu,1 Li wklv sod|hu grhv qrw djuhh wr sd| wkh euleh/ wkhq kh vwloo zlqv wkh remhfw dqg sd|v dffruglqjo| wr wkh dxfwlrq uxohv h1j1/ klv elg zkhq d uvw sulfh dxfwlrq lv xvhg1 Lq wkh vhfrqg fdvh/ wkh dxfwlrqhhu dssurdfkhv wkh orvhu dqg rhuv klp wr lqfuhdvh klv elg wr pdwfk wkh kljkhvw elg lq h{fkdqjh iru d euleh1 Li wklv sod|hu djuhhv/ kh zlqv wkh dxfwlrq dqg sd|v wklv kljkhvw elg1 Rwkhuzlvh/ wklv sod|hu orvhv wkh dxfwlrq1 Qrwh wkdw wkh eulehu| duudqjhphqw lv frpprq nqrzohgjh1 Pruhryhu/ erwk eulehu| duudqjhphqwv duh frqvlvwhqw zlwk wkh hylghqfh ri olwwoh glvshuvlrq dprqjvw revhuyhg elgv dv glvfxvvhg lq wkh suhylrxv vhfwlrq ehfdxvh xqghu fruuxswlrq wkh wzr revhuyhg elgv zloo eh lghqwlfdo1

Lq wklv sdshu wkh vl}h ri wkh euleh// lv {hg1 Iru h{dpsoh/pljkw kdyh ehhq ghwhuplqhg e| d vrfldo frqyhqwlrq ru qrup1 Dowkrxjk d pruh frpsohwh prgho zrxog kdyh dv hqgrjhqrxv/ shukdsv dv d iudfwlrq ri wkh zlqqlqj elg/ wkh dqdo|vlv ri {hg douhdg| |lhogv vhyhudo lqwhuhvwlqj uhvxowv1 Zh dovr ljqruh wkh sulqflsdo0djhqw uhodwlrqvkls ehwzhhq wkh vhoohu dqg wkh dxfwlrqhhu1 Dv phqwlrqhg hduolhu/ rxu h{dpsoh lv wkdw ri d jryhuqphqw r!fldo zkr kdv d fkrlfh ehwzhhq dq rudo dxfwlrq dqg d uvw sulfh dxfwlrq wr vhoo d jryhuqphqw0

7_{Lq wkh dsshqgl{ zh frqvlghu d vhwxs zlwk} _q _{elgghuv dqg d jhqhudo glvwulexwlrq ri} ydoxhv zkhq wkh dxfwlrqhhu dssurdfkhv wkh zlqqhu1 Wr rewdlq forvh irup vroxwlrqv iru wkh htxloleulxp elgglqj vwudwhjlhv dqg wr frpsxwh wkh vhoohu*v h{shfwhg uhyhqxh dqg wkh h{shfwhg euleh zh olplw rxuvhoyhv wr wkh wzr0elgghu xqlirupo|0glvwulexwhg ydoxhv fdvh1

(8)

rzqhg jrrg1 Qh{w zh dujxh wkdw dq rudo dxfwlrq lv fruuxswlrq surri18 Lq wkh qh{w vhfwlrq zh lqyhvwljdwh xqghu zklfk frqglwlrqv fruuxswlrq zloo rffxu lq d uvw sulfh dxfwlrq1

Sursrvlwlrq 4 Xqghu wkh wzr dowhuqdwlyh eulehu| duudqjhphqwv/ rudo dxf0 wlrqv duh fruuxswlrq iuhh1

