StrategicVotingwithAlmostPerfectSignals Venturini,Andrea MunichPersonalRePEcArchive

Munich Personal RePEc Archive

Strategic Voting with Almost Perfect Signals

Venturini, Andrea

Università del Piemonte Orientale

19 November 2015

Online at https://mpra.ub.uni-muenchen.de/71216/

MPRA Paper No. 71216, posted 11 May 2016 13:42 UTC

❙tr❛t❡❣✐❝ ❱♦t✐♥❣ ✇✐t❤ ❆❧♠♦st P❡r❢❡❝t ❙✐❣♥❛❧s

❆♥❞r❡❛ ❱❡♥t✉r✐♥✐

❯P❖

❉r❛❢t ❱❡rs✐♦♥✿ ◆♦✈❡♠❜❡r ✶✾✱ ✷✵✶✺

❆❜str❛❝t

❆ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ♦❢ str❛t❡❣✐❝ ✈♦t✐♥❣ ✐s t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ s✐❣♥❛❧s✳ ❊❛❝❤

❥✉r♦r ♦❜s❡r✈❡s ❛ s✐❣♥❛❧ ❛t t❤❡ ✐♥t❡r✐♠ st❛❣❡ ♦❢ t❤❡ ❣❛♠❡✳ ❚❤❡♥ s❤❡ ✈♦t❡s ❛❝❝♦r❞✐♥❣ t♦ ❤❡r ♣r✐✈❛t❡

✐♥❢♦r♠❛t✐♦♥ ✐♥ ♦r❞❡r t♦ ♠❛①✐♠✐③❡ ❤❡r ❡①♣❡❝t❡❞ ✉t✐❧✐t②✳ ❚❤✐s ✇♦r❦ ✐♥tr♦❞✉❝❡s ❛ ❞❡♣❡♥❞❡♥❝② ❜❡t✇❡❡♥

s✐❣♥❛❧s✱ r❡✢❡❝t✐♥❣ ❛ ♠♦r❡ r❡❛❧✐st✐❝ s✐t✉❛t✐♦♥✱ ✐♥ ✇❤✐❝❤ ❡✈✐❞❡♥❝❡s ❝❛♥ ❜❡ ✐♥❝♦♥tr♦✈❡rt✐❜❧❡✳ ❲❡ ❣✐✈❡ ❛

❢✉❧❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ s②♠♠❡tr✐❝ ❡q✉✐❧✐❜r✐❛ ✐♥ ♥♦♥✲✇❡❛❦❧② ❞♦♠✐♥❛t❡❞ str❛t❡❣✐❡s ❛♥❞ ✇❡ ♣r♦✈✐❞❡ ❛

❜❡♥❝❤♠❛r❦ ❜❡t✇❡❡♥ t❤❡ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ❛♥❞ t❤✐s ♥❡✇ ♦♥❡✳

❏❡❧ ❈❧❛ss✐❢✐❝❛t✐♦♥✿ ❈✼✷✱ ❉✼✷

✶ ■♥tr♦❞✉❝t✐♦♥

❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❥✉r②✱ ❤♦✇ ❥✉r♦rs ✈♦t❡ ❛♥❞ ✇❤❡♥ ❛ ❝♦❧❧❡❝t✐✈❡ ❝❤♦✐❝❡ ✐s ❜❡tt❡r t❤❛♥ ❛♥ ✐♥❞✐✈✐❞✉❛❧ ♦♥❡ ❤❛s

❜❡❡♥ ♦❜❥❡❝t ♦❢ st✉❞② ✐♥ ♣♦❧✐t✐❝❛❧ s❝✐❡♥❝❡ ❛♥❞ ✐♥ st❛t✐st✐❝s ❢♦r ❛ ❧♦♥❣ t✐♠❡✳ ❙t❛rt✐♥❣ ❢r♦♠ ❈♦♥❞♦r❝❡t✬s ❢❛♠♦✉s

❚❤❡♦r❡♠ ✭✶✼✽✺✮✱ t❤❛t ❝❛♥ ❜❡ st❛t❡❞ ❛s ✏✉♥❞❡r t❤❡ ♠❛❥♦r✐t② r✉❧❡✱ ❣r♦✉♣s ♦❢ ♣❡♦♣❧❡ ♠❛❦❡ ❜❡tt❡r ❞❡❝✐s✐♦♥s t❤❛♥

s✐♥❣❧❡ ✐♥❞✐✈✐❞✉❛❧s ❛♥❞ ❧❛r❣❡ ❡❧❡❝t♦r❛t❡s ❛❞♦♣t t❤❡ ❝♦rr❡❝t ❞❡❝✐s✐♦♥ ✇✐t❤ ✈❡r② ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✑✱ ♠❛♥② s❝❤♦❧❛rs

