Fuzzy Profit Shifting: A Model for Optimal Tax-induced Transfer Pricing with Fuzzy Arm’s Length Parameter

Munich Personal RePEc Archive

Fuzzy Profit Shifting: A Model for

Optimal Tax-induced Transfer Pricing with Fuzzy Arm’s Length Parameter

Rathke, Alex A.T.

FEA-RP, University of São Paulo, Brazil

12 January 2019

Online at https://mpra.ub.uni-muenchen.de/91425/

MPRA Paper No. 91425, posted 16 Jan 2019 14:39 UTC

❋✉③③② Pr♦✜t ❙❤✐❢t✐♥❣✿ ❆ ▼♦❞❡❧ ❢♦r ❖♣t✐♠❛❧ ❚❛①✲✐♥❞✉❝❡❞

❚r❛♥s❢❡r Pr✐❝✐♥❣ ✇✐t❤ ❋✉③③② ❆r♠✬s ▲❡♥❣t❤ P❛r❛♠❡t❡r

❆❧❡① ❆✳❚✳ ❘❛t❤❦❡^∗

❋❊❆✲❘P✱ ❯♥✐✈❡rs✐t② ♦❢ ❙ã♦ P❛✉❧♦✱ ❇r❛③✐❧

❏❛♥✉❛r② ✶✷✱ ✷✵✶✾

❆❜str❛❝t

❚❤✐s ♣❛♣❡r ♣r♦♣♦s❡s ❛ ♠♦❞❡❧ ♦❢ ♦♣t✐♠❛❧ t❛①✲✐♥❞✉❝❡❞ tr❛♥s❢❡r ♣r✐❝✐♥❣ ✇✐t❤ ❛ ❢✉③③② ❛r♠✬s

❧❡♥❣t❤ ♣❛r❛♠❡t❡r✳ ❋✉③③② ♥✉♠❜❡rs ♣r♦✈✐❞❡ ❛ s✉✐t❛❜❧❡ str✉❝t✉r❡ ❢♦r ♠♦❞❡❧❧✐♥❣ t❤❡ ❛♠❜✐❣✉✐t② t❤❛t ✐s ✐♥tr✐♥s✐❝ t♦ t❤❡ ❛r♠✬s ❧❡♥❣t❤ ♣❛r❛♠❡t❡r✳ ❋♦r t❤❡ ✉s✉❛❧ ❝♦♥❞✐t✐♦♥s r❡❣❛r❞✐♥❣ t❤❡ ❛♥t✐✲

s❤✐❢t✐♥❣ ♠❡❝❤❛♥✐s♠s✱ t❤❡ ♦♣t✐♠❛❧ tr❛♥s❢❡r ♣r✐❝❡ ❜❡❝♦♠❡s ❛ ♠❛①✐♠✐s✐♥❣ α✲❝✉t ♦❢ t❤❡ ❢✉③③②

❛r♠✬s ❧❡♥❣t❤ ♣❛r❛♠❡t❡r✳ ◆♦♥❡t❤❡❧❡ss✱ ✇❡ s❤♦✇ t❤❛t ✐t ✐s ♣r♦✜t❛❜❧❡ ❢♦r ✜r♠s t♦ ❝❤♦♦s❡ ❛♥②

♠❛①✐♠✐s✐♥❣ tr❛♥s❢❡r ♣r✐❝❡ ✐❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❛① ❛✉❞✐t ✐s s✉✣❝✐❡♥t❧② ❧♦✇✱ ❡✈❡♥ ✐❢ t❤❡ ❝❤♦s❡♥

♣r✐❝❡ ✐s ❝♦♥s✐❞❡r❡❞ ❛ ❝♦♠♣❧❡t❡❧② ♥♦♥✲❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡ ❜② t❛① ❛✉t❤♦r✐t✐❡s✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡

❞❡r✐✈❡ t❤❡ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s t♦ ♣r❡✈❡♥t t❤✐s ❡①tr❡♠❡ s❤✐❢t✐♥❣ str❛t❡❣②✳

❑❡②✇♦r❞s✿ ❢✉③③② ♣r♦✜t s❤✐❢t✐♥❣✱ tr❛♥s❢❡r ♣r✐❝✐♥❣✱ t❛① ❡✈❛s✐♦♥✱ t❛① ❡♥❢♦r❝❡♠❡♥t✱ t❛① ♣❡♥❛❧t②✳

❏❊▲ ❈❧❛ss✐✜❝❛t✐♦♥✿ ❋✷✸✱ ❍✷✻✱ ❑✸✹

✶ ■♥tr♦❞✉❝t✐♦♥

❚❛① ❧✐t❡r❛t✉r❡ ❢r❡q✉❡♥t❧② ❞r❛✇s ❛tt❡♥t✐♦♥ t♦ t❤❡ ❛♠❜✐❣✉✐t② ❜❡t✇❡❡♥ ❛ t♦❧❡r❛♥t t❛① ❛✈♦✐❞❛♥❝❡

❜❡❤❛✈✐♦✉r ✈s✳ t❛① ❡✈❛s✐♦♥✳ ❚❤✐s ❛♠❜✐❣✉✐t② ✐s ❡s♣❡❝✐❛❧❧② r❡❧❡✈❛♥t ♦♥ t❤❡ ❛♥❛❧②s✐s ♦❢ ♣r♦✜t s❤✐❢t✐♥❣

str❛t❡❣✐❡s✱ ✇❤❡r❡ ♠✉❧t✐♥❛t✐♦♥❛❧ ❡♥t❡r♣r✐s❡s ✕ ▼◆❊ ❝❛rr② ✐♥tr❛✲✜r♠ tr❛♥s❛❝t✐♦♥s ❜❡t✇❡❡♥ r❡❧❛t❡❞

♣❛rt✐❡s ❢r♦♠ ❞✐✛❡r❡♥t ❥✉r✐s❞✐❝t✐♦♥s✱ s♦ t♦ ❛❞❥✉st t❤❡ tr❛♥s❢❡r ♣r✐❝❡s ✐♥ ♦r❞❡r t♦ r❡❛❧❧♦❝❛t❡ t❛①❛❜❧❡

♣r♦✜ts ❢r♦♠ ❤✐❣❤✲t❛① t♦ ❧♦✇✲t❛① ❧♦❝❛t✐♦♥s^✶✳ ❆♥t✐✲s❤✐❢t✐♥❣ r✉❧❡s r❡q✉✐r❡ t❤❛t t❤❡ tr❛♥s❢❡r ♣r✐❝❡s

❝♦♠♣❧② ✇✐t❤ t❤❡ s♦ ❝❛❧❧❡❞ ❛r♠✬s ❧❡♥❣t❤ ♣r✐♥❝✐♣❧❡ ✭❖❊❈❉✱ ✷✵✶✼✮✱ ✇❤✐❝❤ st❛t❡s t❤❛t ✐♥tr❛✲✜r♠

♣r✐❝❡s ♠✉st ❜❡ ❝♦♥s✐st❡♥t ✇✐t❤ ♦♥❡s t❤❛t ✇♦✉❧❞ ❤❛✈❡ ❜❡❡♥ ❡st❛❜❧✐s❤❡❞ ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t ✉♥r❡❧❛t❡❞

♣❛rt✐❡s✳ ■❢ t❤❡ ❛r♠✬s ❧❡♥❣t❤ ❝♦♥❞✐t✐♦♥ ✐s ♥♦t s❛t✐s✜❡❞✱ t❛① ❛✉t❤♦r✐t✐❡s r❡q✉✐r❡ t❤❡ ♣❛②♠❡♥t ♦❢ t❛①❡s

♦✈❡r t❤❡ s❤✐❢t❡❞ ♣r♦✜ts✱ ❛♥❞ ❛ t❛① ♣❡♥❛❧t② ✉s✉❛❧❧② ❛♣♣❧✐❡s✳

❚❤❡ ❛r♠✬s ❧❡♥❣t❤ ❝♦♥❞✐t✐♦♥ ✐s ❛ ❢✉③③② ❝♦♥❝❡♣t✱ s✐♥❝❡ ✐♥❞❡♣❡♥❞❡♥t ♣r✐❝❡s ❛r❡ ✐♥✢✉❡♥❝❡❞ ❜②

❧❡❣✐t✐♠❛t❡ ❞✐✛❡r❡♥❝❡s ✐♥ tr❛♥s❛❝t✐♦♥s✬ ❝♦♥❞✐t✐♦♥s ✭❇❡❝❦❡r✱ ❉❛✈✐❡s✱ ✫ ❏❛❦♦❜s✱ ✷✵✶✼❀ ❊❞❡♥✱ ✷✵✵✶❀

∗❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s✱ ❇✉s✐♥❡ss ❛♥❞ ❆❝❝♦✉♥t✐♥❣ ❛t ❘✐❜❡✐rã♦ Pr❡t♦✱ ❯♥✐✈❡rs✐t② ♦❢ ❙ã♦ P❛✉❧♦✱ ❇r❛③✐❧✳ ❊✲♠❛✐❧✿

❛❧❡①✳r❛t❤❦❡❅✉s♣✳❜r

✶❊①✐st✐♥❣ st✉❞✐❡s ♣r♦✈✐❞❡ r❡❧❡✈❛♥t ❡✈✐❞❡♥❝❡s ♦❢ ♣r♦✜t s❤✐❢t✐♥❣ ❜② ♠❡❛♥s ♦❢ ❞✐r❡❝t tr❛♥s❢❡r ♣r✐❝✐♥❣ ❛❞❥✉st♠❡♥ts

✭❉❛✈✐❡s✱ ▼❛rt✐♥✱ P❛r❡♥t✐✱ ✫ ❚♦✉❜❛❧✱ ✷✵✶✽❀ ❈r✐st❡❛ ✫ ◆❣✉②❡♥✱ ✷✵✶✻❀ ❇❡r♥❛r❞✱ ❏❡♥s❡♥✱ ✫ ❙❝❤♦tt✱ ✷✵✵✻❀ ❖✈❡r❡s❝❤✱

✷✵✵✻❀ ❇❛rt❡❧s♠❛♥ ✫ ❇❡❡ts♠❛✱ ✷✵✵✸❀ ❈❧❛✉s✐♥❣✱ ✷✵✵✸❀ ❙✇❡♥s♦♥✱ ✷✵✵✶✮✳

✶

❖❊❈❉✱ ✷✵✶✼✮✳ ■t ♠❡❛♥s t❤❛t tr❛♥s❢❡r ♣r✐❝❡s ❛r❡ ♥♦t ❛tt❛✐♥❡❞ t♦ ❛ ✉♥✐q✉❡ tr✉❡ ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡✱

