Do we go shopping downtown or in the ‘burbs?
Philip Ushchev2015, Journal of Urban Economics
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35 pages
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We combine spatial and monopolistic competition to study market interactions between downtown retailers and an outlying shopping mall. Consumers shop at either marketplace or at both, and buy each variety in volume. The market solution stems from the interplay between the market expansion eect generated by consumers seeking more opportunities, and the competition eect. Firms' prots increase (decrease) with the entry of local competitors when the former (latter) dominates. Downtown retailers swiftly vanish when the mall is large. A predatory but ecient mall need not be regulated, whereas the regulator must restrict the size of a mall accommodating downtown retailers.
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st♦r❡ ✐s ♦♥❡ ♦❢ t❤❡ ♠❛✐♥ r❡❛s♦♥s t❤❛t t❤❡ ❝❡♥tr❛❧ ❜✉s✐♥❡ss ❞✐str✐❝t✱ ♦♥❝❡ t❤❡ ♠❡❝❝❛ ❢♦r s❤♦♣♣❡rs✱
❞♦❡s ❧❡ss t❤❛♥ ✺ ♣❡r❝❡♥t ♦❢ t❤❡ r❡t❛✐❧ tr❛❞❡ ♦❢ ♠❡tr♦♣♦❧✐t❛♥ ❛r❡❛s ❡✈❡r②✇❤❡r❡ ❜✉t ✐♥ ◆❡✇ ❨♦r❦✱
◆❡✇ ❖r❧❡❛♥s✱ ❛♥❞ ❙❛♥ ❋r❛♥❝✐s❝♦✳✑
❚❤♦s❡ ✐ss✉❡s ❤❛✈❡ ❜❡❡♥ t❛❝❦❧❡❞ ♠❛✐♥❧② ❢r♦♠ t❤❡ ✉r❜❛♥ ♣❧❛♥♥✐♥❣ ✈✐❡✇♣♦✐♥t✱ ❛♥ ❛♣♣r♦❛❝❤ t❤❛t
t②♣✐❝❛❧❧② ❞✐sr❡❣❛r❞s t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❛♥❞ ✇❡❧❢❛r❡ ❛s♣❡❝ts✳
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❛✐♠ t♦ ❞❡✈❡❧♦♣ ❛
♥❡✇ ❢r❛♠❡✇♦r❦ t❤❛t ❝♦♠❜✐♥❡s s♣❛t✐❛❧ ❝♦♠♣❡t✐t✐♦♥ ✭❍♦t❡❧❧✐♥❣✱ ✶✾✷✾✮ ❛♥❞ ♠♦♥♦♣♦❧✐st✐❝ ❝♦♠♣❡t✐t✐♦♥
✭❉✐①✐t ❛♥❞ ❙t✐❣❧✐t③✱ ✶✾✼✼✮✳ ■♥ ♦✉r ❢r❛♠❡✇♦r❦✱ t❤❡ ❝✐t② ❤❛s t✇♦ s❤♦♣♣✐♥❣ ❧♦❝❛t✐♦♥s✿ ♦♥❡ ✐s ❞♦✇♥t♦✇♥
✭s❤♦♣♣✐♥❣ str❡❡t✮ ❛♥❞ ✐s ❝♦♠♣♦s❡❞ ♦❢ ♠❛♥② r❡t❛✐❧❡rs❀ t❤❡ ♦t❤❡r ✐s s❡t ✉♣ ❛t t❤❡ ❝✐t② ♦✉ts❦✐rts
✶
✭s❤♦♣♣✐♥❣ ♠❛❧❧✮ ❛♥❞ ✐s r✉♥ ❜② ❛ ❞❡✈❡❧♦♣❡r✳
❊❛❝❤ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡r ✐s t♦♦ s♠❛❧❧ t♦ ❛✛❡❝t t❤❡
♠❛r❦❡t ♦✉t❝♦♠❡ ❛♥❞ t❤✉s tr❡❛ts ✐ts ❝♦♠♣❡t✐t♦rs✬ str❛t❡❣✐❡s ❛s ❛ ❣✐✈❡♥✳
❞❡✈❡❧♦♣❡r ✐s ❛ ❜✐❣ ♣❧❛②❡r ✇❤♦ ♠❛♥✐♣✉❧❛t❡s t❤❡ ♠❛r❦❡t✳
■♥ ❝♦♥tr❛st✱ t❤❡ ♠❛❧❧
❈♦♥s✉♠❡rs ❛r❡ ❡①♦❣❡♥♦✉s❧② ❞✐s♣❡rs❡❞
❜❡t✇❡❡♥ t❤❡ t✇♦ ♠❛r❦❡t♣❧❛❝❡s ❛♥❞ ❜❡❛r s♣❡❝✐✜❝ tr❛✈❡❧ ❝♦sts t♦ ❛❝q✉✐r❡ t❤❡ ✈❛r✐❡t✐❡s ❛✈❛✐❧❛❜❧❡ ✐♥
❡❛❝❤ ♦♥❡✳
❚❤♦✉❣❤ ❝♦♥s✉♠❡rs✬ s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r ❤❛s s❡✈❡r❛❧ ❞❡t❡r♠✐♥❛♥ts ✭❚❡❧❧❡r✱ ✷✵✵✽✮✱ ✇❡ ❢♦❝✉s ♦♥ t✇♦
✜rst✲♦r❞❡r ❢♦r❝❡s ❛✛❡❝t✐♥❣ ❝♦♥s✉♠❡r ❛tt✐t✉❞❡s✿
❧♦✈❡ ❢♦r ✈❛r✐❡t②✱
❛♥❞
tr❛✈❡❧❧✐♥❣ ❝♦sts✳
❚❤❡ ❢♦r♠❡r
❡①♣r❡ss❡s ❝♦♥s✉♠❡rs✬ ❞❡s✐r❡ t♦ ❛❝❝❡ss ❛ ❜r♦❛❞ r❛♥❣❡ ♦❢ ❝❤♦✐❝❡s❀ t❤❡ ❧❛tt❡r r❡✢❡❝ts t❤❡ ❡❝♦♥♦♠✐❝ ✐♠✲
♣♦rt❛♥❝❡ ♦❢ s❤♦♣♣✐♥❣ ❝♦sts ✐♥ ❝♦♥s✉♠❡rs✬ ❜✉❞❣❡t✳ ❋♦r ❡①❛♠♣❧❡✱ ❆♠❡r✐❝❛♥ ❤♦✉s❡❤♦❧❞s s♣❡♥❞ ❛❧♠♦st
t❤❡ s❛♠❡ ❛♠♦✉♥t ♦❢ t❤❡✐r ♦✈❡r❛❧❧ tr❛✈❡❧ t✐♠❡ t♦ ❣♦ t♦ ✇♦r❦ ❛♥❞ t♦ ❣♦ s❤♦♣♣✐♥❣✱ ✐✳❡✳✱ ✷✸✳✻ ♣❡r❝❡♥t
✈❡rs✉s ✷✶✳✽ ♣❡r❝❡♥t ✭❈♦✉t✉r❡ ❡t ❛❧✳✱ ✷✵✶✷✮✳ ❇② ❝♦♠❜✐♥✐♥❣ s♣❛t✐❛❧ ❛♥❞ ♠♦♥♦♣♦❧✐st✐❝ ❝♦♠♣❡t✐t✐♦♥✱ ✇❡
❛r❡ ❛❜❧❡ t♦ ❞✐s❡♥t❛♥❣❧❡ t❤❡ ✈❛r✐♦✉s ❡✛❡❝ts tr✐❣❣❡r❡❞ ❜② t❤❡s❡ t✇♦ ♦♣♣♦s✐♥❣ ❢♦r❝❡s✳ ❆❧t❤♦✉❣❤ ♣r✐❝❡
❞✐✛❡r❡♥❝❡ ✐s ❛ ♠❛❥♦r ❞❡t❡r♠✐♥❛♥t ♦❢ ❝♦♥s✉♠❡r ❜❡❤❛✈✐♦r✱ ♦✉r ✐♥✐t✐❛❧ s❡tt✐♥❣ ❛ss✉♠❡s t❤❛t ♣r✐❝❡s ❛r❡
t❤❡ s❛♠❡ ✐♥ t❤❡ t✇♦ s❤♦♣♣✐♥❣ ♣❧❛❝❡s✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ❝❛♣t✉r❡ ✐♥ ❛ s✐♠♣❧❡ ✇❛② ❛ ✇✐❞❡ r❛♥❣❡ ♦❢
✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ t❤❡s❡ ♣❧❛❝❡s✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ s❤♦✇ ❧❛t❡r t❤❛t ♦✉r ❛♣♣r♦❛❝❤ ❞✐s♣❧❛②s ❡♥♦✉❣❤
✶ ❚❤❡
r❡❛s♦♥s ✇❤② r❡t❛✐❧❡rs ❛r❡ ❛❣❣❧♦♠❡r❛t❡❞ ❛t t❤❡ ❝✐t② ❝❡♥t❡r ❛r❡ ♥♦t ❛❞❞r❡ss❡❞ ✐♥ t❤✐s ♣❛♣❡r✳ ❍♦✇❡✈❡r✱ ✐t ✐s
✇❡❧❧ ❦♥♦✇♥ t❤❛t✱ ✇❤❡♥ tr❛✈❡❧ ❝♦sts ❛r❡ ♥♦t t♦♦ ❤✐❣❤✱ ❜❡✐♥❣ ❝❧✉st❡r❡❞ ✐♥ ❛ ❢❡✇ ♣❧❛❝❡s ❛❧❧♦✇s ✜r♠s t♦ ❜❡♥❡✜t ❢r♦♠
❛❣❣❧♦♠❡r❛t✐♦♥ ❡❝♦♥♦♠✐❡s t❤❛t ♦✈❡rs❤❛❞♦✇ t❤❡ ♠❛r❦❡t ❝r♦✇❞✐♥❣ ❡✛❡❝t ✭❲♦❧✐♥s❦②✱ ✶✾✽✸❀ ❙t❛❤❧✱ ✶✾✽✼❀ ❋✉❥✐t❛ ❛♥❞
❚❤✐ss❡✱ ✷✵✶✸✮✳
✷
✈❡rs❛t✐❧✐t② t♦ ❛❝❝♦✉♥t ❢♦r ♣r✐❝❡ ❞✐✛❡r❡♥❝❡s ❛s ✇❡❧❧ ❛s ✈❛r✐♦✉s t②♣❡s ♦❢ ❛s②♠♠❡tr✐❡s ❜❡t✇❡❡♥ t❤❡ t✇♦
♠❛r❦❡t♣❧❛❝❡s✳
❖♥❡✲st♦♣ s❤♦♣♣✐♥❣ ✐s ❝❤❡❛♣❡r ❛♥❞ t❤✉s ♠♦r❡ ❛♣♣❡❛❧✐♥❣ t♦ ❝♦♥s✉♠❡rs t❤❛♥ t✇♦✲st♦♣ s❤♦♣♣✐♥❣✳
❍♦✇❡✈❡r✱ s✐♥❝❡ ❝♦♥s✉♠❡rs ❧♦✈❡ ✈❛r✐❡t②✱ s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r ❞❡♣❡♥❞s ♥♦t ♦♥❧② ♦♥ t❤❡ tr❛✈❡❧ ❝♦sts ❜✉t
❛❧s♦ ♦♥ t❤❡ r❛♥❣❡ ♦❢ ❣♦♦❞s ❛✈❛✐❧❛❜❧❡ ✐♥ ❡❛❝❤ s❤♦♣♣✐♥❣ ♣❧❛❝❡✳ ❚❤✐s ❤❛s ❛♥ ✐♠♣♦rt❛♥t ❝♦♥s❡q✉❡♥❝❡✿
✉♥❧✐❦❡ st❛♥❞❛r❞ ♠♦❞❡❧s ♦❢ s♣❛t✐❛❧ ❝♦♠♣❡t✐t✐♦♥✱ ♠❛r❦❡t ❛r❡❛s ♦✈❡r❧❛♣✳ ❚❤❡ ❡①t❡♥t ♦❢ ♦✈❡r❧❛♣♣✐♥❣
❤✐♥❣❡s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ✜r♠s ❧♦❝❛t❡❞ ✐♥ ❡❛❝❤ s❤♦♣♣✐♥❣ ♣❧❛❝❡ ❛♥❞ t❤❡ tr❛✈❡❧ ❝♦sts✳ ▼♦r❡♦✈❡r✱ ❜♦t❤
t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❝♦♥s✉♠♣t✐♦♥ ♦❢ ❛ ✈❛r✐❡t② ❛♥❞ t❤❡ ♣r♦❞✉❝t r❛♥❣❡ ❝♦♥s✉♠❡❞ ❛❧s♦ ❝❤❛♥❣❡ ✇✐t❤ t❤❡
❝♦♥s✉♠❡r✬s s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r✳ ❚❤❡s❡ ❛r❡ ✐♠♣♦rt❛♥t ❝♦♥s✐❞❡r❛t✐♦♥s t❤❛t ❤❛✈❡ ❜❡❡♥ ♥❡❣❧❡❝t❡❞ ✐♥
t❤❡ ❧✐t❡r❛t✉r❡ ✇❤❡r❡ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❞❡♠❛♥❞ ❢♦r ❛ ✈❛r✐❡t② ✐s ♣❡r❢❡❝t❧② ✐♥❡❧❛st✐❝✳
❖✉r ♠❛✐♥ ✜♥❞✐♥❣s ♠❛② ❜❡ s✉♠♠❛r✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❋✐rst✱ t❤❡ ♠❛r❦❡t ♦✉t❝♦♠❡ st❡♠s ❢r♦♠ t❤❡
✐♥t❡r♣❧❛② ❜❡t✇❡❡♥ t✇♦ ♦♣♣♦s✐♥❣ ❢♦r❝❡s✳ ❚❤❡ ✜rst✱ t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t✱ ✐s ❣❡♥❡r❛t❡❞ ❜②
❝♦♥s✉♠❡rs s❡❡❦✐♥❣ ♠♦r❡ ♦♣♣♦rt✉♥✐t✐❡s✳ ❆s ❛ r❡s✉❧t✱ t❤❡ ❡♥tr② ♦❢ ❝♦♠♣❡t✐t♦rs ❣❡♥❡r❛t❡s ❛ ♥❡t✇♦r❦
❡✛❡❝t t❤❛t ♠❛❦❡s ❛ s❤♦♣♣✐♥❣ ❧♦❝❛t✐♦♥ ♠♦r❡ ❛ttr❛❝t✐✈❡✳✷ ❚❤❡ s❡❝♦♥❞ ❢♦r❝❡ ❛♠♦✉♥ts t♦ ❛ ♠❛r❦❡t
❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝t✿ ❝♦♥s✉♠♣t✐♦♥ ♦❢ ❛ ✈❛r✐❡t② ♦❜t❛✐♥❡❞ ❢r♦♠ ❡❛❝❤ s❤♦♣ ❞❡❝r❡❛s❡s ❛s t❤❡ s✐③❡ ♦❢
t❤❡ ♠❛r❦❡t♣❧❛❝❡ ✐♥❝r❡❛s❡s✱ ❛♥❞ ❛s t❤❡ ♥✉♠❜❡r ♦❢ ♠❛r❦❡t♣❧❛❝❡s ✈✐s✐t❡❞ ❜② t❤❡ ❝♦♥s✉♠❡r ❣♦❡s ✉♣✳
❲❤❡♥ t❤❡ ♥❡t✇♦r❦ ❡✛❡❝t ✐s s✉✣❝✐❡♥t❧② str♦♥❣✱ ✇❡ s❤♦✇ t❤❛t ❝♦♠♣❡t✐t✐♦♥ ✇✐t❤✐♥ ❛ ♠❛r❦❡t♣❧❛❝❡
✐s ♣r♦✜t✲✐♥❝r❡❛s✐♥❣✳✸ ❊q✉❛❧❧② ✐♠♣♦rt❛♥t ✐s t❤❡ ❢❛❝t t❤❛t ❝♦♥s✉♠❡rs ❛r❡ ❛ttr❛❝t❡❞ ❜② ♠❛r❦❡t♣❧❛❝❡s
t❤❛t ❛r❡ ❝❧♦s❡ ❜② ❜❡❝❛✉s❡ ♦❢ tr❛✈❡❧ ❝♦sts✳ ❈♦♠❜✐♥✐♥❣ t❤❡ ❞✐st❛♥❝❡ ❛♥❞ ♥❡t✇♦r❦ ❡✛❡❝ts ✐♠♣❧✐❡s t❤❛t
❝♦♥s✉♠❡rs✬ s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r ✐s ❞r✐✈❡♥ ❜② ❣r❛✈✐t❛t✐♦♥❛❧ ❢♦r❝❡s ✇❤♦s❡ ✐♥t❡♥s✐t② ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡
s✐③❡ ♦❢ ❛ ♠❛r❦❡t♣❧❛❝❡ ❛♥❞ ❞❡❝r❡❛s❡s ❛s t❤❡ ❞✐st❛♥❝❡ ❣r♦✇s ❜❡t✇❡❡♥ ❝♦♥s✉♠❡rs ❛♥❞ t❤❡ ♠❛r❦❡t♣❧❛❝❡✳
❆s ♣r❡❞✐❝t❡❞ ❜② t❤❡ ◆♦rt❤ ❉❡❝❛t✉r ❛♥t✐✲❲❛❧♠❛rt ❛❝t✐✈✐sts✱ t❤❡ ♥✉♠❜❡r ♦❢ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs
s❤r✐♥❦s ❛s t❤❡ ♦✉t❧②✐♥❣ s❤♦♣♣✐♥❣ ♠❛❧❧ ❣r♦✇s✳ ▲❡ss ❡①♣❡❝t❡❞✱ ♣❡r❤❛♣s✱ ✐s t❤❛t t❤❡ ❝✐t② ❝♦♠♠❡r❝✐❛❧
❝❡♥t❡r s✇✐❢t❧② ✈❛♥✐s❤❡s ✇❤❡♥ t❤❡ s✐③❡ ♦❢ t❤❡ ♦✉t✲♦❢✲t♦✇♥ s❤♦♣♣✐♥❣ ♠❛❧❧ ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡✳ ❚❤✐s
r❡s✉❧t ❤✐❣❤❧✐❣❤ts t❤❡ ♠❛✐♥ ❢♦r❝❡s ❛t ✇♦r❦ ✐♥ ♦✉r s❡tt✐♥❣ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❞❡s❡r✈❡s ♠♦r❡ ❡①♣❧❛♥❛t✐♦♥✳
❊st❛❜❧✐s❤✐♥❣ ♥❡✇ st♦r❡s ❛t t❤❡ ❝✐t② ♦✉ts❦✐rts ❞✐✈❡rts ❝♦♥s✉♠❡rs ❢r♦♠ ✈✐s✐t✐♥❣ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs✳
❚❤✐s ✐♥ t✉r♥ ❧❡❛❞s t♦ ❛ ❝♦♥tr❛❝t✐♦♥ ♦❢ t❤❡ ❝❡♥tr❛❧ ❝♦♠♠❡r❝✐❛❧ ❞✐str✐❝t t❤r♦✉❣❤ t❤❡ ❡①✐t ♦❢ r❡t❛✐❧❡rs✱
✇❤✐❝❤ ♠❛❦❡s t❤✐s ♠❛r❦❡t♣❧❛❝❡ ❡✈❡♥ ❧❡ss ❛ttr❛❝t✐✈❡✳ ❚❤❡ ♦✈❡r❛❧❧ ❡✛❡❝t ✐s t♦ ❢✉rt❤❡r r❡❞✉❝❡ t❤❡
♥✉♠❜❡r ♦❢ ❝✉st♦♠❡rs✱ ✇❤✐❝❤ ❝✉ts ❞♦✇♥ t❤❡ ♥✉♠❜❡r ♦❢ r❡t❛✐❧❡rs ♦♥❝❡ ♠♦r❡✳ ❲❤❡♥ t❤❡ r❡❧❛t✐✈❡ s✐③❡
♦❢ t❤❡ ❞♦✇♥t♦✇♥ s❤♦♣♣✐♥❣ ❛r❡❛ ✐s s♠❛❧❧ ❡♥♦✉❣❤✱ t❤✐s ❦❡❡♣s ❣♦✐♥❣ ♦♥ ✉♥t✐❧ ♥♦ ✜r♠ ♦♣❡r❛t❡s ✐♥ t❤❡
❝✐t② ❝❡♥t❡r✳
❙❡❝♦♥❞✱ ✇❡ ❝♦♥✜r♠ t❤❡ ✇❡❧❧✲❞♦❝✉♠❡♥t❡❞ ❢❛❝t t❤❛t t❤❡ ❞✐s❛♣♣❡❛r❛♥❝❡ ♦❢ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs ✐s
♠♦r❡ ❧✐❦❡❧② t♦ ❛r✐s❡ ✐♥ s♠❛❧❧ ♦r ♣♦♦r ❝✐t✐❡s✳ ■♥ ❛ ❧❛r❣❡ ♦r r✐❝❤ ❝✐t②✱ t❤❡ ♠❛❧❧ ❞❡✈❡❧♦♣❡r ♠✉st ❜✉✐❧❞
❛ ❧❛r❣❡ ❝❛♣❛❝✐t② t♦ ❛ttr❛❝t ❛❧❧ ❝✉st♦♠❡rs✳ ❚♦ ❜❡ ♣r❡❝✐s❡✱ t❤❡ ♠❛❧❧ s✐③❡ ♥❡❡❞❡❞ t♦ ❞r✐✈❡ ❞♦✇♥t♦✇♥
r❡t❛✐❧❡rs ♦✉t ♦❢ ❜✉s✐♥❡ss ♠❛② ❜❡ t♦♦ ❜✐❣ ❢♦r ❛ ♣r❡❞❛t♦r② str❛t❡❣② t♦ ❜❡ ♣r♦✜t❛❜❧❡✳ ■♥ t❤✐s ❡✈❡♥t✱ t❤❡
✷ ❚❤✐s
✐s r❡♠✐♥✐s❝❡♥t ♦❢ ❈❤♦✉ ❛♥❞ ❙❤② ✭✶✾✾✵✮ ✇❤♦ ❛♥❛❧②③❡ ❡♥❞♦❣❡♥♦✉s ♥❡t✇♦r❦ ❡✛❡❝ts ✇✐t❤♦✉t ♥❡t✇♦r❦ ❡①t❡r♥❛❧✲
✐t✐❡s✱ ❜✉t t❤❡ s♦✉r❝❡s ♦❢ s✉❝❤ ❡✛❡❝ts ❛r❡ ✈❡r② ❞✐✛❡r❡♥t✳
✸ ◆♦t❡ t❤❛t ❙❝❤✉❧③ ❛♥❞ ❙t❛❤❧ ✭✶✾✾✻✮ ❛♥❞ ❈❤❡♥ ❛♥❞ ❘✐♦r❞❛♥ ✭✷✵✵✽✮ ♦❜t❛✐♥ ♣r♦✜t✲✐♥❝r❡❛s✐♥❣ ❝♦♠♣❡t✐t✐♦♥ ✐♥ ❞✐✛❡r❡♥t✱
❜✉t r❡❧❛t❡❞✱ ♠♦❞❡❧s✳
✸
❞❡✈❡❧♦♣❡r ❝❤♦♦s❡s t♦ ❛❝❝♦♠♠♦❞❛t❡ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs✳ ❚❤✐s s✐t✉❛t✐♦♥ ♠❛② ❛❧s♦
❛r✐s❡ ❜❡❝❛✉s❡ ❧♦❝❛❧ ❣♦✈❡r♥♠❡♥ts ❛❞♦♣t ♣♦❧✐❝✐❡s ❛✐♠❡❞ ❛t ♠❛✐♥t❛✐♥✐♥❣ t❤❡ ❝✐t② ❝❡♥t❡r✬s ❛ttr❛❝t✐✈❡♥❡ss
t❤r♦✉❣❤ t❤❡ s✉♣♣❧② ♦❢ ✈❛r✐♦✉s t②♣❡s ♦❢ ✉r❜❛♥ ❛♠❡♥✐t✐❡s✳ ❖♥ t❤❡ ❝♦♥tr❛r②✱ ❧❛r❣❡✲s❝❛❧❡ ❡❝♦♥♦♠✐❡s ✐♥
❧♦❣✐st✐❝s ❤❛✈❡ ❛❧❧♦✇❡❞ ♦✉t❧②✐♥❣ s❤♦♣♣✐♥❣ ♠❛❧❧s t♦ ❜❡❝♦♠❡ ♠♦r❡ ❛ttr❛❝t✐✈❡ t❤r♦✉❣❤ ❧♦✇❡r ♣r✐❝❡s✳
❚❤✐r❞✱ t❤❡ q✉❡st✐♦♥ ✇❤❡t❤❡r ♠❛❧❧ ❞❡✈❡❧♦♣❡rs s❤♦✉❧❞ ❜❡ r❡❣✉❧❛t❡❞ ♦r ♥♦t ✐s ❛ ♣r✐♦r✐ ✉♥❝❧❡❛r✳ ❨❡t
s♣❡❝✉❧❛t✐♦♥ ♦♥ t❤✐s ✐ss✉❡ ❤❛s ♥❡✈❡r ❜❡❡♥ ✐♥ s❤♦rt s✉♣♣❧②✱ ❛♥❞ t❤✐s ✐s ♦♥❡ ♦❢ t❤❡ ♠❛✐♥ q✉❡st✐♦♥s t❤❛t
❧♦❝❛❧ ❞❡❝✐s✐♦♥ ♠❛❦❡rs ✇♦✉❧❞ ❧✐❦❡ ❛❞❞r❡ss❡❞✳ ❖✉r ❛♥❛❧②s✐s s✉❣❣❡sts ❛ q✉❛❧✐✜❡❞ ❛♥s✇❡r✳ ❘❡❣✉❧❛t✐♦♥ ✐s
♥❡❡❞❡❞ ✇❤❡♥ t❤❡ ♠❛❧❧ ❞❡✈❡❧♦♣❡r ✐s ♥♦t ❡✣❝✐❡♥t ❡♥♦✉❣❤ t♦ ❝❛♣t✉r❡ t❤❡ ✇❤♦❧❡ ♠❛r❦❡t✿ t❤❡ r❡❣✉❧❛t♦r
❛❧✇❛②s s❡❧❡❝ts ❛ s♠❛❧❧❡r s✐③❡ ❢♦r t❤❡ ♠❛❧❧ t❤❛♥ t❤❡ s✐③❡ t❤❛t ✇♦✉❧❞ ❡♠❡r❣❡ ✉♥❞❡r ❢r❡❡ ❝♦♠♣❡t✐t✐♦♥✳
■♥ ❝♦♥tr❛st✱ ✇❤❡♥ t❤❡ ♠❛❧❧ ❞❡✈❡❧♦♣❡r ✐s ✈❡r② ❡✣❝✐❡♥t✱ t❤❡ r❡❣✉❧❛t♦r s❤♦✉❧❞ ♥♦t ✐♥t❡r✈❡♥❡✿
t❤❡
♠❛r❦❡t ♦✉t❝♦♠❡ ✐s s♦❝✐❛❧❧② ♦♣t✐♠❛❧ ❞❡s♣✐t❡ t❤❡ ❞✐s❛♣♣❡❛r❛♥❝❡ ♦❢ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs✳ ❍♦✇❡✈❡r✱ ✇❡
❛❧s♦ s❤♦✇ t❤❛t ❛ ♠❛❥♦r✐t② ♦❢ ❝♦♥s✉♠❡rs ❛r❡ ♦♣♣♦s❡❞ t♦ ❛♥ ♦✉t❧②✐♥❣ s❤♦♣♣✐♥❣ ♠❛❧❧✱ ❡✈❡♥ ✇❤❡♥ t❤❡
♦♣❡♥✐♥❣ ♦❢ s✉❝❤ ❛ ♠❛❧❧ ✐s s♦❝✐❛❧❧② ❞❡s✐r❛❜❧❡✳ ❚❤✐s ♣r♦✈✐❞❡s ❛ r❛t✐♦♥❛❧❡ ❢♦r t❤❡ ✇✐❞❡s♣r❡❛❞ ❤♦st✐❧❡
❛tt✐t✉❞❡ ♦❢ ❝✐t✐③❡♥ ❣r♦✉♣s ❛♥❞ ❝✐t② ❝♦✉♥❝✐❧s t♦✇❛r❞ ❜✐❣✲❜♦① st♦r❡s ❝♦♠✐♥❣ t♦ t♦✇♥✳
▲❛st✱ ✇❡ ❝♦♥s✐❞❡r ❢♦✉r ❡①t❡♥s✐♦♥s ♦❢ t❤❡ ❜❛s❡❧✐♥❡ ♠♦❞❡❧✳ ■♥ t❤❡ ✜rst ♦♥❡✱ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs
❛♥❞ ♠❛❧❧ st♦r❡s ❛r❡ ❤❡t❡r♦❣❡♥❡♦✉s✱ ❝❤❛r❣❡ ❞✐✛❡r❡♥t ♣r✐❝❡s✱ ❛♥❞ ♦✛❡r ❞✐✛❡r❡♥t ❛♠❡♥✐t✐❡s t♦ t❤❡✐r
❝✉st♦♠❡rs✳
❲❡ s❤♦✇ ❤♦✇ t❤❡s❡ ❛s②♠♠❡tr✐❡s ❛✛❡❝t t❤❡ ❜❡❤❛✈✐♦r ❛♥❞ s✐③❡ ♦❢ t❤❡ ♠❛r❦❡t♣❧❛❝❡s✳
■♥ t❤❡ s❡❝♦♥❞ ❡①t❡♥s✐♦♥✱ ✇❡ str❡ss t❤❛t ❛ ❧❛r❣❡ ❛♥❞✴♦r r✐❝❤ ❝✐t② ✐s ♠♦r❡ ❧✐❦❡❧② t♦ ❦❡❡♣ ❛ ❧✐✈❡❧②
❞♦✇♥t♦✇♥ t❤❛♥ ❛ s♠❛❧❧ ❛♥❞✴♦r ♣♦♦r ❝✐t②✳ ■♥ t❤❡ t❤✐r❞ ❡①t❡♥s✐♦♥✱ ✇❡ s❤♦✇ t❤❛t ✉r❜❛♥ s♣r❛✇❧ r❛✐s❡s
t❤❡ ❝♦♠♣❡t✐t✐✈❡ ❛❞✈❛♥t❛❣❡ ♦❢ ♦✉t✲♦❢✲t♦✇♥ ♠❛❧❧s ❛t t❤❡ ❡①♣❡♥s❡ ♦❢ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs✳
❖✉r ❧❛st
❡①t❡♥s✐♦♥ st✉❞✐❡s t❤❡ ✐♠♣❛❝t ♦❢ t❤❡ ♠❛❧❧ ❧♦❝❛t✐♦♥✳
❘❡❧❛t❡❞ ❧✐t❡r❛t✉r❡
✳
■♥ t❤❡ ❧✐t❡r❛t✉r❡ ♦♥ ❝♦♥s✉♠❡rs✬ s❡❛r❝❤✱ ✐t ✐s t②♣✐❝❛❧❧② ❛ss✉♠❡❞ t❤❛t
✈✐s✐t✐♥❣ ❛ ♥❡✇ s❤♦♣ ❣❡♥❡r❛t❡s t❤❡ s❛♠❡ ❣✐✈❡♥ ❝♦st ❛s ❛♥② ♦t❤❡r ♦♥❡ ✭▼❝▼✐❧❧❛♥ ❛♥❞ ❘♦t❤s❝❤✐❧❞
✶✾✾✹✮✳ ❲♦❧✐♥s❦② ✭✶✾✽✸✮ ✐s ❛ ♥♦t✐❝❡❛❜❧❡ ❡①❝❡♣t✐♦♥ ✐♥ ✇❤✐❝❤ s❡❛r❝❤ ❝♦sts ✈❛r② ✇✐t❤ t❤❡ ❧♦❝❛t✐♦♥ ♦❢
s❤♦♣s✳ ❙❝❤✉❧③ ❛♥❞ ❙t❛❤❧ ✭✶✾✾✻✮ ❝♦♥s✐❞❡r ❛ ♠❛r❦❡t ❢♦r ❞✐✛❡r❡♥t✐❛t❡❞ ♣r♦❞✉❝ts ✐♥ ✇❤✐❝❤ ✐♠♣❡r❢❡❝t❧②
✐♥❢♦r♠❡❞ ❛❣❡♥ts ❡♥❣❛❣❡ ✐♥ ❝♦st❧② s❡❛r❝❤ ❢♦r t❤❡ ❜❡st ✈❛r✐❡t②✲♣r✐❝❡ ❝♦♠❜✐♥❛t✐♦♥✱ ✇❤✐❧❡ ❆r♠str♦♥❣
❛♥❞ ❱✐❝❦❡rs ✭✷✵✶✵✮ ❛❧❧♦✇ ❢♦r t✇♦✲st♦♣ s❤♦♣♣✐♥❣✳
❯♥❧✐❦❡ ✉s✱ t❤❡s❡ ✈❛r✐♦✉s ❛✉t❤♦rs ❛ss✉♠❡ t❤❛t
t❤❡ ✈♦❧✉♠❡s ❛♥❞ ♣r♦❞✉❝t r❛♥❣❡s ❝♦♥s✉♠❡❞ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ✜r♠s✬ str❛t❡❣✐❡s ❛♥❞ ✐♥❞✐✈✐❞✉❛❧
s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r✳ ▼♦r❡♦✈❡r✱ t❤❡② ❞♦ ♥♦t ❝♦♥s✐❞❡r ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ ♠❛r❦❡t♣❧❛❝❡s ♦❢ ❞✐✛❡r❡♥t
s✐③❡ ❛♥❞ ❞✐✛❡r❡♥t ♦r❣❛♥✐③❛t✐♦♥❛❧ str✉❝t✉r❡✳ ❙♠✐t❤ ❛♥❞ ❍❛② ✭✷✵✵✺✮ st✉❞② ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ t✇♦
s♣❛t✐❛❧❧② s❡♣❛r❛t❡ ♠❛r❦❡t♣❧❛❝❡s ❜✉t t❤❡r❡ ❛r❡ t❤r❡❡ ♠❛❥♦r ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡✐r ❛♣♣r♦❛❝❤ ❛♥❞
♦✉rs✳ ❋✐rst✱ t❤❡✐r ❝♦♥s✉♠❡rs ❛r❡ ♦♥❡✲st♦♣ s❤♦♣♣❡rs✳ ❙❡❝♦♥❞✱ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❞❡♠❛♥❞ ❢♦r ❛ ✈❛r✐❡t② ✐s
♣❡r❢❡❝t❧② ✐♥❡❧❛st✐❝✳ ❚❤✐r❞✱ t❤❡✐r ♠❛r❦❡t♣❧❛❝❡s ❤❛✈❡ t❤❡ s❛♠❡ ♦r❣❛♥✐③❛t✐♦♥❛❧ str✉❝t✉r❡✳ ❖✉r ♣❛♣❡r✱
❜❡❝❛✉s❡ ✐t ❝♦♠❜✐♥❡s ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ s♠❛❧❧ ❜✉s✐♥❡ss❡s ❛♥❞ ♦♥❡ ❜✐❣ ❞❡✈❡❧♦♣❡r✱ ❛❧s♦ ❝♦♥tr✐❜✉t❡s t♦
t❤❡ r❡❝❡♥t ❧✐t❡r❛t✉r❡ t❤❛t ❢♦❝✉s❡s ♦♥ ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ ✏❧❛r❣❡✑ ❛♥❞ ✏s♠❛❧❧✑ ✜r♠s✱ ♥❛♠❡❧② ✜r♠s
t❤❛t ❞✐✛❡r ✐♥ ♥❛t✉r❡✱ ♥♦t ❥✉st ✐♥ t②♣❡ ✭❙❤✐♠♦♠✉r❛ ❛♥❞ ❚❤✐ss❡✱ ✷✵✶✷✮✳
❋✐♥❛❧❧②✱ ♦✉r ♠♦❞❡❧ ✐s r❡❧❛t❡❞ t♦ t❤❡ ❧✐t❡r❛t✉r❡ ♦♥ ✈❡rt✐❝❛❧ r❡❧❛t✐♦♥s ✇❤❡r❡ ❛ t②♣✐❝❛❧ q✉❡st✐♦♥ ✐s ❤♦✇
❞✐✛❡r❡♥t t②♣❡s ♦❢ ❝♦♥tr❛❝ts ❜❡t✇❡❡♥ ✉♣str❡❛♠ ❛♥❞ ❞♦✇♥str❡❛♠ ✜r♠s ❛✛❡❝t t❤❡ ♠❛r❦❡t str✉❝t✉r❡
✹
❛♥❞ s♦❝✐❛❧ ✇❡❧❢❛r❡✳ ❲❡ ❛❞❞r❡ss ❛ s✐♠✐❧❛r ✐ss✉❡ ❜✉t ♦✉r ❛♣♣r♦❛❝❤ ❤❛s ❢❡❛t✉r❡s t❤❛t ❞✐st✐♥❣✉✐s❤ ✐t
❢r♦♠ t❤❡ ❡①✐st✐♥❣ ❧✐t❡r❛t✉r❡✿ t❤❡ ✉♣str❡❛♠ ✜r♠ ✭t❤❡ ♠❛❧❧ ❞❡✈❡❧♦♣❡r✮ ♣r♦❞✉❝❡s ❛ s❝❛r❝❡ ✐♥♣✉t ✭s❧♦ts
❢♦r s❤♦♣s✮ ❛♥❞ ❝❤♦♦s❡s ❜♦t❤ t❤❡ ♥✉♠❜❡r ♦❢ ❞♦✇♥str❡❛♠ ✜r♠s ✭t❤❡ ♠❛❧❧ st♦r❡s✮ t❤❛t ✇✐❧❧ ❣❡t ❛ s❧♦t