❡①t❡♥❞❡❞ t❤✐s r❡s✉❧t ✉s✐♥❣ ♠♦r❡ r❡❧❛①❡❞ st❛t✐st✐❝❛❧ ❛ss✉♠♣t✐♦♥s✳ ❚❤❡ ♠❛✐♥ ❛r❣✉♠❡♥t ✐s t❤❛t ❡❛❝❤ ❥✉r♦r ❤❛s ❛

♣r♦❜❛❜✐❧✐t②p∈ ¹₂,1♦❢ ❝❤♦♦s✐♥❣ t❤❡ ❝♦rr❡❝t ♦♣t✐♦♥ ❛♥❞ s❤❡ ✇❛♥ts t♦ ❞♦ ✐t✳ ❚❤r♦✉❣❤ ❛ ♣r♦♣❡r ❛❣❣r❡❣❛t✐♦♥

♠❡t❤♦❞ ♦❢ t❤❡ ✈♦t❡s t❤❡ st❛t❡♠❡♥t ❤♦❧❞s✳ ❚❤❡ ✈❛❧✉❡ ♦❢p❞❡♣❡♥❞s ♦♥ t❤❡ q✉❛❧✐t✐❡s ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧s✱ ✐✳❡✳

t❤❡✐r ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ❞❡❜❛t❡❞ ♠❛tt❡r ♦r t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❝♦♥s❡q✉❡♥❝❡s✳ ❚❤❡ ❛❣❣r❡❣❛t✐♦♥ ♠❡t❤♦❞

❞❡♣❡♥❞s ♦♥ t❤❡ t②♣❡ ♦❢ ✐♥st✐t✉t✐♦♥ ❝♦♥s✐❞❡r❡❞✳ ❉❡s♣✐t❡ t❤❡ ❡✛♦rts t♦ ❡♥r✐❝❤ t❤❡ ♠♦❞❡❧✱ ✐✳❡✳ ✐♥tr♦❞✉❝✐♥❣ t❤❡

❞❡♣❡♥❞❡♥❝② ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ✭s❡❡✱ ❛♠♦♥❣ ♦t❤❡rs✱ ❇♦❧❛♥❞✱ ✶✾✽✾❀ ❇❡r❣✱ ✶✾✾✸❀ ▲❛❞❤❛✱ ✶✾✾✺✱ ✶✾✾✼❀ ❇❡r❡♥❞

❛♥❞ ❙❛♣✐r✱ ✷✵✵✼✮ ♦r ❛❞❞✐♥❣ ❤❡t❡r♦❣❡♥❡✐t② ✐♥ t❤❡ ❝❤❛♥❝❡ ♦❢ ♠❛❦✐♥❣ t❤❡ ❝♦rr❡❝t ❝❤♦✐❝❡ ✭❇♦❧❛♥❞✱ ✶✾✽✾✮✱ ❛❧❧

t❤❡s❡ ❛♣♣r♦❛❝❤❡s ❝♦♥s✐❞❡r ❥✉r♦rs ❛s ✐❢ t❤❡② ✇❡r❡ ❝♦♠♠✐tt❡❞ t♦ ✈♦t❡ ✐♥❢♦r♠❛t✐✈❡❧②✳ ❚❤❡r❡ ✐s ♥♦ r♦♦♠ ❢♦r ❛♥②

♦t❤❡r t②♣❡ ♦❢ ❡✈❛❧✉❛t✐♦♥✱ r❛t❤❡r t❤❛♥ t❤❡ st❛t✐st✐❝❛❧ str✉❝t✉r❡✳ ❙♦ t❤❡ ✐♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥ ✐s t❤❛t ❡❛❝❤ ❥✉r♦r

❜❡❤❛✈❡s ❛s ✐❢ s❤❡ ✇❛s ❛❧♦♥❡✳ ❊✈❡♥ ✐❢ s❤❡ ❦♥♦✇s t❤❛t t❤❡ ✜♥❛❧ ♦✉t❝♦♠❡ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ✉♥✐q✉❡❧② ❜② ❤❡r

❝❤♦✐❝❡✳ ❚❤✐s ❝♦♥tr❛❞✐❝ts t❤❡ ♥♦t✐♦♥ ♦❢ r❛t✐♦♥❛❧✐t②✳ ❋♦r t❤✐s r❡❛s♦♥ t❤✐s s✉❜❥❡❝t ❤❛s ❜❡❡♥ ❛♥❛❧②③❡❞ ❛❧s♦ ❢r♦♠

t❤❡ ❣❛♠❡ t❤❡♦r❡t✐❝ ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❚❤❡ ❜❛s✐❝ ❛ss✉♠♣t✐♦♥ ✐s t❤❛t✱ ❡✈❡♥ ✐❢ ❥✉r♦rs s❤❛r❡ t❤❡ s❛♠❡ ♦❜❥❡❝t✐✈❡✱

t❤❡② ✈♦t❡ str❛t❡❣✐❝❛❧❧② ✐♥ ♦r❞❡r t♦ ♠❛①✐♠✐③❡ t❤❡✐r ❡①♣❡❝t❡❞ ✉t✐❧✐t②✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛❦✐♥❣ t❤❡ ❝♦rr❡❝t