❜✉t r❛t❤❡r t♦ ❛ r❛♥❣❡ ♦❢ ♦❜s❡r✈❛❜❧❡ ♣❛r❛♠❡t❡r ♣r✐❝❡s ✇✐t❤ ❞✐✛❡r❡♥t ❞❡❣r❡❡s ♦❢ ❛♣♣r♦♣r✐❛t❡♥❡ss

✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛r♠✬s ❧❡♥❣t❤ ❝♦♥❞✐t✐♦♥✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛ t❛① ❛✉❞✐t✱ t❤❡ t❛① ❛✉t❤♦r✐t② ❤❛s t♦ ❛ss❡ss ✐❢ t❤❡ tr❛♥s❢❡r ♣r✐❝❡s ❛♣♣❧✐❡❞ ❜② t❤❡ ▼◆❊ s❛t✐s❢② t❤❡ ❛r♠✬s ❧❡♥❣t❤ ❝♦♥❞✐t✐♦♥✱ ♦r ✐❢ t❤❡

❞❡✈✐❛t✐♦♥s ❢r♦♠ t❤❡ ❝♦r❡ ♦❢ t❤❡ ❛r♠✬s ❧❡♥❣t❤ r❛♥❣❡ r❡♣r❡s❡♥t ❡✈✐❞❡♥❝❡s ♦❢ ♣r♦✜t s❤✐❢t✐♥❣✳ ❚❤✐s ✐s

♥♦ ♠♦r❡ t❤❛♥ ❛♥ ❛♠❜✐❣✉♦✉s ❞❡❝✐s✐♦♥ t♦ ❜❡ t❛❦❡♥ ❜② t❤❡ t❛① ❛✉t❤♦r✐t②✱ t❤✉s ✐t ✐♠♣❧✐❡s ✐♥ ❛❞❞✐t✐♦♥❛❧

✉♥❝❡rt❛✐♥t✐❡s ❢♦r t❤❡ ▼◆❊✳

❚❤✐s ♣❛♣❡r ❞❡r✐✈❡s ❛ ♠♦❞❡❧ ❢♦r ♦♣t✐♠❛❧ t❛①✲✐♥❞✉❝❡❞ tr❛♥s❢❡r ♣r✐❝✐♥❣ s✉❜❥❡❝t❡❞ t♦ ❛ ❢✉③③②

❛r♠✬s ❧❡♥❣t❤ ♣❛r❛♠❡t❡r✳ ❲❡ ❛♣♣❧② ❢✉③③② ♥✉♠❜❡rs✱ ✇❤✐❝❤ ✇❡r❡ ✜rst ♣r♦♣♦s❡❞ ❜② ✭❩❛❞❡❤ ❡t ❛❧✳✱

✶✾✻✺✮ ❛♥❞ ❞❡✈❡❧♦♣❡❞ ❢✉rt❤❡r ❜② s❡✈❡r❛❧ r❡s❡❛r❝❤❡rs ✭❩✐♠♠❡r♠❛♥♥✱ ✶✾✾✶❀ ❑❧✐r ✫ ❨✉❛♥✱ ✶✾✾✺❀

❱❡r❞❡❣❛②✱ ✶✾✽✷✮✱ t❤✉s t♦ ♠♦❞❡❧ t❤❡ ✐♠♣❛❝t ♦❢ t❤❡ ✉♥❝❡rt❛✐♥t② t❤❛t ✐s ✐♥tr✐♥s✐❝ t♦ t❤❡ ❛r♠✬s

❧❡♥❣t❤ ♣❛r❛♠❡t❡r ♦✈❡r t❤❡ ♣r♦✜t✲♠❛①✐♠✐s❛t✐♦♥ str❛t❡❣②✳ ❖✉r ♠♦❞❡❧ ❢♦❧❧♦✇s t❤❡ ❝♦♥❝❡❛❧♠❡♥t

❝♦sts ❛♣♣r♦❛❝❤ t❤❛t ✐s tr❛❞✐t✐♦♥❛❧ ✐♥ ♣r♦✜t s❤✐❢t✐♥❣ ❧✐t❡r❛t✉r❡ ✭❆❧❧✐♥❣❤❛♠ ✫ ❙❛♥❞♠♦✱ ✶✾✼✷❀ ❑❛♥t✱

✶✾✽✽❀ ❍✐♥❡s ❏r ✫ ❘✐❝❡✱ ✶✾✾✹✮✱ ❤♦✇❡✈❡r ✇❡ ❞❡s✐❣♥ ✐t ✐♥ ❛ ❣❡♥❡r❛❧✐s❡❞ t❛① ❝♦♥❞✐t✐♦♥✱ ✇❤✐❝❤ ❛❧❧♦✇s

❢♦r t❤❡ ♠❛①✐♠✐s❛t✐♦♥ ❛♥❛❧②s✐s ✇✐t❤♦✉t ❝♦♥str❛✐♥ts ♦♥ t❤❡ s❤✐❢t✐♥❣ ❞✐r❡❝t✐♦♥✳ ❚❤❡ ♠♦❞❡❧ t❛❦❡s t❤❡ ❛r♠✬s ❧❡♥❣t❤ ♣❛r❛♠❡t❡r ❛s ❛ ❢✉③③② ♥✉♠❜❡r✱ t❤❡r❡❢♦r❡ t❤❡ ♠❛①✐♠✐s❛t✐♦♥ ♦❜❥❡❝t ✐s ❛❧s♦ ❛ ❢✉③③②

♦❜❥❡❝t✳

❇❛s❡❧✐♥❡ ❛♥❛❧②s✐s s❤♦✇s t❤❛t t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢✉③③② ♠❛①✐♠✐s❛t✐♦♥ ♦❜❥❡❝t ✉♥❞❡r ✉s✉❛❧ ❝♦♥✲

❞✐t✐♦♥s ✐s ❛α✲❝✉t ♦❢ t❤❡ ❢✉③③② ❛r♠✬s ❧❡♥❣t❤ ♣❛r❛♠❡t❡r✱ ❛♥❞ ❛♥② ❛❞❥✉st♠❡♥ts ♦♥ t❤❡ tr❛♥s❢❡r ♣r✐❝❡

✉♣ t♦ t❤❡ ♦♣t✐♠❛❧ ❧❡✈❡❧ ♣r♦✈✐❞❡ ❛ ♣r♦✜t✲s❤✐❢t✐♥❣ ❣❛✐♥ ❢♦r t❤❡ ▼◆❊✳ ◆♦♥❡t❤❡❧❡ss✱ ✇❡ s❤♦✇ t❤❛t t❤❡ ▼◆❊ ♠❛② ❝♦♠♣❧❡t❡❧② ❞✐sr❡❣❛r❞ t❤❡ ❛r♠✬s ❧❡♥❣t❤ ♣❛r❛♠❡t❡r ✐❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❛① ❛✉❞✐ts ✐s s✉✣❝✐❡♥t❧② ❧♦✇✳ ■t ♠❡❛♥s t❤❛t ✐t ✐s ♣r♦✜t❛❜❧❡ t♦ ❝❤♦♦s❡ ❛♥② ♠❛①✐♠✐s✐♥❣ tr❛♥s❢❡r ♣r✐❝❡ ✐❢ t❤❡ ▼◆❊

❤❛s ❧♦✇ ❝❤❛♥❝❡s ♦❢ ❜❡✐♥❣ ❛✉❞✐t❡❞✱ ❡✈❡♥ ✐❢ t❤❡ ♠❛①✐♠✐s✐♥❣ tr❛♥s❢❡r ♣r✐❝❡ ✐s ❝♦♥s✐❞❡r❡❞ ❛ ❝♦♠♣❧❡t❡❧②

♥♦♥✲❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡✳ ■♥ t❤✐s s❡♥s❡✱ ✇❡ ❞❡r✐✈❡ t❤❡ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s t♦ ♣r❡✈❡♥t t❤✐s ❡①tr❡♠❡ s❤✐❢t✐♥❣ ❝❛s❡✳

❚❤❡ r❡♠❛✐♥✐♥❣ ♦❢ t❤✐s ♣❛♣❡r ✐s str✉❝t✉r❡❞ ❛s ❢♦❧❧♦✇s✳ ❙❡❝t✐♦♥ ✷ ♣r❡s❡♥ts t❤❡ ❜❛s✐❝ ♥♦t✐♦♥s

♦❢ ❢✉③③② s❡ts ❛♥❞ ❢✉③③② ♥✉♠❜❡rs✳ ❙❡❝t✐♦♥ ✸ ❞❡r✐✈❡s t❤❡ ❣❡♥❡r❛❧ ♠♦❞❡❧✳ ❙❡❝t✐♦♥ ✹ s♦❧✈❡s t❤❡

❢✉③③② ♠❛①✐♠✐s❛t✐♦♥ ♦❜❥❡❝t✱ ♣r❡s❡♥ts t❤❡ s❡♥s✐t✐✈✐t② ❛♥❛❧②s❡s✱ ❛♥❞ ❞❡r✐✈❡s t❤❡ ✐♠♣❛❝t ♦❢ ❛ ❣❡♥❡r❛❧

t❛① ❡♥❢♦r❝❡♠❡♥t ❡✛❡❝t r❡❣❛r❞✐♥❣ t❤❡ ❝♦✉♥tr②✲❧❡✈❡❧ ❛♥t✐✲s❤✐❢t✐♥❣ ✈❛r✐❛❜❧❡s✳ ❙❡❝t✐♦♥ ✺ ❞r❛✇s s♦♠❡

❝♦♥❝❧✉❞✐♥❣ ❝♦♠♠❡♥ts✳

✷ ❇❛s✐❝s ♦♥ ❋✉③③② ❙❡ts

❋✉③③② s❡ts ✇❡r❡ ✜rst ✐♥tr♦❞✉❝❡❞ ❜② s❡♠✐♥❛❧ ♣❛♣❡r ♦❢ ✭❩❛❞❡❤ ❡t ❛❧✳✱ ✶✾✻✺✮ ❛♥❞ ❣❡♥❡r❛❧✐s❡ t❤❡

❝❧❛ss✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝r✐s♣ s❡ts✳ ❋✉③③② s❡ts ❛r❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❡❧❡♠❡♥ts ✐♥ ❛ ✉♥✐✈❡rs❡ ✇❤❡r❡ t❤❡

❜♦✉♥❞❛r② ♦❢ t❤❡ s❡t ✐s ♥♦t ❝❧❡❛r❧② ❞❡✜♥❡❞✳ ❚❤❡ ❛♠❜✐❣✉✐t② ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❜♦✉♥❞s ♦❢ t❤❡ ❢✉③③② s❡t A˜ ✐♥ ❛ ✉♥✐✈❡rs❡ X ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛s µ_A˜(x) : R → [0,1]✱ x∈X✱ ❢♦rµ_A˜(x)♠❡❛s✉r❡s t❤❡ ❣r❛❞❡ ♦❢ ♠❡♠❜❡rs❤✐♣ ♦❢ ❡❧❡♠❡♥tx✐♥A˜✳ ■❢ t❤❡ ❣r❛❞❡ ♦❢ ♠❡♠❜❡rs❤✐♣