❛♥❞ ✐ts ♣r✐❝❡✳ ❙✐♥❝❡ t❤❡ s❧♦t ♣r✐❝❡ ♠❛② ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ ✜①❡❞ ❝♦st ♦❢ ❞♦✇♥str❡❛♠ ✜r♠s✱ t❤❡ ♠❛❧❧
❞❡✈❡❧♦♣❡r ❞♦❡s ♥♦t ❛✛❡❝t ❞✐r❡❝t❧② t❤❡s❡ ✜r♠s✬ ♣r✐❝❡ ❛♥❞ ♦✉t♣✉ts✳ ❍♦✇❡✈❡r✱ ❛s ✐♥ ❈❤❡♠❧❛ ✭✷✵✵✸✮✱
✐t ❞♦❡s s♦ ✐♥❞✐r❡❝t❧② t❤r♦✉❣❤ t❤❡ ♥✉♠❜❡r ♦❢ ♦♣❡r❛t✐♥❣ ❞♦✇♥str❡❛♠ ✜r♠s✳ ■t t❤✉s s❡❡♠s ❢❛✐r t♦ s❛②
t❤❛t ♦✉r ♠♦❞❡❧ t✐❡s t♦❣❡t❤❡r ❞✐✛❡r❡♥t str❛♥❞s ♦❢ ❧✐t❡r❛t✉r❡✳
❚❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❚❤❡ ♠♦❞❡❧ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✱ ❢♦❝✉s✐♥❣ ♦♥ ❛ s❤♦♣♣✐♥❣
♠❛❧❧✳ ❍♦✇❡✈❡r✱ ✇❡ s❤♦✇ t❤❛t ♠♦st ♦❢ ♦✉r ❛♥❛❧②s✐s st✐❧❧ ❤♦❧❞s ❢♦r t❤❡ ♠♦r❡ t❡❝❤♥✐❝❛❧❧② ✐♥✈♦❧✈❡❞ ❝❛s❡
♦❢ ❛ s✉♣❡r♠❛r❦❡t s✉❝❤ ❛s ❲❛❧♠❛rt✳ ■♥ ❙❡❝t✐♦♥ ✸✱ ✇❡ st✉❞② ❤♦✇ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs ❛r❡ ❛✛❡❝t❡❞
❜② t❤❡ s✐③❡ ♦❢ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧✱ ❛♥❞ ❤♦✇ ♠❛❧❧ st♦r❡s ❛r❡ ❛✛❡❝t❡❞ ❜② t❤❡ s✐③❡ ♦❢ t❤❡ ❞♦✇♥t♦✇♥
s❤♦♣♣✐♥❣ ❞✐str✐❝t✳ ■♥ ❙❡❝t✐♦♥ ✹✱ t❤❡ ♠❛r❦❡t ♦✉t❝♦♠❡ ✐s ❞❡t❡r♠✐♥❡❞ ❛s t❤❡ ♣❡r❢❡❝t ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠
♦❢ ❛ s❡q✉❡♥t✐❛❧ ❣❛♠❡ ✐♥✈♦❧✈✐♥❣ ❛ ♠❛❧❧ ❞❡✈❡❧♦♣❡r✱ ✇❤♦ ❝❤♦♦s❡s t❤❡ ♥✉♠❜❡r ♦❢ s❧♦ts ❤❡ s❡❧❧s✴r❡♥ts t♦
st♦r❡s❀ ♠❛❧❧ st♦r❡s✱ ✇❤✐❝❤ ❜✉②✴r❡♥t ❛ s❧♦t ✐♥ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧❀ ❛♥❞ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ r❡t❛✐❧❡rs✱
✇❤✐❝❤ ❧♦❝❛t❡ ✐♥ t❤❡ ❝✐t② ❝❡♥t❡r✳ ■♥ ❙❡❝t✐♦♥ ✺✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ✇❡❧❢❛r❡ ✐♠♣❧✐❝❛t✐♦♥s ♦❢ r❡❣✉❧❛t✐♥❣
s❤♦♣♣✐♥❣ ♠❛❧❧s ❛♥❞ ♣r♦✈✐❞❡ ❛ ♣♦❧✐t✐❝❛❧ ❡❝♦♥♦♠② ❛♥❛❧②s✐s ♦❢ ❜✐❣ r❡t❛✐❧❡rs✬ ❡♥tr②✳ ❙❡❝t✐♦♥ ✻ st✉❞✐❡s
❡①t❡♥s✐♦♥s ♦❢ t❤❡ ❜❛s❡❧✐♥❡ ♠♦❞❡❧✱ ♥❛♠❡❧②✱ ❝♦st ❛♥❞ q✉❛❧✐t② ❛s②♠♠❡tr✐❡s ❜❡t✇❡❡♥ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs
❛♥❞ ♠❛❧❧ st♦r❡s❀ t❤❡ ✐♠♣❛❝t ♦❢ ❝✐t② s✐③❡ ❛♥❞ ♣♦♣✉❧❛t✐♦♥ ❞❡♥s✐t②❀ ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s❤♦♣♣✐♥❣
♠❛❧❧✳ ❙❡❝t✐♦♥ ✼ ❝♦♥❝❧✉❞❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ s❤♦✇ ❤♦✇ ♦✉r s❡tt✐♥❣ ❝❛♥ ❜❡ r❡✐♥t❡r♣r❡t❡❞ t♦ ❞❡s❝r✐❜❡
❝♦♥s✉♠❡rs ❤❛✈✐♥❣ ❤❡t❡r♦❣❡♥❡♦✉s t❛st❡s ❢♦r ❞✐✛❡r❡♥t s❤♦♣♣✐♥❣ ❡♥✈✐r♦♥♠❡♥ts✳
✷
✷✳✶
❚❤❡ ▼♦❞❡❧ ❛♥❞ Pr❡❧✐♠✐♥❛r② ❘❡s✉❧ts
❈♦♥s✉♠❡rs ❛♥❞ s❡❧❧❡rs
❈♦♥s✐❞❡r ❛ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝✐t② ♣♦♣✉❧❛t❡❞ ✇✐t❤ ❛ ✉♥✐t ♠❛ss ♦❢ ❝♦♥s✉♠❡rs ❞✐str✐❜✉t❡❞ ✇✐t❤ ❛
✉♥✐❢♦r♠ ❞❡♥s✐t② ♦✈❡r [0, 1]✳ ❚❤❡ ❝❛s❡s ♦❢ ❛ ✈❛r✐❛❜❧❡ ❝✐t② s✐③❡ ❛♥❞ ❛ ♥♦♥✲✉♥✐❢♦r♠ ♣♦♣✉❧❛t✐♦♥ ❞❡♥s✐t②
❛r❡ ❞✐s❝✉ss❡❞ ✐♥ ❙❡❝t✐♦♥ ✻✳ ▲❡t x ∈ [0, 1] ❞❡♥♦t❡ ❛ ❝♦♥s✉♠❡r✬s ❧♦❝❛t✐♦♥ ❛♥❞ ❤❡r ❞✐st❛♥❝❡ t♦ t❤❡
❙❙❚✳ ❚❤❡ ❝✐t② ❤❛s t✇♦ ♠❛r❦❡t♣❧❛❝❡s✳ ❚❤❡ ✜rst ♦♥❡ ✐s ❛ s❤♦♣♣✐♥❣ str❡❡t ✭❙❙❚✮ ❧♦❝❛t❡❞ ✐♥ t❤❡ ❝❡♥tr❛❧
❜✉s✐♥❡ss ❞✐str✐❝t ❛t x = 0✳ ❚❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❛ s❤♦♣♣✐♥❣ ♠❛❧❧ ✭❙▼✮ ❧♦❝❛t❡❞ ❛t x = 1✳ ❚❤❡ ❙▼ ✐s
❧♦❝❛t❡❞ ❛t t❤❡ ❝✐t② ❧✐♠✐t ❜❡❝❛✉s❡ s✉❝❤ ❛ ❧♦❝❛t✐♦♥ ❛❧❧♦✇s ♦✛❡r✐♥❣ t❤❡ ❝✉st♦♠❡rs ✈❛r✐♦✉s ❢❛❝✐❧✐t✐❡s✱ ❡✳❣✳
♣❛r❦✐♥❣✱ ✇❤✐❝❤ ❝❛♥ ❤❛r❞❧② ❜❡ ♣r♦✈✐❞❡❞ ❛t t❤❡ ❝✐t② ❝❡♥t❡r ✇❤❡r❡ t❤❡ ♣r✐❝❡ ♦❢ ❧❛♥❞ ✐s ♠✉❝❤ ❤✐❣❤❡r✳
■♥ ✇❤❛t ❢♦❧❧♦✇s✱ t❤❡ ❙❙❚ ❛♥❞ t❤❡ ❙▼ ❛r❡ r❡❢❡rr❡❞ t♦ ❛s s❤♦♣♣✐♥❣ ♦r ♠❛r❦❡t♣❧❛❝❡s✳
❚❤❡r❡ ❛r❡ t✇♦ ❣♦♦❞s✱ t❤❛t ✐s✱ ❛ ❤♦r✐③♦♥t❛❧❧② ❞✐✛❡r❡♥t✐❛t❡❞ ❣♦♦❞ ❛♥❞ ❛♥ ♦✉ts✐❞❡ ❣♦♦❞✳ ❚❤❡
❞✐✛❡r❡♥t✐❛t❡❞ ❣♦♦❞ ✐s s✉♣♣❧✐❡❞ ❜② ❛ ❧❛r❣❡ ♥✉♠❜❡r ✭❢♦r♠❛❧❧②✱ ❛ ❝♦♥t✐♥✉✉♠✮ ♦❢ ♣r♦✜t✲♠❛①✐♠✐③✐♥❣
✜r♠s❀ ❡❛❝❤ ♦♥❡ ✐s ❢r❡❡ t♦ ❝❤♦♦s❡ ✐♥ ✇❤✐❝❤ ♠❛r❦❡t♣❧❛❝❡ t♦ ❧♦❝❛t❡ ✐ts s❤♦♣✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ r❡❢❡r
t♦ t❤❡ ✜r♠s ❧♦❝❛t❡❞ ✐♥ t❤❡ ❙❙❚ ❛s r❡t❛✐❧❡rs✱ ✇❤✐❧❡ t❤♦s❡ ❛❝❝♦♠♠♦❞❛t❡❞ ❜② t❤❡ ❙▼ ❛r❡ ❝❛❧❧❡❞ st♦r❡s✳
❊❛❝❤ ✜r♠ s✉♣♣❧✐❡s ❛ s✐♥❣❧❡ ✈❛r✐❡t② ❛♥❞ ❡❛❝❤ ✈❛r✐❡t② ✐s s✉♣♣❧✐❡❞ ❜② ❛ s✐♥❣❧❡ ✜r♠✳ ❍❡♥❝❡✱ ✈❛r✐❡t✐❡s
✺
❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ t✇♦ s❤♦♣♣✐♥❣ ♣❧❛❝❡s ❛r❡ ❞✐✛❡r❡♥t✐❛t❡❞✳ ❚❤❡ ♦✉ts✐❞❡ ❣♦♦❞ ✐s s✉♣♣❧✐❡❞ ❜② ♣❡r❢❡❝t❧②
❝♦♠♣❡t✐t✐✈❡ ✜r♠s ❧♦❝❛t❡❞ ❛t x = 0 ❛♥❞ x = 1❀ ✐t ✐s ❝❤♦s❡♥ ❛s t❤❡ ♥✉♠❡r❛✐r❡✳
❈♦♥s✉♠❡rs✳ ❈♦♥s✉♠❡rs s❤❛r❡ t❤❡ s❛♠❡ ❈❊❙ ♣r❡❢❡r❡♥❝❡s ♦✈❡r t❤❡ ❞✐✛❡r❡♥t✐❛t❡❞ ❣♦♦❞✱ ✇❤✐❝❤
❛r❡ ♥❡st❡❞ ✐♥ ❛ q✉❛s✐✲❧✐♥❡❛r ✉t✐❧✐t②✿✹
ˆ N
ˆ n
1
ρ
ρ
Qj dj + A
qi di + I(SM )
U ≡ ln I(SST )
ρ
0
0
✭✶✮
✇❤❡r❡ n ✭N ✮ ✐s t❤❡ ♠❛ss ♦❢ ✈❛r✐❡t✐❡s s✉♣♣❧✐❡❞ ✐♥ t❤❡ ❙❙❚ ✭❙▼✮✱ qi ✭Qj ✮ t❤❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ t❤❡ it❤
✈❛r✐❡t② ✭j t❤ ✈❛r✐❡t②✮ ✈❛r✐❡t② ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❙❙❚ ✭❙▼✮✱ 0 < ρ < 1✱ ❛♥❞ A t❤❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ t❤❡
♥✉♠❡r❛✐r❡✳ ❚❤❡ ✉♥✐t ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛t❡❞ ❣♦♦❞ ✐s ❝❤♦s❡♥ ❢♦r t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ❧♦❣❛r✐t❤♠ t♦ ❜❡
❡q✉❛❧ t♦ 1/ρ✳ ■♥ ✭✶✮✱ I(k) ✐s t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿
1 ✇❤❡♥ t❤❡ ❝♦♥s✉♠❡r ✈✐s✐ts k ∈ {SST, SM }
I(k) ≡
0 ♦t❤❡r✇✐s❡✳
❇❡❝❛✉s❡ ♣r❡❢❡r❡♥❝❡s ❛r❡ s②♠♠❡tr✐❝✱ t❤❡ ✉t✐❧✐t② ♦❢ ❝♦♥s✉♠✐♥❣ ❛ ✈❛r✐❡t② ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❙❙❚ ✐s t❤❡
s❛♠❡ ❛s t❤❡ ✉t✐❧✐t② ♦❢ ❛ ✈❛r✐❡t② s✉♣♣❧✐❡❞ ✐♥ t❤❡ ❙▼✳ ●✐✈❡♥ ✭✶✮✱ ✐t ✐s r❡❛❞✐❧② ✈❡r✐✜❡❞ t❤❛t ❛ ❝♦♥s✉♠❡r
❜✉②s ❛ str✐❝t❧② ♣♦s✐t✐✈❡ ❛♠♦✉♥t ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛t❡❞ ❣♦♦❞✱ ❢♦r ♦t❤❡r✇✐s❡ ✐ts ✉t✐❧✐t② ❡q✉❛❧s −∞✳ ❆s
❛ ❝♦♥s❡q✉❡♥❝❡✱ ❡✈❡r② ❝♦♥s✉♠❡r ✈✐s✐ts ❛t ❧❡❛st ♦♥❡ s❤♦♣♣✐♥❣ ❧♦❝❛t✐♦♥✳ ❙✐♥❝❡ ❝♦♥s✉♠❡rs ❤❛✈❡ ❛ ❧♦✈❡
❢♦r ✈❛r✐❡t②✱ t❤❡② ❤❛✈❡ ❛♥ ✐♥❝❡♥t✐✈❡ t♦ ✈✐s✐t t❤❡ t✇♦ ♦❢ t❤❡♠✳ ❍♦✇❡✈❡r✱ tr❛✈❡❧❧✐♥❣ t♦ ❛ s❤♦♣♣✐♥❣
♣❧❛❝❡ r❡q✉✐r❡s ✐s ❝♦st❧②✳ ■♥ st❛♥❞❛r❞ ♠♦❞❡❧s ♦❢ s♣❛t✐❛❧ ❝♦♠♣❡t✐t✐♦♥✱ tr❛✈❡❧ ❝♦sts ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦
t❤❡ ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s s❤✐♣♣❡❞✳ ❚❤✐s ❛♣♣r♦❛❝❤ ✐s ♥♦t s✉✐t❛❜❧❡ t♦ st✉❞② s❤♦♣♣✐♥❣
♣❧❛❝❡s ❜❡❝❛✉s❡✱ ♦♥❝❡ ❛ ❝♦♥s✉♠❡r ✐s ✐♥ t❤❡ ❙❚❚ ♦r t❤❡ ❙▼✱ ✐t ❞♦❡s ♥♦t ❤❛✈❡ t♦ ♣❛② ❛❞❞✐t✐♦♥❛❧ ❝♦sts
t♦ ✈✐s✐t ❛❧❧ t❤❡ s❡❧❧❡rs ❡st❛❜❧✐s❤❡❞ t❤❡r❡✐♥✳ ❇② ✐♠♣❧✐❝❛t✐♦♥✱ t❤❡r❡ ❛r❡ s❝♦♣❡ ❡❝♦♥♦♠✐❡s ✐♥ s❤♦♣♣✐♥❣✳
❚❤❡② ❛r❡ ❝❛♣t✉r❡❞ ❜② ❛ss✉♠✐♥❣ t❤❛t ❝♦♥s✉♠❡rs ❜❡❛r tr❛✈❡❧ ❝♦sts ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ q✉❛♥t✐t✐❡s ♦❢
❣♦♦❞s t❤❡② ♣✉r❝❤❛s❡✱ ❜✉t ❧✐♥❡❛r ✐♥ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ s❤♦♣♣✐♥❣ ♣❧❛❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐♥❞✐✈✐❞✉❛❧
♦✉t❧❛②s ♦♥ tr❛♥s♣♦rt❛t✐♦♥ ❛r❡ ❧✉♠♣✲s✉♠ ✭❙t❛❤❧✱ ✶✾✽✷✱ ✶✾✽✼✮✳ ▲❡t τ > 0 ❜❡ t❤❡ tr❛✈❡❧ ❝♦st ♣❡r ✉♥✐t
❞✐st❛♥❝❡✳ ❍❡♥❝❡✱ ❛ ❝♦♥s✉♠❡r r❡s✐❞✐♥❣ ❛t x ❢❛❝❡s t❤❡ ❜✉❞❣❡t ❝♦♥str❛✐♥t✿
I(SST )
ˆ
0
n
pi qi di + τ x + I(SM )
ˆ
0
N
Pj Qj dj + τ (1 − x) + A ≤ Y
✭✷✮
✇❤❡r❡ pi ✭Pj ✮ t❤❡ ♣r✐❝❡ ♦❢ t❤❡ it❤ ✭j t❤✮ ✈❛r✐❡t② s✉♣♣❧✐❡❞ ✐♥ t❤❡ ❙❙❚ ✭❙▼✮✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡
✐♥❝♦♠❡ Y ✐s s✉✣❝✐❡♥t❧② ❤✐❣❤ ❢♦r t❤❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ t❤❡ ♥✉♠❡r❛✐r❡ t♦ ❜❡ ♣♦s✐t✐✈❡ ✐♥ ❡q✉✐❧✐❜r✐✉♠✳
■❢ ❛ ❝♦♥s✉♠❡r ✈✐s✐ts t❤❡ ❙❙❚ ✭❙▼✮ ♦♥❧②✱ ✐ts ✐♥✈❡rs❡ ❞❡♠❛♥❞ ❢♦r ❛ ✈❛r✐❡t② i ✭j ✮ ✐s ❣✐✈❡♥ ❜②
e1 Qρ−1
j
Pj =
A1 (Q)
e0 qiρ−1
pi =
A0 (q)
!
✭✸✮
✹ ❚❤❡ ♣❛rt✐❛❧ ❡q✉✐❧✐❜r✐✉♠ ❛♣♣r♦❛❝❤ ✉s❡❞ ❤❡r❡ ✐s ♠♦r❡ ✐♥ t❤❡ s♣✐r✐t ♦❢ ✐♥❞✉str✐❛❧ ♦r❣❛♥✐③❛t✐♦♥✳ ❆♥ ✉r❜❛♥ ❡❝♦♥♦♠✐❝s
❛♣♣r♦❛❝❤ ✇♦✉❧❞ ❢❛✈♦r ❛ ❣❡♥❡r❛❧ ❡q✉✐❧✐❜r✐✉♠ s❡tt✐♥❣ ✐♥ ✇❤✐❝❤ s❤♦♣♣✐♥❣ ❡①♣❡♥❞✐t✉r❡ ❡q✉❛❧s ✐♥❝♦♠❡✳ ❆t t❤❡ ❡①♣❡♥s❡s
♦❢ ❛ ♠♦r❡ t❡❝❤♥✐❝❛❧ ❛♥❛❧②s✐s✱ ♦✉r s❡tt✐♥❣ ❝❛♥ ❜❡ ♠♦❞✐✜❡❞ t♦ ❛❝❝♦✉♥t ❢♦r ✐♥❝♦♠❡ ❡✛❡❝ts✳
✻
✇❤❡r❡❛s ✐ts r❡s♣❡❝t✐✈❡ ✐♥✈❡rs❡ ❞❡♠❛♥❞s ❛r❡ ❣✐✈❡♥ ❜②
(e0 + e1 )Qρ−1
j
Pj =
A(q, Q)
(e0 + e1 )qiρ−1
pi =
A(q, Q)
✭✹✮
✇❤❡♥ ✐t ✈✐s✐ts ❜♦t❤ ♠❛r❦❡t♣❧❛❝❡s✳
■♥ t❤❡s❡ ❡①♣r❡ss✐♦♥s✱ A0 (q)✱ A1 (Q) ❛♥❞ A(q, Q) ❛r❡ ♠❛r❦❡t ❛❣❣r❡❣❛t❡s ❣✐✈❡♥ ❜②
A0 (q) ≡
ˆ
0
n
qkρ dk
A(q, Q) ≡
ˆ
0
A1 (Q) ≡
n
qkρ dk
+
ˆ
0
ˆ
N
0
Qρl dl
N
Qρl dl
✇❤✐❧❡ e0 ✭e1 ✮ ✐s t❤❡ ❝♦♥s✉♠❡r✬s ❡①♣❡♥❞✐t✉r❡ ✐♥ t❤❡ ❙❙❚ ✭❙▼✮✳ ■t ❢♦❧❧♦✇s ❢r♦♠ ✭✶✮ ❛♥❞ ✭✷✮ t❤❛t e0 = 1
❛♥❞ e1 = 0 ✐❢ t❤❡ ❝♦♥s✉♠❡r ♣❛tr♦♥✐③❡s t❤❡ ❙❙❚ ♦♥❧②❀ e0 = 0 ❛♥❞ e1 = 1 ✐❢ ✐t ✈✐s✐ts t❤❡ ❙▼ ♦♥❧②❀
❛♥❞ e0 + e1 = 1 ✐❢ ✐t ❣♦❡s t♦ ❜♦t❤ ♣❧❛❝❡s✳ ❚❤✉s✱ r❡❣❛r❞❧❡ss ♦❢ ✐ts s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r✱ t❤❡ ❝♦♥s✉♠❡r
s♣❡♥❞s ♦♥❡ ✉♥✐t ♦❢ t❤❡ ♥✉♠❡r❛✐r❡ ♦♥ t❤❡ ❞✐✛❡r❡♥t✐❛t❡❞ ❣♦♦❞✱ ❜✉t t❤❡ ✈❛❧✉❡s ♦❢ e0 ❛♥❞ e1 ❞❡♣❡♥❞
♦♥ t❤❡ ❝♦♥s✉♠❡r✬s s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r✳
❙❡❧❧❡rs✳ ❯♣ t♦ ❙❡❝t✐♦♥ ✻✱ ✇❡ ❛ss✉♠❡ t❤❛t ❜♦t❤ r❡t❛✐❧❡rs ❛♥❞ st♦r❡s s❤❛r❡ t❤❡ s❛♠❡ ♠❛r❣✐♥❛❧
♣r♦❞✉❝t✐♦♥ ❝♦st c✳ ❇❡❝❛✉s❡ ❛ s❡❧❧❡r ✐s ♥❡❣❧✐❣✐❜❧❡ t♦ t❤❡ ♠❛r❦❡t✱ ✐ts ♦✇♥ ♣r✐❝❡ ❝❤♦✐❝❡ ❤❛s ♥♦ ✐♠♣❛❝t ♦♥
t❤❡ ♠❛r❦❡t ❛❣❣r❡❣❛t❡s A0 ✱ A1 ❛♥❞ A ✐♥ ✭✸✮ ❛♥❞ ✭✹✮✳ ❋♦r t❤❡ s❛♠❡ r❡❛s♦♥✱ ❛ s❡❧❧❡r✬s ♣r✐❝❡ ❝❤♦✐❝❡ ❛❧s♦
❤❛s ♥♦ ✐♠♣❛❝t ❡✐t❤❡r ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝✉st♦♠❡rs ✈✐s✐t✐♥❣ t❤❡ s❤♦♣♣✐♥❣ ♣❧❛❝❡ ✐♥ ✇❤✐❝❤ t❤❡ s❡❧❧❡r ✐s
s❡t ✉♣✳ ❚❤❡r❡❢♦r❡✱ ❡❛❝❤ s❡❧❧❡r ❢❛❝❡s ❛♥ ✐s♦❡❧❛st✐❝ ❞❡♠❛♥❞✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t t❤❡ ♣r♦✜t✲♠❛①✐♠✐③✐♥❣
♣r✐❝❡s ❛r❡ t❤❡ s❛♠❡ ❛❝r♦ss ❛❧❧ ✈❛r✐❡t✐❡s ❛♥❞ ❣✐✈❡♥ ❜②
c
p∗ = P ∗ = .
ρ
✭✺✮
❈♦♥s❡q✉❡♥t❧②✱ ❛s ✐♥ ❉✐①✐t ❛♥❞ ❙t✐❣❧✐t③ ✭✶✾✼✼✮✱ ♣r♦✜t✲♠❛①✐♠✐③✐♥❣ ♣r✐❝❡s ❛r❡ ✉♥❛✛❡❝t❡❞ ❜② t❤❡
♥✉♠❜❡r ♦❢ ❝♦♠♣❡t✐t♦rs ❛♥❞ ♠❛r❦❡t s✐③❡✳ ❆❞♠✐tt❡❞❧②✱ t❤❡s❡ ♣r♦♣❡rt✐❡s ❛r❡ ✈❡r② r❡str✐❝t✐✈❡✳ ❖✉r ❧✐♥❡
♦❢ ❞❡❢❡♥❝❡ ✐♥✈♦❧✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❣✉♠❡♥ts✳ ❋✐rst✱ ♦✉r ♠♦❞❡❧ ❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ❝♦♣❡ ✇✐t❤
❞✐s❝♦✉♥t r❡t❛✐❧✐♥❣✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ st♦r❡s ❛r❡ ♠♦r❡ ❡✣❝✐❡♥t t❤❛♥ r❡t❛✐❧❡rs✱ ✐✳❡✳ c1 < c0 ✱ ✈❛r✐❡t✐❡s ❛r❡
❝❤❡❛♣❡r ✐♥ t❤❡ ❙▼ t❤❛♥ ✐♥ t❤❡ ❙❙❚ ❜✉t t❤❡ ♣r✐❝❡ r❛t✐♦ p∗ /P ∗ > 1 r❡♠❛✐♥s ✐♥❞❡♣❡♥❞❡♥t ♦❢ n ❛♥❞
N ✳ ❊✈❡r②t❤✐♥❣ ❡❧s❡ ❜❡✐♥❣ ❡q✉❛❧✱ t❤✐s ♠❛❦❡s t❤❡ ❙▼ r❡❧❛t✐✈❡❧② ♠♦r❡ ❛ttr❛❝t✐✈❡ t❤❛♥ t❤❡ ❙❙❚✳ ❲❡
r❡t✉r♥ t♦ t❤✐s ✐ss✉❡ ✐♥ ❙❡❝t✐♦♥ ✻✳✶✳ ❙❡❝♦♥❞✱ ❛s ❛r❣✉❡❞ ✐♥ t❤❡ ❝♦♥❝❧✉❞✐♥❣ s❡❝t✐♦♥✱ ♠♦st ♦❢ ♦✉r r❡s✉❧ts
r❡♠❛✐♥ ✈❛❧✐❞ ❢♦r ❛ ✇✐❞❡r ❝❧❛ss ♦❢ ♣r❡❢❡r❡♥❝❡s t❤❛t ❛❝❝♦✉♥ts ❢♦r ♣r✐❝❡ ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ s❡❧❧❡rs✳
❲❡ ❤❛✈❡ ❝❤♦s❡♥ t♦ ✇♦r❦ ✇✐t❤ t❤❡ ❈❊❙ ❜❡❝❛✉s❡ ✐t ❛❧❧♦✇s ✉s t♦ ♦❜t❛✐♥ s✐♠♣❧❡ ❝❧♦s❡❞✲❢♦r♠ s♦❧✉t✐♦♥s
❛s ✇❡❧❧ ❛s t♦ ❞✐s❡♥t❛♥❣❧❡ s✐③❡ ❛♥❞ ❞✐st❛♥❝❡ ❡✛❡❝ts ❢r♦♠ ♣r✐❝❡ ❡✛❡❝ts✳
❚❤✐r❞✱ t❤❡ ♦✉t✲♦❢✲t♦✇♥ s❤♦♣♣✐♥❣ ♣❧❛❝❡ ✐s ❡st❛❜❧✐s❤❡❞ ❜② ❛ ♣r♦✜t✲♠❛①✐♠✐③✐♥❣ ❞❡✈❡❧♦♣❡r ✇❤♦ r✉♥s
❛ s❤♦♣♣✐♥❣ ♠❛❧❧ ✲ t❤❡ ❞❡✈❡❧♦♣❡r ❝❤♦♦s✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥❞❡♣❡♥❞❡♥t s❤♦♣s t❤❛t ✇✐❧❧ ❜❡ ✐♥❝❧✉❞❡❞ ✐♥
t❤❡ ♠❛❧❧ ❛♥❞ ❛ r❡♥t✐♥❣ ♣r✐❝❡✳ ❆❧t❡r♥❛t✐✈❡❧②✱ t❤❡ ❞❡✈❡❧♦♣❡r ❝♦✉❧❞ r✉♥ ❛ s✉♣❡r♠❛r❦❡t✱ ✇❤✐❝❤ ♠❡❛♥s
✼
t❤❛t t❤❡ ❞❡✈❡❧♦♣❡r s❡❧❡❝ts t❤❡ ♥✉♠❜❡r ♦❢ st♦r❡s ❛♥❞ ❛❧s♦ t❤❡ ♣r✐❝❡ ❛t ✇❤✐❝❤ t❤❡② s❡❧❧ t❤❡✐r ♣r♦❞✉❝t✳✺
❚❤✉s✱ ❛ s✉♣❡r♠❛r❦❡t ♠❛② ❜❡ ✈✐❡✇❡❞ ❛s ❛ ✈❡rt✐❝❛❧❧② ✐♥t❡❣r❛t❡❞ ✜r♠✱ ✇❤❡r❡❛s ❛ s❤♦♣♣✐♥❣ ♠❛❧❧ ❝❛♥ ❜❡
❞❡s❝r✐❜❡❞ ❜② ❛ ✈❡rt✐❝❛❧ r❡❧❛t✐♦♥s❤✐♣ ✐♥✈♦❧✈✐♥❣ ♦♥❡ ✉♣str❡❛♠ ♣r♦❞✉❝❡r ❛♥❞ ❛ ♠②r✐❛❞ ♦❢ ❞♦✇♥str❡❛♠
s❤♦♣✲❦❡❡♣❡rs✳ ■❢ t❤❡ ❞❡✈❡❧♦♣❡r ♦♣❡r❛t❡s ❛ s✉♣❡r♠❛r❦❡t✱ t❤❡ s②♠♠❡tr② ♦❢ ♣r❡❢❡r❡♥❝❡s ✐♠♣❧✐❡s t❤❛t
❛❧❧ ✈❛r✐❡t✐❡s ❛r❡ s♦❧❞ ❛t t❤❡ s❛♠❡ ♣r♦✜t✲♠❛①✐♠✐③✐♥❣ ♣r✐❝❡ P̄ ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ r❡❞♦ t❤❡ s❛♠❡
❛♥❛❧②s✐s ❜② r❡♣❧❛❝✐♥❣ P ∗ ✇✐t❤ P̄ ✳ ❍♦✇ t❤❡ s✉♣❡r♠❛r❦❡t ❝❤♦♦s❡s P̄ ✐s ✐♠♠❛t❡r✐❛❧ ❢♦r ♦✉r ♣✉r♣♦s❡✳
▲❛st✱ ✐♥ ❙❡❝t✐♦♥ ✻✳✶✱ ✇❡ s❤♦✇ t❤❛t ❝♦♥s✉♠❡rs ♠❛② ❤❛✈❡ ❛ ♣r❡❢❡r❡♥❝❡ ❢♦r ♦♥❡ t②♣❡ ♦❢ ♠❛r❦❡t♣❧❛❝❡
♦✈❡r t❤❡ ♦t❤❡r✳
❆❧t❤♦✉❣❤ ✇❡ ❛ss✉♠❡ t❤❛t r❡t❛✐❧❡rs ❛❝t ♥♦♥❝♦♦♣❡r❛t✐✈❡❧②✱ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t s♠❛❧❧✲❜✉s✐♥❡ss ❛s✲
s♦❝✐❛t✐♦♥s ❛✐♠ t♦ ❡①❝❤❛♥❣❡ t❤❡✐r ♣♦❧✐t✐❝❛❧ ✐♥✢✉❡♥❝❡ ❢♦r ❣♦✈❡r♥♠❡♥t❛❧ ♣♦❧✐❝✐❡s t❤❛t ❝♦♠♣❡♥s❛t❡ ❢♦r
t❤❡✐r ✇❡❛❦♥❡ss ✐♥ t❤❡ ♠❛r❦❡t♣❧❛❝❡✳ ■♥ t❤❡ s❛♠❡ ✈❡✐♥✱ ❝♦❛❧✐t✐♦♥s ♦❢ ❧♦❝❛❧ ♠❡r❝❤❛♥ts ❛♥❞ ❝♦♠♠✉♥✐t②
❧❡❛❞❡rs ✇♦r❦ t♦ ✐♠♣r♦✈❡ ❧♦❝❛❧ ♣✉❜❧✐❝ s♣❛❝❡s✱ ✇❤✐❝❤ t❛❦❡s t❤❡ ❝♦♥❝r❡t❡ ❢♦r♠ ♦❢ ✉r❜❛♥ ❛♠❡♥✐t✐❡s ❛♥❞
♣❡❞❡str✐❛♥ ❛r❡❛s✳ ❚❤❡s❡ ✐ss✉❡s ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ❡①t❡♥s✐♦♥s ♦❢ ♦✉r ❜❛s❡❧✐♥❡ ♠♦❞❡❧✳
✷✳✷
❙❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r
❈♦♥s✉♠❡r s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r ✐s ❞r✐✈❡♥ ❜② ✉t✐❧✐t② ♠❛①✐♠✐③❛t✐♦♥✳ ❙✐♥❝❡ ❜♦t❤ r❡t❛✐❧❡rs ❛♥❞ st♦r❡s
❝❤❛r❣❡ t❤❡ s❛♠❡ ♣r✐❝❡ ✇❤✐❧❡ t❤❡ ❞❡❣r❡❡ ♦❢ ♣r♦❞✉❝t ❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s t❤❡ s❛♠❡ ✐♥ t❤❡ t✇♦ s❤♦♣♣✐♥❣
❛r❡❛s✱ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥s✉♠❡rs ❞r❛✇♥ ❜② ❛ ♠❛r❦❡t♣❧❛❝❡ ❤✐♥❣❡s ♦♥❧② ✉♣♦♥ t❤❡ ♠❛ss ♦❢ ✈❛r✐❡t✐❡s
✐t s✉♣♣❧✐❡s r❡❧❛t✐✈❡❧② t♦ t❤❡ ♠❛ss ♦❢ ✈❛r✐❡t✐❡s ♣r♦✈✐❞❡❞ ❜② t❤❡ ♦t❤❡r s❤♦♣♣✐♥❣ ♣❧❛❝❡✳ ❍♦✇❡✈❡r✱ ❛s
✇✐❧❧ ❜❡ s❤♦✇♥ ❜❡❧♦✇✱ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ✉t✐❧✐t② ❧❡✈❡❧ ✈❛r✐❡s ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❡t✐❡s s✉♣♣❧✐❡❞ ✐♥
❡❛❝❤ s❤♦♣♣✐♥❣ ♣❧❛❝❡✱ n ❛♥❞ N ✱ ❛s ✇❡❧❧ ❛s ✇✐t❤ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❡t✐❡s✱ n + N ✳ ❚♦ s✐♠♣❧✐❢②
♥♦t❛t✐♦♥✱ ✇❡ ❝❤♦♦s❡ t❤❡ ✉♥✐t ♦❢ t❤❡ ♥✉♠❡r❛✐r❡ ❢♦r c/ρ t♦ ❜❡ ❡q✉❛❧ t♦ 1✳
❚❤r❡❡ s❤♦♣♣✐♥❣ ♣❛tt❡r♥s ♠❛② ❛r✐s❡✳
✭✐✮ I(SST ) = I(SM ) = 1✳ ❚❤❡ ❝♦♥s✉♠❡r s❤♦♣s ❛t ❜♦t❤ ♠❛r❦❡t♣❧❛❝❡s✳ ❯s✐♥❣ ✭✹✮ ❛♥❞ ✭✺✮✱ ✐t ✐s
r❡❛❞✐❧② ✈❡r✐✜❡❞ t❤❛t s✉❝❤ ❛ ❝♦♥s✉♠❡r ❜✉②s t❤❡ s❛♠❡ q✉❛♥t✐t② ♦❢ ❡❛❝❤ ♦❢ t❤❡ n + N ✈❛r✐❡t✐❡s✿
q=Q=
1
.