Surri1 Zh zloo qrz ghprqvwudwh wkdw lw lv d grplqdqw vwudwhj| wr elg rqh*v wuxh ydoxdwlrq dqg wkdw wkhuh lv qr fruuxswlrq lq htxloleulxp1 Zh zloo frqvlghu wkh wzr eulehu| duudqjhphqwv vhsdudwho|1 Iluvw/ dvvxph wkdw wkh dxfwlrqhhu dssurdfkhv wkh zlqqhu ri wkh dxfwlrq1 Lq wklv fdvh wkh dujxphqw lv frpsohwho| dqdorjrxv wr wkh vwdqgdug fdvh zlwkrxw fruuxswlrq1 Dvvxph wkdw Elgghu 5 elgv dq duelwudu| qxpehu K₂ dqg Elgghu 4 elgv K : 1 Elgghu 4 grhv qrw jdlq dq|wklqj e| elgglqj pruh wkdq klv ydoxh exw pd| dv zhoo orvh li K : K₂ : Vlploduo|/ vxssrvh Elgghu 4 elgv K Elgghu 4 grhv qrw jdlq e| elgglqj ohvv wkdq klv ydoxh exw pd| dv zhoo orvh wkh dxfwlrq +dqg frqvhtxhqwo| wkh remhfw ehfdxvh wkh dxfwlrqhhu dssurdfkhv rqo| wkh zlqqhu, zkhq lw vkrxog kdyh zrq/ wkdw lv/ zkhq : K₂ : KQrwh wkdw wkhuh lv qhyhu fruuxswlrq ehfdxvh wkh zlqqhu dozd|v sd|v wkh vhfrqg kljkhvw elg +zlwkrxw wkh qhhg wr sd| d euleh,1

Qrz vxssrvh wkdw wkh dxfwlrqhhu dssurdfkhv wkh orvhu ri wkh dxfwlrq1 Dvvxph wkdw Elgghu 5 elgv dq duelwudu| qxpehuK₂ dqg Elgghu 4 elgvK : 1 Dv deryh/ Elgghu 4 grhv qrw jdlq dq|wklqj e| elgglqj pruh wkdq klv ydoxh exw pd| dv zhoo orvh li K : K₂ : dqg Elgghu 5 ghflghv qrw wr sd| wkh euleh1 Qrz vxssrvh Elgghu 4 elgv K Zkhq K : K₂ Elgghu 5 lv rhuhg wkh rssruwxqlw| wr pdwfk 4*v elg dqg zlq wkh dxfwlrq1 Lq wklv fdvh/ e| lqfuhdvlqj klv elg +xs wr,/ Sod|hu 4 zrxog lqfuhdvh klv h{shfwhg surwv1 Li : K₂ : K/ wkhq Sod|hu 4 lv rhuhg wr pdwfk 5*v elg dqg sd| wkh euleh1 E| lqfuhdvlqj klv elg/ Sod|hu 4 zrxog lqfuhdvh klv h{shfwhg surwv e| qrw kdylqj wr sd| wkh euleh1 Dv lq wkh suhylrxv fdvh/ wkhuh lv qhyhu fruuxswlrq ehfdxvh wkh zlqqhu dozd|v sd|v wkh vhfrqg kljkhvw elg +zlwkrxw wkh qhhg wr sd| d euleh,1

8_{Lq wklv sdshu zh gr qrw glvwlqjxlvk ehwzhhq vhfrqg0sulfh vhdohg0elg dqg Hqjolvk dxf0} wlrqv> wkh| duh vwudwhjlfdoo| htxlydohqw lq dq lqghshqghqw0sulydwh ydoxhv prgho1

(9)

6 Fruuxswlrq lq Iluvw0Sulfh Dxfwlrqv

Lq wklv vhfwlrq zh lqyhvwljdwh wkh hhfwv ri fruuxswlrq rq elgglqj ehkdylru dqg wkh vhoohu*v h{shfwhg uhyhqxh lq d uvw sulfh dxfwlrq1 Zh dovr frpsxwh wkh suredelolw| ri fruuxswlrq/ lqyhvwljdwh wkh h!flhqf| ri uvw sulfh dxfwlrqv xqghu fruuxswlrq dqg frpsxwh wkh vhoohu*v h{shfwhg uhyhqxh1 Zh h{dplqh wkh wzr eulehu| duudqjhphqwv vhsdudwho|1

614 Wkh zlqqhu lv dssurdfkhg e| wkh dxfwlrqhhu1

Zh frqvlghu wkh fdvh zkhq wkh dxfwlrqhhu dssurdfkhv wkh zlqqhu dqg rhuv klp wkh srvvlelolw| wr pdwfk wkh vhfrqg kljkhvw elg lq h{fkdqjh iru d euleh1 Lq wkh qh{w sursrvlwlrq zh fkdudfwhul}h d v|pphwulf htxloleulxp elgglqj vwudwhjlhv1

Sursrvlwlrq 5

KE '

2 li f 2

2 n I

e32

2 li 2

lv dq htxloleulxp li 2 ns2

c wkdw lv/ li f2b2 Hb 9

Surri1 Li Elgghu 5 sod|vKE ohw xv qg Elgghu 4 ehvw uhvsrqvh1 Li Elgghu 4 elgv %f klv h{shfwhg xwlolw| lv

E% '.