✐s ✵✱ t❤❡♥ t❤❡ ❡❧❡♠❡♥tx❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦A✳ ■❢ t❤❡ ❣r❛❞❡ ♦❢ ♠❡♠❜❡rs❤✐♣ ✐s ✶✱ t❤❡♥ t❤❡ ❡❧❡♠❡♥t˜ x ❝♦♠♣❧❡t❡❧② ❜❡❧♦♥❣s t♦ A˜✳ ■❢ t❤❡ ❣r❛❞❡ ♦❢ ♠❡♠❜❡rs❤✐♣ ✐s ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧ ❬✵✱✶❪✱ t❤❡♥ t❤❡

❡❧❡♠❡♥t x ♦♥❧② ♣❛rt✐❛❧❧② ❜❡❧♦♥❣s t♦ A˜✳ ❚❤❡ ❢✉③③② s❡t A˜ ✐s t❤❡r❡❢♦r❡ ❝❤❛r❛❝t❡r✐s❡❞ ❜② t❤❡ ♣❛✐r {(x, µA˜(x)) :x∈X}✳ ❚✇♦ ❢✉③③② s❡tsA˜❛♥❞ B˜ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❡q✉❛❧ ✐✛ µA˜(x) =µB˜(x)✳

▲❡tA˜={(x, µ_A˜(x)) :x∈X} ❜❡ ❛ ❢✉③③② s❡t ❛♥❞ ❞❡✜♥❡ ❛ ❝♦♥t✐♥✉♦✉s ✐♥t❡r✈❛❧α∈[0,1]✳ ❚❤❡

✷

♦r❞✐♥❛r② ❝r✐s♣ s❡t ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛♥② α∈[0,1]✐s ❝❛❧❧❡❞ α✲❝✉t ♦❢ t❤❡ ❢✉③③② s❡t A˜❛♥❞ ✐s ❞❡✜♥❡❞

❛sAα={x∈X:µ_A˜(x)≥α}✳ ❲❡ ❝❛♥ ✉s❡ α✲❝✉ts t♦ r❡♣r❡s❡♥t ✐♥t❡r✈❛❧s ♦♥ ❢✉③③② s❡ts ❛s

A˜_α = [A^∧_α, A^∨_α]

=h

minx {A},˜ max

x {A}˜ i

: ˜A={(X, µA˜(x)), µA˜(x)≥α}.

❚❤❡ s❡tsAα✱α∈[0,1]r❡❢❡r t♦ ❛ ❞❡❝r❡❛s✐♥❣ s✉❝❝❡ss✐♦♥ ♦❢ s✉❜s❡ts ❝♦♥t✐♥✉❛❀α₁ ≥α₂ ⇔Aα₁ ⊆ Aα2✱α₁✱α₂∈[0,1]✭❑❧✐r ✫ ❨✉❛♥✱ ✶✾✾✺✮✳

❚❤❡♦r❡♠✳ ✭❘❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡♦r❡♠ ✲ ✭❑❧✐r ✫ ❨✉❛♥✱ ✶✾✾✺❀ ❩✐♠♠❡r♠❛♥♥✱ ✶✾✾✶❀ ❱❡r❞❡❣❛②✱

✶✾✽✷✮^✷✮ ❋♦r ❛ ❢✉③③② s❡t A˜ ❛♥❞ ✐tsα✲❝✉ts Aα✱ α∈[0,1]✱ ✇❡ ❤❛✈❡

A˜= [

α∈[0,1]

α·Aα.

■❢ t❤❡ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥ µAα(x) ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❡t Aα

µAα(x) =

( 1, ✐✛ x∈Aα

0, ♦t❤❡r✇✐s❡

t❤❡ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢✉③③② s❡t A˜ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ ✐ts α✲❝✉ts ❛s

µ_A˜(x) = sup

α∈[0,1]

min (α, µ_Aα(x)).

△

❆ ❢✉③③② s❡t A˜ ✐s ❝♦♥✈❡① ✐✛ ✐ts α✲❝✉ts ❛r❡ ❝♦♥✈❡①✳ ❊q✉✐✈❛❧❡♥t❧②✱ A˜ ✐s ❝♦♥✈❡① ✐✛ ∀x₁✱ x₂ ∈ X✱ λ ∈ [0,1] : µA˜(λx1 + (1−λ)x2) ≥ min (µA˜(x1), µA˜(x2))✳ ❆ ❢✉③③② s❡t A˜ ✐s ♥♦r♠❛❧✐s❡❞ ✐✛

sup_x∈Xµ_A˜= 1✳

❆ ❢✉③③② ♥✉♠❜❡r ✐s ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ ❢✉③③② s❡t ♦♥ t❤❡ r❡❛❧ ❧✐♥❡ t❤❛t ✐s ❜♦t❤ ❝♦♥✈❡① ❛♥❞

♥♦r♠❛❧✐③❡❞✳ ■ts ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥ ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞∃x0∈R:µA˜(x0) = 1 ✐s ❝❛❧❧❡❞

✐ts ♠♦❞❡✳ ❙✐♥❝❡ ❢✉③③② s❡ts ❛r❡ ❝♦♠♣❧❡t❡❧② ❞❡✜♥❡❞ ❜② t❤❡✐r ❝♦rr❡s♣♦♥❞✐♥❣ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥s✱

✇❡ r❡❢❡r t♦ ❛ ❢✉③③② ♥✉♠❜❡r ❛s t❤❡ s❡t A˜ ❛s ✇❡❧❧ ❛s t❤❡ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥ µ_A˜(x) ❤❡r❡✐♥❛❢t❡r✳

❋♦r ❛ s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ♥✉♠❜❡rs x^∧ ≤ x¯^∧ ≤ x¯^∨ ≤ x^∨ ∈ R✱ t❤❡ ❢✉③③② ♥✉♠❜❡r A˜ s❛t✐s✜❡s t❤❡

❢♦❧❧♦✇✐♥❣✿

❛✳ µ_A˜(x) = 0 ❢♦r ❡❛❝❤x /∈[x^∧, x^∨]❀

❜✳ µ_A˜(x) ✐s ♥♦♥✲❞❡❝r❡❛s✐♥❣ ✐♥ [x^∧,x¯^∧]❛♥❞ ♥♦♥✲✐♥❝r❡❛s✐♥❣ ✐♥ [¯x^∨, x^∨]❀

❝✳ µ_A˜(x) = 1 ❢♦r ❡❛❝❤x∈[¯x^∧,x¯^∨]❀

✇❤❡r❡ [¯x^∧,x¯^∨] ✐s t❤❡ ♠♦❞❡ ♦❢ t❤❡ ❢✉③③② ♥✉♠❜❡r✱[x^∧,x¯^∧] ✐s t❤❡ ✐♥t❡r✈❛❧ ♦♥ t❤❡ ❧♦✇❡r s✐❞❡ ♦❢ t❤❡

♠♦❞❡ ✇✐t❤ ✇✐❞t❤ x¯^∧−x^∧✱ ❛♥❞[¯x^∨, x^∨]✐s t❤❡ ✐♥t❡r✈❛❧ ♦♥ t❤❡ ✉♣♣❡r s✐❞❡ ♦❢ t❤❡ ♠♦❞❡ ✇✐t❤ ✇✐❞t❤

✷✭❑❧✐r ✫ ❨✉❛♥✱ ✶✾✾✺✮ ❛♥❛❧②s❡ t❤✐s t❤❡♦r❡♠ ✐♥ ❛ s❡t ♦❢ t❤r❡❡ ❉❡❝♦♠♣♦s✐t✐♦♥ ❚❤❡♦r❡♠s ❢♦r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢

❢✉③③② s❡ts ❜② ♠❡❛♥s ♦❢ t❤❡✐rα✲❝✉ts✳

✸

x^∨−x¯^∨✳ ❆ ❢✉③③② ♥✉♠❜❡rA˜✐s ♦❢ t❤❡LR✲t②♣❡ ✐❢ ✐t ❝❛♥ ❜❡ ♣❛r❛♠❡tr✐s❡❞ ❜② s❤❛♣❡ ❢✉♥❝t✐♦♥sf^∧(·)

❛♥❞f^∨(·)♦♥ t❤❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r s✐❞❡s ♦❢ t❤❡ ♠♦❞❡ r❡s♣❡❝t✐✈❡❧②^✸✳ ❆ ♣❧❛♥❡ ❢✉③③② ♥✉♠❜❡r s❛t✐s✜❡s

∃(¯x^∧,x¯^∨)∈R✱x¯^∧ <x¯^∨ :∀x∈[¯x^∧,x¯^∨]→ µA˜(x) = 1✱ ✐✳❡✳ ✐ts ♠♦❞❡ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥t❡r✈❛❧ ✇✐t❤

♠♦r❡ t❤❛♥ ♦♥❡ ❡❧❡♠❡♥t ✭❑❧✐r ✫ ❨✉❛♥✱ ✶✾✾✺❀ ❩✐♠♠❡r♠❛♥♥✱ ✶✾✾✶✮✳ ❆ ❢✉③③② ♥✉♠❜❡r ✐s ❝❛❧❧❡❞ ❛ tr❛♣❡③♦✐❞❛❧ ❢✉③③② ♥✉♠❜❡r ✐✛ ✐t t❛❦❡s t❤❡ ❢♦r♠

µ_A˜(x) =



































x−x^∧

¯

x^∧−x^∧, x^∧ ≤x≤x¯^∧

1, x¯^∧ ≤x≤x¯^∨

x^∨−x

x^∨−x¯^∨, x¯^∨ ≤x≤x^∨

0, ♦t❤❡r✇✐s❡✳

❆ ❢✉③③② ♥✉♠❜❡r ✐s ❝❛❧❧❡❞ ❛ tr✐❛♥❣✉❧❛r ❢✉③③② ♥✉♠❜❡r ✐✛ ✐t t❛❦❡s t❤❡ ❢♦r♠

µ_B˜(x) =























x−x^∧

¯

x^∧−x^∧, x^∧≤x≤x¯^∧ x^∨−x

x^∨−x¯^∧, x¯^∧≤x≤x^∨

0, ♦t❤❡r✇✐s❡✳

❋✐❣✉r❡ ✶✿ ❊①❛♠♣❧❡s ♦❢ ❢✉③③② ♥✉♠❜❡rs✿ ❆ s②♠♠❡tr✐❝ tr❛♣❡③♦✐❞❛❧ ❢✉③③② ♥✉♠❜❡r A˜♦♥ t❤❡ ❧❡❢t ❛♥❞