n+N
✭✻✮
P❧✉❣❣✐♥❣ ✭✻✮ ✐♥t♦ ✭✶✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ✐♥❞✐r❡❝t ✉t✐❧✐t② ♦❢ ❛ ❝♦♥s✉♠❡r ✈✐s✐t✐♥❣ t❤❡ t✇♦ s❤♦♣♣✐♥❣
♣❧❛❝❡s✿
V =
ln (n + N )
− τ + (I − 1)
σ−1
✭✼✮
✇❤❡r❡ σ ≡ 1/(1 − ρ) ✐s t❤❡ ❡❧❛st✐❝✐t② ♦❢ s✉❜st✐t✉t✐♦♥ ❛❝r♦ss ✈❛r✐❡t✐❡s✳ ◆♦t❡ t❤❛t ✭✼✮ ✐s ✐♥❞❡♣❡♥❞❡♥t
♦❢ t❤❡ ❝♦♥s✉♠❡r ❧♦❝❛t✐♦♥ x ❜❡❝❛✉s❡ τ x + τ (1 − x) = τ ✳
✭✐✐✮ I(SST ) = 1 ❛♥❞ I(SM ) = 0✳ ✐♥ t❤✐s ❡✈❡♥t✱ t❤❡ ❝♦♥s✉♠❡r ♣r❡❢❡rs ❛ ♦♥❡✲st♦♣ s❤♦♣♣✐♥❣ ❛t
✺ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ♣r✐❝✐♥❣ r✉❧❡ ✭✺✮ st✐❧❧ ❤♦❧❞s ❢♦r t❤❡ r❡t❛✐❧❡rs✱ ❜✉t ♥♦t ❢♦r t❤❡ st♦r❡s ❜❡❝❛✉s❡ t❤❡ ❞❡✈❡❧♦♣❡r
✐♥t❡r♥❛❧✐③❡s t❤❡ ❝❛♥♥✐❜❛❧✐③❛t✐♦♥ ❡✛❡❝t ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❛❧❡ ♦❢ s✉❜st✐t✉t❡s✳
✽
t❤❡ ❙❙❚✳ ■t ❢♦❧❧♦✇s ❢r♦♠ ✭✸✮ ❛♥❞ ✭✺✮ t❤❛t ✐ts ❞❡♠❛♥❞ ❢♦r ❛ ✈❛r✐❡t② ✐s
q=
1
n
✭✽✮
✇❤✐❧❡ ✐ts ✐♥❞✐r❡❝t ✉t✐❧✐t② ❧❡✈❡❧ ✐s ❡q✉❛❧ t♦
V0 (x) =
ln n
− τ x + (I − 1)
σ−1
✭✾✮
✇❤✐❝❤ ❞❡❝r❡❛s❡s ✇✐t❤ x✳
✭✐✐✐✮ I(SST ) = 0 ❛♥❞ I(SM ) = 1✳ ❲❤❡♥ t❤❡ ❝♦♥s✉♠❡r ♣r❡❢❡rs s❤♦♣♣✐♥❣ ❛t t❤❡ ❙▼ ♦♥❧②✱ ✐ts
❞❡♠❛♥❞ ✐s ❣✐✈❡♥ ❜②
1
✭✶✵✮
Q= .
N
❙✉❝❤ ❛ ❝♦♥s✉♠❡r ❡♥❥♦②s ❛♥ ✐♥❞✐r❡❝t ✉t✐❧✐t② ❧❡✈❡❧ ❣✐✈❡♥ ❜②
V1 (x) =
ln N
− τ (1 − x) + (I − 1)
σ−1
✭✶✶✮
✇❤✐❝❤ ✐♥❝r❡❛s❡s ✇✐t❤ x✳ ❆❧t❤♦✉❣❤ t❤❡ ❞❡✈❡❧♦♣❡r ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ❞✐r❡❝t❧② ✈❛r✐❡t✐❡s✱ ✐t ❢❛❝❡s t❤❡
✐ss✉❡ ♦❢ ❝❛♥♥✐❜❛❧✐③❛t✐♦♥ t❤❛t ❝❤❛r❛❝t❡r✐③❡s ♠✉❧t✐✲♣r♦❞✉❝t ✜r♠s✿ t❤❡ ❧❛r❣❡r t❤❡ ♥✉♠❜❡r ♦❢ st♦r❡s ✐♥
t❤❡ ❙▼✱ t❤❡ s♠❛❧❧❡r t❤❡ q✉❛♥t✐t② s♦❧❞ ❜② ❡❛❝❤ st♦r❡✳
❖❜s❡r✈❡ t❤❛t✱ r❡❣❛r❞❧❡ss ♦❢ t❤❡ ❝♦♥s✉♠❡r✬s s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r✱ ✐ts ✉t✐❧✐t② ❧❡✈❡❧ ✐♥❝r❡❛s❡s ❛t ❛
❞❡❝r❡❛s✐♥❣ r❛t❡ ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❡t✐❡s ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❝❡♥t❡r✭s✮ ✐t ❝❤♦♦s❡s t♦ ✈✐s✐t✳ ❋✐❣✉r❡ ✶
❞❡♣✐❝ts t❤❡ ✉t✐❧✐t② ❧❡✈❡❧ r❡❛❝❤❡❞ ❜② ❝♦♥s✉♠❡rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❧♦❝❛t✐♦♥✳ ❈♦♥s✉♠❡rs ❧♦❝❛t❡❞ ♥❡❛r
t❤❡ ♠✐❞❞❧❡ ♦❢ t❤❡ s❡❣♠❡♥t ❛r❡ ✇♦rs❡✲♦✛ t❤❛♥ t❤♦s❡ ❧♦❝❛t❡❞ ❝❧♦s❡ t♦ t❤❡ s❤♦♣♣✐♥❣ ♣❧❛❝❡s ❜❡❝❛✉s❡ t❤❡②
❜❡❛r ❤✐❣❤❡r tr❛✈❡❧ ❝♦sts✳ ❚❤❡② ♣❛rt✐❛❧❧② ❝♦♠♣❡♥s❛t❡ t❤❡✐r ❧♦❝❛t✐♦♥❛❧ ❞✐s❛❞✈❛♥t❛❣❡ ❜② ❝♦♥s✉♠✐♥❣
t❤❡ ❡♥t✐r❡ r❛♥❣❡ ♦❢ ✈❛r✐❡t✐❡s✳
✾
❋✐❣✉r❡ ✶✳ ❯t✐❧✐t✐❡s ❛♥❞ s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r
❚❤r♦✉❣❤♦✉t t❤❡ ♣❛♣❡r✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✳ ▲❡t ν ≡ N/n ❜❡ t❤❡ r❡❧❛t✐✈❡ s✐③❡ ♦❢
t❤❡ ❙▼ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❙❙❚ ❛♥❞
T ≡ exp[(σ − 1)τ ] > 1.
❆ ❤✐❣❤ ✭❧♦✇✮ ✈❛❧✉❡ ♦❢ T ♠❡❛♥s t❤❛t tr❛✈❡❧ ❝♦sts ❛r❡ ❤✐❣❤ ✭❧♦✇✮ ♦r ✈❛r✐❡t✐❡s ❛r❡ ❣♦♦❞ ✭♣♦♦r✮
s✉❜st✐t✉t❡s✳ ❚❤❡ ♣❛r❛♠❡t❡r T t❤✉s ❝❛♣t✉r❡s t❤❡ tr❛✈❡❧ ❛♥❞ ♣r♦❞✉❝t ❞✐✛❡r❡♥t✐❛t✐♦♥ ❡✛❡❝ts✱ ✇❤✐❝❤
❛r❡ ❝r✐t✐❝❛❧ ✐♥ ♦✉r ❛♣♣r♦❛❝❤ t♦ s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r✳ ❇♦t❤ ♠❛r❦❡t♣❧❛❝❡s ❜❡❝♦♠❡ ♠♦r❡ ❛ttr❛❝t✐✈❡
✇❤❡♥ ❡✐t❤❡r ♦❢ t❤❡s❡ ♣❛r❛♠❡t❡rs ❞❡❝r❡❛s❡s✳ ❋♦r ❛ ❣✐✈❡♥ ✈❛❧✉❡ ♦❢ τ ✱ ♠♦r❡ ❞✐✛❡r❡♥t✐❛t❡❞ ✈❛r✐❡t✐❡s
♠❛❦❡ ❜♦t❤ t❤❡ ❙❙❚ ❛♥❞ ❙▼ ♠♦r❡ ❛ttr❛❝t✐✈❡✱ ✇❤✐❧❡ ❢♦r ❛ ❣✐✈❡♥ σ ✱ ❧♦✇❡r✐♥❣ tr❛✈❡❧ ❝♦sts ♠❛❦❡s ✐t
❡❛s✐❡r t♦ ✈✐s✐t ❜♦t❤ ♠❛r❦❡t♣❧❛❝❡s✳
❚❤❡ s♦❧✉t✐♦♥ t♦ V0 (x) = V1 (x) ✐s ❣✐✈❡♥ ❜②
1
x̄ =
2
ln ν
1−
ln T
s♦ t❤❛t ♠♦r❡ ❝♦♥s✉♠❡rs ✈✐s✐t t❤❡ ❙❙❚ t❤❛♥ t❤❡ ❙▼ ✐❢ t❤❡ s✐③❡ ♦❢ t❤❡ ❢♦r♠❡r ❡①❝❡❡❞s t❤❛t ♦❢ t❤❡
❧❛tt❡r✳ ❚❤✐s ❡①♣r❡ss✐♦♥ s❤♦✇s t❤❛t✱ ❢♦r ❛ ❣✐✈❡♥ T ✱ t❤❡ t✇♦ ♠❛r❦❡t♣❧❛❝❡s ❝❛♥♥♦t ❜❡ t♦♦ ❞✐ss✐♠✐❧❛r
✐♥ s✐③❡✱ ❢♦r ♦t❤❡r✇✐s❡ ❛❧❧ ❝♦♥s✉♠❡rs ✇♦✉❧❞ ♣❛tr♦♥✐③❡ t❤❡ ❜✐❣❣❡r ♠❛r❦❡t♣❧❛❝❡ ♦♥❧②✳
✶✵
❖✉r ❛♣♣r♦❛❝❤ t♦ s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r ✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤r❡❡ ❣r♦✉♣s ♦❢ ❝♦♥s✉♠❡rs✳
✭✐✮ ❆ ❝♦♥s✉♠❡r ❧♦❝❛t❡❞ ❛t x ✈✐s✐ts ❜♦t❤ t❤❡ ❙❙❚ ❛♥❞ t❤❡ ❙▼ ✐❢ ✐t ❤❛s ❛ ❤✐❣❤❡r ✉t✐❧✐t② ✇❤❡♥ ✐t
♣❛tr♦♥✐③❡s ❜♦t❤ ♣❧❛❝❡s r❛t❤❡r t❤❛♥ ❛ s✐♥❣❧❡ ♦♥❡✿
✭✶✷✮
V ≥ max {V0 (x), V1 (x)}
✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦ x̄0 ≤ x ≤ x̄1 ✇❤❡r❡
ln(1 + ν)
x̄0 = max 0, 1 −
ln T
ln (1 + 1/ν)
x̄1 = min 1,
ln T
✭✶✸✮
✭✶✹✮
t❤❡ s❡❝♦♥❞ t❡r♠ ✐♥ ✭✶✸✮ ❛♥❞ ✭✶✹✮ ❜❡✐♥❣ t❤❡ s♦❧✉t✐♦♥ t♦✱ r❡s♣❡❝t✐✈❡❧②✱ V = V0 (x) ❛♥❞ V = V1 (x)✳
❋♦r ❛ ♣♦s✐t✐✈❡ ♠❛ss ♦❢ ❝♦♥s✉♠❡rs t♦ ❜❡ t✇♦✲st♦♣ s❤♦♣♣❡rs✱ ✐t ♠✉st ❜❡ t❤❛t V > V0 (x̄) = V1 (x̄)✳
✭✐✐✮ ❆ ❝♦♥s✉♠❡r ❛t x ♣r❡❢❡rs s❤♦♣♣✐♥❣ ❛t t❤❡ ❙❙❚ ✐❢
V0 (x) > max {V, V1 (x)}
✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦ 0 ≤ x < min {x̄0 , x̄}✳ ❚❤✐s ✐♥t❡r✈❛❧ ✐s ♥♦♥❡♠♣t② ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥s✉♠❡r
❛t x = 0 s❤♦♣s ❛t t❤❡ ❙❙❚ ♦♥❧②✱ ✇❤✐❝❤ ❛♠♦✉♥ts t♦
ν < T − 1.
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ s✐③❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❙▼ ✐s ♥♦t s✉✣❝✐❡♥t❧② ❧❛r❣❡ ❢♦r t❤❡ ❝♦♥s✉♠❡rs ❧♦❝❛t❡❞
♥❡❛r t❤❡ ❙❙❚ t♦ ❜❡❛r t❤❡ ❛❞❞✐t✐♦♥❛❧ tr❛✈❡❧ ❝♦sts t♦ ✈✐s✐t t❤❡ ❙▼✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❝♦♥s✉♠❡r
✐♥❞✐✛❡r❡♥t ❜❡t✇❡❡♥ ♣❛tr♦♥✐③✐♥❣ t❤❡ s♦❧❡ ❙❙❚ ♦r ❜♦t❤ s❤♦♣♣✐♥❣ ♣❧❛❝❡s✱ ✐s ❧♦❝❛t❡❞ ❛t x = min {x̄0 , x̄}✳
✭✐✐✐✮ ❙✐♠✐❧❛r❧②✱ ❛ ❝♦♥s✉♠❡r ❛t x ♣❛tr♦♥✐③❡s ♦♥❧② t❤❡ ❙▼ ✐❢
V1 (x) > max {V, V0 (x)}
✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦ max {x̄1 , x̄} < x ≤ 1✳ ❚❤✐s ✐♥t❡r✈❛❧ ✐s ♥♦♥❡♠♣t② ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥s✉♠❡r
❛t x = 1 s❤♦♣s ❛t t❤❡ ❙▼ ♦♥❧②✱ ✇❤✐❝❤ ❛♠♦✉♥ts t♦
ν>
1
.
T −1
❋♦r ♦✉r ♣✉r♣♦s❡✱ t❤❡ ♠♦st ✐♥t❡r❡st✐♥❣ ♠❛r❦❡t ❝♦♥✜❣✉r❛t✐♦♥ ✐♥✈♦❧✈❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ t❤r❡❡
❣r♦✉♣s ♦❢ ❝♦♥s✉♠❡rs✱ ✇❤✐❝❤ r❡q✉✐r❡s 0 < x̄0 < x̄ < x̄1 < 1✳ ❋♦r t✇♦✲st♦♣ s❤♦♣♣✐♥❣ t♦ ❛r✐s❡✱ ✭✶✷✮
♠✉st ❤♦❧❞ ❛t x̄✱ ✇❤✐❝❤ ✐s ♥♦t s❛t✐s✜❡❞ ✐♥ ❣❡♥❡r❛❧ ✇❤❡♥ T > 4✳ ❲❡ ❛❧s♦ r✉❧❡ ♦✉t t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤
❛❧❧ ❝♦♥s✉♠❡rs ✈✐s✐t t❤❡ t✇♦ ♠❛r❦❡t♣❧❛❝❡s✱ ✐✳❡✳ x̄0 = 0 ❛♥❞ x̄1 = 1✳ ❯s✐♥❣ ✭✶✸✮ ❛♥❞ ✭✶✹✮✱ ✐t ✐s r❡❛❞✐❧②
✈❡r✐✜❡❞ t❤❛t t❤❡s❡ t✇♦ ❡q✉❛❧✐t✐❡s ♥❡✈❡r ❤♦❧❞ s✐♠✉❧t❛♥❡♦✉s❧② ✇❤❡♥ T > 2✳ ❚❤❡r❡❢♦r❡✱ ✐♥ ✇❤❛t ❢♦❧❧♦✇s
✶✶
✇❡ ❢♦❝✉s ♦♥ t❤❡ s♣❡❝✐❛❧✱ ❜✉t r❡❧❡✈❛♥t✱ ❝❛s❡ ✐♥ ✇❤✐❝❤ 2 < T < 4✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss ✐♥ t❤❡ ❝♦♥❝❧✉❞✐♥❣
s❡❝t✐♦♥ ✇❤❛t ♦✉r ♠❛✐♥ r❡s✉❧ts ❜❡❝♦♠❡ ✇❤❡♥ T ≤ 2 ❛♥❞ T ≥ 4✳
❖♥❡ ♦❢ t❤❡ ♠❛✐♥ ❞✐st✐♥❝t✐✈❡ ❢❡❛t✉r❡s ♦❢ ♦✉r ♠♦❞❡❧ t❤❡♥ ❧✐❡s ✐♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❝♦♥t❡♥t✐♦♥
❛r❡❛ [x̄0 , x̄1 ] ❢♦r♠❡❞ ❜② t❤❡ t✇♦✲st♦♣ s❤♦♣♣❡rs ✭s❡❡ ❋✐❣✉r❡ ✶ ❢♦r ❛♥ ✐❧❧✉str❛t✐♦♥✮✳ ❚❤✐s ❛r❡❛ ✇✐❞❡♥s
✭s❤r✐♥❦s✮ ❛s T ❞❡❝r❡❛s❡s ✭✐♥❝r❡❛s❡s✮ ❜❡❝❛✉s❡ ❜♦t❤ ♠❛r❦❡t♣❧❛❝❡s ❜❡❝♦♠❡ ♠♦r❡ ✭❧❡ss✮ ❛ttr❛❝t✐✈❡✳
❈♦♥s✉♠❡rs ❧✐✈✐♥❣ ✐♥ t❤❡ ❝♦♥t❡♥t✐♦♥ ❛r❡❛ s♣❧✐t t❤❡✐r ❡①♣❡♥❞✐t✉r❡ ❜❡t✇❡❡♥ t❤❡ ❙❙❚ ❛♥❞ t❤❡ ❙▼
❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✇❛❧❧❡t s❤❛r❡s 1/(1 + ν) ❛♥❞ ν/(1 + ν)✳ ❲❤❡♥ ν r✐s❡s✱ t❤❡ ❙❙❚ ✭❙▼✮ ❤❛s ❢❡✇❡r
✭♠♦r❡✮ ❝✉st♦♠❡rs ❜❡❝❛✉s❡ ❜♦t❤ x̄0 ❛♥❞ x̄1 ♠♦✈❡ t♦ t❤❡ ❧❡❢t ✭r✐❣❤t✮✱ ❛♥❞ t❤✉s ❢❡✇❡r ✭♠♦r❡✮ ❝♦♥s✉♠❡rs
s❤♦♣ ❞♦✇♥t♦✇♥✳ ■♥ ❛❞❞✐t✐♦♥✱ ❝♦♥s✉♠❡rs ❧♦❝❛t❡❞ ✐♥ t❤❡ ❝♦♥t❡♥t✐♦♥ ❛r❡❛ s♣❡♥❞ ❧❡ss ✭♠♦r❡✮ ✐♥ t❤❡
❙❙❚ ✭❙▼✮✳ ❚❤✉s✱ ✉♥❧✐❦❡ st❛♥❞❛r❞ ♠♦❞❡❧s ♦❢ s♣❛t✐❛❧ ❝♦♠♣❡t✐t✐♦♥✱ ✇❡ ❤❛✈❡ t✇♦ ✲ ✐♥st❡❛❞ ♦❢ ♦♥❡ ✲
♠❛r❣✐♥❛❧ ❝♦♥s✉♠❡rs ❛♥❞ ❛ ✇❛❧❧❡t s❤❛r❡ ❡✛❡❝t t❤❛t ✈❛r✐❡s ✇✐t❤ t❤❡ r❡❧❛t✐✈❡ s✐③❡ ♦❢ t❤❡ ❙❙❚ ❛♥❞ ❙▼✳
❋♦r ❛♥② ❣✐✈❡♥ n ❛♥❞ N ✱ ✐t ❢♦❧❧♦✇s ✐♠♠❡❞✐❛t❡❧② ❢r♦♠ ✭✺✮✱ ✭✻✮✱ ✭✽✮✱ ❛♥❞ ✭✶✵✮ t❤❛t t❤❡ ♦♣❡r❛t✐♥❣
♣r♦✜ts ♦❢ ❛ r❡t❛✐❧❡r ❛r❡ ❣✐✈❡♥ ❜②
πR (n, N ) = (1 − ρ)
✇❤❡r❡❛s t❤♦s❡ ♠❛❞❡ ❜② ❛ st♦r❡ ❛r❡ s✉❝❤ t❤❛t
πS (n, N ) = (1 − ρ)
x̄0 x̄1 − x̄0
+
n
n+N
x̄1 x̄1 − x̄0
+
N
n+N
.
✭✶✺✮
✭✶✻✮
❚❤❡r❡❢♦r❡✱ t❤❡ ♣r♦✜t ❢✉♥❝t✐♦♥s ❛r❡ s②♠♠❡tr✐❝ ✿
✭✶✼✮
πS (n, N ) = πR (N, n).
✸
❈♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ ❙❤♦♣♣✐♥❣ ❈❡♥t❡rs
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ st✉❞② ❤♦✇ ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ s❤♦♣♣✐♥❣ ♣❧❛❝❡s ✐s ❛✛❡❝t❡❞ ❜② ❛♥
❡①♦❣❡♥♦✉s ❝❤❛♥❣❡ ✐♥ t❤❡ s✐③❡ ♦❢ ♦♥❡ ♠❛r❦❡t♣❧❛❝❡✱ ❤❡r❡ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧✳ ❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ♠❛r❦✉♣
❜❡✐♥❣ ❝♦♥st❛♥t✱ ❢♦r ❛♥② ❣✐✈❡♥ n✱ t❤❡ ♦♣❡r❛t✐♥❣ ♣r♦✜ts ♦❢ ❛ st♦r❡ ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ ✐ts ❞❡♠❛♥❞✳
❙✐♥❝❡ x̄0 ❛♥❞ x̄1 ❛r❡ ❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ 0 ✐♥ (n, N )✱ t❤✐s ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ✇♦r❦✐♥❣ ✇✐t❤
ν = N/n ♦♥❧②✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✭✶✼✮ ✐♠♣❧✐❡s t❤❛t t❤❡ s❛♠❡ ❛♥❛❧②s✐s ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ t❡r♠s ♦❢ µ ≡
1/ν = n/N t♦ st✉❞② t❤❡ ✐♠♣❛❝t ♦❢ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ ❙❚❚ s✐③❡✳
❈♦♥s✐❞❡r ❛ ✜①❡❞ ♥✉♠❜❡r n ♦❢ r❡t❛✐❧❡rs ❛♥❞ ❛♥ ❙▼ ✇❤♦s❡ r❡❧❛t✐✈❡ s✐③❡ ✐s ν ✳ ■t ✐s t❤❡♥ r❡❛❞✐❧②
✈❡r✐✜❡❞ t❤❛t t♦t❛❧ ❡①♣❡♥❞✐t✉r❡s ✐♥ t❤❡ ❙▼ ✐♥❝r❡❛s❡ ✇✐t❤ ν ✳ ❍♦✇❡✈❡r✱ t❤✐s ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t t❤❡
q✉❛♥t✐t② s♦❧❞ ❜② ❡❛❝❤ st♦r❡ ❛❧s♦ r✐s❡s ✇✐t❤ ν ❜❡❝❛✉s❡ ♠♦r❡ ✜r♠s ❝♦♠♣❡t❡ ✇✐t❤✐♥ t❤❡ ❙▼✳
❆ st♦r❡ ❞❡♠❛♥❞ ✐s ❣✐✈❡♥ ❜②
1
DS (ν) =
n
1 − x̄1 x̄1 − x̄0
+
ν
1+ν
✶✷
.
✭✶✽✮
❚❤❡ ♥✉♠❜❡r
n
♦❢ r❡t❛✐❧❡rs ❜❡✐♥❣ ❡①♦❣❡♥♦✉s✱ ✭✶✽✮ ✐♠♣❧✐❡s t❤❛t ❛ st♦r❡ ❞❡♠❛♥❞ ❞❡♣❡♥❞s ♦♥❧②
✉♣♦♥ t❤❡ ❙▼✬s r❡❧❛t✐✈❡ s✐③❡✳ ❚♦ s❡❡ ❤♦✇
DS
✈❛r✐❡s ✇✐t❤
ν✱
❝♦♥s✐❞❡r ❛ ♠❛r❣✐♥❛❧ ✐♥❝r❡❛s❡
dν
✐♥ t❤❡
♠❛ss ♦❢ st♦r❡s✿
n dDS = −
1 − x̄1
x̄1 − x̄0
+
2
ν
(1 + ν)2
1
dν −
dx̄0 −
1+ν
1
1
−
ν 1+ν
dx̄1 .