EKE+_{KE+ %3}n E %_%:KE+:%3

+4,

Ohw xv frqvlghu uvw%Wkhq% fdqg wkhuhiruhE% ' E% h E% : KE+ ' E%K3E% Vlqfh % c K3E% ' 2% Wkxv E% ' E% 2% Wklv

ixqfwlrq lv pd{lpl}hg dw % ' ₂ li 2 dqg dw % ' li : 2 Ohw xv qrz frqvlghu : % :f

E% ' .k +

2

_{+ 2E%3}ln E % h E% : KE+: % '

] 2E%3

f

+

2

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9_{Wkh lpsolflw dvvxpswlrq lv wkdw} _E₌₈_{rwkhuzlvh fruuxswlrq zloo qhyhu rffxu dqg zh} zloo kdyh wkh vwdqgdug htxloleulxp elgglqj vwudwhj|e+y, @ y

51

(10)

0 0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8

Iljxuh 4= Judsk riKE E 'e

Wkh ghulydwlyh ri lv

E% ' 2 E%

K3E%2 E%

n E%

K3E%2

'

K3E%2 E%

n E%

K3E% '

K3E%2 E%

n E%

KEK3E%

Zh qrz qrwh wkdw K vdwlvi| wkh glhuhqwldo htxdwlrq

KE ' KE

n 22KE

Wkxv KEK3E% ' _K3K_E%n232%3E%3% 1 Wkhuhiruh

E% 'K3E%%

KEK3E% n

E%

KEK3E% '

K3E%

KEK3E%

Wkxv lv pd{lpl}hg dw % ' KE lq wkh udqjh KE 2 Vlqfh

KE ' 2 li dqg rqo| li '

2 ns2

zh duh grqh1

Jlyhq wkdw : 2b lv d vwurqj uhvwulfwlrq rq wkh euleh c zh surfhhg wr vkrz krz wr qg htxloleuld iru dq| Lw lv dq lwhudwlyh surfhgxuh1 Ghqh

bE '

2 n

s

e2

2 c2

2 ns2 +5, dqg frqvlghu wkh elgglqj ixqfwlrq=

KE '

; ?

=

2 li f 2

bE li 2

2 ns2

E li 2 ns2

(11)

Zh vxssrvhE lqfuhdvlqj dqg2 ns2

' 2 WkxvKElv lqfuhdv0 lqj dqg frqwlqxrxv zlwk udqjhKEdfco ' dfc2o_^

2 ns2

c E

' dfc Eo Li Elgghu 5 elgv dffruglqjo| wr KE wkhq li Elgghu 4 kdv ydoxdwlrq

dqg elgv %fklv h{shfwhg xwlolw| lv

E% '

] K3E%3

f EKE+_+n E% h E% : KE+: % '

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f EKE+_+n E%

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+6,

Wkh ghulydwlyh ri lv jlyhq e|

E% '

K3

E% E%

K3E%K3E%

n E%

K3

E%

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E%'

+7,

K3E%K3E%

n E %

K3

E%'

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KEK3E%

Lq htxloleulxp %'KE dqgEKE ' f Wkxv

K3EKE

n EKE

KE ' f

Ru

KE ' KE

K3EKE +8,

Wkh qdwxuh ri wkh lqgxfwlyh frqvwuxfwlrq lv dv iroorzv1 Zh ehjlq zlwkKE '