❛ s②♠♠❡tr✐❝ tr✐❛♥❣✉❧❛r ❢✉③③② ♥✉♠❜❡r B˜ ♦♥ t❤❡ r✐❣❤t✳ ❇♦t❤ A˜❛♥❞ B˜ ❛r❡ s♣❡❝✐❛❧ ❢✉③③② ♥✉♠❜❡rs

♦❢ t❤❡ LR✲t②♣❡✳

❋✐❣✉r❡ ✶ s❤♦✇s ❡①❛♠♣❧❡s ♦❢ tr❛♣❡③♦✐❞❛❧ ❛♥❞ tr✐❛♥❣✉❧❛r ❢✉③③② ♥✉♠❜❡rs✳ ■t ✐s ❝❧❡❛r t❤❛t ❛ tr❛♣❡③♦✐❞❛❧ ❢✉③③② ♥✉♠❜❡r ✐s ❛♥ ✐♥st❛♥❝❡ ♦❢ ♣❧❛♥❡ ❢✉③③② ♥✉♠❜❡r✱ ❛♥❞ ❛ tr✐❛♥❣✉❧❛r ❢✉③③② ♥✉♠❜❡r r❡❢❡rs t♦ ❛ tr❛♣❡③♦✐❞❛❧ ❢✉③③② ♥✉♠❜❡r ✇✐t❤ x¯^∧ = ¯x^∨✳

✸▲✐t❡r❛t✉r❡ ❝♦♠♠♦♥❧② r❡❢❡r t♦ t❤❡ ❧❡❢t ❛♥❞ r✐❣❤t s✐❞❡s ♦❢ t❤❡ ♠♦❞❡ µA˜(x) = 1✱ ✐✳❡✳ t❤♦✉❣❤ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡

t❡r♠LR✲t②♣❡ ✇✐t❤ s❤❛♣❡ ❢✉♥❝t✐♦♥sL(·)❛♥❞R(·)✳

✹

✸ ❚❤❡ ▼♦❞❡❧

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❞❡r✐✈❡ ❛ ♠♦❞❡❧ t♦ ❛♥❛❧②s❡ t❤❡ ♦♣t✐♠❛❧ t❛①✲✐♥❞✉❝❡❞ tr❛♥s❢❡r ♣r✐❝✐♥❣✳ ❲❡ ✜rst s❡t t❤❡ ❜❛s❡❧✐♥❡ ♥❡t ♣r♦✜t ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ▼◆❊✱ t❤❡♥ ✇❡ ❞❡r✐✈❡ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❢✉③③②

♣r♦✜t s❤✐❢t✐♥❣ ♦♣t✐♠✐s❛t✐♦♥✳

✸✳✶ ❇❛s❡❧✐♥❡ Pr♦✜t ❉❡s✐❣♥

❈♦♥s✐❞❡r ❛ ✈❡rt✐❝❛❧❧② ✐♥t❡❣r❛t❡❞ ▼◆❊ ✇✐t❤ t✇♦ ❞✐✈✐s✐♦♥s✱ t❤❡ ♣❛r❡♥t ❝♦♠♣❛♥② ❧♦❝❛t❡❞ ✐♥ ❈♦✉♥tr②

✶ ❛♥❞ ❛ ✇❤♦❧❧② ♦✇♥❡❞ s✉❜s✐❞✐❛r② ❧♦❝❛t❡❞ ✐♥ ❈♦✉♥tr② ✷✱ i = {1,2}✳ ❇♦t❤ ❞✐✈✐s✐♦♥s^✹ ♣r♦❞✉❝❡

♦✉t♣✉ts x_i ✉♥❞❡r ❝♦sts C_i(x_i)✱ ❜r✐♥❣✐♥❣ r❡✈❡♥✉❡s R_i(s_i) ❜❛s❡❞ ♦♥ ❞♦♠❡st✐❝ s❛❧❡s s_i(x_i)✳ P❛r❡♥t

✜r♠ ❛❧s♦ ❡①♣♦rts ❛ ♣♦rt✐♦♥ m♦❢ ✐ts ♦✉t♣✉t t♦ s✉❜s✐❞✐❛r② ✐♥ ❈♦✉♥tr② ✷✱ r❡❣❛r❞✐♥❣ ❛ s✐♥❣❧❡ t②♣❡ ♦❢

♣r♦❞✉❝t✱ ❝❤❛r❣✐♥❣ ❛ tr❛♥s❢❡r ♣r✐❝❡ p ❡st❛❜❧✐s❤❡❞ ❜② ♠❡❛♥s ♦❢ ❡①❝❧✉s✐✈❡ s❡❧❢✲❞✐s❝r❡t✐♦♥ ♦❢ ▼◆❊✬s

❝❡♥tr❛❧ ♠❛♥❛❣❡♠❡♥t✳ ❲❡ s❡tm=m(s₂) ❛♥❞∂m/∂s₂ >0✱ t❤✉s ✐♥tr❛✲✜r♠ ♦✉t♣✉tm ❞❡♣❡♥❞s ♦♥

t❤❡ ♠❛r❦❡t ❞❡♠❛♥❞ ❢♦r ✜♥❛❧ ♣r♦❞✉❝t ✐♥ ❈♦✉♥tr② ✷✳ ❚❤❡ ♣r❡✲t❛① ♣r♦✜ts ♦❢ ❜♦t❤ ❞✐✈✐s✐♦♥s ❛r❡

π₁ =R₁(s₁)−C₁(s₁+m) +pm;

π₂ =R₂(s₂)−C₂(s₂−m)−pm.

❈♦✉♥tr② ✶ ❛♣♣❧✐❡s t❤❡ s♦✉r❝❡ ♣r✐♥❝✐♣❧❡ ♦♥ t❛①❛t✐♦♥ ♦❢ ❢♦r❡✐❣♥ ♣r♦✜ts✱ ❛♥❞ ✇❡ ❛ss✉♠❡ ♥♦ ✐♥✲

❝r❡♠❡♥t❛❧ ♦♣❡r❛t✐♦♥❛❧ ❝♦st ♦♥ tr❛♥s❢❡rr✐♥❣ ✐♥t❡r♥❛❧ ♦✉t♣✉tmt♦ ❞✐✈✐s✐♦♥ ✷✱ ✐✳❡✳ ∂Ci(m, xi)/∂m=

∂Ci(m, xi)/∂xi✳ ❋♦r ❛♥ ✐♥❝♦♠❡ t❛① r❛t❡ τi∈[0,1]✐♥ ❡❛❝❤ ❝♦✉♥tr②✱ ▼◆❊✬s ❣❧♦❜❛❧ ♥❡t ♣r♦✜ts ❛r❡

Π(τi, si, p, m) = (1−τ₁)π₁+ (1−τ₂)π₂✳ Pr♦✜t s❤✐❢t✐♥❣ ✐♥❝❡♥t✐✈❡s ❛r✐s❡ ✇❤❡♥ t❛① r❛t❡s ❜❡t✇❡❡♥

❞✐✈✐s✐♦♥s ❛r❡ ❞✐✛❡r❡♥t✱ τ1 6=τ2✱ ❛♥❞ t♦t❛❧ ♥❡t ♣r♦✜t Π(·) ✐♥❝r❡❛s❡s ✇❤❡♥ ▼◆❊ ✐s ❛❜❧❡ t♦ ❝❤♦♦s❡

❛ s♣❡❝✐✜❝ tr❛♥s❢❡r ♣r✐❝❡ p s♦ ♣r♦✜ts ❛r❡ tr❛♥s❢❡rr❡❞ ❢r♦♠ t❤❡ ❤✐❣❤✲t❛① ❝♦✉♥tr② t♦ t❤❡ ❧♦✇✲t❛①

❝♦✉♥tr②✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ ∂Π(·)/∂p= (τ₂−τ₁)m ✐♠♣❧✐❡s t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝❛s❡s✿

▲♦✇ ❚r❛♥s❢❡r Pr✐❝❡ ❝❛s❡ ✲ ▲❚P✿ τ₂< τ₁ → ∂Π(·)

∂p <0;

❍✐❣❤ ❚r❛♥s❢❡r Pr✐❝❡ ❝❛s❡ ✲ ❍❚P✿ τ₂> τ₁ → ∂Π(·)

∂p >0.

✭✶✮

■♥ t❤❡ ▲❚P ❝❛s❡✱ t❤❡ ▼◆❊ ❤❛s ✐♥❝❡♥t✐✈❡s t♦ s❤✐❢t ♣r♦✜ts ❢r♦♠ ❞✐✈✐s✐♦♥ ✶ t♦ ❞✐✈✐s✐♦♥ ✷ ❜②

❝❤♦♦s✐♥❣ ❛ ❧♦✇ tr❛♥s❢❡r ♣r✐❝❡p✱ t❤✉s ❤❛r♠✐♥❣ t❛① r❡✈❡♥✉❡s ✐♥ ❈♦✉♥tr② ✶✳ ■♥ t❤❡ ❍❚P ❝❛s❡✱ ▼◆❊

❝❤♦♦s❡s ❛ ❤✐❣❤ ♣r✐❝❡ ps♦ t♦ s❤✐❢t ♣r♦✜ts t♦ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✱ t❤✉s ❤❛r♠✐♥❣ ❈♦✉♥tr② ✷✳

✸✳✷ ❋✉③③✐❢②✐♥❣ t❤❡ ❆r♠✬s ▲❡♥❣❤t Pr✐❝❡

❆ss✉♠❡ t❤❛t ❜♦t❤ ❝♦✉♥tr✐❡s ✐♠♣♦s❡ ❛ ♥♦♥✲♥❡❣❧✐❣✐❜❧❡ ❛♥❞ ♥♦♥✲❞❡❞✉❝t✐❜❧❡ t❛① ♣❡♥❛❧t② zi > 0 ✐❢

♣r♦✜t s❤✐❢t✐♥❣ ✐s ❞❡t❡❝t❡❞✱ ✇❤✐❝❤ ✐s ❝♦♠♣✉t❡❞ ❛s ❛ ♣♦rt✐♦♥ ♦❢ t❤❡ ❛♠♦✉♥t ♦❢ ❡✈❛❞❡❞ t❛①❡s✳ ■t

♠❡❛♥s t❤❛t t❤❡ t❛① ♣❡♥❛❧t② zi ✐s ✐♠♣♦s❡❞ ✐❢ t❤❡ ❤❛r♠❡❞ ❈♦✉♥tr② i ♦❜s❡r✈❡s t❤❛t t❤❡ tr❛♥s❢❡r

✹❋♦r s✐♠♣❧✐✜❝❛t✐♦♥✱ ✇❡ ❛♣♣❧② s✉❜s❝r✐♣t i ❢♦r t❤❡ r❡❢❡r❡♥❝❡ ♦❢ ❜♦t❤ ❝♦✉♥tr✐❡s ❛♥❞ t♦ ❡❛❝❤ ▼◆❊✬s ❞✐✈✐s✐♦♥s