✭✶✾✮
❚❤❡ ✜rst t❡r♠ ♦❢ t❤✐s ❡①♣r❡ss✐♦♥ st❛♥❞s ❢♦r t❤❡ ✏❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝t✑ ✇✐t❤✐♥ t❤❡ ❙▼ ✇❤❡♥ t❤❡
♠❛r❦❡t ❜♦✉♥❞❛r✐❡s
x̄0 ❛♥❞ x̄1 ❛r❡ ✉♥❝❤❛♥❣❡❞✳
❚❤✐s t❡r♠✱ ✇❤✐❝❤ st❡♠s ❢r♦♠ t❤❡ ❞❡❡♣❡r ❢r❛❣♠❡♥t❛t✐♦♥
♦❢ ❞❡♠❛♥❞ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ❧♦❝❛❧ ✈❛r✐❡t✐❡s✱ ✐s ❛❧✇❛②s ♥❡❣❛t✐✈❡✳ ❚❤❡ s❡❝♦♥❞ ❛♥❞
t❤✐r❞ t❡r♠s ♦❢ ✭✶✾✮ r❡♣r❡s❡♥t t❤❡ ✏♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t✑ ❣❡♥❡r❛t❡❞ ❜② ❛ ❜✐❣❣❡r ❙▼✳ ❲❤❡♥
♠♦r❡ ✈❛r✐❡t✐❡s ❛r❡ ♦✛❡r❡❞ ✐♥ t❤❡ ❙▼✱ s♦♠❡ ❝♦♥s✉♠❡rs ❧♦❝❛t❡❞ ❝❧♦s❡ t♦ t❤❡ ❙❙❚ ❝❤♦♦s❡ t♦ ✈✐s✐t
❜♦t❤ ♠❛r❦❡t♣❧❛❝❡s ✐♥st❡❛❞ ♦❢ s❤♦♣♣✐♥❣ ❛t t❤❡ ❙❙❚ ♦♥❧② ✭x̄0 ❞❡❝r❡❛s❡s✮✱ ✇❤❡r❡❛s ♠♦r❡ ❝♦♥s✉♠❡rs
❡st❛❜❧✐s❤❡❞ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ❙▼ ♥♦✇ ❝❤♦♦s❡ t♦ ✈✐s✐t t❤❡ s♦❧❡ ❙▼ ✭x̄1 ❞❡❝r❡❛s❡s✮✳ ❚♦ ❜❡ ♣r❡❝✐s❡✱
✉s✐♥❣ ✭✶✸✮ ❛♥❞ ✭✶✹✮✱ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t
dx̄0
❛♥❞
dx̄1
❛r❡ ❜♦t❤ ♥❡❣❛t✐✈❡✳ ❆s ❛ r❡s✉❧t✱ t❤❡ s❡❝♦♥❞
❛♥❞ t❤✐r❞ t❡r♠s ♦❢ ✭✶✾✮ ❛r❡ ❛❧✇❛②s ♣♦s✐t✐✈❡✳
❲❤✐❝❤ ❡✛❡❝t ❞♦♠✐♥❛t❡s ❞❡♣❡♥❞s ♦♥ t❤❡ r❡❧❛t✐✈❡ s✐③❡ ♦❢ t❤❡ t✇♦ s❤♦♣♣✐♥❣ ❛r❡❛s ❛s ✇❡❧❧ ❛s ♦♥
t❤❡ ✈❛❧✉❡ ♦❢
T✳
❚❤✐s ✐♥t❡r❛❝t✐♦♥ ❞✐st✐♥❣✉✐s❤❡s ♦✉r ♠♦❞❡❧ ❢r♦♠ t❤❡ ❡①✐st✐♥❣ ❧✐t❡r❛t✉r❡ ✐♥ ✇❤✐❝❤
✐♥❞✐✈✐❞✉❛❧ ❞❡♠❛♥❞s ❛r❡ ♣❡r❢❡❝t❧② ✐♥❡❧❛st✐❝ ✭❙❝❤✉❧③ ❛♥❞ ❙t❛❤❧✱ ✶✾✾✻❀ ●❡❤r✐❣✱ ✶✾✾✽❀ ❙♠✐t❤ ❛♥❞ ❍❛②✱
✷✵✵✺✮✳ ❍❡r❡ ✐♥❝r❡❛s✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✜r♠s ✐♥ ❛ s❤♦♣♣✐♥❣ ❝❡♥t❡r ❛✛❡❝ts ❛ s❡❧❧❡r✬s ❞❡♠❛♥❞ t❤r♦✉❣❤
t❤❡ ♥✉♠❜❡r ♦❢ ❝✉st♦♠❡rs
❛♥❞
t❤❡✐r ✐♥❞✐✈✐❞✉❛❧ ❝♦♥s✉♠♣t✐♦♥✳
❚❤r❡❡ ❝❛s❡s ♠❛② ❛r✐s❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✈❛❧✉❡ ♦❢
ν✳
✶✳ ❈♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❙▼ ✐s s♠❛❧❧✱ ❛♥❞ t❤✉s
❛ st♦r❡ ✐s ❣✐✈❡♥ ❜②
DS (ν) =
❜❡❝❛✉s❡
ν < 1/(T − 1) < 1✳
❚❤❡ ❞❡♠❛♥❞ ❢❛❝❡❞ ❜②
1 1 − x̄0
.
n 1+ν
❉✐✛❡r❡♥t✐❛t✐♥❣ t❤✐s ❡①♣r❡ss✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦
n
0 < x̄0 < x̄1 = 1✳
ν
❛♥❞ ✉s✐♥❣ ✭✶✸✮✱ ✇❡ ♦❜t❛✐♥
dDS
1 − ln(1 + ν)
>0
=
dν
(1 + ν)2 ln T
■♥ t❤✐s ❝❛s❡✱ t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t ❞♦♠✐♥❛t❡s t❤❡ ❝♦♠♣❡t✐t✐♦♥
❡✛❡❝t✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ t❤❡ ❜♦✉♥❞❛r②
x̄0 ✐s ✈❡r② s❡♥s✐t✐✈❡ t♦ ❛♥ ✐♥❝r❡❛s❡ ✐♥ ν ✱ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ♠❛r❦❡t
❡①♣❛♥s✐♦♥ ❡✛❡❝t str♦♥❣❡r✱ ✇❤❡r❡❛s t❤❡r❡ ❛r❡ ♦♥❧② ❛ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ❝♦♥s✉♠❡rs ✈✐s✐t✐♥❣ t❤❡ ❙▼✱
❛♥❞ t❤✉s t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝t ✐s ✇❡❛❦✳ ❍❡♥❝❡✱ ✇❤❡♥ t❤❡ ❙▼ ✐s s♠❛❧❧✱ t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t
✐s s✉✣❝✐❡♥t❧② str♦♥❣ t♦ s❤✐❢t ✉♣✇❛r❞s t❤❡ ❞❡♠❛♥❞ ❢♦r t❤❡ ✈❛r✐❡t✐❡s s✉♣♣❧✐❡❞ ❜② t❤❡ ✐♥❝✉♠❜❡♥ts✳ ❆s
❛ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡s❡ st♦r❡s ❡❛r♥ ❤✐❣❤❡r ♣r♦✜ts ✇❤❡♥ t❤❡② ❢❛❝❡ ♠♦r❡ ❝♦♠♣❡t✐t✐♦♥ ✇✐t❤✐♥ t❤❡ ❙▼✳
■♥ ♦t❤❡r ✇♦r❞s✱
t❤❡r❡ ✐s ♣r♦✜t✲✐♥❝r❡❛s✐♥❣ ❝♦♠♣❡t✐t✐♦♥✳
✷✳ ❲❡ ♥♦✇ ❝♦♠❡ t♦ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥
0 < x̄0 < x̄1 < 1✱
✇❤✐❝❤ ❛r✐s❡s ✇❤❡♥ t❤❡ ❙▼ ✐s ❧❛r❣❡r ❜✉t
♥♦t t♦♦ ❧❛r❣❡✳ ❚✇♦ s✉❜❝❛s❡s ♠❛② ❛r✐s❡✳ ■♥ t❤❡ ✜rst ♦♥❡✱ ✇❡ ❤❛✈❡
✶✸
T ≤ 2.39✳
❲❡ s❤♦✇ ✐♥ ❆♣♣❡♥❞✐① ✶
t❤❛t t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t st✐❧❧ ❞♦♠✐♥❛t❡s t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝t ❢♦r ❛❧❧ 1/(T −1) < ν < T −1✳
❚❤❡ ✐♥t✉✐t✐♦♥ ❜❡❤✐♥❞ t❤✐s r❡s✉❧t ✐s t❤❛t ✉♥❞❡r ❛ ❤✐❣❤ ❞❡❣r❡❡ ♦❢ ♣r♦❞✉❝t ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞✴♦r ❧♦✇
tr❛✈❡❧ ❝♦sts✱ ❜♦t❤ t❤❡ ❜♦✉♥❞❛r✐❡s x̄0 ❛♥❞ x̄1 ❛r❡ ✈❡r② s❡♥s✐t✐✈❡ t♦ ❛♥ ✐♥❝r❡❛s❡ ✐♥ ν ✱ t❤✉s ❣❡♥❡r❛t✐♥❣
❛ str♦♥❣ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t✳
■♥ t❤❡ s❡❝♦♥❞ s✉❜❝❛s❡✱ T ❡①❝❡❡❞s 2.39✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✈❛r✐❡t✐❡s ❛r❡ ❧❡ss ❞✐✛❡r❡♥t✐❛t❡❞ ❛♥❞✴♦r
tr❛✈❡❧ ❝♦sts ❛r❡ ❤✐❣❤❡r✳ ❲❡ s❤♦✇ ✐♥ ❆♣♣❡♥❞✐① ✶ t❤❛t t❤❡r❡ ❡①✐sts ❛ t❤r❡s❤♦❧❞ ν̄ ∈ [1/(T − 1), T − 1]
s✉❝❤ t❤❛t t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t ❦❡❡♣s ❞♦♠✐♥❛t✐♥❣ t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝t ♣r♦✈✐❞❡❞ t❤❛t ν < ν̄ ✳
❖t❤❡r✇✐s❡✱ r❛✐s✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ st♦r❡s ♠❛❦❡s t❤❡ ✐♥❝✉♠❜❡♥ts ✇♦rs❡ ♦✛✳ ■♥❞❡❡❞✱ t❤❡ r❛♥❣❡ ♦❢
✈❛r✐❡t✐❡s ♣r♦✈✐❞❡❞ ❜② t❤❡ ❙▼ ❜❡✐♥❣ ❧❛r❣❡r✱ t❤❡ ♠❛r❣✐♥❛❧ ✉t✐❧✐t② ♦❢ ♥❡✇ ✈❛r✐❡t✐❡s ✐s ❧♦✇❡r✳ ❲❡ ❢❛❧❧
❜❛❝❦ ❤❡r❡ ♦♥ t❤❡ st❛♥❞❛r❞ r❡s✉❧t ♦❢ ♣r♦✜t✲❞❡❝r❡❛s✐♥❣ ❝♦♠♣❡t✐t✐♦♥✳
✸✳ ■t r❡♠❛✐♥s t♦ ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ t❤❡ ❙▼ ✐s ✈❡r② ❧❛r❣❡✳ ❙✐♥❝❡ x̄0 = 0 < x̄1 < 1✱ t❤❡
❞❡♠❛♥❞ ❢❛❝❡❞ ❜② ❛ st♦r❡ ❧♦❝❛t❡❞ ✐♥ t❤❡ ❙▼ ✐s ♥♦✇ ❣✐✈❡♥ ❜②
1
DS (ν) =
n
x̄1
1 − x̄1
+
ν
1+ν
.
✭✷✵✮
❆❣❛✐♥✱ t✇♦ s✉❜❝❛s❡s ♠❛② ❛r✐s❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✈❛❧✉❡ ♦❢ T ✳ ❋✐rst✱ ✇❤❡♥ T ≤ 2.06✱ ν̂ > T − 1
❡①✐sts s✉❝❤ t❤❛t t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t ❞♦♠✐♥❛t❡s t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝t ❢♦r ❛❧❧ ✈❛❧✉❡s ♦❢
ν ∈]T − 1, ν̂[✳ ❙❡❝♦♥❞✱ ✇❤❡♥ T > 2.06✱ t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝ts ❜❡❝♦♠❡s str♦♥❣❡r t❤❛♥ t❤❡ ♠❛r❦❡t
❡①♣❛♥s✐♦♥ ❡✛❡❝t ❢♦r ❛❧❧ ν > T − 1 ✭s❡❡ ❆♣♣❡♥❞✐① ✶✮✳
❚❤❡ ♥❡①t ♣r♦♣♦s✐t✐♦♥ s✉♠♠❛r✐③❡s t❤❡ ❛❜♦✈❡ r❡s✉❧ts✳
Pr♦♣♦s✐t✐♦♥ ✶✳ ❆ss✉♠❡ t❤❛t n ✐s ✜①❡❞✳ ❚❤❡♥✱ ✐❢ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧ ✐s ♥♦t t♦♦ ❜✐❣ r❡❧❛t✐✈❡❧②
t♦ t❤❡ ♥✉♠❜❡r ♦❢ r❡t❛✐❧❡rs ❡st❛❜❧✐s❤❡❞ ❛t t❤❡ ❙❙❚✱ ❛ st♦r❡✬s ♣r♦✜t ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢
❝♦♠♣❡t✐t♦rs✳ ■♥ ❝♦♥tr❛st✱ t❤❡ ♣r♦✜ts ❡❛r♥❡❞ ❜② ❛ st♦r❡ ❞❡❝r❡❛s❡ ✇✐t❤ N ✇❤❡♥ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧ ✐s
❜✐❣ ❡♥♦✉❣❤✳
■♥ ♦t❤❡r ✇♦r❞s✱ ❛ st♦r❡✬s ♣r♦✜t ❢✉♥❝t✐♦♥ πS (n, N ) ✐s ✉♥✐♠♦❞❛❧ ✐♥ N ✳ ■♥ ❝♦♥tr❛st✱ r❛✐s✐♥❣ t❤❡
♥✉♠❜❡r ♦❢ st♦r❡s ✐♥ t❤❡ ❙▼ ❞❡❝r❡❛s❡s t♦t❛❧ ❡①♣❡♥❞✐t✉r❡ ✐♥ t❤❡ ❙❙❚✱ t❤✉s ✐♠♣❧②✐♥❣ t❤❛t ❛❞❞✐♥❣
st♦r❡s t♦ t❤❡ ❙▼ ✐s ❛❧✇❛②s ❞❡tr✐♠❡♥t❛❧ t♦ t❤❡ r❡t❛✐❧❡rs✳ ❚❤❡r❡❢♦r❡✱ ❛ r❡t❛✐❧❡r✬s ♣r♦✜t ❢✉♥❝t✐♦♥
πR (n, N ) ❞❡❝r❡❛s❡s ✇✐t❤ N ✳ ■t ❢♦❧❧♦✇s ❢r♦♠ ✭✶✼✮ t❤❛t πR (n, N ) ✐s ✉♥✐♠♦❞❛❧ ✐♥ n ✇❤✐❧❡ πS (n, N )
❞❡❝r❡❛s❡s ✇✐t❤ n ✇❤❡♥ N ✐s ✜①❡❞✳
❚❤❡ st❛♥❞❛r❞ t❤♦✉❣❤t ❡①♣❡r✐♠❡♥t ♦❢ s♣❛t✐❛❧ ❝♦♠♣❡t✐t✐♦♥ t❤❡♦r② ✐s t♦ st✉❞② t❤❡ ✐♠♣❛❝t ♦❢ ❜❡tt❡r
tr❛♥s♣♦rt❛t✐♦♥ ❢❛❝✐❧✐t✐❡s ♦♥ t❤❡ ♠❛r❦❡t ♦✉t❝♦♠❡✳ ❇❡❝❛✉s❡ ♦❢ s②♠♠❡tr②✱ ✇❡ r❡str✐❝t ♦✉rs❡❧✈❡s t♦ t❤❡
❝❛s❡ ✐♥ ✇❤✐❝❤ t❤❡ ❙❙❚ ✐s s♠❛❧❧❡r t❤❛♥ t❤❡ ❙▼ ✭n < N ✮✳ ❲❤❡♥ n < N/(T − 1)✱ t❤❡ ❙❙❚ ❛ttr❛❝ts
♠♦r❡ ❝✉st♦♠❡rs ♣✉r❝❤❛s✐♥❣ t❤❡ ✇❤♦❧❡ ❛rr❛② ♦❢ ✈❛r✐❡t✐❡s✱ ✇❤✐❧❡ t❤❡ ❙▼ r❡t❛✐♥s t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢
❝✉st♦♠❡rs✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ r❡t❛✐❧❡rs ❜❡♥❡✜t ❢r♦♠ ❧♦✇❡r tr❛✈❡❧ ❝♦sts✳ ❚❤❡ ♣❛tt❡r♥ ❝❤❛♥❣❡s ✇❤❡♥
n ❡①❝❡❡❞s N/(T − 1)✳ ❖✇✐♥❣ t♦ t❤❡ ❜❡tt❡r ❛❝❝❡ss✐❜✐❧✐t② ❢r♦♠ ❡✈❡r②✇❤❡r❡ ✐♥ t❤❡ ❝✐t②✱ ❜♦t❤ s❤♦♣♣✐♥❣
♣❧❛❝❡s ❣❛✐♥ ❝✉st♦♠❡rs✳ ❍♦✇❡✈❡r✱ ❜❡❝❛✉s❡ n < N ✱ t❤♦s❡ ✇❤♦ ♣❛tr♦♥✐③❡❞ t❤❡ ❙❙❚ ❜✉t ♥♦✇ s❤♦♣ ✐♥
❜♦t❤ ♣❧❛❝❡s s♣❡♥❞ ♠♦r❡ ✐♥ t❤❡ ❙▼ t❤❛♥ ✐♥ t❤❡ ❙❙❚✳ ■♥ ❝♦♥tr❛st✱ t❤♦s❡ ✇❤♦ ✉s❡❞ t♦ ❜✉② ❢r♦♠ t❤❡
❙▼ ♦♥❧② ❦❡❡♣ s♣❡♥❞✐♥❣ ♠♦r❡ ✐♥ t❤❡ ❙▼✳ ❆♣♣❡♥❞✐① ✷ s❤♦✇s t❤❛t✱ ✐♥ t❤✐s ❡✈❡♥t✱ ❧♦✇❡r tr❛✈❡❧ ❝♦sts
✶✹
♠❛❦❡ t❤❡ r❡t❛✐❧❡rs ✇♦rs❡ ♦✛✳ ❚❤❡r❡❢♦r❡✱
✐s ♥♦♥✲♠♦♥♦t♦♥✐❝✳
t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ r❡t❛✐❧❡rs✬ ♣r♦✜ts ❛♥❞ tr❛✈❡❧ ❝♦sts
❚❤✐s ✐s t♦ ❜❡ ❝♦♥tr❛st❡❞ ✇✐t❤ st❛♥❞❛r❞ ♠♦❞❡❧s ♦❢ s♣❛t✐❛❧ ❝♦♠♣❡t✐t✐♦♥ ✇❤❡r❡
❧♦✇❡r✐♥❣ tr❛✈❡❧ ❝♦sts ✐s ❞❡tr✐♠❡♥t❛❧ t♦ ❛❧❧ s❡❧❧❡rs✳
❚❤❡ ❛❜♦✈❡ ❛♥❛❧②s✐s ❤❛s ❤✐❣❤❧✐❣❤t❡❞ t❤❡ ♠❛r❦❡t ❡✛❡❝ts ❛t ✇♦r❦ ✐♥ t❤❡ ♣r♦❝❡ss ♦❢ ❝♦♠♣❡t✐t✐♦♥
❜❡t✇❡❡♥ s♣❛t✐❛❧❧② s❡♣❛r❛t❡❞ s❤♦♣♣✐♥❣ ♣❧❛❝❡s✳ ❲♦r❦✐♥❣ ✇✐t❤ ❡①♦❣❡♥♦✉s ♥✉♠❜❡rs ♦❢ r❡t❛✐❧❡rs ❛♥❞
st♦r❡s ❝❛♥ ❜❡ ❥✉st✐✜❡❞ ♦♥ t❤❡ ❣r♦✉♥❞ t❤❛t t❤❡ s✐③❡s ♦❢ t❤❡ ❙❙❚ ❛♥❞ ❙▼ ♠❛② ❜❡ ❞❡t❡r♠✐♥❡❞ t❤r♦✉❣❤
❞✐✛❡r❡♥t ✐♥st✐t✉t✐♦♥❛❧ ♠❡❝❤❛♥✐s♠s✳ ❋♦r ❡①❛♠♣❧❡✱ ❧♦❝❛❧ ♣♦❧✐❝✐❡s ♠❛② r❡str✐❝t r❡t❛✐❧✐♥❣ ❜② ❛❧❧♦❝❛t✐♥❣
❧❛♥❞ t♦ ♦✣❝❡s✱ ❤♦✉s✐♥❣✱ ❛♥❞ tr❛♥s♣♦rt ❢❛❝✐❧✐t✐❡s✱ ✇❤❡r❡❛s ❧❛✇s ♦r ❛♥t✐tr✉st ❛✉t❤♦r✐t✐❡s ♠❛② ✐♠♣♦s❡
❧✐♠✐ts ♦♥ t❤❡ s✐③❡ ♦❢ s❤♦♣♣✐♥❣ ♠❛❧❧s✳
✹
❚❤❡ ❙✐③❡ ♦❢ ❙❤♦♣♣✐♥❣ ❈❡♥t❡rs
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❞❡t❡r♠✐♥❡ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s✐③❡ ♦❢ t❤❡ s❤♦♣♣✐♥❣ ♣❧❛❝❡s✱ ✇❤✐❝❤ ❤❛✈❡ ❞✐✛❡r❡♥t
♦r❣❛♥✐③❛t✐♦♥❛❧ ❢♦r♠s✳ ❚❤❡ t✉r♥♦✈❡r ♦❢ s♠❛❧❧ r❡t❛✐❧❡rs ❜❡✐♥❣ ❤✐❣❤✱ ❢r❡❡ ❡♥tr② ❛♥❞ ❡①✐t ♣r❡✈❛✐❧s ✐♥ t❤❡
❙❙❚✳ ■♥ ❝♦♥tr❛st✱ t❤❡ ❙▼ ✐s ❜✉✐❧t ❜② ❛ ♣r♦✜t✲♠❛①✐♠✐③✐♥❣
t❤❛t s❡tt❧❡ t❤❡r❡✐♥✳
❞❡✈❡❧♦♣❡r
✇❤♦ ❝❤❛r❣❡s ❛ ❢❡❡ t♦ t❤❡ st♦r❡s
❚❤✉s✱ t❤❡ ❞❡✈❡❧♦♣❡r ✐♥t❡r♥❛❧✐③❡s t❤❡ ❜❡♥❡✜ts ❛ss♦❝✐❛t❡❞ ✇✐t❤ s✐③❡✱ ✇❤❡r❡❛s
r❡t❛✐❧❡rs ♠❛①✐♠✐③❡ t❤❡✐r ♦✇♥ ♣r♦✜ts ✇❤✐❧❡ ♥❡❣❧❡❝t✐♥❣ t❤❡ ✐♠♣♦rt❛♥❝❡ t♦ ❛❝t ♦♥ ❛♥ ❛❣❣r❡❣❛t❡ ❧❡✈❡❧✳
■♥❞❡♣❡♥❞❡♥t ♣r♦✜t✲♠❛①✐♠✐③✐♥❣ st♦r❡s ❛r❡ ❢r❡❡ t♦ ❡♥t❡r ❜② ♣❛②✐♥❣ ❛ ♣❡r s❧♦t ♣r✐❝❡ t♦ t❤❡ ❞❡✲
✈❡❧♦♣❡r✳ ❇❡❝❛✉s❡ ❡❛❝❤ st♦r❡ ✐s s♠❛❧❧ ✭♣r❡❝✐s❡❧②✱ ♦❢ ♠❡❛s✉r❡ ③❡r♦✮✱ ✐t ✐s ♥❛t✉r❛❧ t♦ ❛ss✉♠❡ t❤❛t t❤❡
❞❡✈❡❧♦♣❡r ❤❛s t❤❡ ✇❤♦❧❡ ❜❛r❣❛✐♥✐♥❣ ♣♦✇❡r ♦✈❡r t❤❡ ♣❡r s❧♦t ♣r✐❝❡✱ ❛♥❞ t❤✉s ❝❤♦♦s❡s
φ t♦ ♠❛①✐♠✐③❡s
✐ts ♣r♦✜ts ❣✐✈❡♥ ❜②
Π ≡ φ N − B(N D )
✇❤❡r❡
B(N D )
✭✷✶✮
❞❡♥♦t❡s t❤❡ ❞❡✈❡❧♦♣❡r✬s ❝♦st ♦❢ ❜✉✐❧❞✐♥❣ ❛♥ ❙▼ ♦❢ s✐③❡
♦❢ st♦r❡s ❝❤♦♦s✐♥❣ t♦ s❡t ✉♣ ✐♥ t❤❡ ❙▼ ❛♥❞ ♣❛② t❤❡ ❢❡❡
φ✳
N D✱
✇❤✐❧❡
N
✐s t❤❡ ♥✉♠❜❡r
❚❤♦✉❣❤ ❛❞♠✐tt❡❞❧② s✐♠♣❧❡✱ t❤✐s t②♣❡
♦❢ ❝♦♥tr❛❝t ✐s s✉✣❝✐❡♥t ❢♦r ✉s t♦ s❤♦✇ ♦✉r ♠❛✐♥ r❡s✉❧ts✳ ❚♦ ❜❡ s✉r❡✱ ♠♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❝♦♥tr❛❝ts
❜❡t✇❡❡♥ t❤❡ ❞❡✈❡❧♦♣❡r ❛♥❞ st♦r❡s✱ ✇❤✐❝❤ ✇♦✉❧❞ ❛❧❧♦✇ t❤❡ ❞❡✈❡❧♦♣❡r t♦ ❛❝t ❛s ❛ s✉♣❡r♠❛r❦❡t✱ ❝♦✉❧❞
❜❡ ❝♦♥s✐❞❡r❡❞✳ ❍♦✇❡✈❡r✱ ✐♥✈❡st✐❣❛t✐♥❣ s✉❝❤ ✐ss✉❡s ✇♦✉❧❞ t❛❦❡ ✉s ❢❛r ❢r♦♠ t❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s
♣❛♣❡r✳
■t ✐s ❤✐st♦r✐❝❛❧❧② ✇❡❧❧ ❞♦❝✉♠❡♥t❡❞ t❤❛t ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs ❤❛✈❡ ❜❡❡♥ ❛❝t✐✈❡ ❧♦♥❣ ❜❡❢♦r❡ t❤❡
❛♣♣❡❛r❛♥❝❡ ♦❢ s✉❜✉r❜❛♥ ♠❛❧❧s ✭❈♦❤❡♥✱ ✶✾✾✻❀ ❋♦❣❡❧s♦♥✱ ✷✵✵✺✮✳ ❚❤✉s✱ ✇❡ ❛ss✉♠❡ t❤❛t✱
♣r✐♦r t♦ t❤❡
❡♥tr② ♦❢ ❛ ❞❡✈❡❧♦♣❡r✱ t❤❡ ❙❙❚ ❤♦sts ne ≡ (1 − ρ)/f r❡t❛✐❧❡rs✱ t❤❛t ✐s✱ t❤❡ ♥✉♠❜❡r ♦❢ ✜r♠s ♣r❡✈❛✐❧✐♥❣
✉♥❞❡r ❢r❡❡ ❡♥tr② ✐♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❛ s❤♦♣♣✐♥❣ ♠❛❧❧✳ ❚❤❡ ❡♥s✉✐♥❣ ♠❛r❦❡t ♣r♦❝❡ss ✐s t❤❡♥ ❞❡s❝r✐❜❡❞
❜② t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡✲st❛❣❡ ❣❛♠❡✳ ❚❤❡ ❞❡✈❡❧♦♣❡r✱ st♦r❡s ❛♥❞ r❡t❛✐❧❡rs ❛r❡ ✐♥✈♦❧✈❡❞ ✐♥ ❛ str❛t❡❣✐❝
❡♥✈✐r♦♥♠❡♥t ✐♥✈♦❧✈✐♥❣ ♦♥❡ ✏❧❛r❣❡✑ ♣❧❛②❡r ❛♥❞ ❛ ❝♦♥t✐♥✉✉♠ ♦❢ ✏s♠❛❧❧✑ ♣❧❛②❡rs✳ ❚❤❡ t✐♠✐♥❣ ♦❢ t❤❡
N D ♦❢ t❤❡ ❙▼ ❛♥❞ t❤❡ ♣❡r
N D = 0✱ t❤❡ ❞❡✈❡❧♦♣❡r ❞♦❡s
❣❛♠❡ ✐s ❛s ❢♦❧❧♦✇s✳ ■♥ t❤❡ ✜rst st❛❣❡✱ t❤❡ ❞❡✈❡❧♦♣❡r ❝❤♦♦s❡s t❤❡ s✐③❡
s❧♦t ♣r✐❝❡
φ
❤❡ ❝❤❛r❣❡s t♦ t❤❡ st♦r❡s t❤❛t s❡t ✉♣ ✐♥ t❤❡ ❙▼✳ ❲❤❡♥
♥♦t ❡♥t❡r✳ ■♥ t❤❡ s❡❝♦♥❞ st❛❣❡✱ ♦✉t ♦❢ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ♣♦t❡♥t✐❛❧ ✜r♠s ✭❢♦r♠❛❧❧②✱ ❛ ❝♦♥t✐♥✉✉♠✮✱
✶✺
s♦♠❡ ❞❡❝✐❞❡ t♦ ❜✉②✴r❡♥t ❛ s❧♦t ✐♥ t❤❡ ❙▼ ✇❤❡♥ N D > 0✱ s♦♠❡ ♦t❤❡rs ❝❤♦♦s❡ t♦ st❛② ✐♥ t❤❡ ❙❙❚✱
✇❤❡r❡❛s t❤❡ r❡♠❛✐♥✐♥❣ ✜r♠s ❛r❡ ♦✉t ♦❢ ❜✉s✐♥❡ss✳ ▲❛st✱ ✐♥ t❤❡ t❤✐r❞ st❛❣❡✱ ❣✐✈❡♥ t❤❡ ❙❙❚ ❛♥❞ ❙▼
s✐③❡s✱ r❡t❛✐❧❡rs ❛♥❞ st♦r❡s ❝♦♠♣❡t❡ ✐♥ ♣r✐❝❡✳
❙❡✈❡r❛❧ r❡❛s♦♥s ❥✉st✐❢② t❤✐s st❛❣✐♥❣✳ ❋✐rst✱ t❤❡ ❞❡✈❡❧♦♣❡r ♥❡❝❡ss❛r✐❧② ❝♦♠♠✐ts t♦ ❛ ❝❡rt❛✐♥ s✐③❡
✇❤❡♥ ✐t ❜✉✐❧❞s ❛♥ ❙▼✳ ❙❡❝♦♥❞✱ t❤❡ ❛❝t✉❛❧ ♥✉♠❜❡r ♦❢ st♦r❡s ✐♥ t❤❡ ❙▼ ❞❡♣❡♥❞s ♦♥ ❤♦✇ ♠✉❝❤ t❤❡②
❤❛✈❡ t♦ ♣❛② ❢♦r ❛ s❧♦t✳ ❚❤✐r❞✱ t❤❡ ❞❡✈❡❧♦♣❡r ✉♥❞❡rst❛♥❞s t❤❛t t❤❡ ❞❡❝✐s✐♦♥s N D ❛♥❞ φ ❛r❡ ❝❧♦s❡❧②
❧✐♥❦❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ ❜✉✐❧❞✐♥❣ ❛ s♠❛❧❧ ❝❛♣❛❝✐t② ❛♥❞ ❝❤❛r❣✐♥❣ ❛ ❧♦✇ ❢❡❡ ❛r❡ ✐♥❝♦♥s✐st❡♥t ❜❡❝❛✉s❡ t❤❡
❙▼ ❝❛♥♥♦t ❛❝❝♦♠♠♦❞❛t❡ t❤❡ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ st♦r❡s ❛ttr❛❝t❡❞ ❜② ❛ ❧♦✇ ❢❡❡✳ ❆s ❛ r❡s✉❧t✱ ✇❡ ✜♥❞ ✐t
r❡❛s♦♥❛❜❧❡ t♦ ❛ss✉♠❡ t❤❛t t❤❡ ❞❡✈❡❧♦♣❡r ❝❤♦♦s❡s N D ❛♥❞ φ s✐♠✉❧t❛♥❡♦✉s❧② ❛t t❤❡ ✜rst st❛❣❡✳ ▲❛st✱
r❡t❛✐❧❡rs ❛♥❞ st♦r❡s ♠♦✈❡ t♦❣❡t❤❡r ❜❡❝❛✉s❡ t❤❡② ❞✐s♣❧❛② ❛ s✐♠✐❧❛r ✢❡①✐❜✐❧✐t② ✐♥ t❤❡✐r ✐♥✈❡st♠❡♥t ❛♥❞
♣r✐❝❡ ❞❡❝✐s✐♦♥s✳ ❚❤❡✐r ♣❛②♦✛s ❛r❡ ❣✐✈❡♥ r❡s♣❡❝t✐✈❡❧② ❜② ✭✶✺✮ ❛♥❞ ✭✶✻✮✳ ❙✐♥❝❡ ❍♦t❡❧❧✐♥❣ ✭✶✾✷✾✮✱ t❤❡
s❡q✉❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ ♣r✐❝❡ st❛❣❡s ✐s st❛♥❞❛r❞ ✐♥ s♣❛t✐❛❧ ❝♦♠♣❡t✐t✐♦♥ ♠♦❞❡❧s✳
❲❡ s❡❡❦ ❛ s✉❜❣❛♠❡ ♣❡r❢❡❝t ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ s♦❧✈❡ t❤❡ ❣❛♠❡ ❜② ❜❛❝❦✇❛r❞ ✐♥❞✉❝t✐♦♥✳ ❲❡
❤❛✈❡ s❡❡♥ t❤❛t t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡s ❝❤♦s❡♥ ✐♥ t❤❡ t❤✐r❞ st❛❣❡ ❛r❡ ❣✐✈❡♥ ❜② ✭✺✮✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱
✇❡ ❞❡s❝r✐❜❡ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♦✉t❝♦♠❡ ♦❢ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ st❛❣❡s✳
✹✳✶
❚❤❡ s✐③❡ ♦❢ t❤❡ ❙❙❚ ✉♥❞❡r ❢r❡❡ ❡♥tr②
❘❡t❛✐❧❡rs ❜❡❛r ❛♥ ❡①♦❣❡♥♦✉s ❡♥tr② ❝♦st f ✱ ✇❤✐❝❤ ♠❛② ❜❡ ❛ ✜①❡❞ ♣r♦❞✉❝t✐♦♥ ❝♦st ♦r ❛ ❧✉♠♣✲s✉♠ t❛①
t♦ ❜❡ ♣❛✐❞ t♦ t❤❡ ❝✐t② ❣♦✈❡r♥♠❡♥t t♦ s❡t ✉♣ ✐♥ t❤❡ ❙❙❚✳ ❋♦r ❛♥② ❣✐✈❡♥ N ✱ ❛ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠
n∗ (N ) ❛r✐s❡s ✇❤❡♥ t❤❡ ③❡r♦✲♣r♦✜t ❝♦♥❞✐t✐♦♥
πR (n, N ) = f
✭✷✷✮
❤♦❧❞s✳ ❆ss✉♠❡ t❤❛t N D ✐s ❧❛r❣❡ ❡♥♦✉❣❤ ❢♦r st♦r❡s t♦ ❡♥t❡r t❤❡ ❙▼ ✉♥t✐❧ t❤❡✐r ♦♣❡r❛t✐♥❣ ♣r♦✜ts
❡q✉❛❧ t❤❡ ♣❡r s❧♦t ♣r✐❝❡ φ✳ ❯♥❞❡r t❤❡s❡ ❝✐r❝✉♠st❛♥❝❡s✱ t❤❡ ♥✉♠❜❡r N ∗ (n) ♦❢ st♦r❡s ✐♥ t❤❡ ❙▼
s❛t✐s✜❡s t❤❡ ③❡r♦✲♣r♦✜t ❝♦♥❞✐t✐♦♥
πS (n, N ) = φ.
✭✷✸✮
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♥✉♠❜❡r ♦❢ st♦r❡s ✐s ❞❡t❡r♠✐♥❡❞ ❛s ✐❢ ✜r♠s ✇❡r❡ t♦ ♦♣❡r❛t❡ ✉♥❞❡r ♠♦♥♦♣♦❧✐st✐❝
❝♦♠♣❡t✐t✐♦♥ ✇❤✐❧❡ ❢❛❝✐♥❣ t❤❡ ✜①❡❞ ❝♦st φ s❡t ❜② t❤❡ ❞❡✈❡❧♦♣❡r✳ ❚❤❡ ❢✉♥❝t✐♦♥s πR (n, N ) ❛♥❞ πS (n, N )
❜❡✐♥❣ ❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ −1✱ πR (µ, 1) ❛♥❞ πS (1, ν) ❛r❡ ❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ 0✳ ❙✐♥❝❡ t❤❡
❢✉♥❝t✐♦♥s πS (1, ·) ❛♥❞ πR (·, 1) ❛r❡ ✐❞❡♥t✐❝❛❧✱ N ∗ (n) ✐s t❤❡ ♠✐rr♦r ✐♠❛❣❡ ♦❢ n∗ (N )✳ ❆♥ ❡q✉✐❧✐❜r✐✉♠
♦❢ ❛ s❡❝♦♥❞✲st❛❣❡ s✉❜❣❛♠❡ ✐s t❤✉s ❛♥ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥t ♦❢ t❤❡ t✇♦ ❝✉r✈❡s n∗ (N ) ❛♥❞ N ∗ (n)✳
❇❡❝❛✉s❡ ♦❢ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♥❡t✇♦r❦ ❡✛❡❝ts✱ t❤❡r❡ ❡①✐st s❡✈❡r❛❧ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐❛✳ ❋✐rst✱ ❛s
s❤♦✇♥ ❜② ❋✐❣✉r❡ ✷✱ t❤❡r❡ ❛r❡ t✇♦ ❡q✉✐❧✐❜r✐❛ s✉❝❤ t❤❛t n∗ (N ) ✐s str✐❝t❧② ♣♦s✐t✐✈❡✳ ❍♦✇❡✈❡r✱ ❛
❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠ ♥❡❡❞ ♥♦t ❜❡ st❛❜❧❡ ❜❡❝❛✉s❡ t❤❡ ❡♥tr② ✭❡①✐t✮ ♦❢ ❛♥ ❛r❜✐tr❛r✐❧② s♠❛❧❧ ♠❛ss
♦❢ r❡t❛✐❧❡rs ♠❛② tr✐❣❣❡r ❛ ♣r♦✜t ❤✐❦❡ ✭❞r♦♣✮✱ ❤❡♥❝❡ t❤❡ ❡♥tr② ✭❡①✐t✮ ♦❢ ♥❡✇ ✭❡①✐st✐♥❣✮ s❡❧❧❡rs✳ ❆
✶✻
❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠ ✐s st❛❜❧❡ ✐❢ t❤❡ ❡♥tr② ♦❢ ❛❞❞✐t✐♦♥❛❧ s❡❧❧❡rs r❡❞✉❝❡s ♣r♦✜ts ❜❡❧♦✇ ③❡r♦✿
∂πR ∗
(n , N ) < 0.