2cf 2 Wkh glhuhqwldo htxdwlrq deryh pdnhv vhqvh li lv vxfk wkdw

KE ₅dfc ozklfk lv wkh udqjh riK_m_dfc2o Li zh ghqhbE 'K_m_d2c/ozh kdyh wkdw

bE ' bE

K3EbE '

bE

2 EbE

zlwk lqlwldo frqglwlrqbE2 'Wklv htxdwlrq fdq eh h{solflwo| vroyhg dqg lv jlyhq e| +5,1 Wkxv zh irxqgK_m_d_fc_E_2nI₂_oWr qgzh frqvlghu wkh lqlwldo

(12)

frqglwlrq 2 ns2

' 2 dqg

E ' E

K3EE '

E

b3EE

Wkhuh h{lvwv dq / : 2 ns2

vxfk wkdwE/ ' Li/ zh duh grqh1 Li qrw/ zh iroorz wkh lqgxfwlyh dssurdfk rxwolqhg deryh1

Qrwh wkdw wkh htxloleulxp elgglqj vwudwhjlhv ghvfulehg deryh duh lqfuhdv0 lqj lq wkh lqglylgxdov* ydoxhv1 Wkhuhiruh/ wkh zlqqhu ri wkh dxfwlrq lv wkh lqglylgxdo zlwk wkh kljkhvw ydoxdwlrq1 Jlyhq wkdw wkh dxfwlrqhhu rqo| ds0 surdfkhv wkh zlqqhu ri wkh dxfwlrq/ zh fdq frqfoxgh wkdw wkh remhfw +hyhq zlwk fruuxswlrq, dozd|v jrhv wr wkh lqglylgxdo zlwk wkh kljkhvw ydoxdwlrq1

Fruroodu| 4 Lq wkh htxloleulxp fkdudfwhul}hg deryh uvw sulfh dxfwlrqv duh h!flhqw1

Wkh deryh fruroodu| dqg wkh uhfrjqlwlrq wkdw wkh lqglylgxdo zlwk wkh orz0 hvw ydoxdwlrq kdv }hur h{shfwhg surwv doorz xv wr frqfoxgh wkdw wkh Uhyhqxh Htxlydohqfh Wkhruhp krogv lq wklv vhwwlqj1 Wkhuhiruh/ wkh zlqqhu*v h{shfwhg wrwdo sd|phqw lq wklv vfhqdulr lv lghqwlfdo wr klv h{shfwhg sd|phqw xqghu dq| phfkdqlvp wkdw doorfdwhv wkh jrrg wr wkh kljkhvw elgghu vxfk dv wkh vwdqgdug uvw sulfh dxfwlrq1 Zlwk wzr elgghuv zlwk ydoxdwlrqv xqlirupo| glv0 wulexwhg rq wkh lqwhuydo ^3/4`/ wkh zlqqhu*v wrwdo h{shfwhg sd|phqw lv htxdo wr 4261 Lq wkh vhwwlqj deryh/ wklv sd|phqw lv htxdo wr h{shfwhg euleh soxv wkh sd|phqw wr wkh vhoohu1 Qh{w zh frpsxwh wkh suredelolw| ri fruuxswlrq rffxuulqj1

Sursrvlwlrq 6 Iru 2b2 D/ wkh suredelolw| ri fruuxswlrq lv htxdo wr

22

Surri1 Vxssrvh Sod|hu 4 kdv d kljkhu ydoxdwlrq1 Wkdw lv/ : 2 Fru0 uxswlrq zloo rffxu zkhqhyhu 4*v ydoxh dqg Sod|hu 5*v elg lv vxfk wkdw lw lv ihdvleoh iru Sod|hu 4 wr sd| wkh euleh soxv wkh uhylvhg elg/ wkdw lv/ zkhqhyhu

KE₂ n 1

Ghqh ' 4@ _ic ₂_j dqg2 ' 4?_ic ₂_jZh fdq qrz zulwh wkh sured0 elolw| ri fruuxswlrq dv

hdFruuxswlrqo ' hdKE2 n o

(13)

Wkdw lv

hdFruuxswlrqo ' 2 hd : ₂c KE₂ n o

Xvlqj KE zh fdq zulwh

hd : ₂c KE₂ n o ' hd₂ 2c2

2 n o n

hd : ₂ :2c KE₂ n o

Qrwh wkdw

hd₂ 2c2

2 n o '

hd₂ 2c 2o n hd₂ 2c2

2 n c 2o

Wkdw lv

hd₂ 2c2

2 n o ' 2

2

Zh fdq fkhfn wkdw KE₂ n ₂ iru hyhu| ₂ :2 Wkxv

hd : ₂ :2c KE₂ n o ' hd : ₂ :2o '