❤❡r❡✐♥❛❢t❡r✳

✺

♣r✐❝❡ p✐s ❞✐✛❡r❡♥t ❢r♦♠ ❛ ♣❛r❛♠❡t❡r ♣r✐❝❡p¯❡st❛❜❧✐s❤❡❞ ✉♥❞❡r ❛r♠✬s ❧❡♥❣t❤ ❝♦♥❞✐t✐♦♥s^✺ ❛♥❞ t❤✐s

♣r✐❝❡ ❣❛♣ r❡s✉❧ts ✐♥ t❤❡ ♦✉t✢♦✇ ♦❢ t❛①❛❜❧❡ ♣r♦✜ts ❢r♦♠ ❈♦✉♥tr② i✳ ❚❤❡ ♣❛r❛♠❡t❡r ♦❢ ❛♥ ❛r♠✬s

❧❡♥❣t❤ ♣r✐❝❡ ✐s ❛ ❢✉③③② ❝♦♥❝❡♣t✱ s✐♥❝❡ ✐♥❞❡♣❡♥❞❡♥t ♣r✐❝❡s ✈❛r② ❛❝❝♦r❞✐♥❣ t♦ ❧❡❣✐t✐♠❛t❡ ❞✐✛❡r❡♥❝❡s

✐♥ tr❛♥s❛❝t✐♦♥s✬ ❝♦♥❞✐t✐♦♥s✳ ❚❤❡r❡❢♦r❡✱ ❝♦✉♥tr✐❡s r❛t❤❡r ♦❜s❡r✈❡ ❛ ❢✉③③② s❡t ♦❢ ♣❛r❛♠❡t❡r ♣r✐❝❡sP˜✱

❛❧❧ ♦❢ ✇❤✐❝❤ ❤❛✈❡ ❞✐✛❡r❡♥t ❞❡❣r❡❡s ♦❢ ❛♣♣r♦♣r✐❛t❡♥❡ss ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛r♠✬s ❧❡♥❣t❤ ♣r✐♥❝✐♣❧❡^✻✳

❉❡✜♥❡ t❤❡ ❢✉③③② s❡t ♦❢ ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡sP˜ ={(pj, µP˜(pj)) :pj ∈P}✱j6=i✱P ∈R₊✱ ✇❤❡r❡

P ✐s t❤❡ ✉♥✐✈❡rs❡ ♦❢ ❛❧❧ ♦❜s❡r✈❛❜❧❡ ✐♥❞❡♣❡♥❞❡♥t ♣r✐❝❡s✱ ✉♥✐✈❡rs❡ P ✐s ❝♦♥✈❡①✱ ❛♥❞ µ_P˜(p_j) ✐s t❤❡

♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢✉③③② s❡t P˜✳ ❋♦r ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ♣r✐❝❡s p^∧ ≤p¯^∧ ≤p¯^∨ ≤ p^∨∈P✱ t❤❡ ❢✉③③② s❡tP˜ s❛t✐s✜❡s t❤❡ ✉s✉❛❧ ❝♦♥❞✐t✐♦♥s

µ_P˜(pj) = 0 ❢♦r ∀pj ∈/ [p^∧, p^∨]; ✭✷✮

µ_P˜(p_j)✐s ♥♦♥✲❞❡❝r❡❛s✐♥❣ ❢♦r ∀p_j ∈[p^∧,p¯^∧]; ✭✸✮

µ_P˜(pj) ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❢♦r ∀pj ∈[¯p^∨, p^∨]. ✭✹✮

❚❤❡ ♠♦❞❡ ♦❢ t❤❡ ❢✉③③② s❡t P˜ s❛t✐s✜❡s ∀pj ∈P :µ_P˜(pj) = 1✱ ✇❤✐❝❤ ♣r♦✈✐❞❡s t❤❡ ✐♥t❡r✈❛❧ ♦❢

♣r✐❝❡s t❤❛t ❝♦♠♣❧❡t❡❧② s❛t✐s❢② t❤❡ ❛r♠✬s ❧❡♥❣t❤ ♣r✐♥❝✐♣❧❡✱ µ_P˜(pj) = 1 ❢♦r ∀pj ∈ [¯p^∧,p¯^∨]✳ ❍❡♥❝❡✱

t❤❡ ❝❤♦✐❝❡ ♦❢ ❛♥② str✐❝t ♣❛r❛♠❡t❡r ♣r✐❝❡ p¯♠✉st ❧✐❡ ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧ ♦❢ ♣r✐❝❡s t❤❛t ❞❡✜♥❡ t❤❡

♠♦❞❡ ♦❢ t❤❡ ❢✉③③② s❡tP˜✱ ✐✳❡✳ p¯∈[¯p^∧,p¯^∨]✳ ❊q✳ ✷ ❞❡✜♥❡s t❤❡ ❧✐♠✐t✐♥❣ ✐♥t❡r✈❛❧[p^∧, p^∨]♦✉t ♦❢ ✇❤✐❝❤

❛♥② ♣r✐❝❡ p✐s ❝♦♥s✐❞❡r❡❞ ❛ ❝♦♠♣❧❡t❡❧② ♥♦♥✲❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡✳

❯♥❞❡r t❤❡s❡ ❝♦♥❞✐t✐♦♥s✱ t❤❡ ❢✉③③② s❡tP˜ ❜❡❝♦♠❡s ❛ ❢✉③③② ♥✉♠❜❡r ♦❢ t❤❡LR✲t②♣❡✳ ❈❛❧❧P˜ t❤❡

❢✉③③② ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡✳ ❲❡ ❞❡✜♥❡ ❛ st❛♥❞❛r❞ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢✉③③② ♥✉♠❜❡r P˜ ❛s

❢♦❧❧♦✇s✿

µ_P˜(pj) =

































 f^∧

pj−p^∧

¯ p^∧−p^∧

, p^∧≤pj ≤p¯^∧

1, p¯^∧≤pj ≤p¯^∨

f^∨

pj−p^∨

¯ p^∨−p^∨

, p¯^∨≤pj ≤p^∨

0, ♦t❤❡r✇✐s❡

✭✺✮

✇✐t❤ ❜♦t❤ ❢✉♥❝t✐♦♥s f^∧(·) ❛♥❞ f^∧(·) ♠♦♥♦t♦♥❡ ❝♦♥t✐♥✉♦✉s✳ ■♥ ❊q✳ ✺✱ ✇❡ ❛❧❧♦✇ ❢♦r t❤❡ ❢✉③③②

❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡ P˜ t♦ ❜❡ ❛s②♠♠❡tr✐❝✳ ❚❤✐s ❛s②♠♠❡tr② ♠❛② ❜❡ ❞✉❡ t♦ ❛ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡

✇✐❞t❤sp¯^∧−p^∧ ❛♥❞ p^∨−p¯^∨ ♦♥ t❤❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r s✐❞❡s ♦❢ t❤❡ ❢✉③③② ♥✉♠❜❡r P˜ r❡s♣❡❝t✐✈❡❧②✱ ❛s

✇❡❧❧ ❛s ❢♦r ❞✐✛❡r❡♥❝❡s ✐♥ ❣r❛❞❡s ♦❢ ♠❡♠❜❡rs❤✐♣ ❞❡♥♦t❡❞ ❜② ❢✉♥❝t✐♦♥s f^∧(·) ❛♥❞ f^∨(·)✳ ■♥ ❡✛❡❝t✱

t❤❡ ❛s②♠♠❡tr② ✐♥ t❤❡ ❢✉③③② ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡ P˜ ✐s ✉s❡❢✉❧ t♦ ❞❡s❝r✐❜❡ ❤♦✇ ❈♦✉♥tr✐❡s ✶ ❛♥❞ ✷

❞✐✛❡r ✐♥ t❤❡✐r t♦❧❡r❛♥❝❡ ❢♦r ❛ tr❛♥s❢❡r ♣r✐❝❡p ❢❛rt❤❡r ❢r♦♠ t❤❡ ♣❛r❛♠❡t❡r ♣r✐❝❡ p¯✳

✺❚❤❡ tr❛♥s❢❡r ♣r✐❝✐♥❣ ❣✉✐❞❡❧✐♥❡s ♣r❡♣❛r❡❞ ❜② ✭❖❊❈❉✱ ✷✵✶✼✮ ❤❛✈❡ ❜❡❝♦♠❡ t❤❡ ♠❛✐♥ ❝r✐t❡r✐♦♥ ❛❞♦♣t❡❞ ❜② ♠♦st

❝♦✉♥tr✐❡s ✇♦r❧❞✇✐❞❡ ❢♦r ❡✈❛❧✉❛t✐♦♥ ♦❢ ✐♥tr❛✲✜r♠ ♣r✐❝❡s✳ ❚❤❡ ❣✉✐❞❡❧✐♥❡s ❛r❡ ❜✉✐❧t ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ ❛r♠✬s ❧❡♥❣t❤

♣r✐♥❝✐♣❧❡ ❛s t❤❡ ❢✉♥❞❛♠❡♥t❛❧s ❢♦r t❛①✲❝♦♠♣❧✐❛♥t tr❛♥s❢❡r ♣r✐❝✐♥❣✳

✻■♥ t❤✐s ❧✐♥❡✱ ❛♥t✐✲s❤✐❢t✐♥❣ r✉❧❡s ✉s✉❛❧❧② ❡st❛❜❧✐s❤ ❛♥ ❛r♠✬s ❧❡♥❣t❤ r❛♥❣❡ ♦❢ ❛♣♣r♦♣r✐❛t❡ tr❛♥s❢❡r ♣r✐❝❡s✳ ❚❤❡

❛r♠✬s ❧❡♥❣t❤ r❛♥❣❡ ✐s ✉s✉❛❧❧② s❡t ❛s ❛♥ ✐♥t❡rq✉❛rt✐❧❡ r❛♥❣❡ ✇✐t❤✐♥ t❤❡ ❝♦♠♣❧❡t❡ s❡t ♦❢ ❝♦♠♣❛r❛❜❧❡ ♣r✐❝❡s ✭❖❊❈❉✱

✷✵✶✼✮✳

✻

❋♦r t❤❡ ▲❚P ❝❛s❡ ✐♥ ❊q✳ ✶✱ ❈♦✉♥tr② ✶ ✐s ❧❡ss t♦❧❡r❛♥t ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❧♦✇ tr❛♥s❢❡r ♣r✐❝❡ ❝❧♦s❡

t♦ p^∧✱ ✇❤✐❧❡ ✐t ❛❝❝❡♣ts ♣r✐❝❡s ♥❡❛r ♦r ❤✐❣❤❡r t❤❛♥ t❤❡ ♣❛r❛♠❡t❡r ♣r✐❝❡ p¯✳ ❚❤❡r❡❢♦r❡✱ ❈♦✉♥tr② ✶ ✐s