∂n
❍♦✇ ❞♦❡s t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♥✉♠❜❡r ♦❢ r❡t❛✐❧❡rs n∗ (N ) ✈❛r② ✇✐t❤ t❤❡ s✐③❡ ♦❢ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧❄
❈❧❡❛r❧②✱ ✇❡ ❤❛✈❡
∂πR ∂πR dn∗
+
= 0.
∂N
∂n dN
❚❤❡ ✜rst t❡r♠ ♦❢ t❤✐s ❡①♣r❡ss✐♦♥ ✐s ♥❡❣❛t✐✈❡ ❜❡❝❛✉s❡ t♦t❛❧ ❡①♣❡♥❞✐t✉r❡ ✐♥ t❤❡ ❙❙❚ ❛❧✇❛②s
❞❡❝r❡❛s❡s ✇✐t❤ N ✭s❡❡ ❙❡❝t✐♦♥ ✸✮✳ ❚❤❡r❡❢♦r❡✱ dn∗ /dN < 0 ✭> 0✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ n∗ (N ) ✐s ❛ st❛❜❧❡
✭✉♥st❛❜❧❡✮ ❡q✉✐❧✐❜r✐✉♠✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♥✉♠❜❡r ♦❢ r❡t❛✐❧❡rs ❞❡❝r❡❛s❡s ✭✐♥❝r❡❛s❡s✮
✇✐t❤ t❤❡ s✐③❡ ♦❢ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧ ✇❤❡♥ t❤❡ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠ ✐s st❛❜❧❡ ✭✉♥st❛❜❧❡✮✳✻
❯s✐♥❣ ✭✶✸✮ ❛♥❞ ✭✶✹✮✱ ✐t ✐s r❡❛❞✐❧② ✈❡r✐✜❡❞ t❤❛t ♥♦ ❝♦♥s✉♠❡r ♣❛tr♦♥✐③❡s t❤❡ ❙❙❚ ✇❤❡♥ N ✐s
❛r❜✐tr❛r✐❧② ❧❛r❣❡✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❦♥♦✇ t❤❛t πS (1, ν) ✐s ✉♥✐♠♦❞❛❧ ✐♥ ν ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s❡t ♦❢
❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐❛ ♠❛② ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❢♦❧❧♦✇s✳
Pr♦♣♦s✐t✐♦♥ ✷✳ ❚❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ t❤r❡s❤♦❧❞ N̄ s✉❝❤ t❤❛t ✭✐✮ ✐❢ N < N̄ ✱ t❤❡r❡ ❡①✐st t✇♦ ❢r❡❡✲
❡♥tr② ❡q✉✐❧✐❜r✐❛✱
n∗ > n∗∗ ✱
✇❤❡r❡ t❤❡ ❢♦r♠❡r ✐s st❛❜❧❡ ❛♥❞ t❤❡ ❧❛tt❡r ✉♥st❛❜❧❡ ❛♥❞ ✭✐✐✮ ✐❢
t❤❡ ❙❙❚ ✐♥✈♦❧✈❡s ♥♦ r❡t❛✐❧❡rs✳
N ≥ N̄ ✱
◆♦t❡ t❤❛t✱ ✇❤❡♥❡✈❡r N > 0✱ n∗ = 0 ✐s ❛❧s♦ ❛ st❛❜❧❡ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠ ❜❡❝❛✉s❡ t❤❡ ❙❙❚
✐s t♦♦ s♠❛❧❧ ❢♦r t❤❡ ♦♣❡r❛t✐♥❣ ♣r♦✜ts ♠❛❞❡ ❜② ❛ ❢❡✇ s❡❧❧❡rs t♦ ❝♦✈❡r t❤❡✐r ❡♥tr② ❝♦st✳ ❚❤✉s✱ ❢♦r
N < N̄ ✱ t❤❡r❡ ❡①✐st t❤r❡❡ ❡q✉✐❧✐❜r✐❛✱ t✇♦ ❛r❡ st❛❜❧❡ ❛♥❞ ♦♥❡ ✐s ✉♥st❛❜❧❡✱ ❛ ❝♦♥✜❣✉r❛t✐♦♥ ✇❤✐❝❤ ✐s
t②♣✐❝❛❧ ♦❢ ♠♦❞❡❧s ✇✐t❤ ♥❡t✇♦r❦ ❡✛❡❝ts✳ ❚❤❡ ♣❛tt❡r♥ ♦❢ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐❛ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ N ✐s
❞❡s❝r✐❜❡❞ ✐♥ ❋✐❣✉r❡ ✷✳
❋✐❣✉r❡ ✷✳ ❋r❡❡ ❡♥tr② ❡q✉✐❧✐❜r✐❛
✻ ❚❤❡
❡①✐st❡♥❝❡ ♦❢ t❤❡ ✉♥st❛❜❧❡ ❢r❡❡ ❡♥tr② ❡q✉✐❧✐❜r✐✉♠ ✐s ❛♥ ❛rt❡❢❛❝t st❡♠♠✐♥❣ ❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❛ ❝♦♥t✐♥✉✉♠
♦❢ r❡t❛✐❧❡rs✳ ■❢ ❛ r❡t❛✐❧❡r ✇❡r❡ ♦❢ ❛ ✈❡r② s♠❛❧❧ ❜✉t ♣♦s✐t✐✈❡ s✐③❡ ǫ✱ t❤❡ ✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠ ✇♦✉❧❞ ❝❡❛s❡ t♦ ❜❡ ❛ ◆❛s❤
❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤❡ ❡♥tr② ❣❛♠❡✳ ■♥ t❤✐s ❝❛s❡✱ ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ♠❛ss ♦❢ ❛❝t✐✈❡ ✜r♠s ❜② ǫ ✇♦✉❧❞ r❡♥❞❡r ♣r♦✜ts ♣♦s✐t✐✈❡✱
❛♥❞ t❤✉s ♥♦♥✲❡♥tr❛♥t r❡t❛✐❧❡rs ✇♦✉❧❞ ✜♥❞ ✐t ♣r♦✜t❛❜❧❡ t♦ ❡♥t❡r✳
✶✼
❚✇♦ ♠♦r❡ ❝♦♠♠❡♥ts ❛r❡ ✐♥ ♦r❞❡r✳ ❋✐rst✱ ✇❤❡♥ ✐ts s✐③❡ ✐s ❧❛r❣❡ ✭N ≥ N̄ ✮✱ t❤❡ ❙▼ ✐s s✉✣❝✐❡♥t❧②
❛ttr❛❝t✐✈❡ t♦ ❤♦❧❧♦✇ ♦✉t t❤❡ ❝✐t② ❝❡♥t❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❙▼ ✐s s♦ ❜✐❣ t❤❛t t❤❡ ♠❛r❣✐♥❛❧ ✉t✐❧✐t②
♦❢ t❤❡ ❛❞❞✐t✐♦♥❛❧ ✈❛r✐❡t✐❡s t❤❛t t❤❡ r❡t❛✐❧❡rs ❝♦✉❧❞ s✉♣♣❧② ✐s ❧♦✇❡r t❤❛t t❤❡ tr❛✈❡❧ ❝♦sts ❝♦♥s✉♠❡rs✱
❡✈❡♥ t❤♦s❡ ❧♦❝❛t❡❞ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ❝✐t② ❝❡♥t❡r✱ ♠✉st ❜❡❛r t♦ ❣♦ t♦ t❤❡ ❙❙❚✳ ❇② ❛♥❛❧♦❣② ✇✐t❤
t❤❡ ❝♦♥❝❡♣t ♦❢ ❧✐♠✐t ♣r✐❝❡✱ ✇❡ r❡❢❡r t♦ N̄ ❛s t❤❡ ❧✐♠✐t s✐③❡ ♦❢ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧✳ ❆s ❛ r❡s✉❧t✱ ✇❤❡♥
N < N̄ ✱ t❤❡ ❙▼ ♥❡✈❡r ❛ttr❛❝ts t❤❡ ✇❤♦❧❡ ❝✐t② ♣♦♣✉❧❛t✐♦♥ ❛♥❞ t❤❡ ❙❙❚ s✉r✈✐✈❡s ❛s ❛ ♠❛r❦❡t♣❧❛❝❡✳
■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛❧❧ ❝♦♥s✉♠❡rs t♦ ✈✐s✐t t❤❡ ❙▼✱ ✐t ♠✉st ❜❡ t❤❛t ❛❧❧ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs ❛r❡ ♦✉t
♦❢ ❜✉s✐♥❡ss✳
❙❡❝♦♥❞✱ ❛♥ ❙▼ ❤❛✈✐♥❣ t❤❡ ❧✐♠✐t s✐③❡ tr✐❣❣❡rs t❤❡ ❝♦♠♣❧❡t❡ ❛♥❞ s✉❞❞❡♥ ❞✐s❛♣♣❡❛r❛♥❝❡ ♦❢ t❤❡
❙❙❚✳ ❋♦r N < N̄ ✱ Pr♦♣♦s✐t✐♦♥ ✷ ✐♠♣❧✐❡s t❤❛t n∗ (N ) ❞❡❝r❡❛s❡s ✇✐t❤ N ✳ ❍♦✇❡✈❡r✱ n∗ (N ) ❞♦❡s ♥♦t
❞❡❝r❡❛s❡ s♠♦♦t❤❧② t♦ 0✳ ■♥❞❡❡❞✱ n∗ (N ) ❡①♣❡r✐❡♥❝❡s ❛ ❞♦✇♥✇❛r❞ ❥✉♠♣ t♦ n∗ (N̄ ) = 0✳ ❚❤❡ ✐♥t✉✐t✐♦♥
❜❡❤✐♥❞ t❤✐s r❡s✉❧t ✐s ❛s ❢♦❧❧♦✇s✳ ❲❤❡♥ N = N̄ ✱ t❤❡ r❡♠❛✐♥✐♥❣ r❡t❛✐❧❡rs ♠❛❦❡ ♥❡❣❛t✐✈❡ ♣r♦✜ts✱ ❛♥❞
t❤✉s s♦♠❡ ♦❢ t❤❡♠ ♠✉st ❡①✐t t❤❡ ♠❛r❦❡t✳ ❚❤✐s ✐♥ t✉r♥ ♠❛❦❡s t❤❡ ❙❙❚ ❧❡ss ❛ttr❛❝t✐✈❡ t♦ ❝♦♥s✉♠❡rs
s♦ t❤❛t ❢❡✇❡r ♦❢ t❤❡♠ ✈✐s✐t t❤❡ ❙❙❚✳ ❚❤✐s ✉♥r❛✈❡❧❧✐♥❣ ♣r♦❝❡ss ❦❡❡♣s ❣♦✐♥❣ ♦♥ ✉♥t✐❧ ♥♦ r❡t❛✐❧❡rs ❛r❡
✐♥ ❜✉s✐♥❡ss✳ ❋♦r s✉❝❤ ❛ ♣r♦❝❡ss t♦ ❜❡ s✉st❛✐♥❛❜❧❡✱ t❤❡ r❡❧❛t✐✈❡ s✐③❡ ♦❢ t❤❡ ❙❙❚ ♠✉st ❜❡ s✉✣❝✐❡♥t❧②
s♠❛❧❧✱ ✇❤✐❝❤ ❡①♣❧❛✐♥s ✇❤② ✐t ♦❝❝✉rs ✇❤❡♥ t❤❡ ❙▼ ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡✳
❙✐♥❝❡ N ∗ (n) ❛♥❞ n∗ (N ) ❛r❡ ♠✐rr♦r ✐♠❛❣❡s✱ Pr♦♣♦s✐t✐♦♥ ✷ ❝❛♥ ❜❡ r❡st❛t❡❞ ✐♥ t❡r♠s ♦❢ n ❛♥❞
µ = 1/ν ✱ t❤❛t ✐s✱ n̄ > 0 ❡①✐sts s✉❝❤ t❤❛t t❤❡r❡ ✐s ♥♦ ❙▼ ✐❢ t❤❡ s✐③❡ ♦❢ t❤❡ ❙❙❚ ❡①❝❡❡❞s n̄✳ ❆s ❛
r❡s✉❧t✱ ❢♦r ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs ❛♥❞ ❛ s❤♦♣♣✐♥❣ ♠❛❧❧ t♦ ❝♦❡①✐st ✇✐t❤✐♥ t❤❡ s❛♠❡ ❝✐t②✱ ♦♥❡ s❤♦♣♣✐♥❣
♣❧❛❝❡ ❝❛♥♥♦t ❜❡ ♠✉❝❤ ❧❛r❣❡r t❤❛♥ t❤❡ ♦t❤❡r✳
✹✳✷
❈♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ st♦r❡s ❛♥❞ r❡t❛✐❧❡rs
❲❡ ♥♦✇ ❞❡t❡r♠✐♥❡ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♦✉t❝♦♠❡ (n∗ , N ∗ ) ♦❢ t❤❡ s❡❝♦♥❞✲st❛❣❡ s✉❜❣❛♠❡ ❣❡♥❡r❛t❡❞ ❜②
(N D , φ)✳ ❇❡❝❛✉s❡ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ❡q✉✐❧✐❜r✐❛ ❢♦r ❛❧❧ s✉❜❣❛♠❡s ✐s ❧♦♥❣ ❛♥❞ t❡❞✐♦✉s✱ ✇❡ ✜♥❞ ✐t
❝♦♥✈❡♥✐❡♥t t♦ ♣r♦✈❡ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ r❡s✉❧t t❤❛t r✉❧❡s ♦✉t ❛ ♣r✐♦r✐ s♦♠❡ ♣❛✐rs (n, N )✳
D
❈❧❛✐♠ ✶✳ ❆t ❛♥② ♣❡r❢❡❝t ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ s✉❝❤ t❤❛t N
> 0✱ t❤❡r❡ ✐s ♥♦ ✐❞❧❡ ❝❛♣❛❝✐t②✳
∗
∗
❋✉rt❤❡r♠♦r❡✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣❡r s❧♦t ♣r✐❝❡ ✐s ❡q✉❛❧ t♦ πS (n , N )✳
✭✐✮ ❆ss✉♠❡ t❤❛t N D > N ∗ ✳ ❚❤❡♥✱ ❜② s❧✐❣❤t❧② r❡❞✉❝✐♥❣ N D ✱ t❤❡ ❞❡✈❡❧♦♣❡r s❛✈❡s ♦♥ ❜✉✐❧❞✐♥❣
❝♦sts ✇✐t❤♦✉t r❡❞✉❝✐♥❣ ❤✐s r❡✈❡♥✉❡✳ ■♥❞❡❡❞✱ N ∗ s♦❧✈❡s t❤❡ ❡q✉❛t✐♦♥ πS (n∗ , N ) = φ✱ ✇❤✐❝❤ ✐s
✐♥❞❡♣❡♥❞❡♥t ♦❢ N D ✳ ✭✐✐✮ ■❢ φ < πS (n∗ , N ∗ ) ❛♥❞ N ∗ = N D ✱ t❤❡ ❞❡✈❡❧♦♣❡r ❝♦✉❧❞ ✐♥❝r❡❛s❡ ❤✐s ♣r♦✜t
❜② ❝❤❛r❣✐♥❣ ❛ ❤✐❣❤❡r ❢❡❡ ❜❡❝❛✉s❡ st♦r❡s ❡❛r♥ ♣♦s✐t✐✈❡ ♣r♦✜ts✳ ■❢ φ > πS (n∗ , N ∗ )✱ st♦r❡s ✇♦✉❧❞ ♠❛❦❡
♥❡❣❛t✐✈❡ ♣r♦✜ts✳
❲❡ ❤❛✈❡ s❡❡♥ t❤❛t✱ ❢♦r ❛♥② N < N̄ ✱ t❤❡r❡ ❡①✐st t❤r❡❡ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐❛✳ ❲❡ ❢♦❧❧♦✇ t❤❡
❧✐t❡r❛t✉r❡ ❛♥❞ ❞✐sr❡❣❛r❞ t❤❡ ✉♥st❛❜❧❡ ♦✉t❝♦♠❡✳ ❚❤✉s✱ ✇❡ ❡♥❞ ✉♣ ✇✐t❤ t✇♦ st❛❜❧❡ ❡q✉✐❧✐❜r✐❛✳ ❆r❡
t❤❡② ❡q✉❛❧❧② ❧✐❦❡❧②❄ ❆s s❛✐❞ ❛❜♦✈❡✱ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs ✇❡r❡ ❛❝t✐✈❡ ❧♦♥❣ ❜❡❢♦r❡ t❤❡ ❛♣♣❡❛r❛♥❝❡ ♦❢
s✉❜✉r❜❛♥ ♠❛❧❧s✳ ❚❤❡r❡❢♦r❡✱ ✐t s❡❡♠s ♥❛t✉r❛❧ t♦ ❡①♣❡❝t t❤❡ ❙❙❚ t♦ s❤r✐♥❦ ❢r♦♠ ne ✉♥t✐❧ ✐t ❣❡ts ✐♥t♦
t❤❡ ❜❛s✐♥ ♦❢ ❛ttr❛❝t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠ n∗ = N/ν ∗ < ne ✇❤❡r❡ t❤❡ ♠❡❧t✐♥❣ ♦❢ t❤❡
✶✽
❙❙❚ ✇♦✉❧❞ st♦♣✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠ n∗ = N/ν ∗ ✐s ♠♦r❡ ❧✐❦❡❧② t♦ ❡♠❡r❣❡ t❤❛♥
n∗ = 0 ❛s ❛ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠✳ ❚❤✉s✱ ✇❡ ✜♥❞ t❤✐s ❡q✉✐❧✐❜r✐✉♠ ✐♠♣❧❛✉s✐❜❧❡ ❛♥❞ ❡①❝❧✉❞❡ ✐t✳
❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ❝❛♥ ❛♣♣❡❛❧ t♦ t❤❡ ❝♦♥❝❡♣t ♦❢ ❝♦❛❧✐t✐♦♥✲♣r♦♦❢ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ t♦ ❥✉st✐❢② t❤✐s
❝❤♦✐❝❡✳ ❇❡❝❛✉s❡ ❛ r❡t❛✐❧❡r ♠❛❦✐♥❣ ③❡r♦✲♣r♦✜ts ♣r❡❢❡rs t♦ ❜❡ ✐♥ t❤❛♥ ♦✉t ♦❢ ❜✉s✐♥❡ss✱ t❤❡ ♦♥❧②
❝♦❛❧✐t✐♦♥✲♣r♦♦❢ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ✐s t❤❡ st❛❜❧❡ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠ ✐♥✈♦❧✈✐♥❣ n∗ > 0 s❡❧❧❡rs✳ ■♥✲
❞❡❡❞✱ t❤❡ ✐♥❝✉♠❜❡♥t r❡t❛✐❧❡rs ❛r❡ ♥♦t ✇✐❧❧✐♥❣ t♦ ❡①✐t t❤❡ ♠❛r❦❡t ❛s ❧♦♥❣ ❛s t❤❡② ♠❛❦❡ ③❡r♦ ♣r♦✜ts✱
✇❤❡r❡❛s t❤❡ ❡♥tr② ♦❢ ❛ ♣♦s✐t✐✈❡ ♠❛ss ♦❢ r❡t❛✐❧❡rs ✇♦✉❧❞ r❡s✉❧t ✐♥ ♥❡❣❛t✐✈❡ ♣r♦✜ts✳ ❚❤❡ ♦t❤❡r t✇♦
❡q✉✐❧✐❜r✐❛ ❛r❡ ♥♦t ❝♦❛❧✐t✐♦♥✲♣r♦♦❢ ❜❡❝❛✉s❡ ❛ ♣♦s✐t✐✈❡ ♠❛ss ♦❢ r❡t❛✐❧❡rs s✉❝❤ t❤❛t t❤❡ s✐③❡ ♦❢ t❤❡ ❙❙❚
✐s n∗ ✇♦✉❧❞ ❝❤♦♦s❡ t♦ ❡♥t❡r t❤❡ ❙❙❚✳
❇❛s❡❞ ♦♥ t❤✐s ❛r❣✉♠❡♥t ❛♥❞ ❈❧❛✐♠ ✶✱ ✇❡ ✜♥❞ ✐t ❧❡❣✐t✐♠❛t❡ t♦ r❡str✐❝t t❤❡ ❛♥❛❧②s✐s t♦ ❡q✉✐❧✐❜✲
r✐❛ (n∗ , N ∗ ) ♦❢ t❤❡ s❡❝♦♥❞✲st❛❣❡ s✉❜❣❛♠❡ s✉❝❤ t❤❛t ✭✐✮ N ∗ = N D ✱ ✭✐✐✮ πS (n∗ , N ∗ ) = φ✱ ✭✐✐✐✮ ✐❢
N ∗ < N̄ ✱ t❤❡♥ n∗ > 0✳ ❙✉❝❤ ❡q✉✐❧✐❜r✐❛ ❛r❡ ❝❛❧❧❡❞ ♣❧❛✉s✐❜❧❡✳ ■♥ t❤❡ ♥❡①t ❝❧❛✐♠✱ ✇❡ ♣r♦✈✐❞❡ ❛ ❢✉❧❧
❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❧❛✉s✐❜❧❡ ❡q✉✐❧✐❜r✐❛✳
❈❧❛✐♠ ✷✳ ❆ss✉♠❡ t❤❛t
s✐❜❧❡ ❡q✉✐❧✐❜r✐✉♠✳
ND > 0
❋✉rt❤❡r♠♦r❡✱ ✐t ✐s
πS (n∗ (N D ), N D ) = φ✳
D
✉♥✐q✉❡✳ ■❢ N
< N̄ ✱ t❤❡♥
❛♥❞
❚❤❡♥✱
n∗ (N D ), N D
✐s ❛ ♣❧❛✉✲
❜♦t❤ s❤♦♣♣✐♥❣ ♣❧❛❝❡s ❛r❡ ❛❝t✐✈❡✳
✳
✭✐✮ ❘❡t❛✐❧❡rs ❤❛✈❡ ♥♦ ✐♥❝❡♥t✐✈❡s t♦ ❡♥tr②✴❡①✐t ❢♦r n∗ (N D ) ✐s ❛ st❛❜❧❡ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠✳ ❋✉r✲
t❤❡r♠♦r❡✱ t❤❡ ❝❛♣❛❝✐t② ♦❢ t❤❡ ❙▼ ♣r❡✈❡♥ts ❢✉rt❤❡r ❡♥tr② ✐♥ t❤❡ ❙▼✱ ✇❤✐❧❡ st♦r❡s ❤❛✈❡ ♥♦ ✐♥❝❡♥t✐✈❡s
t♦ ❡①✐t s✐♥❝❡ t❤❡✐r ♣r♦✜ts ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡✳ ❚❤✉s✱ n∗ (N D ), N D ✐s ❛ ♣❧❛✉s✐❜❧❡ ❡q✉✐❧✐❜r✐✉♠✳ ❙✐♥❝❡
N D ✐s ❣✐✈❡♥✱ ✉♥✐q✉❡♥❡ss ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t n∗ (·) ✐s ❛ s✐♥❣❧❡✲✈❛❧✉❡❞ ♠❛♣♣✐♥❣✳ ✭✐✐✮ ❲❡ ❦♥♦✇
❢r♦♠ Pr♦♣♦s✐t✐♦♥ ✷ t❤❛t n∗ (N D ) > 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ N D < N̄ ✳
❆♠♦♥❣ ♦t❤❡r t❤✐♥❣s✱ ❈❧❛✐♠ ✷ ❛❧❧♦✇s ❞❡s❝r✐❜✐♥❣ t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ t❤❡ ♣❧❛✉s✐❜❧❡ ❡q✉✐❧✐❜r✐✉♠
t❤r♦✉❣❤ ❛♥ ❛✉❝t✐♦♥ ✉♥❞❡rt❛❦❡♥ ❜② t❤❡ ❞❡✈❡❧♦♣❡r✳ ❆ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ st♦r❡s ❝♦♠♣❡t❡ ❜② ❜✐❞❞✐♥❣ t♦
❣❡t ❛ s❧♦t ❢r♦♠ t❤❡ r❛♥❣❡ s✉♣♣❧✐❡❞ ❜② t❤❡ ❞❡✈❡❧♦♣❡r✳ ■♥ t❤✐s ❛✉❝t✐♦♥✱ ❡❛❝❤ st♦r❡ ✉♥❞❡rst❛♥❞s t❤❛t
t❤❡ ❤✐❣❤❡st ❜✐❞ ✐t ❝❛♥ ♦✛❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ st♦r❡s t❤❛t ✇✐❧❧ ❣❡t ❛ s❧♦t ❛s ✇❡❧❧ ❛s ♦♥ t❤❡
♥✉♠❜❡r ♦❢ r❡t❛✐❧❡rs t❤❛t ✇✐❧❧ ❧♦❝❛t❡ ✐♥ t❤❡ ❙❙❚✳ ❚❤✐s ♠❛❦❡s t❤❡ ❛✉❝t✐♦♥ ♠✉❝❤ ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥
✐♥ st❛♥❞❛r❞ ♠♦❞❡❧s ❜❡❝❛✉s❡ t❤❡ ✐♥❞✐✈✐❞✉❛❧ s✉r♣❧✉s ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡❝✐s✐♦♥s ♠❛❞❡ ❜② t❤❡ ♦t❤❡r
♣❧❛②❡rs✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t♦ ✜♥❞ ✐ts ♠❛①✐♠✉♠ ❜✐❞ ❡❛❝❤ st♦r❡ ♠✉st ❣✉❡ss ✇❤❛t t❤❡ ❡q✉✐❧✐❜r✐✉♠ ✈❛❧✉❡s
♦❢ N ❛♥❞ n ✇✐❧❧ ❜❡✳ ❙✉❝❤ s❡tt✐♥❣s ❛r❡ t②♣✐❝❛❧❧② ♣❧❛❣✉❡❞ ✇✐t❤ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠✉❧t✐♣❧❡ ❡q✉✐❧✐❜r✐❛✳
❇❡❝❛✉s❡ t❤❡ ♣❧❛✉s✐❜❧❡ ❡q✉✐❧✐❜r✐✉♠ ✐s ✉♥✐q✉❡✱ t❤❡ ❛✉❝t✐♦♥ ❡♥❞s ✉♣ ❤❡r❡ ✇✐t❤ ❛ s✐♥❣❧❡ ♥✉♠❜❡r ♦❢
st♦r❡s✱ ✇❤✐❝❤ ✐s ❡q✉❛❧ t♦ N ∗ (n∗ )✳
❖t❤❡r✇✐s❡✱ ♦♥❧② t❤❡ ❙▼ ✐s ❛❝t✐✈❡
✹✳✸
❚❤❡ s✐③❡ ♦❢ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧
■t r❡♠❛✐♥s t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s✐③❡ ❛♥❞ ❢❡❡ ❝❤♦s❡♥ ❜② t❤❡ ❞❡✈❡❧♦♣❡r ❛t t❤❡ ✜rst st❛❣❡
♦❢ t❤❡ ❣❛♠❡✳ ❚❤❡ ✈❛❧✉❡ ♦❢ N st❡♠s ❢r♦♠ t❤❡ ❝♦❧❧❡❝t✐✈❡ ❞❡❝✐s✐♦♥s ♠❛❞❡ ❜② st♦r❡s ✐♥ t❤❡ s❡❝♦♥❞
st❛❣❡ ♦❢ t❤❡ ❣❛♠❡✳ ■t t❤❡♥ ❢♦❧❧♦✇s ❢r♦♠ ❈❧❛✐♠ ✷ t❤❛t t❤❡ ❞❡✈❡❧♦♣❡r ❛❧✇❛②s ❝❤♦♦s❡s N D = N ❛♥❞
✶✾
φ = πS (n∗ (N ), N )✳ ❆s ❛ r❡s✉❧t✱ ❤✐s ♣r♦✜t ❢✉♥❝t✐♦♥ ✭✷✶✮ ♠❛② ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿
Π(n∗ (N ), N ) = N πS (n∗ (N ), N ) − B(N ).
✭✷✹✮
❋♦r N < N̄ , t❤❡ ❢✉♥❝t✐♦♥ n∗ (N ) ✐s ❞❡✜♥❡❞ ✐♠♣❧✐❝✐t❧② ❜② t❤❡ ❢r❡❡✲❡♥tr② ❛♥❞ st❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s
✐♥ t❤❡ ❙❚❚✿
πR (n, N ) = f
∂πR
≤0
∂n
✇❤❡r❡❛s ✇❡ ❤❛✈❡ n∗ (N ) = 0 ❢♦r N ≥ N̄ ✳ ❙✐♥❝❡ n∗ (N ) ❡①❤✐❜✐ts ❛ ❞♦✇♥✇❛r❞ ❥✉♠♣ ❛t N = N̄
✇❤❡r❡ n∗ = 0✱ t❤❡ ❞❡✈❡❧♦♣❡r✬s ♣r♦✜t ❢✉♥❝t✐♦♥ ❤❛s ❛♥ ✉♣✇❛r❞ ❥✉♠♣ ❛t N = N̄ ✳ ■♥❞❡❡❞✱ ❜❡❝❛✉s❡
st♦r❡s✬ ♣r♦✜ts ❞❡❝r❡❛s❡ ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ❙❙❚ r❡t❛✐❧❡rs✱ ✇❡ ❤❛✈❡ πS (n∗ (N ), N̄ ) < πS (0, N̄ )✱ ✇❤✐❝❤
✇♦✉❧❞ ❛❧❧♦✇ t❤❡ ❞❡✈❡❧♦♣❡r t♦ ✐♥❝r❡❛s❡ t❤❡ ♣❡r s❧♦t ♣r✐❝❡ φ ❜② ❛♥ ✐♥❝r❡♠❡♥t ∆φ > 0 ✇❤❡♥ N = N̄ ✳
❋✉rt❤❡r♠♦r❡✱ t❤❡ ❞❡✈❡❧♦♣❡r ♥❡✈❡r ❝❤♦♦s❡s ❛ s✐③❡ ❡①❝❡❡❞✐♥❣ N̄ ❜❡❝❛✉s❡ ❤✐s ♣r♦✜ts ❛r❡ ❡q✉❛❧ t♦
Π = 1 − ρ − B(N )
✇❤✐❝❤ ❛❧✇❛②s ❞❡❝r❡❛s❡s ✇✐t❤ N ✳
❇❡❝❛✉s❡ t❤❡ ❢✉♥❝t✐♦♥ Π(n∗ (N ), N ) ✐s ❜♦✉♥❞❡❞ ❛❜♦✈❡ ♦♥ ]0, N̄ [✱ t❤❡ ❞❡✈❡❧♦♣❡r✬s ♠❛①✐♠✉♠ ♣r♦✜t
✐s ❣✐✈❡♥ ❜②
(
Π∗ = max 0,
)
sup Π, (1 − ρ) − B(N̄ ) .