22n 2

2

Wkhuhiruh/

hdFruuxswlrqo '

22n

2 2n 2

2_{2 '}₂2

0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82

0.3 0.32 0.34 0.36 0.38 0.4

B

Suredelolw| ri Fruuxswlrq

(14)

Lq wklv fdvh/ wkh suredelolw| ri fruuxswlrq lv txlwh kljk1 Wkh qh{w uhvxow iroorzv iurp wkh Uhyhqxh Htxlydohqfh Wkhruhp jlyhq wkdw wkh h{shfwhg euleh lv vlpso| wkh euleh wlphv wkh suredelolw| ri fruuxswlrq1

Fruroodu| 5 Iru 2b2 D/ wkh vhoohu*v h{shfwhg uhyhqxh lv htxdo wr

E22

0.065 0.07 0.075 0.08 0.085

0.3 0.32 0.34 0.36 0.38 0.4_B 0.42 0.44 0.46 0.48 0.5

Wkh vhoohu*v h{shfwhg uhyhqxh

Wkdw lv/ zkhq wkh dxfwlrqhhu dssurdfkhv wkh zlqqhu/ wkh vhoohu*v h{shfwhg uhyhqxh lv uhgxfhg txlwh gudpdwlfdoo| iru vx!flhqwo| odujh ydoxhv ri 1 Wklv ghprqvwudwhv wkh srwhqwldo ghydvwdwlqj hhfwv rq wkh jryhuqphqw*v uhyhqxh lq yhu| fruuxsw hfrqrplhv1

Ilqdoo|/ qrwh wkdw fkdqjlqj wkh zd| wkh dxfwlrqhhu lv frpshqvdwhg/ vd| iurp d {hg vdodu| wr d {hg shufhqwdjh ri wkh vdohv uhyhqxh dv frpplvvlrq/ pd| qrw holplqdwh ru hyhq uhgxfh fruuxswlrq1 Fruuxswlrq fdq rqo| eh holpl0 qdwhg li wkh frpplvvlrq lv vx!flhqwo| odujh vr wkdw wkh h{shfwhg frpplvvlrq lv odujhu wkdq wkh h{shfwhg eulehu|1 Wklv qglqj frqwudvwv zlwk wkdw ri wkh fodvvlfdo hfrqrplf wkhru| ri fruuxswlrq h1j1/ Ehfnhu dqg Vwljohu +4<:7, exw lw lv frqvlvwhqw zlwk uhfhqw qglqjv +Prrnkhumhh dqg Sqj/ 4<<8, wkdw wr zlsh rxw fruuxswlrq d odujh glvfuhwh mxps lq wkh r!fldo*v frpshqvdwlrq lv uhtxluhg1

615 Wkh orvhu lv dssurdfkhg e| wkh dxfwlrqhhu

Zh qrz dvvxph wkdw wkh dxfwlrqhhu dssurdfkhv wkh elgghu zkr orvw wkh dxf0 wlrq wr rhu klp wkh srvvlelolw| ri uhylvlqj klv elg wr pdwfk wkh kljkhvw elg

(15)

dqg zlq wkh dxfwlrq lq h{fkdqjh iru d euleh1 Lq wkh qh{w sursrvlwlrq zh fkdudfwhul}h d v|pphwulf htxloleulxp elgglqj vwudwhj| ri wklv dxfwlrq jdph1

Sursrvlwlrq 7

KE '

2 li 2

li 2

lv d v|pphwulf htxloleulxp li

0 0.1 0.2 0.3 0.4

0.2 0.4 _v 0.6 0.8 1

Iljxuh 5= Judsk ri KE + 'e,

Surri1 Li Elgghu 5 elgv dffruglqj wr KE wkhq li Elgghu 4 kdv d ydoxdwlrq

dqg elgv % f klv h{shfwhg xwlolw| li wkhuh lv qr srvvlelolw| ri d wlh lv htxdo wr