♦♥❧② ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ ❧♦✇❡r s✐❞❡ f^∧(·)♦❢ t❤❡ ❢✉③③② ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡ P✳ ❚❤❡ ♦♣♣♦s✐t❡ ♦❝❝✉rs˜

❢♦r t❤❡ ❍❚P ❝❛s❡ ✐♥ ❊q✳ ✶✱ s✐♥❝❡ ❈♦✉♥tr② ✷ ✐s ♦♥❧② ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ ❤✐❣❤❡r s✐❞❡ f^∨(·) ♦❢P˜✳ ■❢

✇❡ ❞✐✈✐❞❡ t❤❡ ❢✉③③② ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡P˜ ✐♥t♦ t✇♦ ♠❡♠❜❡rs❤✐♣ s❡❝t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ ❧♦✇❡r s✐❞❡

f^∧(·) ❛♥❞ ✉♣♣❡r s✐❞❡f^∨(·)✱ ✇❡ ♦❜t❛✐♥ t✇♦ ❢✉③③② ♥✉♠❜❡rs P˜^∧ ❛♥❞ P˜^∨ s❛t✐s❢②✐♥❣ t❤❡ ❛❞❞✐t✐♦♥❛❧

❝♦♥❞✐t✐♦♥s✿

µ_P˜∧(pj) =

( µP˜(pj), pj ≤p¯^∨

1, p_j >p¯^∨. ✭✻✮

µ_P˜∨(pj) =

( µ_P˜(p_j), p_j ≥p¯^∧

1, pj <p¯^∧. ✭✼✮

P˜ = ˜P^∧∩P˜^∨. ✭✽✮

■t ✐s ❝❧❡❛r t❤❛t t❤❡ ❢✉③③② ♥✉♠❜❡rsP˜^∧❛♥❞P˜^∨ r❡❢❡r t♦ t❤❡ ❢✉③③② ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡s t❛❦❡♥ ✐♥t♦

❛❝❝♦✉♥t ❜② ❈♦✉♥tr✐❡s ✶ ❛♥❞ ✷ r❡s♣❡❝t✐✈❡❧②^✼✳ ❲❡ ✐♥❞✐❝❛t❡ t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ t❤❡ ❢✉③③② ❛r♠✬s

❧❡♥❣t❤ ♣r✐❝❡s s❛t✐s❢②✐♥❣ ❝♦♥❞✐t✐♦♥s ✐♥ ❊q✳ ✻✲✽ ❛sP˜^c✱c={∧,∨}✳ ❚❤❡ ♠♦❞❡ ♦❢ t❤❡ ❢✉③③② ♥✉♠❜❡rs P˜^c s❛t✐s✜❡s t❤❡ st❛♥❞❛r❞ ❝♦♥❞✐t✐♦♥∀p_j ∈P :µ_P˜c(p_j) = 1✳ ❚❤❡ ❜♦✉♥❞ ♦❢ t❤❡ ♠♦❞❡ ♦❢ t❤❡ ❢✉③③②

♥✉♠❜❡rs P˜^c ✐s ❞❡✜♥❡❞ ✐♥ st❛♥❞❛r❞ ❢♦r♠^✽ ❛sp¯^c✳ ❍❡♥❝❡✱ ❜♦t❤ ♣r♦✜t s❤✐❢t✐♥❣ ❝❛s❡s ✐♥ ❊q✳ ✶ ✐♠♣❧②

▲❚P→ {i= 1, c=∧}✱ ❍❚P → {i= 2, c=∨}✳

✸✳✸ ❚❛① ❆✉❞✐ts ❛♥❞ ❚❛① P❡♥❛❧t✐❡s

❇♦t❤ ❝♦✉♥tr✐❡s ♣❡r❢♦r♠ t❛① ❛✉❞✐ts ✐♥ ♦r❞❡r t♦ ♣r❡✈❡♥t t❤❡ ♣r♦✜t s❤✐❢t✐♥❣✳ ■♥ t❤❡ ✉♥✐✈❡rs❡ ♦❢ ❛❧❧

t❛①♣❛②❡rs✱ ✇❡ ❛ss✉♠❡ t❤❛t ❝♦✉♥tr✐❡s ❛r❡ ♥♦t ❛❜❧❡ t♦ ❝♦♥t✐♥✉♦✉s❧② ♦❜s❡r✈❡ ❛❧❧ ▼◆❊ ✐♥ ❛❜s♦❧✉t❡

❝♦♠♣❧❡t❡♥❡ss✱ ❜✉t t❤❡② ❤❛✈❡ t♦ ❡① ❛♥t❡ s❡❧❡❝t ✇❤✐❝❤ ▼◆❊ ❛r❡ ❣♦✐♥❣ t♦ ❜❡ ❛✉❞✐t❡❞✳ ■♥ s♣❡❝✐❛❧✱

❜♦t❤ ❝♦✉♥tr✐❡s ❤❛✈❡ ♥♦ ♣r✐♦r ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t t❤❡ ❡①✐st❡♥❝❡ ♦❢ ✐♥tr❛✲✜r♠ tr❛♥s❛❝t✐♦♥spm✱ t❤♦✉❣❤

t❤✐s ❦♥♦✇❧❡❞❣❡ ❞❡♣❡♥❞s ♦♥ ❛♥ ✐♥✐t✐❛❧ ♣✐❝❦✳ ❋♦❧❧♦✇✐♥❣ ✭▲❡✈❛❣❣✐ ✫ ▼❡♥♦♥❝✐♥✱ ✷✵✶✸✮✱ ✇❡ s❡t t❤❡

❛✉❞✐t s❡❧❡❝t✐♦♥ ✐♥ ❈♦✉♥tr②i❛s ❛ P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ ✐♥t❡♥s✐t② r❛t❡λ_i>0❤♦♠♦❣❡♥❡♦✉s t❤r♦✉❣❤

t❤❡ t♦t❛❧ ♣❡r✐♦❞ ❞❡t❡r♠✐♥❡❞ ✐♥ t❤❡ ❧❡❣❛❧ st❛t✉t❡ ♦❢ ❧✐♠✐t❛t✐♦♥s✳ ❘❛t❡ λi r❡❢❡rs t♦ t❤❡ t❛① ❛✉❞✐t

✐♥t❡♥s✐t② ✐♥ ❈♦✉♥tr② i✳ ■❢ t❤❡ ▼◆❊ ✐s s❡❧❡❝t❡❞✱ ❈♦✉♥tr② i ✇✐❧❧ ♦❜s❡r✈❡ pm✱ t❤✉s tr✐❣❣❡r✐♥❣ ❛

❝❤❛♥❝❡ ❢♦r t❛① ♣❡♥❛❧t② z_i✳

■❢ t❤❡ ♥✉♠❜❡r ♦❢ t❛① ❛✉❞✐ts ♣❡r❢♦r♠❡❞ ❜② ❈♦✉♥tr② i✐sq ∈N✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❡①❛❝t q=k t❛① ❛✉❞✐ts ✐s P(q =k, λi) =λ^k_ie^−λi/k!✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❝✉♠✉❧❛t✐✈❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❈♦✉♥tr② it♦

♣❡r❢♦r♠ ✉♣ t♦ k ❛✉❞✐ts✱P(0≤q ≤k, λ_i) ✐s ❝♦♠♣✉t❡❞ ❛s

P(0≤q ≤k, λi) =

k

X

q=0

P(q, λi) = Γ(k+ 1, λi)

Γ(k+ 1) ✭✾✮

✼■t ✐s ❛❧s♦ ❝❧❡❛r t❤❛t t❤❡ ❢✉③③② ♥✉♠❜❡rsP˜^∧ ❛♥❞P˜^∨❛r❡ ♦❢ t❤❡L✲t②♣❡ ❛♥❞R✲t②♣❡ r❡s♣❡❝t✐✈❡❧②✳

✽❚❤❡ ❜♦✉♥❞p¯^c ♦❢ t❤❡ ♠♦❞❡ ♦❢ t❤❡ ❢✉③③② ♥✉♠❜❡rP˜^c❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s

¯

p^c:µP˜^c(¯p^c+ ∆p)<1, lim

∆p→0µP˜^c(¯p^c+ ∆p) = 1

✇✐t❤ ❞❡✈✐❛t✐♦♥∆p∈R✳

✼

✇❤❡r❡ Γ(k) ✐s t❤❡ ❣❛♠♠❛ ❢✉♥❝t✐♦♥ ❛♥❞ Γ(k, λ) ✐s t❤❡ ✉♣♣❡r ❣❛♠♠❛ ❢✉♥❝t✐♦♥^✾✳ ❘❡♠❛r❦ t❤❛t ♥♦

♣❡♥❛❧✐s❛t✐♦♥ ✇✐❧❧ ❜❡ ✐♠♣♦s❡❞ ✐❢ t❤❡r❡ ✐s ♥♦ t❛① ❛✉❞✐t✱ q = 0✳ ▼♦r❡♦✈❡r✱ ❡✈❡♥ ✇✐t❤ ❛♥ ❡st✐♠❛t❡

♦❢ t❤❡ ♥✉♠❜❡r ♦❢ t❛① ❛✉❞✐ts E(q =k, λi) = λi✱ t❤❡ ▼◆❊ ❝❛♥ ❜❡ s❡❧❡❝t❡❞ ✉♥❞❡r ❛♥② ♥✉♠❜❡r q

❞✐✛❡r❡♥t ❢r♦♠ k✳ ■♥ s✉♠♠❛r②✱ ▼◆❊ ❤❛s ❛ ❝❤❛♥❝❡ ♦❢ ❜❡✐♥❣ s❡❧❡❝t❡❞ ❢♦r t❛① ❛✉❞✐t ✐❢ ❈♦✉♥tr② i

♣❡r❢♦r♠s ❛t ❧❡❛st ♦♥❡ ❛✉❞✐t✳ ❚❤❡r❡❢♦r❡✱ t❤❡ t♦t❛❧ ♣r♦❜❛❜✐❧✐t② ♦❢ t❛① ❛✉❞✐t ❢♦r t❤❡ ▼◆❊ ✐s P(q >0, λ_i) = 1−P(q= 0, λ_i) = 1−Γ(1, λi)