N ∈]0,N̄ [
■❢ Π∗ = 0✱ t❤❡ ❞❡✈❡❧♦♣❡r ❞♦❡s ♥♦t ❧❛✉♥❝❤ ❛♥ ❙▼ ❛♥❞ ❝♦♥s✉♠❡rs ✇✐❧❧ s❤♦♣ ❛t t❤❡ ❙❙❚ ♦♥❧②✳
■❢ Π∗ = sup Π > 0✱ t❤❡ ❞❡✈❡❧♦♣❡r ♦♣❡♥s ❛ s❤♦♣♣✐♥❣ ♠❛❧❧ ❛♥❞ ❛❝❝♦♠♠♦❞❛t❡s r❡t❛✐❧❡rs ❧♦❝❛t❡❞ ✐♥
t❤❡ ❝❡♥tr❛❧ ❜✉s✐♥❡ss ❞✐str✐❝t✳ ❋✐♥❛❧❧②✱ ✐❢ Π∗ = (1 − ρ) − B(N̄ )✱ t❤❡ ❞❡✈❡❧♦♣❡r ❝❤♦♦s❡s t♦ ❧❛✉♥❝❤ ❛♥
❙▼ ❤❛✈✐♥❣ t❤❡ ❧✐♠✐t s✐③❡ N̄ ✱ ❛♥❞ t❤✉s tr✐❣❣❡rs t❤❡ ❡①✐t ♦❢ ❛❧❧ r❡t❛✐❧❡rs✳ ❍❡♥❝❡✱ ❛ s✉❜❣❛♠❡ ♣❡r❢❡❝t
◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♦❢ ♦✉r t❤r❡❡✲st❛❣❡ ❣❛♠❡ ❛❧✇❛②s ❡①✐sts✳ ❲❤❡♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s✐③❡ N ∗ ♦❢ t❤❡ ❙▼
✐s s♠❛❧❧❡r t❤❛♥ ✐ts ❧✐♠✐t s✐③❡✱ ✇❡ s❛② t❤❛t t❤❡ ❞❡✈❡❧♦♣❡r ❛❝❝♦♠♠♦❞❛t❡s t❤❡ ♣r❡s❡♥❝❡ ♦❢ r❡t❛✐❧❡rs❀
♦t❤❡r✇✐s❡✱ t❤❡ ♠❛r❦❡t ✐♥✈♦❧✈❡s ❛ ♣r❡❞❛t♦r② ❞❡✈❡❧♦♣❡r✳
■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❜✉✐❧❞✐♥❣ ❝♦st ✐s ❣✐✈❡♥ ❜②
B(N ) ≡ F +
mN
✐❢ 0 < N < N E
M (N − N E ) + mN E
✐❢ N E ≤ N ✳
✭✷✺✮
✇❤❡r❡ F ✱ m✱ M ✱ ❛♥❞ N E ❛r❡ ♣♦s✐t✐✈❡ ♣❛r❛♠❡t❡rs s✉❝❤ t❤❛t m < M ❛♥❞ F < (M − m)N E ✳ ❚❤❡
❧❛tt❡r ✐♥❡q✉❛❧✐t② ✐♠♣❧✐❡s t❤❛t ✭✷✺✮ ❞✐s♣❧❛②s ✐♥❝r❡❛s✐♥❣ ✭❞❡❝r❡❛s✐♥❣✮ r❡t✉r♥s ❢♦r N s♠❛❧❧❡r ✭❧❛r❣❡r✮
t❤❛♥ N E ✳ ■t ✐s r❡❛❞✐❧② ✈❡r✐✜❡❞ t❤❛t F < (M − m)N E ✐♠♣❧✐❡s t❤❛t t❤❡ ❡✣❝✐❡♥t s✐③❡ ♦❢ t❤❡ ❙▼✱
✇❤✐❝❤ ♠✐♥✐♠✐③❡s t❤❡ ❛✈❡r❛❣❡ ❜✉✐❧❞✐♥❣ ❝♦st✱ ✐s ❡q✉❛❧ t♦ N E ✳ ❲❤❡♥ F > (M − m)N E ✱ t❤❡ ❡✣❝✐❡♥t
s✐③❡ ♦❢ t❤❡ ❙▼ ✐s ❛r❜✐tr❛r✐❧② ❧❛r❣❡✳
✷✵
❚❤❡ ❞❡✈❡❧♦♣❡r✬s r❡✈❡♥✉❡ ✐s ❡q✉❛❧ t♦ N πS (n∗ (N ), N ) = πS (µ∗ (N ), 1)✳ ❙✐♥❝❡ πS (µ, 1) ❞❡❝r❡❛s❡s
✇✐t❤ µ ❛♥❞ µ∗ (N ) ❞❡❝r❡❛s❡s ✇✐t❤ N ✱ t❤❡ r❡✈❡♥✉❡ ❢✉♥❝t✐♦♥ ❛❧✇❛②s ✐♥❝r❡❛s❡s ♦✈❡r [0, N̄ ]✳ ❆s ❛ r❡s✉❧t✱
t❤❡ ❞❡✈❡❧♦♣❡r ❛❧✇❛②s ❝❤♦♦s❡s ✐ts ❧✐♠✐t s✐③❡ ✇❤❡♥ N E > N̄ ❛♥❞ s❡❝✉r❡s t❤❡ ✇❤♦❧❡ ♠❛r❦❡t✳ ■❢ N E ≤ N̄
❛♥❞ M ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡✱ t❤❡ ❞❡✈❡❧♦♣❡r ❝❤♦♦s❡s ❤✐s ❡✣❝✐❡♥t s✐③❡ ❛♥❞ ❛❝❝♦♠♠♦❞❛t❡s t❤❡ ♣r❡s❡♥❝❡
♦❢ r❡t❛✐❧❡rs✳ ❇② ❝❤❛♥❣✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ M ✱ t❤❡ ♣r♦✜t✲♠❛①✐♠✐③✐♥❣ s✐③❡ ♦❢ t❤❡ ❙▼ ❝❛♥ ✐♥❝r❡❛s❡ ❢r♦♠
0 t♦ t❤❡ ❧✐♠✐t s✐③❡ N̄ ✱ ✇❤❡r❡❛s t❤❡ ❡q✉✐❧✐❜r✐✉♠ s✐③❡ ♦❢ t❤❡ ❙❙❚ s❤r✐♥❦s ❢r♦♠ ne t♦ 0✳ ◆♦t❡ ❛❧s♦ t❤❛t
t❤❡ ✜①❡❞ ❝♦st F ❝❛♥♥♦t ❜❡ ✈❡r② ❧❛r❣❡ ❢♦r t❤❡ ❞❡✈❡❧♦♣❡r t♦ ❡♥t❡r✳
✺
❙❤♦✉❧❞ t❤❡ ❙✐③❡ ♦❢ t❤❡ ❙❤♦♣♣✐♥❣ ▼❛❧❧ ❇❡ ❘❡❣✉❧❛t❡❞❄
❙❡✈❡r❛❧ ❝♦✉♥tr✐❡s ❤❛✈❡ ♣❛ss❡❞ ❧❛✇s r❡str✐❝t✐♥❣ t❤❡ s✐③❡ ♦r ❡♥tr② ♦❢ ❜✐❣✲❜♦①❡s ❜❡❝❛✉s❡ t❤❡ ♣r❡s❡♥❝❡ ♦❢
s♠❛❧❧ ❜✉s✐♥❡ss❡s ✇♦✉❧❞ ❛❧❧♦✇ ❝♦♥s✉♠❡rs t♦ ❜❡♥❡✜t ❢r♦♠ ❛ ✇✐❞❡r ❛rr❛② ♦❢ ✈❛r✐❡t✐❡s ❛♥❞ s❡r✈✐❝❡s✳ ■t ✐s✱
t❤❡r❡❢♦r❡✱ ✇♦rt❤ st✉❞②✐♥❣ ❤♦✇ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❡t✐❡s ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❝✐t②✱ N ≡ N + n∗ (N )✱
✈❛r✐❡s ✇✐t❤ t❤❡ s✐③❡ N ♦❢ t❤❡ ❙▼✳ ❙✐♥❝❡ t❤❡ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠ n∗ (N ) ❞❡❝r❡❛s❡s ✇✐t❤ N ✱ ✐t
✐s ❛ ♣r✐♦r✐ ✉♥❝❧❡❛r ✇❤❡t❤❡r N ✐♥❝r❡❛s❡s ♦r ❞❡❝r❡❛s❡s ✇✐t❤ N ✳ ❨❡t✱ ✐t t✉r♥s ♦✉t t♦ ❜❡ ♣♦ss✐❜❧❡ t♦
❝❤❛r❛❝t❡r✐③❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ N ✐♥ ❛ ♣r❡❝✐s❡ ✇❛②✳ ❚❤❡ ❛r❣✉♠❡♥t ❣♦❡s ❛s ❢♦❧❧♦✇s✳
❚❤❡ ❢✉♥❝t✐♦♥ πR (n; N ) ❜❡✐♥❣ ❤♦♠♦❣❡♥♦✉s ♦❢ ❞❡❣r❡❡ −1✱ t❤❡ ③❡r♦✲♣r♦✜t ❝♦♥❞✐t✐♦♥ πR (n; N ) = f
❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿
π(µ) ≡ πR (µ, 1) = f N
✭✷✻✮
✇❤❡r❡ µ = n/N ✳ ❙✐♥❝❡ πR (n, N ) = πR (µ, 1)/N ✱ Pr♦♣♦s✐t✐♦♥ ✶ ✐♠♣❧✐❡s t❤❛t t❤❡ ❢✉♥❝t✐♦♥ πR (µ; 1) ✐s
✉♥✐♠♦❞❛❧ ✐♥ µ✳ ▼✉❧t✐♣❧②✐♥❣ ❜♦t❤ s✐❞❡s ♦❢ ✭✷✻✮ ❜② 1 + µ✱ ✇❡ ♦❜t❛✐♥✿
1+µ
π(µ) = N + n∗ (N ).
f
✭✷✼✮
❲❡ s❤♦✇ ✐♥ ❆♣♣❡♥❞✐① ✸ t❤❛t t❤✐s ❡①♣r❡ss✐♦♥ ✐s ✉♥✐♠♦❞❛❧ ✐♥ µ ✇✐t❤ ❛ ✉♥✐q✉❡ ♠❛①✐♠✐③❡r ❛t
µ0 > 0✳ ❍❡♥❝❡✱ N ✐♥❝r❡❛s❡s ✭❞❡❝r❡❛s❡s✮ ✇✐t❤ N ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✭✷✼✮ ✐s ❞❡❝r❡❛s✐♥❣ ✭✐♥❝r❡❛s✐♥❣✮ ❛t
µ∗ (N )✳ ❙✐♥❝❡ µ∗ (N ) ❞❡❝r❡❛s❡s ✇✐t❤ N ✱ N ✐♥❝r❡❛s❡s ✇✐t❤ N ✐❢ ❛♥❞ ♦♥❧② ✐❢ N ≤ N0 ≡ π(µ0 )/f ✳
❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ✐s ❛ s✉♠♠❛r②✳
Pr♦♣♦s✐t✐♦♥ ✸✳ ❚❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❡t✐❡s ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❝✐t②✱ ✜rst✱ ✐♥❝r❡❛s❡s ❛♥❞✱ t❤❡♥✱
❞❡❝r❡❛s❡s ✇✐t❤ t❤❡ s✐③❡ ♦❢ t❤❡ ❙▼✳
❚❤✉s✱ t❤❡ ♠❛ss ♦❢ ✈❛r✐❡t✐❡s r❡❛❝❤❡s ✐ts ♠❛①✐♠✉♠ ✇❤❡♥ t❤❡ ❝✐t② ❛❝❝♦♠♠♦❞❛t❡s ❜♦t❤ ❛♥ ❙❙❚
❛♥❞ ❛♥ ❙▼✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❡♥tr② ♦❢ ❛♥ ❙▼ ♥❡❡❞ ♥♦t ❜❡ ❞❡tr✐♠❡♥t❛❧ t♦ ♣r♦❞✉❝t ❞✐✈❡rs✐t②✿ t❤❡
❝✐t② ✐♥✈♦❧✈❡s ♠♦r❡ ✈❛r✐❡t✐❡s ❛s ❧♦♥❣ ❛s t❤❡ s✐③❡ ♦❢ t❤❡ ❙▼ ✐s ♥♦t ❧❛r❣❡ ❡♥♦✉❣❤ ❢♦r N t♦ ❜❡ s♠❛❧❧❡r
t❤❛♥ ne ✳ ❚❤✐s s✉❣❣❡st t❤❛t t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛♥ ❙▼ ❝♦✉❧❞ ✇❡❧❧ ❜❡ ❜❡♥❡✜❝✐❛❧ t♦ ❝♦♥s✉♠❡rs✳
❙✉❝❤ ❛♥ ❛r❣✉♠❡♥t ✐s ✐♥❝♦♠♣❧❡t❡✱ ❤♦✇❡✈❡r✱ ❜❡❝❛✉s❡ ✐t ♦✈❡r❧♦♦❦s t❤❡ ❢❛❝t t❤❛t ❝♦♥s✉♠❡rs ♠✉st
❜❡❛r s♣❡❝✐✜❝ tr❛✈❡❧ ❝♦sts t♦ ✈✐s✐t t❤❡ s❤♦♣♣✐♥❣ ♣❧❛❝❡s✳ ❖✇✐♥❣ t♦ t❤❡ ❡①✐t ♦❢ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs✱ t❤❡
❝♦♥s✉♠❡rs ✐♥ [0, x̄0 ] ❤❛✈❡ ❛❝❝❡ss t♦ ❛ ♥❛rr♦✇❡r ❛rr❛② ♦❢ ✈❛r✐❡t✐❡s✳ ❚❤❡ ❝♦♥s✉♠❡rs r❡s✐❞✐♥❣ ✐♥ [x̄0 , y]✱
✷✶
✇❤❡r❡ y ≥ x̄0 ✐s t❤❡ s♦❧✉t✐♦♥ t♦
(1 − ρ) ln ne − τ y = (1 − ρ) ln N − τ (1 − y)
✭✷✽✮
♥♦✇ ❝♦♥s✉♠❡ t❤❡ ✇❤♦❧❡ r❛♥❣❡ ♦❢ ✈❛r✐❡t✐❡s ❜✉t t❤❡② ❜❡❛r ❛ ❤✐❣❤❡r tr❛✈❡❧ ❝♦st✳ ❖♥❧② t❤❡ ❝♦♥s✉♠❡rs
❧♦❝❛t❡❞ ✐♥ t❤❡ ✐♥t❡r✈❛❧ [y, 1]✱ ✐❢ ❛♥②✱ ❛r❡ ❜❡tt❡r ♦✛ ✇❤❡♥ ❛♥ ❙▼ ♦❢ s✐③❡ N ✐s ❧❛✉♥❝❤❡❞✳ ❚❤❡r❡❢♦r❡✱
✇❤❡♥ N ✐s s✉✣❝✐❡♥t❧② s♠❛❧❧✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ ✭✷✽✮ t❤❛t y ✐s ❝❧♦s❡ ♦r ❡q✉❛❧ t♦ 1✱ ❛♥❞ t❤✉s t❤❡ ♠❛❥♦r✐t②
♦❢ ❝♦♥s✉♠❡rs ✐s ✇♦rs❡ ♦✛ ✇❤❡♥ t❤❡ ❙▼ ✐s ❧❛✉♥❝❤❡❞✳
❉♦❡s t❤✐s ❡①♣❧❛✐♥ ✇❤② ❝✐t✐③❡♥ ❣r♦✉♣s ❛♥❞✴♦r ❧♦❝❛❧ ❣♦✈❡r♥♠❡♥ts ♦❢t❡♥ ♦❜❥❡❝t t♦ t❤❡ ❡♥tr② ♦❢ ❛
❜✐❣✲❜♦① ✐♥ t❤❡✐r ❛r❡❛❄ ❋♦r t❤❡ ♠❛❥♦r✐t② ♦❢ ❝♦♥s✉♠❡rs t♦ ❛❣r❡❡ ✇✐t❤ t❤❡ ❧❛✉♥❝❤✐♥❣ ♦❢ ❛♥ ❙▼✱ ✐t
♠✉st ❜❡ t❤❛t t❤❡ ❝♦♥s✉♠❡r ❧♦❝❛t❡❞ ❛t x = 1/2 ✐s ❜❡tt❡r ♦✛ ❛❢t❡r ❡♥tr②✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❢♦❧❧♦✇✐♥❣
❝♦♥❞✐t✐♦♥ ♠✉st ❤♦❧❞✿
(1 − ρ) ln N(N ) − τ ≥ (1 − ρ) ln ne − τ /2
✇❤✐❝❤ ❛♠♦✉♥ts t♦
N(N ) ≥
√
T ne .
✭✷✾✮
❋✐❣✉r❡ ✸✱ ✇❤✐❝❤ ❞❡♣✐❝ts t❤❡ ♣❧♦t ♦❢ t❤❡ ♠❛①✐♠✉♠ ♣r♦❞✉❝t r❛♥❣❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ T ✱ s❤♦✇s
t❤❛t t❤✐s ✐♥❡q✉❛❧✐t② ♥❡✈❡r ❤♦❧❞s✳ ❚❤✉s✱ r❡❣❛r❞❧❡ss ♦❢ t❤❡ ✈❛❧✉❡ ♦❢ N s♠❛❧❧❡r t❤❛♥ t❤❡ ❧✐♠✐t s✐③❡✱ ❛
♠❛❥♦r✐t② ♦❢ ❝♦♥s✉♠❡rs ✈♦t❡ ❛❣❛✐♥st t❤❡ ❡♥tr② ♦❢ ❛♥ ❙▼✳ ❚❤❛t ❛ ♠❛❥♦r✐t② ♦❢ ❝♦♥s✉♠❡rs ✐s ❛❣❛✐♥st
t❤❡ ❡♥tr② ♦❢ t❤❡ ❙▼ ✇❤❡♥ N ✐s ♠❛①✐♠✐③❡❞ ♠❛② ❝♦♠❡ ❛s ❛ s✉r♣r✐s❡ s✐♥❝❡ N(N0 ) ❡①❝❡❡❞s ne ✳ ❚❤✐s
✐s ❜❡❝❛✉s❡ t❤❡ ❛❞❞✐t✐♦♥❛❧ tr❛✈❡❧ ❝♦st t❤❡ ❝♦♥s✉♠❡r ❛t x = 1/2 ♠✉st ❜❡❛r t♦ ❝♦♥s✉♠❡ t❤❡ ✈❛r✐❡t✐❡s
s✉♣♣❧✐❡❞ ❜② t❤❡ ❙▼ ✐s ♥♦t ❝♦♠♣❡♥s❛t❡❞ ❜② t❤❡ ✈❛r✐❡t② ✐♥❝r❡❛s❡✳ ❖❢ ❝♦✉rs❡✱ t❤✐s r❡s✉❧t ❞❡♣❡♥❞s
♦♥ t❤❡ ❧♦✇❡r ❜♦✉♥❞ T > 2✳ ❲❤❡♥ T t❛❦❡s ♦♥ s✉✣❝✐❡♥t❧② ❧♦✇ ✈❛❧✉❡s✱ ✭✷✾✮ ✇✐❧❧ ❜❡ s❛t✐s✜❡❞ ❛♥❞
t❤❡ ❧❛✉♥❝❤✐♥❣ ♦❢ ❛♥ ❙▼ ❛t t❤❡ ❝✐t② ♦✉ts❦✐rts ✇✐❧❧ ❣❡t t❤❡ s✉♣♣♦rt ♦❢ ❛ ♠❛❥♦r✐t② ♦❢ ❝♦♥s✉♠❡rs✳ ■♥
♦t❤❡r ✇♦r❞s✱ ♣❡♦♣❧❡✬s ❛tt✐t✉❞❡ t♦✇❛r❞ t❤❡ ❡♥tr② ♦❢ ❛ s❤♦♣♣✐♥❣ ♠❛❧❧ ✐♥✢✉❡♥❝❡❞ ❜② t❤❡ q✉❛❧✐t② ♦❢ t❤❡
tr❛♥s♣♦rt❛t✐♦♥ s②st❡♠✳
✷✷
❋✐❣✉r❡ ✸✳ ▼❛①✐♠✉♠ ♣r♦❞✉❝t r❛♥❣❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢
T
❚❤✐s ♣♦❧✐t✐❝❛❧ ❡❝♦♥♦♠② ❛r❣✉♠❡♥t ❝♦✉❧❞ ❡①♣❧❛✐♥ ✇❤② ✐♥ s❡✈❡r❛❧ ❝♦✉♥tr✐❡s ♦r ❥✉r✐s❞✐❝t✐♦♥s t❤❡ s✐③❡
♦❢ s❤♦♣♣✐♥❣ ♠❛❧❧s ✐s r❡❣✉❧❛t❡❞ ❜② ♣✉❜❧✐❝ ❛✉t❤♦r✐t✐❡s✳ ■s t❤✐s ❞❡❝✐s✐♦♥ ❥✉st✐✜❡❞ ♦♥ ❡✣❝✐❡♥❝② ❣r♦✉♥❞s❄
❚♦ ❛♥s✇❡r t❤✐s q✉❡st✐♦♥✱✇❡ ❝♦♥s✐❞❡r ❛ ✇❡❧❢❛r❡✲♠❛①✐♠✐③✐♥❣ r❡❣✉❧❛t♦r ✇❤♦ ❝❤♦♦s❡s t❤❡ s✐③❡ N ♦❢ t❤❡
❙▼ t❤❛t ♣r❡✈❡♥ts t❤❡ ❝❡♥tr❛❧ ❜✉s✐♥❡ss ❞✐str✐❝t t♦ ❜❡❝♦♠❡ ❛ ❣❤♦st t♦✇♥✳ ■♥ ❝♦♥tr❛st✱ t❤❡ ♥✉♠❜❡r
♦❢ r❡t❛✐❧❡rs ✐s ✉♥r❡❣✉❧❛t❡❞ ❛♥❞ ❞❡t❡r♠✐♥❡❞ ❜② ❢r❡❡ ❡♥tr② ❛♥❞ ❡①✐t✳
❙✐♥❝❡ ♣r❡❢❡r❡♥❝❡s ❛r❡ q✉❛s✐✲❧✐♥❡❛r✱ t❤❡ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s t♦t❛❧ ❝♦♥s✉♠❡r s✉r♣❧✉s
♣❧✉s ✜r♠s✬ ♣r♦✜ts✿
W =
ˆ
0
1
V (x)dx + n [πR (n, N ) − f ] + N [πS (n, N ) − φ] + Π(N ).
✭✸✵✮
❚❤❡ r❡❣✉❧❛t♦r s❡❡❦s t❤❡ ❙▼ s✐③❡ t❤❛t ♠❛①✐♠✐③❡s ✭✸✵✮ s✉❜❥❡❝t t♦ n = n∗ (N ) ❛♥❞ N < N̄ ✳ ❚❤❡
❢♦r♠❡r ❝♦♥str❛✐♥t ✐s t❤❡ ❢r❡❡✲❡♥tr② ❝♦♥❞✐t✐♦♥ ❛t t❤❡ ❙❙❚✳ ❚❤❡ ❧❛tt❡r ♠❡❛♥s t❤❛t t❤❡ ❝✐t② ❝❡♥t❡r
♠✉st r❡♠❛✐♥ ❛ s❤♦♣♣✐♥❣ ♣❧❛❝❡ ✇❤♦s❡ s✐③❡ ✐s ❡♥❞♦❣❡♥♦✉s❧② ❞❡t❡r♠✐♥❡❞ ❜② ❡♥tr② ❛♥❞ ❡①✐t✳
❯s✐♥❣ ✭✼✮ ✕ ✭✶✶✮✱ ✭✸✵✮ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s ✭✉♣ t♦ ❛ ❝♦♥st❛♥t✮✿
h
i
τ 2
✭✸✶✮
x̄0 + (1 − x̄1 )2 + [N πS (n∗ (N ), N ) − B(N )]
W = (1 − ρ) ln (N + n∗ (N )) +
2
✇❤❡r❡ x̄0 ❛♥❞ x̄1 ❛r❡ t❤❡ ♠❛r❣✐♥❛❧ ❝♦♥s✉♠❡rs✬ ❧♦❝❛t✐♦♥s ❣✐✈❡♥ ❜② ✭✶✸✮ ✕ ✭✶✹✮✳
■t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ s❤♦✇ t❤❛t t❤❡ ✜rst t❡r♠ ✐♥ ✭✸✶✮ ❞❡❝r❡❛s❡s ✇✐t❤ N ❢♦r ❛❧❧ N < N̄ ✱ ✇❤❡r❡❛s
t❤❡ s❡❝♦♥❞ ✐s ❣✐✈❡♥ ❜② ✭✷✹✮✳ ❙✐♥❝❡ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s✐③❡ N ∗ ♠❛①✐♠✐③❡s t❤❡ ❞❡✈❡❧♦♣❡r✬s ♣r♦✜ts ✉♥❞❡r
n∗ (N )✱ ✇❡ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✳
Pr♦♣♦s✐t✐♦♥ ✹✳ ▲❡t N ∗ ❛♥❞ N o ❜❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❙▼ ❛t t❤❡ ♠❛r❦❡t ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ s♦❝✐❛❧
o
♦♣t✐♠✉♠✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ N
≤ N ∗ ✳ ❋✉rt❤❡r♠♦r❡✱ ✇❤❡♥ 0 < N ∗ < N̄ ✱ ❛ ✇❡❧❢❛r❡✲
o
∗
♠❛①✐♠✐③✐♥❣ r❡❣✉❧❛t♦r ❛❧✇❛②s ❝❤♦♦s❡s N < N ✳
✷✸
❚❤❡r❡❢♦r❡✱ t❤❡ r❡❣✉❧❛t♦r ❝❤♦♦s❡s ❛ s✐③❡ ❢♦r t❤❡ ❙▼ s♠❛❧❧❡r t❤❛♥ t❤❡ s✐③❡ t❤❛t ✇♦✉❧❞ ❡♠❡r❣❡ ✉♥❞❡r
✉♥❧❡❛s❤❡❞ ❝♦♠♣❡t✐t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❡♥ t❤❡ ♠❛r❦❡t ♦✉t❝♦♠❡ ✐♥✈♦❧✈❡s ❛♥ ❛❝❝♦♠♠♦❞❛t✐♥❣
❞❡✈❡❧♦♣❡r✱ r❡❣✉❧❛t✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ ❙▼ ✐s ✇❡❧❢❛r❡✲❡♥❤❛♥❝✐♥❣✳ ❍♦✇❡✈❡r✱ ❜❛♥♥✐♥❣ t❤❡ ❡♥tr② ♦❢ ❛♥
❙▼ ✐s ❣❡♥❡r❛❧❧② ♥♦t ♦♣t✐♠❛❧✳
❙❤♦✉❧❞ ❛ ♣r❡❞❛t♦r② ❞❡✈❡❧♦♣❡r ❜❡ r❡❣✉❧❛t❡❞ t♦♦❄ ❚❤❡ ❛♥s✇❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡✈❡❧♦♣❡r✬s ❡✣✲
❝✐❡♥❝②✳ ❚♦ ✐❧❧✉str❛t❡✱ ❝♦♥s✐❞❡r t❤❡ ❜✉✐❧❞✐♥❣ ❝♦st ❢✉♥❝t✐♦♥ ✭✷✺✮ ❛♥❞ s❡t m = 0.1 ❛♥❞ N E = 0.3 ❛s
✇❡❧❧ ❛s σ = 2✱ τ = 0.564 ❛♥❞ f = 0.5✳ ◆✉♠❡r✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥s s❤♦✇ t❤❛t t❤❡ ❞❡✈❡❧♦♣❡r ❛❞♦♣ts
❛ ♣r❡❞❛t♦r② ❜❡❤❛✈✐♦r ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ♠❛r❣✐♥❛❧ ❝♦st ♦❢ ❛♥ ❛❞❞✐t✐♦♥❛❧ st♦r❡ M ✐s ♥♦t t♦♦ ❤✐❣❤
✭M ≤ 4.71✮✱ ✇❤✐❧❡ ✐t ✐s ♦♣t✐♠❛❧ t♦ r❡❣✉❧❛t❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❙▼ ✇❤❡♥ M ✐s ♥♦t t♦♦ ❧♦✇ ✭M ≥ 0.357✮✳
❍❡♥❝❡✱ ✇❤❡♥ M ✐s ✈❡r② s♠❛❧❧✱ t❤❡ ❞❡✈❡❧♦♣❡r ✐s s♦ ❡✣❝✐❡♥t t❤❛t t❤❡ ♣❧❛♥♥❡r ❝❤♦♦s❡s ♥♦t t♦ r❡❣✉❧❛t❡
t❤❡ ❙▼✱ ✇❤✐❝❤ ❝❤♦♦s❡s ✐t ❧✐♠✐t s✐③❡ ✭N o = N̄ ✮✳ ❚❤❡ ❞✐s❛♣♣❡❛r❛♥❝❡ ♦❢ t❤❡ ❙❙❚ ✐s ❝♦♠♣❡♥s❛t❡❞ ❜②
t❤❡ ✈❡r② ❧❛r❣❡ ♥✉♠❜❡r ♦❢ st♦r❡s ❤♦st❡❞ ❜② t❤❡ ❙▼✳ ■♥ ❝♦♥tr❛st✱ ✇❤❡♥ M = 3✱ t❤❡ ❙▼ ✐s ♠✉❝❤ ❧❡ss
❡✣❝✐❡♥t✳ ❚❤✐s ❧❡❛❞s t❤❡ r❡❣✉❧❛t♦r t♦ ✐♥t❡r✈❡♥❡ ❛♥❞ t♦ ❝❤♦♦s❡ t❤❡ s✐③❡ N o = 0.5✱ ❛♥❞ t❤✉s t❤❡ ❙❙❚
❤❛s ❛ s✐③❡ n∗(N o) ≈ 1.57✳ ■♥ s✉♠✱ ❛ ✈❡r② ❡✣❝✐❡♥t ❞❡✈❡❧♦♣❡r ♠✉st ♥♦t ❜❡ r❡❣✉❧❛t❡❞✳ ❖t❤❡r✇✐s❡✱ t❤❡
r❡❣✉❧❛t♦r ✐♥❝r❡❛s❡s s♦❝✐❛❧ ✇❡❧❢❛r❡ ❜② ❝❤♦♦s✐♥❣ ❛ s✐③❡ ❢♦r t❤❡ ❙▼ s♠❛❧❧❡r t❤❛♥ ✐ts ❡q✉✐❧✐❜r✐✉♠ s✐③❡✳
❍♦✇❡✈❡r✱ ✐♥ ❜♦t❤ ❝❛s❡s✱ ❛ ♠❛❥♦r✐t② ♦❢ ❝♦♥s✉♠❡rs r❡♠❛✐♥s ❤♦st✐❧❡ t♦ t❤❡ ❙▼✬s ❡♥tr②✳ ❚❤✐s ♣r♦✈✐❞❡s
❛ ♥❡❛t ✐❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ♣♦ss✐❜❧❡ ❞✐s❝r❡♣❛♥❝② ❜❡t✇❡❡♥ t❤❡ ❡✣❝✐❡♥t ❛♥❞ ✈♦t✐♥❣ ♦✉t❝♦♠❡s✳
■♥ ❋r❛♥❝❡✱ t❤❡ ❘♦②❡r ▲❛✇ ✐♠♣♦s❡s r❡str✐❝t✐♦♥s ♦♥ ❞❡♣❛rt♠❡♥t st♦r❡s ✇❤♦s❡ ❝r❡❛t✐♦♥ ❤❛s t♦
❜❡ ❛♣♣r♦✈❡❞ ❜② ❛ ❧♦❝❛❧ ❜♦❛r❞ ❝♦♠♣♦s❡❞ ♦❢ s❤♦♣✲♦✇♥❡rs✱ ❝♦♥s✉♠❡r r❡♣r❡s❡♥t❛t✐✈❡s✱ ❛♥❞ ❧♦❝❛❧❧②
❡❧❡❝t❡❞ ♣♦❧✐t✐❝✐❛♥s✳ ❇❡t✇❡❡♥ ✶✾✼✹ ❛♥❞ ✶✾✾✽✱ t❤❡ ❧♦❝❛❧ ❜♦❛r❞s ❛♣♣r♦✈❡❞ ♦♥❧② ❛❜♦✉t 40 ♣❡r❝❡♥t ♦❢
t❤❡ ❛♣♣❧✐❝❛t✐♦♥s✳ ❇❡rtr❛♥❞ ❛♥❞ ❑r❛♠❛r③ ✭✷✵✵✷✮ ❤❛✈❡ s❤♦✇❡❞ t❤❛t t❤❡ ❡♥❢♦r❝❡♠❡♥t ♦❢ t❤✐s ❧❛✇ ❤❛s
❤❛❞ ❛ ♥❡❣❛t✐✈❡ ✐♠♣❛❝t ♦♥ ❥♦❜ ❝r❡❛t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❡♥tr② ♦❢ ❛ ♥❡✇ ❙▼✱ ✇❤✐❝❤ ♠❛② ❜❡
❞❡s✐r❛❜❧❡ ✐♥ t❡r♠s ♦❢ ❥♦❜ ❝r❡❛t✐♦♥✱ ✐s ♦♣♣♦s❡❞ ❜② ❛ ♠❛❥♦r✐t② ♦❢ ❜♦❛r❞ ♠❡♠❜❡rs✳ ❚❤✐s s❡❡♠s t♦ ❜❡
✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ t❤❡ ❛❜♦✈❡ r❡s✉❧ts✳
✻
❍❡t❡r♦❣❡♥❡♦✉s ▼❛r❦❡t♣❧❛❝❡s ❛♥❞ t❤❡ ❈✐t②
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❞✐s❝✉ss t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①t❡♥s✐♦♥s ♦❢ t❤❡ ❜❛s❡❧✐♥❡ ♠♦❞❡❧✿ ✭✐✮ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝♦st
❛♥❞ q✉❛❧✐t② ❛s②♠♠❡tr✐❡s ❜❡t✇❡❡♥ r❡t❛✐❧❡rs ❛♥❞ st♦r❡s❀ ✭✐✐✮ ❛ ❝❤❛♥❣❡ ✐♥ ♠❛r❦❡t✴❝✐t② s✐③❡❀ ✭✐✐✐✮ ❛
♥♦♥✲✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦❢ ❝♦♥s✉♠❡rs ❛❝r♦ss t❤❡ ❝✐t②❀ ✭✐✈✮ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ❙▼ ❧♦❝❛t✐♦♥ ❜② t❤❡
❞❡✈❡❧♦♣❡r✳
✻✳✶
❍❡t❡r♦❣❡♥❡♦✉s s❤♦♣♣✐♥❣ ♣❧❛❝❡s
❲❡ ❤❛✈❡ ❛r❣✉❡❞ ✐♥ ❙❡❝t✐♦♥ ✷ t❤❛t ✈❛r✐♦✉s t②♣❡s ♦❢ ❛s②♠♠❡tr✐❡s ❜❡t✇❡❡♥ t❤❡ t✇♦ s❤♦♣♣✐♥❣ ♣❧❛❝❡s
♠❛② ❜❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✐♥ ♦✉r s❡tt✐♥❣✳ ❚♦ ✐❧❧✉str❛t❡✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s♣❡❝✐❛❧✱ ❜✉t
r❡❧❡✈❛♥t✱ ❝❛s❡✿ r❡t❛✐❧❡rs ❛r❡ ❧❡ss ❡✣❝✐❡♥t t❤❛♥ st♦r❡s ❜✉t s✉♣♣❧② ❜❡tt❡r q✉❛❧✐t② ✈❛r✐❡t✐❡s✳ ■♥ ♦t❤❡r
✇♦r❞s✱ t❤❡ r❡t❛✐❧❡rs s❤❛r❡ t❤❡ ♠❛r❣✐♥❛❧ ❝♦st c✱ ✇❤❡r❡❛s st♦r❡s ❤❛✈❡ t❤❡ ♠❛r❣✐♥❛❧ ❝♦st C = λc✱
✷✹
✇❤❡r❡ λ < 1✳ ❚❤❡ ♣❛r❛♠❡t❡r λ ❝❛♣t✉r❡s t❤❡ ❡✣❝✐❡♥❝② ❤❡t❡r♦❣❡♥❡✐t② ❜❡t✇❡❡♥ st♦r❡s ❛♥❞ r❡t❛✐❧❡rs✱
✇❤✐❝❤ ♠❛② st❡♠ ❢r♦♠ ❛❞❞✐t✐♦♥❛❧ ❢❛❝✐❧✐t✐❡s s✉♣♣❧✐❡❞ ❜② t❤❡ ❞❡✈❡❧♦♣❡r t♦ ❤✐s t❡♥❛♥ts✳ ■♥ t❤❡ s❛♠❡
✈❡✐♥✱ ❜❡✐♥❣ ❞❡s✐❣♥❡❞✱ ❞❡✈❡❧♦♣❡❞ ❛♥❞ ♠❛♥❛❣❡❞ ❛s ❛ s✐♥❣❧❡ ✉♥✐t✱ ❛ s✉♣❡r♠❛r❦❡t ❜❡♥❡✜ts ❢r♦♠ s❝♦♣❡
❡❝♦♥♦♠✐❡s ❛♥❞ ✐s ♦❢t❡♥ ❛❜❧❡ t♦ ❜✉② ✐♥♣✉ts ✐♥ ❜✉❧❦ ❛t ♣r✐❝❡s ❧♦✇❡r t❤❛♥ r❡t❛✐❧❡rs✳
❈♦♥s✉♠❡rs✬ ♣r❡❢❡r❡♥❝❡s ❛r❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿
ˆ N
ˆ n
1
ρ
ρ
Qj dj + A
U ≡ ln I(SST ) α
qi di + I(SM )
ρ
0
0
✇❤❡r❡ t❤❡ ♣❛r❛♠❡t❡r α ❝❛♣t✉r❡s t❤❡ q✉❛❧✐t② ❤❡t❡r♦❣❡♥❡✐t② ❜❡t✇❡❡♥ t❤❡ t✇♦ t②♣❡s ♦❢ ✈❛r✐❡t✐❡s✳ ■♥
✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ ❢♦❝✉s ♦♥ t❤❡ ❝❛s❡ ✇❤❡r❡ α > 1✳
❯♥❞❡r t❤❡s❡ ❝✐r❝✉♠st❛♥❝❡s✱ t❤❡ ♣r✐❝❡s ❝❤❛r❣❡❞ ❜② t❤❡ r❡t❛✐❧❡rs ❛♥❞ st♦r❡s ❛r❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❣✐✈❡♥
❜②
p∗ =
c
ρ
P∗ =
λc
ρ
❛♥❞ t❤✉s t❤❡ ♣r✐❝❡ r❛t✐♦ ✐s ❡q✉❛❧ t♦ λ < 1✳ ❘❡t❛✐❧❡rs✬ ❛♥❞ st♦r❡s✬ ♣r♦✜t ❢✉♥❝t✐♦♥s ❛r❡ ♥♦✇ ❣✐✈❡♥ ❜②
πR (n, N ) = (1 − ρ)
πS (n, N ) = k(1 − ρ)
x̄0 x̄1 − x̄0
+
n
n + kN
1 − x̄1 x̄1 − x̄0
+
kN
n + kN
✭✸✷✮
✭✸✸✮
✇❤❡r❡ k ≡ λ−(σ−1) α−σ ✳ ◆♦t❡ t❤❛t k ✐s ❧❛r❣❡r ✭s♠❛❧❧❡r✮ t❤❛♥ 1 ✇❤❡♥ ♦♥❧② ❡✣❝✐❡♥❝② ✭q✉❛❧✐t②✮
❤❡t❡r♦❣❡♥❡✐t② ♠❛tt❡rs✳ ❲❤❡♥ ❜♦t❤ ❦✐♥❞s ♦❢ ❤❡t❡r♦❣❡♥❡✐t② ❛r❡ ♣r❡s❡♥t✱ k ❝❛♥ ❜❡ ❡✐t❤❡r s♠❛❧❧❡r ♦r
❧❛r❣❡r t❤❛♥ 1✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤✐❝❤ ♦❢ t❤❡ t✇♦ t②♣❡s ♦❢ ❤❡t❡r♦❣❡♥❡✐t② ❞♦♠✐♥❛t❡s✳ ❲❤❡♥ k > 1✱
❡✈❡r②t❤✐♥❣ ✇♦r❦ ❛s ✐❢ t❤❡ ♥✉♠❜❡r ♦❢ st♦r❡s ✇❡r❡ ❧❛r❣❡r ❛♥❞ ❣✐✈❡♥ ❜② kN ✐♥st❡❛❞ ♦❢ N ✳ ❚❤✐s ✐♥
t✉r♥ ♠❛❦❡s t❤❡ ❙▼ ♠♦r❡ ❛ttr❛❝t✐✈❡✱ ❛♥❞ t❤✉s ❡①♣❡♥❞✐t✉r❡ ✐♥ t❤❡ ❙▼ ❛❧s♦ ❣❡ts ❤✐❣❤❡r ❛♥❞ ❡q✉❛❧ t♦
k > 1✳ ❲❤❡♥ k < 1✱ t❤❡ ❛r❣✉♠❡♥t ✐s r❡✈❡rs❡❞✳ ❚❤✉s✱ t❤❡ t✇♦ t②♣❡s ♦❢ ❤❡t❡r♦❣❡♥❡✐t② ❜❡t✇❡❡♥ t❤❡
❝❡♥t❡rs ❛r❡ ❢♦r♠❛❧❧② ❡q✉✐✈❛❧❡♥t✳
■t ✐s ❡❛s✐❧② ✈❡r✐✜❡❞ t❤❛t t❤❡ ♠❛r❣✐♥❛❧ ❝♦♥s✉♠❡rs ❛r❡ ♥♦✇ ❣✐✈❡♥ ❜②
ln (1 + kN/n)
x̄0 = max 0, 1 −
ln T
ln (1 + n/kN )
x̄1 = min 1,
ln T
✭✸✹✮
✭✸✺✮
✇❤❡r❡ N ✐s ❛❣❛✐♥ r❡♣❧❛❝❡❞ ✇✐t❤ kN ✳
❚❤❡ ❡①♣r❡ss✐♦♥s ✭✸✷✮✲✭✸✺✮ s❤♦✇ t❤❛t t❤❡ ♣r♦✜t ❢✉♥❝t✐♦♥s πR (n, N ) ❛♥❞ πS (n, N ) ❛r❡ t❤❡ s❛♠❡
❛s t❤♦s❡ ♦❜t❛✐♥❡❞ ✇❤❡♥ k = 1 ✉♣ t♦ r❡♣❧❛❝✐♥❣ N ✇✐t❤ kN ❛♥❞ ♠✉❧t✐♣❧②✐♥❣ πS ❜② k ✳ ❚❤❡r❡❢♦r❡✱
t❤❡ r❡s✉❧ts t❤❛t ❤♦❧❞ ✉♥❞❡r k > 1 ❛r❡ q✉❛❧✐t❛t✐✈❡❧② t❤❡ s❛♠❡ ❛s t❤♦s❡ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ❢♦r❡❣♦✐♥❣
✷✺
t✇♦ s❡❝t✐♦♥s✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t ✏❡✛❡❝t✐✈❡✑ ♥✉♠❜❡r ♦❢ st♦r❡s✱
♥✉♠❜❡r✱
N✳
kN ✱
❡①❝❡❡❞s t❤❡ ✏❛❝t✉❛❧✑
■♥ ♣❛rt✐❝✉❧❛r✱ s❡❧❧✐♥❣ ❛t ❛ ❧♦✇❡r ♣r✐❝❡ tr❛♥s❧❛t❡s ✐♥t♦ ❛ s♠❛❧❧❡r ❧✐♠✐t s✐③❡ ❢♦r t❤❡ ❙▼✱
✇❤✐❝❤ ✐s ❧❡ss ❝♦st❧② ❢♦r t❤❡ ❞❡✈❡❧♦♣❡r t♦ ✐♠♣❧❡♠❡♥t✳
❖t❤❡r ❤❡t❡r♦❣❡♥❡✐t✐❡s s✉❝❤ ❛s ❛ ❤✐❣❤❡r ❞❡❣r❡❡ ♦❢ ♣r♦❞✉❝t ❞✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ t❤❡ ❙❙❚ t❤❛♥ ✐♥ t❤❡
❙▼✱ ✇❤✐❝❤ s✉♣♣❧✐❡s ♠♦r❡ st❛♥❞❛r❞✐③❡❞ ❣♦♦❞s✱ ❛♥❞ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ✉r❜❛♥ ❛♠❡♥✐t✐❡s ❛✈❛✐❧❛❜❧❡ ❛t t❤❡
❝✐t② ❝❡♥t❡r✱ ❝❛♥ s✐♠✐❧❛r❧② ❜❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ❛s ❧♦♥❣ ❛s t❤❡ ♣r✐❝❡ r❛t✐♦ r❡♠❛✐♥s ❝♦♥st❛♥t✳
✻✳✷
❈✐t② s✐③❡
❖✇✐♥❣ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♥❡t✇♦r❦ ❡✛❡❝ts✱ ✐t ✐s ✇♦rt❤ st✉❞②✐♥❣ ❤♦✇ ❛♥ ❡①♦❣❡♥♦✉s s❤♦❝❦ ♦♥ t❤❡
❝✐t②✬s ♣♦♣✉❧❛t✐♦♥ s✐③❡
L
❛✛❡❝ts t❤❡ ♠❛r❦❡t ♦✉t❝♦♠❡✳ ❈♦♥s✉♠❡rs✬ ♣r❡❢❡r❡♥❝❡s ❛r❡ st✐❧❧ ❣✐✈❡♥ ❜② ✭✶✮✱
✇❤✐❧❡ r❡t❛✐❧❡rs✬ ❛♥❞ st♦r❡s✬ ♦♣❡r❛t✐♥❣ ♣r♦✜ts ❛r❡ ❣✐✈❡♥ ❜②
πl
❜❡✐♥❣ ❣✐✈❡♥ ❜② ✭✶✺✮ ✕ ✭✶✻✮✳
Lπl (n, N ) ✇✐t❤ l ∈ {R, S}✱
t❤❡ ❢✉♥❝t✐♦♥s
❲❡ ♠❛② t❛❝❦❧❡ t❤✐s ♣r♦❜❧❡♠ ❢r♦♠ t✇♦ ❞✐✛❡r❡♥t ❛♥❣❧❡s✳ ■♥ t❤❡ ✜rst ♦♥❡✱ ✇❡ st✉❞② t❤❡ ✐♠♣❛❝t ♦❢
❛ ❣r♦✇✐♥❣ ❝✐t② ♦♥ t❤❡ s✐③❡
n∗ (N ; L)
♦❢ t❤❡ ❙❙❚ ✇❤❡♥
N
✐s ❣✐✈❡♥ ❛♥❞ ❞❡t❡r♠✐♥❡❞ t❤r♦✉❣❤ ❞✐✛❡r❡♥t
✐♥st✐t✉t✐♦♥❛❧ ♠❡❝❤❛♥✐s♠s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛♥❛❧②s✐s ♦❢ ❙❡❝t✐♦♥ ✹✳✶ ✐s ❛♣♣❧✐❝❛❜❧❡✳
❋r❡❡ ❡♥tr② ✐♥ t❤❡ ❙❙❚ ❛♥❞ ❤♦♠♦❣❡♥❡✐t② ♦❢
πR
πR
✐♠♣❧✐❡s
n N
,
L L
=f
✭✸✻✮
✇❤✐❝❤ ②✐❡❧❞s
∗
n (N ; L) = L n
❜❡✐♥❣ ❞❡❝r❡❛s✐♥❣ ✐♥
N✱
N
;1 .