E% ' E% h E% : KE+c %n : + n

]

% KE+

EKE+n_+ +9,

Iru wklv htxloleulxp zh kdyh wr frqvlghu wkh srvvlelolw| ri d wlh1 Khuh zh vxssrvh wkdw lq wkh fdvh ri d wlh wkdw wkhuh lv d 83 shufhqw ri hdfk elgghu ehlqj fkrvhq wr zlq wkh remhfw sd|lqj wkh euleh dqg klv elg Iru d elg

% ₉' Elgghu 4*v h{shfwhg xwlolw| lv E% dv deryh1 Li %' zh uhghqh

dvG

E ' E 2 n2 2

(16)

Qrwh dovr wkdw

E '

3

n 2 2

Vxssrvh uvw wkdw Elgghu 4 elgv % : Wkhq % : KE+ lv dozd|v wuxh1 Vr E% ' E% E%n Vlqfh E% ' 2% 2%

2 E% f wkh h{shfwhg xwlolw| lv qhyhu pd{lpl}hg zlwk % : Vlqfh

En E zh kdyh wkdw 4@ _iE%c %_j ' EVxssrvh qrz wkdw f% Wkhq

E% ' E% 2%n

]

2%E

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2 E % li %

e

2 %

li %

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f Vlqfh E E ' E n 32₂ wkh pd{lpxp ri zloo eh dw

% ' L1h1 4@ _iE% (%_j'Eli 2 Frqvlghu qrz 2 Li

% c E% ' 2 E% : f Wkxv wr pd{lpl}h / % : Vlqfh

%' ₂ lv vxfk wkdw% f zh kdyh wkdw lv pd{lpl}hg iruf%

zlwk % ' ₂ Wkxv 4@ _iE% (%_j ' 4@

₂c E

li 2 Lw uhpdlqv wr frpsduh₂' ₂2 zlwk E ' E 2n32₂ Vroylqj wkh lqhtxdolw|

2

2 E 2n

2

zh rewdlq 2 Wklv suryhv wkdwKE lv dq htxloleulxp1

Sursrvlwlrq 8 Wkh remhfw grhv qrw jr wr wkh elgghu zlwk wkh kljkhvw ydoxd0 wlrq zlwk suredelolw| E32₂ 2

Surri1 Ohw xv vxssrvh wkdw : / Wkh remhfw zloo jr wr wkh elgghu zlwk wkh orzhvw ydoxdwlrq zlwk suredelolw| ₂ li dqg rqo| li / 2 Wkxv wkh suredelolw| wkdw wkh dxfwlrq lv lqh!flhqw lv E32₂ 2

Dv fruuxswlrq rffxuv rqo| li erwk sod|huv kdyh ydoxdwlrq deryh 2/ wkh qh{w uhvxow iroorzv lpphgldwho|1

(17)

Sursrvlwlrq 9 Wkh suredelolw| ri fruuxswlrq lv htxdo wr E22

0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.3 0.32 0.34 0.36 0.38 0.4

B

Suredelolw| ri Fruuxswlrq

Wkh vhoohu*v uhyhqxh lv htxdo wr li dw ohdvw rqh ri wkh sod|huv kdv d ydoxh juhdwhu wkdq 2 wkh suredelolw| ri wklv hyhqw lv E

E22

Li wkh wzr sod|huv kdyh ydoxhv ehorz 2 wkh vhoohu*v uhyhqxh lv htxdo wr wkh pd{lpxp ri _i

2c

₂

2j E| wdnlqj wkh h{shfwhg ydoxh zh rewdlq wkh vhoohu*v h{shfwhg

uhyhqxh dv hvwdeolvkhg lq wkh qh{w sursrvlwlrq1

Sursrvlwlrq : Wkh h{shfwhg euleh lv htxdo wr E22 dqg wkh vhoohu*v h{shfwhg uhyhqxh lv htxdo wr HnEe2

0.27 0.28 0.29 0.3 0.31 0.32 0.33

0.3 0.32 0.34 0.36 0.38 0.4_B 0.42 0.44 0.46 0.48 0.5

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(18)

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(19)

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(20)

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(21)

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(22)

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