Γ(1) = 1−e^−λi. ✭✶✵✮

■♥ t❤❡ ❝❛s❡ ♦❢ ❛✉❞✐t s❡❧❡❝t✐♦♥✱ ❈♦✉♥tr②i♦❜s❡r✈❡s t❤❡ ✐♥tr❛✲✜r♠ tr❛♥s❛❝t✐♦♥spm❛♥❞ ❝♦♠♣❛r❡s t❤❡ tr❛♥s❢❡r ♣r✐❝❡ p ✇✐t❤ t❤❡ ❛r♠✬s ❧❡♥❣t❤ ♣❛r❛♠❡t❡r p¯✳ ■❢ t❤❡ ❤❛r♠❡❞ ❈♦✉♥tr② i ❝♦♥❝❧✉❞❡s t❤❛t t❤❡ ▼◆❊ ✐s s❤✐❢t✐♥❣ t❛①❛❜❧❡ ♣r♦✜ts ❛✇❛②✱ t❤❡ ▼◆❊ ✐s r❡q✉✐r❡❞ t♦ ♣❛② t❤❡ ❛♠♦✉♥t ♦❢

❡✈❛❞❡❞ t❛①❡s ♣❧✉s ❛ ♣❡♥❛❧t② zi ❧❡✈✐❡❞ ♦✈❡r t❤✐s ❛♠♦✉♥t✳ ■♥ t❤✐s ❝❛s❡✱ t❛① ♣❡♥❛❧t② ✐s ❝♦♠♣✉t❡❞

❛s Zi(zi, τi, p,p, m) = (1 +¯ zi)·sgn(τ₂−τ₁)τi·(p−p)m¯ ≥ 0✱ ✇❤❡r❡ sgn(·) ✐s t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥

❛♥❞ t❛① r❛t❡s ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡✱ τi ∈[0,1] ✳ ❖❜s❡r✈❡ t❤❛t t❤❡ t♦t❛❧ t❛① ♣❡♥❛❧t② ✐s ♥♦♥✲♥❡❣❛t✐✈❡

Zi(·)≥0❢♦r ❜♦t❤ ▲❚P ❛♥❞ ❍❚P ❝❛s❡s^✶✵✳

◆♦♥❡t❤❡❧❡ss✱ t❤❡ ❛ss❡ss♠❡♥t ♦❢ t❤❡ tr❛♥s❢❡r ♣r✐❝❡p❜② ❈♦✉♥tr② i✐s ❜❛s❡❞ ♦♥ t❤❡ ❢✉③③② ❛r♠✬s

❧❡♥❣t❤ ♣❛r❛♠❡t❡rP˜^c✱c={∧,∨}✳ ❋♦r♠❛❧❧②✱ t❤✐s ❛ss❡ss♠❡♥t ✐s ♠❛❞❡ ❜② t❛❦✐♥❣ t❤❡ ❢✉③③② ♥✉♠❜❡r P˜^c = {(pj, µ_P˜c(pj)) :pj ∈ P} ❛♥❞ s❡tt✐♥❣ t❤❡ ❡q✉❛❧✐t② p =pj✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❢✉③③② ♣r✐❝❡ ❣❛♣

∆p =p^−p¯^c✱ ✇❤❡r❡ p¯^c ✐s t❤❡ ❜♦✉♥❞ ♦❢ t❤❡ ♠♦❞❡ ♦❢ t❤❡ ❢✉③③② ♥✉♠❜❡r P˜^c✳ ❚❤❡ ❢✉③③② ♣r✐❝❡ ❣❛♣

∆p ✐s ❞❡✜♥❡❞ s✉❝❤ ❛s t♦ s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥ p ={(¯p^c+ ∆p, µ_P˜c(¯p^c + ∆p)) : p ∈P}✳ ❋♦r t❤❡

❤❛r♠❡❞ ❈♦✉♥tr② i✱ ♣r♦✜t s❤✐❢t✐♥❣ ♠❛② ❡①✐st ✐✛ µ_P˜c(p)<1✱ ✐✳❡✳ ✐✛ t❤❡ ❢✉③③② ♣r✐❝❡ ❣❛♣∆p♣✉s❤❡s t❤❡ tr❛♥s❢❡r ♣r✐❝❡ p ❛✇❛② ❢r♦♠ t❤❡ ♠♦❞❡ ♦❢ P˜^c✱∀pj :µP˜^c(pj) = 1✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ♦r✐❣✐♥❛❧ t❛①

♣❡♥❛❧t② Z_i(·)≥0 t✉r♥s ✐♥t♦ ❛ ❢✉③③② t❛① ♣❡♥❛❧t② ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛♥❞❛r❞ ❢♦r♠✿

Z˜i(zi, τi,∆p, m) =

( 0, µP˜^c(p) = 1

(1 +z_i)·sgn(τ₂−τ₁)τ_i·(p^−p¯^c)m, ♦t❤❡r✇✐s❡✳ ✭✶✶✮

■t ♠❡❛♥s t❤❛t t❤❡ ❤❛r♠❡❞ ❈♦✉♥tr②i❤❛s t❤❡ t❛s❦ t♦ ❛ss❡ss ✐❢ t❤❡ ♣r✐❝❡ ❣❛♣ ∆p✐s ❛ t♦❧❡r❛❜❧❡

✈❛r✐❛♥❝❡ ✉♥❞❡r t❤❡ ❢✉③③② ❛r♠✬s ❧❡♥❣t❤ ❝♦♥❞✐t✐♦♥s ♦r ✐❢ ✐t ✐s ❛♥ ❡✈✐❞❡♥❝❡ ♦❢ ♣r♦✜t s❤✐❢t✐♥❣✳

✹ ❖♣t✐♠❛❧ ❚r❛♥s❢❡r Pr✐❝✐♥❣

❚❤❡ ▼◆❊ ❛✐♠s ❝❤♦♦s❡ ❛ tr❛♥s❢❡r ♣r✐❝❡ ps♦ t♦ ♠❛①✐♠✐s❡ ❣❧♦❜❛❧ ♥❡t ♣r♦✜tsΠ(·)✱ ❤♦✇❡✈❡r ✐t ❢❛❝❡s t❤❡ ❝❤❛♥❝❡ ♦❢ t❛① ♣❡♥❛❧✐s❛t✐♦♥ ✐❢ t❤❡ ❤❛r♠❡❞ ❈♦✉♥tr② i ✜♥❞s ♦✉t t❤❡ ❡①✐st❡♥❝❡ ♦❢ ✐♥tr❛✲✜r♠

tr❛♥s❛❝t✐♦♥s pm ❛♥❞ ❞❡❝✐❞❡s t❤❛t ✐t r❡♣r❡s❡♥ts ❛ ♣r♦✜t s❤✐❢t✐♥❣ str❛t❡❣②✳ ■♥ t❤✐s ❧✐♥❡✱ ❛ss✉♠✐♥❣

t❤❛t t❤❡ ♦♣t✐♠❛❧ tr❛♥s❢❡r ♣r✐❝❡ p^∗ ✐♠♣❧✐❡s µ_P˜c(p^∗) < 1✱ t❤❡ ▼◆❊ ❤❛s ❛ ♠❛①✐♠✐s❛t✐♦♥ ♦❜❥❡❝t s♣❡❝✐✜❡❞ ❛s ❢♦❧❧♦✇s✿

✾❉❡r✐✈❛t✐♦♥ ♦❢ ❊q✳ ✾ ✐♥ ❆♣♣❡♥❞✐①✳

✶✵❚♦t❛❧ t❛① ♣❡♥❛❧t②Zi(·)≥0✐s ♥♦♥✲♥❡❣❛t✐✈❡ s✐♥❝❡ t❤❡ s✐❣♥s ♦❢ ❜♦t❤ t❤❡ t❛① ❞✐✛❡r❡♥t✐❛❧τ2−τ1 ❛♥❞ t❤❡ ♣r✐❝❡

❣❛♣p−p¯❝❛rr② ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ s❤✐❢t✐♥❣ ❞✐r❡❝t✐♦♥❀ ❍❚P ✐♠♣❧✐❡sτ2−τ1>0✱p−p >¯ 0✱ ✇❤✐❧❡ ▲❚P ✐♠♣❧✐❡s τ2−τ1<0✱p−p <¯ 0✳

✽

maxp∈P

E( ˜Π(·)) = Π(τi, si, p, m)−E( ˜Zi(zi, τi,∆p, m))

= (1−τ1)π1+ (1−τ2)π2

−(1−e^−λi)·(1 +zi)·sgn(τ₂−τ₁)τi·(p^−p¯^c)m.

✭✶✷✮

❙✐♥❝❡ t❤❡ ❡①♣❡❝t❡❞ t❛① ♣❡♥❛❧t② E( ˜Zi(·)) ✐s ❛ ❢✉③③② ♥✉♠❜❡r✱ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✐♥ ❊q✳ ✶✷

❜❡❝♦♠❡s ❛ ❢✉③③② ♦❜❥❡❝t✐✈❡✱ ❛♥❞ ♣r♦✜t ♠❛①✐♠✐s❛t✐♦♥ ♠✉st t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❢✉③③✐♥❡ss ♦❢ t❤❡

♣r✐❝❡ ❣❛♣ ∆p=p^−p¯^c✳

❈♦♥❞✐t✐♦♥s ✐♥ ❊q✳ ✻✲✽ s❤♦✇ t❤❛t t❤❡ st❛♥❞❛r❞✲❢♦r♠ ❢✉③③② ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡ P˜^c r❡♣r❡s❡♥ts ❛

♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦ ❝♦rr❡s♣♦♥❞❡♥❝❡ µP˜^c(pj) :R → [0,1] ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ ♦❢

✐♥t❡r❡stpj ∈[p^c,p¯^c]✳ ❚❤❡r❡❢♦r❡✱ ✇❡ s♦❧✈❡ ❊q✳ ✶✷ ❜② ❛♣♣❧②✐♥❣ t❤❡ ♣r♦❝❡❞✉r❡ ❢♦r ❢✉③③② ♦♣t✐♠✐s❛t✐♦♥

❞❡✈❡❧♦♣❡❞ ✐♥ t❤❡ ❝❧❛ss✐❝❛❧ ✇♦r❦ ♦❢ ✭❱❡r❞❡❣❛②✱ ✶✾✽✷✮✳

❋♦r t❤❡ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥µ_P˜c(p_j)✱p_j ∈[p^c,p¯^c]✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ α✲❝✉ts ❛r❡ P_α^c ={p_j ∈ [p^c,p¯^c] : µ_P˜c(pj) ≥α}✳ ❋r♦♠ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r ❢✉③③② s❡ts✱ ❊q✳ ✶✷ ✐s ❡①♣r❡ss❡❞ ✐♥

t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❢♦r♠✿

α∈[0,1]max

p∈P_α^c

E( ˜Π(·)) = (1−τ₁)π₁+ (1−τ₂)π₂

−(1−e^−λi)·(1 +zi)·sgn(τ₂−τ₁)τi·(p−p¯^c)f(α)m ✭✶✸✮

✇✐t❤ α ∈ [0,1]✱ ✇❤❡r❡ f(α) : [0,1] → P ∈ R₊✱ f(α) =µ⁻¹_˜

P^c(α) ✐s t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡

♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥µ_P˜c(p_j)✳ ❙✐♠♣❧② st❛t❡❞✱ ✐❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❊q✳ ✶✸ ✐sp^∗(α)✱ t❤❡♥ t❤❡ s♦❧✉t✐♦♥