L
✭✸✼✮
L ❛♠♦✉♥ts t♦ ❞❡❝r❡❛s✐♥❣ N ✳ ■t t❤❡♥
❢♦❧❧♦✇s ❢r♦♠ ✭✸✼✮ t❤❛t✱ ✇❤❡♥ t❤❡ ❙▼ ❤❛s ❛ ❣✐✈❡♥ s✐③❡ N ✱ ❛ ♣♦♣✉❧❛t✐♦♥ ❤✐❦❡ tr✐❣❣❡rs t❤❡ ❡♥tr② ♦❢ ❛
♠♦r❡ t❤❛♥ ♣r♦♣♦rt✐♦♥❛t❡ ♥✉♠❜❡r ♦❢ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs✳ ❚❤❡ ✐♥t✉✐t✐♦♥ ❢♦r t❤✐s ❛ ♣r✐♦r✐ ✉♥❡①♣❡❝t❡❞
❋✉rt❤❡r♠♦r❡✱
n∗ (N ; 1)
∗
r❛✐s✐♥❣
r❡s✉❧t ✐s ❛s ❢♦❧❧♦✇s✳ ❲❤❡♥ t❤❡r❡ ✐s ♥♦ ❙▼✱ t❤❡ s✐③❡ ♦❢ t❤❡ ❙❙❚ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♣♦♣✉❧❛t✐♦♥
s✐③❡✿
ne (L) = L(1 − ρ)/f ✳
❙✐♥❝❡
❙❡❝t✐♦♥ ✹✳✶✳ ❚❤✉s✱ ❢♦r ❛♥② ❣✐✈❡♥
N/L
n✱
❞❡❝r❡❛s❡s ✇✐t❤
L✱
t❤✐♥❣s ✇♦r❦ ❛s ✐❢
N
✇❡r❡ ❞❡❝r❡❛s✐♥❣ ✐♥
♣r♦✜ts ♦❢ ❞♦✇♥t♦✇♥ r❡t❛✐❧❡rs ❛r❡ s❤✐❢t❡❞ ✉♣✇❛r❞✱ ✇❤✐❝❤ ✐♥✈✐t❡s
❛❞❞✐t✐♦♥❛❧ ❡♥tr②✳
■♥ t❤❡ s❡❝♦♥❞ ♦♥❡✱ ✇❡ st✉❞② t❤❡ ✐♠♣❛❝t ♦❢ ❛ ❣r♦✇✐♥❣ ❝✐t② ♦♥ t❤❡ ❞❡✈❡❧♦♣❡rs✬ ❧✐♠✐t s✐③❡✳ ■t ❢♦❧❧♦✇s
❢r♦♠ ✭✸✻✮ t❤❛t
π(µ) =
❙✐♥❝❡
π(µ)
✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢
L✱
t❤❡ ❧✐♠✐t s✐③❡
Nf
.
L
N̄ (L)
N̄ (L) =
✷✻
✭✸✽✮
♦❢ t❤❡ ❙▼ ✐s ❣✐✈❡♥ ❜②
L
π(µ̄)
f
✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ ❧✐♠✐t s✐③❡ ♦❢ t❤❡ ❙▼ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝✐t②✬s ♣♦♣✉❧❛t✐♦♥ s✐③❡✿ N̄ (L) =
L N̄ (1)✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐♥ ❛ ❜✐❣❣❡r ❝✐t② ✐t ✐s ❤❛r❞❡r ❢♦r t❤❡ ❞❡✈❡❧♦♣❡r t♦ s❡❝✉r❡ t❤❡ ✇❤♦❧❡ ♠❛r❦❡t✳
❚❤❡s❡ r❡s✉❧ts t♦❣❡t❤❡r s✉♣♣♦rt t❤❡ ✐❞❡❛ t❤❛t ❛ ❧❛r❣❡r ❝✐t② ✐s ♠♦r❡ ❧✐❦❡❧② t♦ r❡t❛✐♥ ❞♦✇♥t♦✇♥
r❡t❛✐❧❡rs t❤❛♥ ❛ s♠❛❧❧❡r ✭♦r ♣♦♦r❡r✮ ❝✐t②✳ ❚♦ s♦♠❡ ❡①t❡♥t✱ t❤✐s ❡①♣❧❛✐♥s ✇❤② ❧❛r❣❡ ❛♥❞ r✐❝❤ ❝✐t✐❡s✱ s✉❝❤
❛s ◆❡✇ ❨♦r❦✱ ❙❛♥ ❋r❛♥❝✐s❝♦✱ P❛r✐s ♦r ▼✐❧❛♥✱ ❤❛✈❡ ♠❛✐♥t❛✐♥❡❞ ❛ ✈✐❜r❛♥t ❞♦✇♥t♦✇♥✱ ▲♦s ❆♥❣❡❧❡s
❜❡✐♥❣ ❛ ♣r♦♠✐♥❡♥t ❝♦✉♥t❡r✲❡①❛♠♣❧❡✳ ❚❤❡ ❤♦❧❧♦✇✐♥❣✲♦✉t ♦❢ ✉r❜❛♥ ❝❡♥t❡rs ❝❤❛r❛❝t❡r✐③❡s ♠❛✐♥❧②
s♠❛❧❧ ❛♥❞ ♠❡❞✐✉♠ s✐③❡ ❝✐t✐❡s✱ ❡s♣❡❝✐❛❧❧② t❤♦s❡ ✇❤✐❝❤ ❞♦ ♥♦t ❤❛✈❡ ❤✐st♦r✐❝❛❧ ❛♠❡♥✐t✐❡s ❣❡♥❡r❛t❡❞ ❜②
♠♦♥✉♠❡♥ts✱ ❜✉✐❧❞✐♥❣s✱ ♣❛r❦s✱ ❛♥❞ ♦t❤❡r ✉r❜❛♥ ✐♥❢r❛str✉❝t✉r❡ ❢r♦♠ ♣❛st ❡r❛s t❤❛t ❛r❡ ❛❡st❤❡t✐❝❛❧❧②
♣❧❡❛s✐♥❣ t♦ ♣❡♦♣❧❡✳
✻✳✸
P♦♣✉❧❛t✐♦♥ ❞❡♥s✐t②
■♥ ♠♦♥♦❝❡♥tr✐❝ ❝✐t✐❡s✱ ♣♦♣✉❧❛t✐♦♥ ✐s ❝♦♥❝❡♥tr❛t❡❞ ❛r♦✉♥❞ t❤❡ ❝❡♥tr❛❧ ❜✉s✐♥❡ss ❞✐str✐❝t ✭❋✉❥✐t❛ ❛♥❞
❚❤✐ss❡✱ ✷✵✶✸✮✳ ▲❡t g(x) ❛♥❞ G(x) ❜❡✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❞❡♥s✐t② ❛♥❞ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐✲
❜✉t✐♦♥ ❢✉♥❝t✐♦♥s✳ ❖❜s❡r✈❡✱ ✜rst✱ t❤❛t t❤❡ ♠❛r❣✐♥❛❧ ❝♦♥s✉♠❡rs x̄0 ❛♥❞ x̄1 ❣✐✈❡♥ ❜② ✭✶✸✮ ❛♥❞ ✭✶✹✮
❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ G✳ ❍❡♥❝❡✱ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ♣❡♦♣❧❡ ✇❤♦ ❝❤♦♦s❡ ♦♥❡✲st♦♣ s❤♦♣♣✐♥❣ ❛t t❤❡ ❙❙❚
✭❙▼✮ ❡q✉❛❧s G(x̄0 ) ✭1 − G(x̄1 )✮✱ ✇❤❡r❡❛s G(x̄1 ) − G(x̄0 ) ✐s t❤❡ s❤❛r❡ ♦❢ ❝♦♥s✉♠❡rs ✇❤♦ s❤♦♣ ✐♥
❜♦t❤ ♠❛r❦❡t♣❧❛❝❡s✳ ❋✉rt❤❡r✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ♦❢ ❡❛❝❤ ✈❛r✐❡t② ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s ❜♦✉❣❤t ❜②
❝♦♥s✉♠❡rs ❛r❡ ❛❧s♦ ✐♥❞❡♣❡♥❞❡♥t ♦❢ G✳ ❍❡♥❝❡✱ r❡t❛✐❧❡rs✬ ♦♣❡r❛t✐♥❣ ♣r♦✜ts ❛r❡ ❣✐✈❡♥ ❜②
G(x̄0 ) G(x̄1 ) − G(x̄0 )
+
.
πR (n, N ) = (1 − ρ)
n
n+N
✭✸✾✮
❚❤❡ ❛♥❛❧②s✐s ♦❢ ❙❡❝t✐♦♥s ✸ ❛♥❞ ✹ st✐❧❧ ❤♦❧❞s ❢♦r ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❞✐str✐❜✉t✐♦♥s G(x)✳ ❇❛s✐❝❛❧❧②✱ ❢♦r
♠♦st ♦❢ ♦✉r ❛♥❛❧②s✐s✱ ✇❤❛t ✇❡ ♥❡❡❞ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝♦♥❞✐t✐♦♥s✿ πR (0, N ) = πR (∞, N ) = 0
❢♦r N > 0✳ ❚♦ s❤♦✇ ❤♦✇ t❤✐s ✇♦r❦s✱ ❝♦♥s✐❞❡r ❛ ♥❡❣❛t✐✈❡ ❡①♣♦♥❡♥t✐❛❧ ♣♦♣✉❧❛t✐♦♥ ❞❡♥s✐t②✱ ✇❤✐❝❤ ✐s
❦♥♦✇♥ t♦ ♣r♦✈✐❞❡ ❛ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❝✐t② ♣♦♣✉❧❛t✐♦♥ ❞❡♥s✐t✐❡s ✭❆♥❛s ❡t ❛❧✳✱ ✶✾✾✽✮✿
g(x) ≡
1 − e−αx
αe−αx
G(x)
=
1 − e−α
1 − e−α
✭✹✵✮
✇❤❡r❡ α > 0 ♠❡❛s✉r❡s t❤❡ ♣♦♣✉❧❛t✐♦♥ s❦❡✇♥❡ss t♦✇❛r❞s t❤❡ ❙❙❚✳ ❲❤❡♥ α = 0✱ t❤❡ ❞❡♥s✐t② ✐s
✉♥✐❢♦r♠✳
❙✐♠✉❧❛t✐♦♥s s❤♦✇ t❤❛t ✭✐✮ ❢♦r ❡❛❝❤ T t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ ♣♦s✐t✐✈❡ t❤r❡s❤♦❧❞ ✈❛❧✉❡ ᾱ(T )✱ s✉❝❤ t❤❛t
πR ✐s ✉♥✐♠♦❞❛❧ ✐♥ n ✐❢ ❛♥❞ ♦♥❧② ✐❢ α ≤ ᾱ(T )✱ ✭✐✐✮ πS ✐s ✉♥✐♠♦❞❛❧ ✐♥ N ❀ ❛♥❞ ✭✐✐✐✮ ✇❤❡♥ α ≤ ᾱ(T )✱ t❤❡
❞❡✈❡❧♦♣❡r✬s r❡✈❡♥✉❡ ✐♥❝r❡❛s❡s ✐♥ N ✳ ❚❤✐s ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣❧✐❝❛t✐♦♥s✳ ❋✐rst✱ ✇❤❡♥ t❤❡ ♣♦♣✉❧❛t✐♦♥
✐s ♥♦t t♦♦ ♠✉❝❤ ❝♦♥❝❡♥tr❛t❡❞ t♦✇❛r❞ t❤❡ ❙❙❚✱ N̄ ❡①✐sts s✉❝❤ t❤❛t ❢♦r ❡❛❝❤ N ∈]0, N̄ [ t❤❡r❡ ✐s ❛
✉♥✐q✉❡ ♣❧❛✉s✐❜❧❡ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠❀ ♦t❤❡r✇✐s❡✱ s❡✈❡r❛❧ ♣❧❛✉s✐❜❧❡ ❡q✉✐❧✐❜r✐❛ ♠❛② ❡①✐st✳ ❙❡❝♦♥❞✱
✐❢ α ✐s ♥♦t t♦♦ ❧❛r❣❡✱ ♥❡✐t❤❡r t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t ♥♦r t❤❡ ♠❛r❦❡t ❝r♦✇❞✐♥❣ ❡✛❡❝t ❛❧✇❛②s
❞♦♠✐♥❛t❡s t❤❡ ♦t❤❡r✳ ❚❤✐r❞✱ ✉♥❞❡r t❤❡ s❛♠❡ ❝♦♥❞✐t✐♦♥ ♦♥ α✱ t❤❡ ♠❛r❦❡t ♦✉t❝♦♠❡ ✐s ❞❡s❝r✐❜❡❞ ❜②
❛ ♣❡r❢❡❝t ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ✇❤♦s❡ str✉❝t✉r❡ ✐s s✐♠✐❧❛r t♦ t❤❛t st✉❞✐❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳ ❚❤✉s✱ ✇❤❡♥ t❤❡
✷✼
s❦❡✇♥❡ss ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ✐s ♥♦t t♦♦ ❤✐❣❤✱ t❤❡ ❦❡② r❡s✉❧ts ♦❜t❛✐♥❡❞ ✐♥ ❙❡❝t✐♦♥s ✸ ❛♥❞
✳
✹ r❡♠❛✐♥ q✉❛❧✐t❛t✐✈❡❧② t❤❡ s❛♠❡
✻✳✹
❚❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s❤♦♣♣✐♥❣ ♠❛❧❧
■t r❡♠❛✐♥s t♦ ❞✐s❝✉ss ✇❤❛t ♦✉r ❛♥❛❧②s✐s ❜❡❝♦♠❡s ✇❤❡♥ t❤❡ ❙▼ ✐s ❜✉✐❧t ❛t xSM ∈ [0, 1[✳ ❋♦r t❤❡
r❡❛s♦♥s ♠❡♥t✐♦♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✶✱ t❤❡ ❙❙❚ ✐s ❧♦❝❛t❡❞ ❛t x = 0✳
❚♦ s❡❡ ❤♦✇ ❝♦♥s✉♠❡rs✬ s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r ✐s ❛✛❡❝t❡❞ ❜② ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ ❙▼ ❧♦❝❛t✐♦♥✱ ✇❡ ❣♦
❜❛❝❦ t♦ t❤❡ ✐♥❞✐r❡❝t ✉t✐❧✐t✐❡s ❣✐✈❡♥ ✐♥ ❙❡❝t✐♦♥ ✷✳✷✳ ■t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ s❤♦✇ t❤❛t ✭✼✮✱ ✭✾✮ ❛♥❞
✭✶✶✮ ❜❡❝♦♠❡
V0 (x, xSM ) = (1 − ρ) ln n − τ x
V1 (x, xSM ) = (1 − ρ) ln N − τ |x − xSM |
V (x, xSM ) = (1 − ρ) ln (n + N ) − τ max{x, xSM }.
❚❤❡ ❦❡② ❞✐✛❡r❡♥❝❡ ✇✐t❤ t❤❡ ❝❛s❡ ✇❤❡r❡ xSM = 1 ✐s t❤❛t t❤❡ ✇❡❧❢❛r❡ ♦❢ ❝♦♥s✉♠❡rs ♣❛tr♦♥✐③✐♥❣
❜♦t❤ s❤♦♣♣✐♥❣ ♣❧❛❝❡s ♥♦✇ ❞❡♣❡♥❞s ♦♥ t❤❡✐r ❧♦❝❛t✐♦♥s ✭s❡❡ ❋✐❣✉r❡ ✹ ❢♦r ❛♥ ✐❧❧✉str❛t✐♦♥✮✳ ❚♦ ❜❡
♣r❡❝✐s❡✱ V ❞❡❝r❡❛s❡s ✇✐t❤ t❤❡ ❞✐st❛♥❝❡ x t♦ t❤❡ ❙▼ ❢♦r t❤♦s❡ ❝♦♥s✉♠❡rs ✇❤♦ ❛r❡ ❧♦❝❛t❡❞ t♦ t❤❡
r✐❣❤t ♦❢ xSM ✳ ▼♦r❡♦✈❡r✱ V str✐❝t❧② ✐♥❝r❡❛s❡s ✇❤❡♥ t❤❡ ❙▼ ❣❡ts ❝❧♦s❡r t♦ t❤❡ ❙❙❚✳ ■♥ ♦t❤❡r ✇♦r❞s✱
t❤❡ ❝❧♦s❡r ❙▼ t♦ t❤❡ ❙❙❚✱ t❤❡ ❧❛r❣❡r t❤❡ ♥✉♠❜❡r ♦❢ t✇♦✲st♦♣ s❤♦♣♣❡rs✳
❋✐❣✉r❡ ✹✳ ❙❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r ✉♥❞❡r ✈❛r✐❛❜❧❡ ❧♦❝❛t✐♦♥ ♦❢ ❙▼
✷✽
■t ✐s r❡❛❞✐❧② ✈❡r✐✜❡❞ t❤❛t
ln(1 + 1/ν)
ln T
✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ t✇♦ ♠❛r❣✐♥❛❧ ❝♦♥s✉♠❡rs ❧♦❝❛t❡❞ x̂0 ❛♥❞ x̂1 ✱ ✇❤✐❝❤ s♦❧✈❡✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡
❡q✉❛t✐♦♥s V0 = V ❛♥❞ V1 = V ✿
xSM ≥
x̂0 = max 0, xSM
ln(1 + ν)
−
ln T
ln (1 + 1/ν)
x̂1 = min 1,
ln T
✇❤✐❝❤ ❛r❡ ✐❞❡♥t✐❝❛❧ t♦ ✭✶✸✮ ❛♥❞ ✭✶✹✮ ✇❤❡♥ xSM = 1✳
❆s ✐♥ ❙❡❝t✐♦♥ ✷✳✷✱ ❛ ❝♦♥s✉♠❡r ❧♦❝❛t❡❞ ❛t x ✈✐s✐ts t❤❡ ❙❙❚ ♦♥❧② ✭❜♦t❤ s❤♦♣♣✐♥❣ ♣❧❛❝❡s✱ ♦r t❤❡
❙▼ ♦♥❧②✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ x < x̂0 ✭x̂0 ≤ x ≤ x̂1 ✱ ♦r x > x̂1 ✮✳ ◆♦t❡ t❤❛t x̂1 ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❙▼
❧♦❝❛t✐♦♥✱ ✇❤✐❧❡ x̂0 ♠♦✈❡s t♦❣❡t❤❡r ✇✐t❤ xSM ✳ ❚❤✉s✱ ✐❢
xSM ≤
ln(1 + ν)
ln T
✭✹✶✮
t❤❡♥ x̂0 = 0✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t ♥♦ ❝♦♥s✉♠❡r ❝❤♦♦s❡s t♦ ✈✐s✐t ♦♥❧② t❤❡ ❙❙❚✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢
xSM <
ln(1 + 1/ν)
ln T
✭✹✷✮
t❤❡♥ ❛❧❧ ❝♦♥s✉♠❡rs ❧♦❝❛t❡❞ t♦ t❤❡ r✐❣❤t ♦❢ x̂0 ❛r❡ t✇♦✲st♦♣ s❤♦♣♣❡rs✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡r❡ ✐s ❛
❞♦✇♥✇❛r❞ ❥✉♠♣ ✐♥ t❤❡ st♦r❡s✬ r❡✈❡♥✉❡ ❛t xSM = x̂1 ✳
■♥ s✉♠✱ ✐❢
xSM ≤ min
ln(1 + ν) ln(1 + 1/ν)
,
ln T
ln T
t❤❡♥ ❛❧❧ ❝♦♥s✉♠❡rs ❛r❡ t✇♦✲st♦♣ s❤♦♣♣❡rs✳ ❚❤✉s✱ ❝❤♦♦s✐♥❣ ❛ ❧♦❝❛t✐♦♥ ❝❧♦s❡r t♦ t❤❡ ❙❙❚ ❧❡❛✈❡s
st♦r❡s✬ r❡✈❡♥✉❡s✱ ❤❡♥❝❡ t❤❡ ❞❡✈❡❧♦♣❡r✬s r❡✈❡♥✉❡✱ ✉♥❝❤❛♥❣❡❞✳ ❙✐♥❝❡ t❤❡ ❞❡✈❡❧♦♣❡r✬s ❜✉✐❧❞✐♥❣ ❝♦st
❢✉♥❝t✐♦♥ B xSM , N D ✐s ❧✐❦❡❧② t♦ ❞❡❝r❡❛s❡ ✇✐t❤ xSM ❜❡❝❛✉s❡ t❤❡ ❧❛♥❞ r❡♥t ❛t ❛ ❧♦❝❛t✐♦♥ ❝❧♦s❡r t♦
t❤❡ ❝✐t② ❝❡♥t❡r ✐s ♠✉❝❤ ❤✐❣❤❡r ✭❋✉❥✐t❛ ❛♥❞ ❚❤✐ss❡✱ ✷✵✶✸✮✱ ❝❤♦♦s✐♥❣ ❛ ❧♦❝❛t✐♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡
❙❙❚ r❡❞✉❝❡s t❤❡ ❞❡✈❡❧♦♣❡r✬s ♣r♦✜ts✳ ❚❤✉s✱ t❤❡ ❞❡✈❡❧♦♣❡r ✐s ✐♥❝❡♥t✐✈✐③❡❞ t♦ ❝❤♦♦s❡ ❛ ❧♦❝❛t✐♦♥ ❢♦r
t❤❡ ❙▼ ❞✐st❛♥t ❢r♦♠ t❤❡ ❙❙❚✳ ❍♦✇ ❢❛r ❛r❡ t❤❡ t✇♦ s❤♦♣♣✐♥❣ ♣❧❛❝❡s ❞❡♣❡♥❞s ♦♥ t❤❡ ♣❛r❛♠❡t❡rs
♦❢ t❤❡ ♠❛r❦❡t✳ ❚♦ ❛ ❝❡rt❛✐♥ ❡①t❡♥t✱ ♦✉r ❛♥❛❧②s✐s ❤✐❣❤❧✐❣❤ts ✇❤② s❤♦♣♣✐♥❣ ♠❛❧❧s ♦r s✉♣❡r♠❛r❦❡ts
❤❛✈❡ ❝❤♦s❡♥ t♦ s❡t ✉♣ ❛t t❤❡ ❝✐t② ♦✉ts❦✐rts ♦♥❝❡ tr❛✈❡❧ ❝♦sts ❜❡❝❛♠❡ s✉✣❝✐❡♥t❧② ❧♦✇ t❤r♦✉❣❤ t❤❡
✇✐❞❡s♣r❡❛❞ ✉s❡ ♦❢ ❝❛rs ✭❋♦❣❡❧s♦♥✱ ✷✵✵✺✮✳
✼
❈♦♥❝❧✉❞✐♥❣ ❘❡♠❛r❦s
❲❡ ❤❛✈❡ ❞❡✈❡❧♦♣❡❞ ❛ ♠♦❞❡❧ ♦❢ ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ t✇♦ s♣❛t✐❛❧❧② s❡♣❛r❛t❡❞ s❤♦♣♣✐♥❣ ❛r❡❛s ❤❛✈✐♥❣
❞✐✛❡r❡♥t ♦r❣❛♥✐③❛t✐♦♥❛❧ ❢♦r♠s ❛♥❞ s❤♦✇❡❞ ❤♦✇ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ ❧♦✈❡ ❢♦r ✈❛r✐❡t② ❛♥❞ tr❛✈❡❧ ❝♦sts
❛✛❡❝ts ❝♦♥s✉♠❡rs✬ s❤♦♣♣✐♥❣ ❜❡❤❛✈✐♦r✳ ❚❤❡ st❛♥❞❛r❞ ❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝t ✐s s✉♣♣❧❡♠❡♥t❡❞ ❜② ❛
✷✾
♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t✱ ✇❤✐❝❤ st❡♠s ❢r♦♠ t❤❡ ❤✐❣❤❡r ❛ttr❛❝t✐✈❡♥❡ss ♦❢ ❛ s❤♦♣♣✐♥❣ ❛r❡❛ ♦✛❡r✐♥❣
❛ ✇✐❞❡r r❛♥❣❡ ♦❢ ♣r♦❞✉❝ts✳ ❖✉r r❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢♦r t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ 2 < T < 4✳
❲❤❛t ❞♦ t❤❡② ❜❡❝♦♠❡ ✇❤❡♥ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s ♥♦t s❛t✐s✜❡❞❄ ❙✐♠✉❧❛t✐♦♥s s✉❣❣❡st t❤❛t t❤❡② r❡♠❛✐♥
q✉❛❧✐t❛t✐✈❡❧② t❤❡ s❛♠❡ ❢♦r ❛ ❜r♦❛❞❡r ❞♦♠❛✐♥ ♦❢ ♣❛r❛♠❡t❡r ✈❛❧✉❡s✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ❛♥♦t❤❡r ❞♦♠❛✐♥
✐♥ ✇❤✐❝❤ ♠✉❧t✐♣❧❡ ✐♥t❡r✐♦r st❛❜❧❡ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐❛ ❛r✐s❡ ❢♦r ❜♦t❤ T < 2 ❛♥❞ T > 4✳ ❚❤❡ r❡❛s♦♥ ✐s
t❤❛t t❤❡ ❢✉♥❝t✐♦♥ π(µ) ♠❛② ❜❡❝♦♠❡ ❜✐♠♦❞❛❧✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t n∗ (N ) ❡①❤✐❜✐ts t✇♦ ❞✐s❝♦♥t✐♥✉✐t✐❡s✳
❯s✐♥❣ t❤❡ s❛♠❡ ❛r❣✉♠❡♥t ❛s ✐♥ ❙❡❝t✐♦♥ ✹✳✷✱ ✇❡ ♠❛② ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❡①♣❡❝t❡❞ ♦✉t❝♦♠❡ ✐s ❣✐✈❡♥
t❤❡ ❧❛r❣❡st st❛❜❧❡ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐✉♠✱ t❤❡ s✐③❡ ♦❢ ✇❤✐❝❤ ✐s s♠❛❧❧❡r t❤❛♥ ne ✳ ❚❤✉s✱ ❞❡❛❧✐♥❣ ✇✐t❤
t❤❡ ❞♦♠❛✐♥s T < 2 ❛♥❞ T > 4 r❡q✉✐r❡s ❧♦♥❣❡r ❞❡✈❡❧♦♣♠❡♥ts✱ ❜✉t ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ♥❛t✉r❡ ♦❢
♦✉r r❡s✉❧ts✳ ▼♦r❡♦✈❡r✱ ✇❤❡t❤❡r ❝♦♥s✉♠❡rs ❛r❡ ♦♥❡✲st♦♣ ♦r t✇♦✲st♦♣ s❤♦♣♣❡rs st✐❧❧ ❞❡♣❡♥❞s ♦♥ t❤❡
r❡❧❛t✐✈❡ s✐③❡ ♦❢ t❤❡ t✇♦ s❤♦♣♣✐♥❣ ♣❧❛❝❡s✳
❘❡❧❛①✐♥❣ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❈❊❙ ♣r❡❢❡r❡♥❝❡s ✐♠♣❧✐❡s t❤❛t ❝♦♥s✉♠❡rs✬ ❡①♣❡♥❞✐t✉r❡s ♦♥ t❤❡ ❞✐❢✲
❢❡r❡♥t✐❛t❡❞ ❣♦♦❞ ❛♥❞ s❡❧❧❡rs✬ ♠❛r❦✉♣s ❛r❡ ✈❛r✐❛❜❧❡ ❛♥❞ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❡t✐❡s
❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ t✇♦ ♠❛r❦❡t♣❧❛❝❡s✳ ❚②♣✐❝❛❧❧②✱ ❝♦♥s✉♠❡rs ❡①♣❡♥❞✐t✉r❡s ✐♥ ❛ s❤♦♣♣✐♥❣ ❛r❡❛ ✐♥❝r❡❛s❡s
✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❡t✐❡s s✉♣♣❧✐❡❞ t❤❡r❡✐♥✱ ✇❤✐❧❡ s❡❧❧❡rs✬ ♠❛r❦✉♣s ❞❡❝r❡❛s❡ ✇✐t❤ t❤❡ ♥✉♠❜❡r
♦❢ ❧♦❝❛❧ ❝♦♠♣❡t✐t♦rs✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❛♥❞ ♠❛r❦❡t ❡①♣❛♥s✐♦♥
❡✛❡❝ts ❜❡❝♦♠❡s ♠✉❝❤ ♠♦r❡ ✐♥✈♦❧✈❡❞✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤❡ s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥❛❧ ❢♦r♠s ♦❢ πR ❛♥❞ πS ❛r❡
♥♦t ♥❡❡❞❡❞ ❢♦r t❤❡ ♠❛✐♥ ✜♥❞✐♥❣s ♦❢ ❙❡❝t✐♦♥s ✸ ❛♥❞ ✹ t♦ ❤♦❧❞✳ ❲❡ ♥❡❡❞ ♦♥❧② t❤❡s❡ ❢✉♥❝t✐♦♥s t♦
❞✐s♣❧❛② ❛ ❢❡✇ r❡❣✉❧❛r✐t✐❡s ✇❡ ♥♦✇ s✉♠♠❛r✐③❡✳ ❋✐rst✱ πR ♠✉st ❜❡ ✉♥✐♠♦❞❛❧ ✐♥ n t♦ ❣❡t t❤❡ ♣❛tt❡r♥
♦❢ ❢r❡❡✲❡♥tr② ❡q✉✐❧✐❜r✐❛ ❞❡s❝r✐❜❡❞ ✐♥ Pr♦♣♦s✐t✐♦♥ ✷✳ ❙❡❝♦♥❞✱ πR ♠✉st ❜❡ ✉♥✐♠♦❞❛❧ ✐♥ n ❛♥❞ πS ✐♥
N ❢♦r t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t t♦ ♦✈❡r❝♦♠❡ ✭❜❡ ❞♦♠✐♥❛t❡❞ ❜②✮ t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝t ✇❤❡♥
t❤❡ s❤♦♣♣✐♥❣ ❛r❡❛ ✐s s♠❛❧❧ ✭❧❛r❣❡✮✳ ▲❛st✱ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❞❡✈❡❧♦♣❡r ❞❡s❝r✐❜❡❞ ✐♥ s✉❜s❡❝t✐♦♥ ✹✳✸
❤✐♥❣❡s ♦♥ t❤❡ ❢❛❝t t❤❛t t❤❡ ❞❡✈❡❧♦♣❡r✬s r❡✈❡♥✉❡ ✐♥❝r❡❛s❡s ✐♥ t❤❡ ❙▼ s✐③❡✳
❖✉r ♣❛♣❡r ❣❡♥❡r❛❧✐③❡s t❤❡ ❧❛✇ ♦❢ r❡t❛✐❧ ❣r❛✈✐t❛t✐♦♥ ♣r♦♣♦s❡❞ ❜② ❘❡✐❧❧② ✭✶✾✸✶✮✱ ✇❤✐❝❤ ❤❛s ❜❡❡♥
❡①t❡♥s✐✈❡❧② st✉❞✐❡❞ ✐♥ s♣❛t✐❛❧ ✐♥t❡r❛❝t✐♦♥ t❤❡♦r②✳ ❆❝❝♦r❞✐♥❣ t♦ t❤✐s ❧❛✇✱ t✇♦ ❝✐t✐❡s ❛ttr❛❝t ❝♦♥s✉♠❡rs
❧✐✈✐♥❣ ✐♥ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ ♣❧❛❝❡ ✐♥ ❞✐r❡❝t ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥s ♦❢ t❤❡ t✇♦ ❝✐t✐❡s ❛♥❞ ✐♥ ✐♥✈❡rs❡
♣r♦♣♦rt✐♦♥ t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡s t♦ t❤❡s❡ t✇♦ ❝✐t✐❡s✳ ❲❡ s❤♦✇ t❤❛t ❝♦♥s✉♠❡rs ❝❧♦s❡ t♦ ♦♥❡
❝✐t② ♥❡❡❞ ♥♦t ✈✐s✐t ❜♦t❤ ♦❢ t❤❡♠✱ ✇❤✐❧❡ t❤♦s❡ ❧♦❝❛t❡❞ ✐♥ ✐♥t❡r♠❡❞✐❛t❡ ♣❧❛❝❡s s♣❧✐t t❤❡✐r ❡①♣❡♥❞✐t✉r❡s
❜❡t✇❡❡♥ t❤❡ t✇♦ ❝✐t✐❡s ❛❝❝♦r❞✐♥❣ t♦ ❛ r✉❧❡ ♠♦r❡ ❣❡♥❡r❛❧ t❤❛♥ ❘❡✐❧❧②✬s✳ ■♥ t❤❡ s❛♠❡ ✈❡✐♥✱ ♦✉r ♠♦❞❡❧
✐s ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡ ❣r❛✈✐t② ❡q✉❛t✐♦♥ ✉s❡❞ ✐♥ tr❛❞❡ t❤❡♦r② ✭❆♥❞❡rs♦♥ ❛♥❞ ✈❛♥ ❲✐♥❝♦♦♣✱ ✷✵✵✹✮✱
❜② ♣r♦✈✐❞✐♥❣ ♠✐❝r♦s♣❛t✐❛❧ ❢♦✉♥❞❛t✐♦♥s t♦ t❤✐s ❡q✉❛t✐♦♥✳
❋✐♥❛❧❧②✱ ✐♥ t❤❡ ✇❛❦❡ ♦❢ ❍♦t❡❧❧✐♥❣ ✭✶✾✷✾✮✱ ✇❡ ❝❛♥ r❡✐♥t❡r♣r❡t ♦✉r ♠♦❞❡❧ t♦ ❞❡s❝r✐❜❡ ❛ ♣♦♣✉❧❛t✐♦♥
♦❢ ❝♦♥s✉♠❡rs ❤❡t❡r♦❣❡♥❡♦✉s ✐♥ t❤❡✐r ❛tt✐t✉❞❡ t♦✇❛r❞ t❤❡ ♦r❣❛♥✐③❛t✐♦♥❛❧ ❢♦r♠ ✉s❡❞ t♦ s✉♣♣❧② ❛
❞✐✛❡r❡♥t✐❛t❡❞ ❣♦♦❞ ✲ t❤✐♥❦ ♦❢ ❜❛❦❡r✐❡s✱ ❜r❡✇❡r✐❡s✱ ♦r ❜♦♦❦st♦r❡s✳ ■♥ t❤❡ ✜rst ♦♥❡✱ t❤❡ ❣♦♦❞ ✐s
♣r♦❞✉❝❡❞ ❛♥❞ s♦❧❞ ❜② ❛ ❧❛r❣❡ ✜r♠❀ ✐♥ t❤❡ s❡❝♦♥❞✱ ✐t ✐s ♣r♦✈✐❞❡❞ ❜② ♠❛♥② s♠❛❧❧ ✜r♠s✳ ❚❤♦✉❣❤
❝♦♥s✉♠❡rs ❧✐❦❡ ✈❛r✐❡t②✱ t❤❡② ♠❛② ❛❧s♦ ❤❛✈❡ ❛ str♦♥❣ ♣r❡❢❡r❡♥❝❡ ❢♦r ♦♥❡ ♦r❣❛♥✐③❛t✐♦♥❛❧ ❢♦r♠ ♦✈❡r t❤❡
♦t❤❡r✳ ❖✉r r❡s✉❧ts t❤❡♥ s❤♦✇ ❤♦✇ t❤❡ t✇♦ ♦r❣❛♥✐③❛t✐♦♥❛❧ ❢♦r♠s ✐♥t❡r❛❝t t♦ s❤❛♣❡ t❤❡ str✉❝t✉r❡ ♦❢
t❤❡ ✐♥❞✉str②✳
✸✵
❘❡❢❡r❡♥❝❡s
❬✶❪ ❆♥❛s✱ ❆✳✱ ❘✳ ❆r♥♦tt✱ ❛♥❞ ❑✳❆✳ ❙♠❛❧❧ ✭✶✾✾✽✮ ❯r❜❛♥ s♣❛t✐❛❧ str✉❝t✉r❡✳
▲✐t❡r❛t✉r❡ ✸✻✿ ✶✹✷✻✲✻✹✳
❬✷❪ ❆♥❞❡rs♦♥✱ ❏✳✱ ❛♥❞ ✈❛♥ ❲✐♥❝♦♦♣✱ ❊✳ ✭✷✵✵✹✮ ❚r❛❞❡ ❝♦sts✳
✻✾✶✲✼✺✶✳
❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝
❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝ ▲✐t❡r❛t✉r❡
✹✷✿
❬✸❪ ❆r♠str♦♥❣✱ ▼✳ ❛♥❞ ❏✳ ❱✐❝❦❡rs ✭✷✵✶✵✮ ❈♦♠♣❡t✐t✐✈❡ ♥♦♥✲❧✐♥❡❛r ♣r✐❝✐♥❣ ❛♥❞ ❜✉♥❞❧✐♥❣✳ ❘❡✈✐❡✇ ♦❢
❊❝♦♥♦♠✐❝ ❙t✉❞✐❡s ✼✼✿ ✸✵✲✻✵✳
❬✹❪ ❇❡rtr❛♥❞✱ ▼✳ ❛♥❞ ❋✳ ❑r❛♠❛r③ ✭✷✵✵✷✮ ❉♦❡s ❡♥tr② r❡❣✉❧❛t✐♦♥ ❤✐♥❞❡r ❥♦❜ ❝r❡❛t✐♦♥❄ ❊✈✐❞❡♥❝❡
❢r♦♠ t❤❡ ❋r❡♥❝❤ r❡t❛✐❧ ✐♥❞✉str②✳ ◗✉❛rt❡r❧② ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s ✶✶✼✿ ✶✸✻✾✲✹✶✸✳
❬✺❪ ❈❤❡♠❧❛✱ ●✳ ✭✷✵✵✸✮ ❉♦✇♥str❡❛♠ ❝♦♠♣❡t✐t✐♦♥✱ ❢♦r❡❝❧♦s✉r❡✱ ❛♥❞ ✈❡rt✐❝❛❧ ✐♥t❡❣r❛t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢
❊❝♦♥♦♠✐❝s ❛♥❞ ▼❛♥❛❣❡♠❡♥t ❙tr❛t❡❣② ✶✷✿ ✷✻✶✲✽✾✳
❬✻❪ ❈❤❡♥✱ ❨✳ ❛♥❞ ▼✳❍✳ ❘✐♦r❞❛♥ ✭✷✵✵✽✮ Pr✐❝❡✲✐♥❝r❡❛s✐♥❣ ❝♦♠♣❡t✐t✐♦♥✳ ❘❆◆❉ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s
✸✾✿ ✶✵✹✷✲✺✽✳
❬✼❪ ❈❤♦✉✱ ❈✳❋✳ ❛♥❞ ❖✳ ❙❤② ✭✶✾✾✵✮ ◆❡t✇♦r❦ ❡✛❡❝ts ✇✐t❤♦✉t ♥❡t✇♦r❦ ❡①t❡r♥❛❧✐t✐❡s✳
❏♦✉r♥❛❧ ♦❢ ■♥❞✉str✐❛❧ ❖r❣❛♥✐③❛t✐♦♥ ✽✿ ✷✺✾✲✼✵✳
■♥t❡r♥❛t✐♦♥❛❧
❬✽❪ ❈♦❤❡♥✱ ▲✳ ✭✶✾✾✻✮ ❋r♦♠ t♦✇♥ ❝❡♥t❡r t♦ s❤♦♣♣✐♥❣ ❝❡♥t❡r✿ ❚❤❡ r❡❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❝♦♠♠✉♥✐t②
♠❛r❦❡t♣❧❛❝❡s ✐♥ P♦st✇❛r ❆♠❡r✐❝❛✳ ❆♠❡r✐❝❛♥ ❍✐st♦r✐❝❛❧ ❘❡✈✐❡✇ ✶✵✶✿ ✶✵✺✵✲✽✳
❬✾❪ ❈♦✉rs❡r✱ ❩✳ ✭✷✵✵✺✮ ❲❛❧✲▼❛rt ❛♥❞ t❤❡ P♦❧✐t✐❝s ♦❢ ❆♠❡r✐❝❛♥ ❘❡t❛✐❧✳ ❲❛s❤✐♥❣t♦♥✱ ❉❈✿ ❈♦♠♣❡t✲
✐t✐✈❡ ❊♥t❡r♣r✐s❡ ■♥st✐t✉t❡✳
❬✶✵❪ ❈♦✉t✉r❡✱ ❱✳✱ ●✳ ❉✉r❛♥t♦♥ ❛♥❞ ▼✳ ❚✉r♥❡r ✭✷✵✶✷✮ ❙♣❡❡❞✳ ❯♥✐✈❡rs✐t② ♦❢ P❡♥♥s②❧✈❛♥✐❛✱ ❲❤❛rt♦♥
❙❝❤♦♦❧✳
❬✶✶❪ ❉✐①✐t✱ ❆✳❑✳ ❛♥❞ ❏✳❊✳ ❙t✐❣❧✐t③ ✭✶✾✼✼✮ ▼♦♥♦♣♦❧✐st✐❝ ❝♦♠♣❡t✐t✐♦♥ ❛♥❞ ♦♣t✐♠✉♠ ♣r♦❞✉❝t ❞✐✈❡rs✐t②✳
❆♠❡r✐❝❛♥ ❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇ ✻✼✿ ✷✾✼✲✸✵✽✳
❬✶✷❪ ❋♦❣❡❧s♦♥✱ ❘✳▼✳ ✭✷✵✵✺✮
Pr❡ss✳
❉♦✇♥t♦✇♥✳ ■ts r✐s❡ ❛♥❞ ❢❛❧❧✱ ✶✽✽✵✲✶✾✺✵✳
◆❡✇ ❍❛✈❡♥✿ ❨❛❧❡ ❯♥✐✈❡rs✐t②
❬✶✸❪ ❋✉❥✐t❛✱ ▼✳ ❛♥❞ ❏✳✲❋✳ ❚❤✐ss❡ ✭✷✵✶✸✮ ❊❝♦♥♦♠✐❝s ♦❢ ❆❣❣❧♦♠❡r❛t✐♦♥✳
❛♥❞ ●❧♦❜❛❧✐③❛t✐♦♥✳ ❈❛♠❜r✐❞❣❡✿ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳
❬✶✹❪ ●❡❤r✐❣✱ ❚✳ ✭✶✾✾✽✮ ❈♦♠♣❡t✐♥❣ ❡①❝❤❛♥❣❡s✳ ❊✉r♦♣❡❛♥
❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇
❬✶✺❪ ❍♦t❡❧❧✐♥❣✱ ❍✳ ✭✶✾✷✾✮ ❙t❛❜✐❧✐t② ✐♥ ❝♦♠♣❡t✐t✐♦♥✳ ❊❝♦♥♦♠✐❝
✸✶
❈✐t✐❡s✱ ■♥❞✉str✐❛❧ ▲♦❝❛t✐♦♥
❏♦✉r♥❛❧
✹✷✿ ✷✼✼✲✸✶✵✳
✸✾✿ ✹✶✲✺✼✳
❬✶✻❪ ▼❝▼✐❧❧❛♥✱ ❏✳✱ ❛♥❞ ▼✳ ❘♦t❤s❝❤✐❧❞ ✭✶✾✾✹✮ ❙❡❛r❝❤✳ ■♥✿ ❘✳❏✳ ❆✉♠❛♥♥ ❛♥❞ ❙✳ ❍❛rt ✭❡❞s✳✮✳ ❍❛♥❞❜♦♦❦
✳ ❆♠st❡r❞❛♠✿ ◆♦rt❤✲❍♦❧❧❛♥❞✱ ✾✵✺✲✷✼✳
♦❢ ●❛♠❡ ❚❤❡♦r② ✇✐t❤ ❊❝♦♥♦♠✐❝ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✉♠❡ ✷
❬✶✼❪ ❘❡✐❧❧②✱ ❲✳❏✳ ✭✶✾✸✶✮
✳ ◆❡✇ ❨♦r❦✿ P✐❧s❜✉r②✳
❚❤❡ ▲❛✇ ♦❢ ❘❡t❛✐❧ ●r❛✈✐t❛t✐♦♥
❬✶✽❪ ❙❝❤✉❧③✱ ◆✳ ❛♥❞ ❑✳ ❙t❛❤❧ ✭✶✾✾✻✮ ❉♦ ❝♦♥s✉♠❡rs s❡❛r❝❤ ❢♦r t❤❡ ❤✐❣❤❡st ♣r✐❝❡❄ ❊q✉✐❧✐❜r✐✉♠ ❛♥❞
♠♦♥♦♣♦❧✐st✐❝ ♦♣t✐♠✉♠ ✐♥ ❞✐✛❡r❡♥t✐❛t❡❞ ♣r♦❞✉❝ts ♠❛r❦❡ts✳
❘❆◆❉ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s
✷✼✿
✺✹✷✲✻✷✳
❬✶✾❪ ❙❤✐♠♦♠✉r❛✱ ❑✳✲■✳ ❛♥❞ ❏✳✲❋✳ ❚❤✐ss❡ ✭✷✵✶✷✮ ❈♦♠♣❡t✐t✐♦♥ ❛♠♦♥❣ t❤❡ ❜✐❣ ❛♥❞ t❤❡ s♠❛❧❧✳
❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s
✹✸✿ ✸✷✾✲✹✼✳
❬✷✵❪ ❙♠✐t❤✱ ❍✳ ❛♥❞ ❉✳ ❍❛② ✭✷✵✵✺✮ ❙tr❡❡ts✱ ♠❛❧❧s✱ ❛♥❞ s✉♣❡r♠❛r❦❡ts✳
▼❛♥❛❣❡♠❡♥t ❙tr❛t❡❣②
❘❆◆❉
❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s ❛♥❞
✶✹✿ ✷✾✲✺✾✳
❬✷✶❪ ❙t❛❤❧✱ ❑✳ ✭✶✾✽✷✮ ▲♦❝❛t✐♦♥ ❛♥❞ s♣❛t✐❛❧ ♣r✐❝✐♥❣ t❤❡♦r② ✇✐t❤ ♥♦♥❝♦♥✈❡① tr❛♥s♣♦rt❛t✐♦♥ ❝♦st
s❝❤❡❞✉❧❡s✳
❇❡❧❧ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s
✶✸✿ ✺✼✺✲✽✷✳
❬✷✷❪ ❙t❛❤❧✱ ❑✳ ✭✶✾✽✼✮ ❚❤❡♦r✐❡s ♦❢ ✉r❜❛♥ ❜✉s✐♥❡ss ❧♦❝❛t✐♦♥✳ ■♥✿ ❊✳❙✳ ▼✐❧❧s ✭❡❞✳✮✳ ❍❛♥❞❜♦♦❦ ♦❢ ❘❡❣✐♦♥❛❧
✳ ❆♠st❡r❞❛♠✿ ◆♦rt❤✲❍♦❧❧❛♥❞✱ ✼✺✾✲✽✷✵✳
❛♥❞ ❯r❜❛♥ ❊❝♦♥♦♠✐❝s✱ ❱♦❧✉♠❡ ✷
❬✷✸❪ ❚❡❧❧❡r✱ ❈ ✳ ✭✷✵✵✽✮ ❙❤♦♣♣✐♥❣ str❡❡ts ✈❡rs✉s s❤♦♣♣✐♥❣ ♠❛❧❧s ✕ ❉❡t❡r♠✐♥❛♥ts ♦❢ ❛❣❣❧♦♠❡r❛t✐♦♥
❢♦r♠❛t ❛ttr❛❝t✐✈❡♥❡ss ❢r♦♠ t❤❡ ❝♦♥s✉♠❡rs ✬ ♣♦✐♥t ♦❢ ✈✐❡✇✳
❉✐str✐❜✉t✐♦♥ ❛♥❞ ❈♦♥s✉♠❡r ❘❡s❡❛r❝❤
■♥t❡r♥❛t✐♦♥❛❧ ❘❡✈✐❡✇ ♦❢ ❘❡t❛✐❧✱
✶✽✿ ✸✽✶✲✹✵✸✳
❬✷✹❪ ❲♦❧✐♥s❦②✱ ❆✳ ✭✶✾✽✸✮ ❘❡t❛✐❧ tr❛❞❡ ❝♦♥❝❡♥tr❛t✐♦♥ ❞✉❡ t♦ ❝♦♥s✉♠❡rs✬ ✐♠♣❡r❢❡❝t ✐♥❢♦r♠❛t✐♦♥✳ ❇❡❧❧
❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s
✶✹✿ ✷✼✺✲✽✷✳
❆♣♣❡♥❞✐①
❆♣♣❡♥❞✐① ✶✳
✭✐✮ ❆ss✉♠❡ t❤❛t T ≤ 2.39✳ ❯s✐♥❣ ✭✶✾✮✱ ✐t ✐s r❡❛❞✐❧② ✈❡r✐✜❡❞ t❤❛t dDS /dν > 0 ❢♦r ❛❧❧ ν ∈
]1/(T − 1), T − 1[ ✐❢ ❛♥❞ ♦♥❧② ✐❢
θ(ν) > ln T
✇❤❡r❡
ν 2 (1 − ln (1 + ν)) + 1
.
1 + 2ν
1
> ln T ❢♦r ❛❧❧ 2 < T < 4✳ ❚❤❡r❡❢♦r❡✱
❚❤❡ ❢✉♥❝t✐♦♥ θ ❞❡❝r❡❛s❡s ✇✐t❤ ν ✱ ✇❤❡r❡❛s θ T −1
dDS /dν > 0 ❢♦r 0 < x̄0 < x̄0 < 1 ✐❢ ❛♥❞ ♦♥❧② ✐❢
1
θ(ν) ≡ ln 1 +
ν
+
✸✷
θ(T − 1) ≥ ln T
✭❆✳✶✮
A(T ) ≡ 1 + (T − 1)2 (1 − ln T ) − (2T − 1) ln (T − 1) ≥ 0.
✭❆✳✷✮
♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱
❚❤❡ ❢✉♥❝t✐♦♥ A ❞❡❝r❡❛s❡s ✇✐t❤ T ♦♥ ]2, 4[✳ ❙✐♥❝❡ A(2) > 0 ❛♥❞ A(4) < 0✱ t❤❡ ❡q✉❛t✐♦♥ A(T ) = 0
❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ✐♥ ]2, 4[✳ ❙♦❧✈✐♥❣ ♥✉♠❡r✐❝❛❧❧② t❤✐s ❡q✉❛t✐♦♥ s❤♦✇s T = 2.39 s♦ t❤❛t t❤❛t ✭❆✳✷✮
❤♦❧❞s ✐❢ ❛♥❞ ♦♥❧② ✐❢ T ≤ 2.39✳
✭✐✐✮ ❆ss✉♠❡ ♥♦✇ t❤❛t T > 2.39✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts ❛ t❤r❡s❤♦❧❞ ν̄ ∈ ]1/(T − 1), T − 1[ s✉❝❤ t❤❛t
❞D/❞ν > 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ ν < ν̄ ✳ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ ❥✉st s❡❡♥ t❤❛t ✭❆✳✶✮ ❞♦❡s ♥♦t ❤♦❧❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢
T > 2.39✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❡q✉❛t✐♦♥ θ(ν) = ln T ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ν̄ ∈ ]1/(T − 1), T − 1[ ❛♥❞
✭❆✳✷✮ ❞♦❡s ♥♦t ❤♦❧❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ν ❡①❝❡❡❞s ν̄ ✳ ◗✳❊✳❉✳
✭✐✐✐✮ ❲❤❡♥ T > 2.06✱ t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝ts ❜❡❝♦♠❡s str♦♥❣❡r t❤❛♥ t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t
❢♦r ❛❧❧ ν > T − 1✳ ■♥❞❡❡❞✱ ❞✐✛❡r❡♥t✐❛t✐♥❣ ✭✷✵✮ ✇✐t❤ r❡s♣❡❝t t♦ ν ②✐❡❧❞s✿
n
x̄1
1 dx̄1 1
1 − x̄1
dDS
−
−
=−
.
dν
ν2
(1 + ν)2 ν dν 1 + ν
❚❤✐s ❡①♣r❡ss✐♦♥ ✐s ♥❡❣❛t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢
✭❆✳✸✮
λ(ν) < ln T
✇❤❡r❡
(1 + 2ν) ln 1 +
λ(ν) ≡
(1 + ν)2
1
ν
+1
.
❚❤❡ ❢✉♥❝t✐♦♥ λ ❞❡❝r❡❛s❡s ✇✐t❤ ν ❢♦r ❛❧❧ ν > 1 ❜❡❝❛✉s❡ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ✐♥❝r❡❛s✐♥❣✱ ✇❤❡r❡❛s
t❤❡ ♥✉♠❡r❛t♦r ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❞❡❝r❡❛s✐♥❣✳ ❆s ❛ r❡s✉❧t✱ ✭❆✳✸✮ ❤♦❧❞s ❢♦r ❛❧❧ ν ≥ T − 1 ✐❢ ❛♥❞ ♦♥❧② ✐❢
λ(T − 1) < ln T
♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱
H(T ) ≡ (T − 1)2 ln T + (2T − 1) ln(T − 1) − 1 > 0.
❚❤❡ ❢✉♥❝t✐♦♥ H(T ) ✐s ✐♥❝r❡❛s✐♥❣ ✐♥ T ✱ ♥❡❣❛t✐✈❡ ❛t T = 2 ❛♥❞ ♣♦s✐t✐✈❡ ❛t T = 4✳ ❚❤❡r❡❢♦r❡✱
H(T ) = 0 ❤❛s ❛ s✐♥❣❧❡ s♦❧✉t✐♦♥ ✐♥ ]2, 4[✳ ❲❡ ✜♥❞ ♥✉♠❡r✐❝❛❧❧② t❤❛t t❤✐s s♦❧✉t✐♦♥ ✐s ❣✐✈❡♥ ❜②
T0 = 2.06✳ ❍❡♥❝❡ ✭❆✳✸✮ ❤♦❧❞s ❢♦r ❛❧❧ ν > T − 1 ✐❢ ❛♥❞ ♦♥❧② ✐❢ T > 2.06✳
✭✐✈✮ ❆ss✉♠❡ t❤❛t T ≤ 2.06✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts ν̂ > T − 1 s✉❝❤ t❤❛t t❤❡ ♠❛r❦❡t ❡①♣❛♥s✐♦♥ ❡✛❡❝t
❞♦♠✐♥❛t❡s t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❡✛❡❝t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ν < ν̂ ✳ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ ❥✉st s❡❡♥ t❤❛t ✭❆✳✸✮ ❞♦❡s
♥♦t ❤♦❧❞ ❢♦r ❛❧❧ ν > T − 1 ✉♥❞❡r T ≤ 2.06✳ ❆s λ(ν) ✐s ❛ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ❛♥❞ λ(∞) = 0✱ t❤❡r❡
✸✸
❡①✐sts ν̂ > T − 1 s✉❝❤ t❤❛t ✭❆✳✸✮ ❤♦❧❞s ♦♥❧② ❢♦r ν < ν̂ ✳ ◗✳❊✳❉✳
❆♣♣❡♥❞✐① ✷✳
✭✐✮ n < N/(T − 1)✳ ❚❤❡♥✱
x̄0 = 0
x̄1 =
ln (1 + n/N )
.
ln T
❍❡♥❝❡✱ ✉s✐♥❣ ✭✶✺✮✱ r❡t❛✐❧❡rs✬ ♦♣❡r❛t✐♥❣ ♣r♦✜t ❜♦✐❧s ❞♦✇♥ t♦
1 − ρ ln (1 + n/N )
ln T
n+N
πR =
✇❤✐❝❤ ✐♥❝r❡❛s❡s ✇❤❡♥ τ ❞❡❝r❡❛s❡s✳
✭✐✐✮ N/(T − 1) ≤ n ≤ N ✳ ■♥ t❤✐s ❝❛s❡✱
x̄0 = 1 −
ln (1 + N/n)
ln T
x̄1 =
ln (1 + n/N )
.
ln T
❍❡♥❝❡✱
1
1 (1 − N/n) ln (1 + N/n) − ln(N/n)
1
.
−
+
πR = (1 − ρ)
n n+N
ln T
n+N
❲❤❡♥ n < N ✱ ✇❡ ❤❛✈❡
N
1−
n
N
ln 1 +
n
❛♥❞ t❤✉s πR ❞❡❝r❡❛s❡s ✇❤❡♥ τ ❞❡❝r❡❛s❡s✳ ◗✳❊✳❉✳
− ln
N
<0
n
❙✐♥❝❡ π(µ) ✐s ✐♥❝r❡❛s✐♥❣ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ ]0, 1/(T − 1)[✱ t❤❡r❡ ✐s ♥♦ st❛❜❧❡ ❢r❡❡✲
❡♥tr② ❡q✉✐❧✐❜r✐✉♠ s✉❝❤ t❤❛t µ < 1/(T − 1)✳ ❈♦♥s✐❞❡r ♥♦✇ t❤❡ ✐♥t❡r✈❛❧ ]1/(T − 1), T − 1[ ❛♥❞ s❤♦✇
t❤❛t N(µ) ✐s ✐♥❝r❡❛s✐♥❣✳ ❯♣ t♦ ❛ ♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥t ✐♥❞❡♣❡♥❞❡♥t ♦❢ µ✱ ✇❡ ❤❛✈❡
❆♣♣❡♥❞✐① ✸✳
N(µ) =
ln T + (µ − 1) ln(1 + µ) + ln µ
.
µ
❉✐✛❡r❡♥t✐❛t✐♥❣ t❤✐s ❡①♣r❡ss✐♦♥ ②✐❡❧❞s
N′ (µ) =
Λ(µ)
µ2
✇❤❡r❡
Λ(µ) ≡
µ2 + 1
− ln T + ln(1 + µ) − ln µ.
µ+1
❚♦ s❤♦✇ t❤❛t Λ(µ) ✐s ♣♦s✐t✐✈❡ ♦✈❡r ]1/(T − 1), T − 1[✱ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ Λ(µ) ❛♥❞ ❣❡t
Λ′ (µ) ≡
(µ − 1)(µ2 + 3µ + 1)
µ(µ + 1)2
✸✹
s♦ t❤❛t µ = 1 ✐s ❛ ♠✐♥✐♠✐③❡r ♦❢ Λ(µ)✳ ❍❡♥❝❡✱ Λ(µ) > 0 ✐❢ Λ(1) > 0✳ ❙✐♥❝❡ T < 4✱ ✇❡ ❤❛✈❡
Λ(1) = 1 + ln 2 − ln T > 1 − ln 2 > 0.
▲❛st✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ N(µ) ❢♦r µ ≥ T − 1✳ ❯♣ t♦ ❛ ♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥t ✐♥❞❡♣❡♥❞❡♥t
♦❢ µ✱ ✇❡ ❤❛✈❡
N(µ) =
(1 + µ) ln T − ln(1 + µ) + ln µ
.
µ
❉✐✛❡r❡♥t✐❛t✐♥❣ t❤✐s ❡①♣r❡ss✐♦♥✱ ✇❡ ♦❜t❛✐♥
N′ (µ) =
K(µ)
.
µ2
✇❤❡r❡
K(µ) ≡
1
− ln T + ln(1 + 1/µ).
1+µ
❈❧❡❛r❧②✱ K(µ) ❞❡❝r❡❛s❡s ✇✐t❤ µ✳ ❋✉rt❤❡r♠♦r❡✱ K(µ)✱ ✇❤❡♥❝❡ N′ (µ)✱ ✐s ♥❡❣❛t✐✈❡ ✉♥❞❡r s✉✣✲
❝✐❡♥t❧② ❧❛r❣❡ ✈❛❧✉❡s ♦❢ µ✳ ❆s ❛ r❡s✉❧t✱ N(µ) ✐s ✉♥✐♠♦❞❛❧ ❛♥❞ ❤❛s ❛ ✉♥✐q✉❡ ♠❛①✐♠✐③❡r µ0 ≥ T − 1✳
◗✳❊✳❉✳
✸✺
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