♦❢ ❊q✳ ✶✷ ✐s t❤❡ ❢✉③③② s❡t p^∗ = {(p(α), α)}✳ ❍❡♥❝❡✱ ♣r♦✜t ♠❛①✐♠✐s❛t✐♦♥ ✐♥ ❊q✳ ✶✷ r❡s✉♠❡s t♦

✜♥❞ t❤❡ ♦♣t✐♠❛❧ α✲❝✉t ❞❡✜♥❡❞ ❜②P_=α^c ={p^∗(α)∈[p^c,p¯^c] :µP˜^c(p^∗(α)) =α}❛t t❤❡ ♠❡♠❜❡rs❤✐♣

❣r❛❞❡ µ_P˜c(p^∗(α)) =α✳

❇❛s❡❞ ♦♥ t❤❡ ❣❡♥❡r❛❧ ❙t♦♥❡✲❲❡✐❡rstr❛ss ❛♣♣r♦①✐♠❛t✐♦♥✱ ❛ss✉♠❡ t❤❛t t❤❡ st❛♥❞❛r❞✲❢♦r♠ s❤❛♣❡

❢✉♥❝t✐♦♥ f^c(·) ✐♥ ❊q✳ ✺ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ❛ s✐♠♣❧❡ ♣♦✇❡r ❢✉♥❝t✐♦♥

f^c

p−p^c

¯ p^c−p^c

=

p−p^c

¯ p^c−p^c

γi

✭✶✹✮

✇✐t❤ γi ∈ (0,1] ❛s ❛ r❡❣✉❧❛r✐s❡❞ ♣❛r❛♠❡t❡r ❢♦r t❤❡ t♦❧❡r❛♥❝❡ ♦❢ ❈♦✉♥tr② i r❡❣❛r❞✐♥❣ ❢✉③③✐♥❡ss

✐♥ t❤❡ ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡✱ ❡✳❣✳ ❛ s❧❛❝❦❡♥ t❛① ❛ss❡ss♠❡♥t ❜② ❈♦✉♥tr② i ✐♠♣❧✐❡s γi → 0✱ ✇❤✐❧❡

❛ t✐❣❤t❡♥ t❛① ❛ss❡ss♠❡♥t ✐♠♣❧✐❡s γ_i → 1✳ ❊q✳ ✶✹ ♣r♦✈✐❞❡s ❛ s♠♦♦t❤ ✈❛r✐❛t✐♦♥ ✐♥ ♠❡♠❜❡rs❤✐♣

❣r❛❞❡ ❛s tr❛♥s❢❡r ♣r✐❝❡p ❣❡ts ❢❛rt❤❡r ❢r♦♠ t❤❡ ❜♦✉♥❞ ♦❢ t❤❡ ♠♦❞❡p¯^c✳ ❋♦r t❤❡ ✐♥t❡r✈❛❧ ♦❢ ✐♥t❡r❡st p∈[p^c,p¯^c]✱ ♣❛r❛♠❡tr✐❝ ♦♣t✐♠✐s❛t✐♦♥ ✐♥ ❊q✳ ✶✸ t❤❡♥ ❜❡❝♦♠❡s

✾

p∈[pmax^c,¯p^c]

E( ˜Π(·)) = (1−τ₁)π₁+ (1−τ₂)π₂

−(1−e^−λi)·(1 +zi)·sgn(τ₂−τ₁)τi·(p−p¯^c)·µ⁻¹_˜

P^c(α)m

= (1−τ₁)π₁+ (1−τ₂)π₂

−(1−e^−λi)·(1 +z_i)·sgn(τ₂−τ₁)τ_i·(p−p¯^c)

1−

p−p^c

¯ p^c−p^c

¹

γim

= (1−τ1)π1+ (1−τ2)π2

−(1−e^−λi)·(1 +z_i)·sgn(τ₂−τ₁)τ_i·(p−p¯^c)

p−p¯^c p^c−p¯^c

¹

γim.

✭✶✺✮

◆♦✇ ✇❡ ❤❛✈❡ t❤❡ ❡①♣❡❝t❡❞ ♥❡t ♣r♦✜ts E( ˜Π(·)) s♣❡❝✐✜❡❞ ❝♦♠♣❧❡t❡❧② ✐♥ t❡r♠s ♦❢ t❤❡ tr❛♥s❢❡r

♣r✐❝❡^✶✶ p✳ ❉✐✛❡r❡♥t✐❛t✐♥❣ ❊q✳ ✶✺ ✇✐t❤ r❡s♣❡❝t t♦ p❛♥❞ s♦❧✈✐♥❣✱ ✇❡ ♦❜t❛✐♥ t❤❡ s♦❧✉t✐♦♥

∂E( ˜Π(·))

∂p = (τ₂−τ₁)m−(1−e^−λi)·(1 +zi)·sgn(τ₂−τ₁)τi·

1 + 1 γ_i

p−p¯^c p^c−p¯^c

¹

γim= 0;

p^∗ = ¯p^c+





τ₂−τ₁

(1−e^−λi)·(1 +zi)·sgn(τ₂−τ₁)τi· 1 +_γ¹

i





γi

(p^c−p¯^c)

= ¯p^c+





|τ₂−τ₁| (1−e^−λi)·(1 +zi)·τi·

1 +_γ¹

i





γi

(p^c−p¯^c)

= P_=α^c ={p^∗ ∈[p^c,p¯^c] :µP˜^c(p^∗) =α}

✭✶✻✮

✇✐t❤ | · | : R → R₊ ❛s t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥^✶✷✳ ❊q✳ ✶✻ s❤♦✇s t❤❛t t❤❡ ♦♣t✐♠❛❧ tr❛♥s❢❡r

♣r✐❝❡ p^∗ ✐s r❡♣r❡s❡♥t❡❞ ❛s ❛ ♠❛①✐♠✐s✐♥❣ α✲❝✉t ♦❢ t❤❡ ❢✉③③② ❛r♠✬s ❧❡♥❣t❤ ♣r✐❝❡ P˜^c ❞❡✜♥❡❞ ❛s P_=α^c ={p^∗ ∈[p^c,p¯^c] : µP˜^c(p^∗) = α}✱ ✐✳❡✳ t❤❡ ♦♣t✐♠❛❧ ♣r✐❝❡ ❣❛♣ ∆p^∗ =p^∗−p¯^c ✐s ❛ s❤❛r❡ ♦❢ t❤❡

♣r✐❝❡ ❞✐✛❡r❡♥❝❡ p^c−p¯^c✳ ❚❤✐s α✲❝✉t ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ s❤❛r❡ ❢✉♥❝t✐♦♥ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ [p^c,p¯^c]✱

✇❤✐❝❤ ✐s ♠❡❛s✉r❡❞ ❛s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ♣r♦✜t s❤✐❢t✐♥❣ ✐♥❝❡♥t✐✈❡ |τ₂ −τ₁| ❛❞❥✉st❡❞ ❜② t❤❡

♠❛r❣✐♥❛❧ ❡①♣❡❝t❡❞ ♣❡♥❛❧✐s❛t✐♦♥ ❡✛❡❝t(1−e^−λi)·(1 +zi)·τi✳ ❚❤❡ s❧♦♣❡ ♦❢ t❤✐s s❤❛r❡ ✐s t❤❡ s❛♠❡

❛s ♦❢ t❤❡ s❤❛♣❡ ❢✉♥❝t✐♦♥ ✐♥ ❊q✳ ✶✹ ❜② ♠❡❛♥s ♦❢ t❤❡ ❡①♣♦♥❡♥t γ_i✳ ■t ❛❧s♦ ❤❛s ❛♥ ❛❞❥✉st♠❡♥t

❡q✉❛❧ t♦ (γi+ 1)/γi✱ ✇❤✐❝❤ ❞❡r✐✈❡s ❢r♦♠ t❤❡ ❡♥❞♦❣❡♥♦✉s s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❢✉③③② ❛r♠✬s ❧❡♥❣t❤

♣r✐❝❡ P˜^c ✐♥ t❡r♠s ♦❢ p ✇✐t❤✐♥ t❤❡ ❡①♣❡❝t❡❞ t❛① ♣❡♥❛❧t② ✐♥ ❊q✳ ✶✺^✶✸✳ ▼♦r❡♦✈❡r✱ t❤❡ ❛♠♦✉♥t ♦❢

✐♥tr❛✲✜r♠ ♦✉t♣✉t m ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ♦♣t✐♠❛❧ tr❛♥s❢❡r ♣r✐❝❡ p^∗ ✐♥ t❤❡ ♠♦❞❡❧✱ ✐✳❡✳ ✐t r❡❢❡rs

✶✶P❛r❛♠❡tr✐❝ ❢♦r♠ ✐♥ ❊q✳ ✶✺ ✐s ♣♦ss✐❜❧❡ s✐♥❝❡ t❤❡ ❛r♠✬s ❧❡♥❣t❤ ♣❛r❛♠❡t❡rsp^c,p¯^c∈P ❛r❡ ❡①♦❣❡♥♦✉s ✇✐t❤ r❡s♣❡❝t t♦Π(·)❛♥❞Z(·)✳˜

✶✷❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt② ✐s ❛♣♣❧✐❡❞✿ ❢♦r ❛♥② r❡❛❧ ♥✉♠❜❡r∀x∈R✱xs❛t✐s✜❡s x=sgn(x)· |x| → |x|= x

sgn(x).

✶✸▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ t❤❡ tr❛♥s❢❡r ♣r✐❝❡p ❛✛❡❝ts ❜♦t❤ t❤❡ tr❛♥s❢❡r ♣r✐❝❡ ❣❛♣∆p=p−p¯^c ❛♥❞ t❤❡ ♠❡♠❜❡rs❤✐♣

r❡❧❛t✐♦♥ µP˜^c(p) s♣❡❝✐✜❡❞ ❜② t❤❡ s❤❛♣❡ ❢✉♥❝t✐♦♥ ✐♥ ❊q✳ ✶✹✱ ❢♦r t❤❡ ❝♦♠❜✐♥❡❞ ♠❛r❣✐♥❛❧ ❡✛❡❝t ♦♥ Z˜(·) ❜❡❝♦♠❡s (γi+ 1)/γi✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ tr❛♥s❢❡r ♣r✐❝❡p❛✛❡❝ts ♠❛r❣✐♥❛❧❧② t❤❡ ♥❡t ♣r♦✜tsΠ(·) ✐♥ ❛ ❞✐r❡❝t ✇❛②✳ ❚❤❡

✶✵