❝❤♦✐❝❡✱ ✇❤✐❝❤ ✐♥❝♦r♣♦r❛t❡s ✐♥❢♦r♠❛t✐♦♥ ❛❝q✉✐s✐t✐♦♥✱ ❝♦♠♣❡t❡♥❝❡ ❛♥❞ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❞❡❜❛t❡❞ ♠❛tt❡r

❛r❡ r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♦❢ ♥♦✐s② s✐❣♥❛❧s✳ ❚❤❡ ❢r❛♠❡✇♦r❦ ❜❡❝♦♠❡s ❛ ❝❧❛ss✐❝❛❧ s❡tt✐♥❣ ♦❢ ❣❛♠❡s

✇✐t❤ ♣r✐✈❛t❡ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡ ✐♥❝❡♥t✐✈❡ t♦ ❧♦♦❦ ❢♦r t❤❡ ♠❛①✐♠✐③❛t✐♦♥ ♦❢ t❤❡ ❡①♣❡❝t❡❞ ✉t✐❧✐t② ❝♦♠❡s ❢r♦♠

t❤❡ ✐♥❢♦r♠❛t✐♦♥✲❜❛s❡❞ ❤❡t❡r♦❣❡♥❡✐t② ♦❢ ✐♥❞✐✈✐❞✉❛❧s ❛t t❤❡ ✐♥t❡r✐♠ st❛❣❡ ♦❢ t❤❡ ❣❛♠❡✳ ❚❤✐s ❛♣♣r♦❛❝❤ ❣✐✈❡s s♦♠❡ ✐♥s✐❣❤ts ❛❜♦✉t t❤❡ ❥✉r♦rs✬ ❜❡❤❛✈✐♦r ✭❆✉st❡♥✲❙♠✐t❤ ❛♥❞ ❇❛♥❦s✱ ✶✾✾✻✮✱ ❡①♣❧❛✐♥s t❤❡ ♣❛rt✐❝✐♣❛t✐♦♥ r❛t❡✱

∗❉✐●❙P❊❙ ✲ ❱✐❛ ❈❛✈♦✉r ✽✹ ✲ ✶✺✶✷✶ ❆❧❡ss❛♥❞r✐❛ ❆▲ ✲ ■t❛❧② ✲ ❤❡✐❞r❡❦r❅❣♠❛✐❧✳❝♦♠

✶

t❤❡ r♦❧❧✲♦✛ ❡✛❡❝t✱ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❞✐s❝❧♦s✉r❡ ✐♥ ❧❛r❣❡ ❡❧❡❝t✐♦♥s ✭❋❡❞❞❡rs❡♥ ❛♥❞ P❡s❡♥❞♦r❢❡r✱ ✶✾✾✻✱ ✶✾✾✼✮

❛♥❞ t❤❡ r♦❜✉st♥❡ss ♦❢ t❤❡ ❛❣❣r❡❣❛t✐♦♥ ♠❡t❤♦❞s ✭❋❡❞❞❡rs❡♥ ❛♥❞ P❡s❡♥❞♦r❢❡r✱ ✶✾✾✽✮✳ ▼♦r❡♦✈❡r✱ ▼②❡rs♦♥

✭✶✾✾✽✮ s❤♦✇s t❤❛t t❤❡ ❈♦♥❞♦r❝❡t✬s ❏✉r② ❚❤❡♦r❡♠ ❤♦❧❞s ❛❧s♦ ✇✐t❤ str❛t❡❣✐❝ ✐♥❞✐✈✐❞✉❛❧s ✇❤❡♥ t❤❡ ♥✉♠❜❡r

♦❢ ♣❛rt✐❝✐♣❛♥ts ❛♥❞ t❤❡ ❝♦rr❡❝t ♦♣t✐♦♥ ❛r❡ ✉♥❝❡rt❛✐♥✳ ❚❤✐s ❛rt✐❝❧❡ ❛♥❛❧②③❡s t❤❡ str❛t❡❣✐❝ ❜❡❤❛✈✐♦r ♦❢ ❛

❥✉r② t❤❛t ♠✉st ✈♦t❡ t♦ ❛❝q✉✐t ♦r ❝♦♥✈✐❝t ❛ ❞❡❢❡♥❞❛♥t ❞✉r✐♥❣ ❛ tr✐❛❧✳ ❚❤❡ s✐t✉❛t✐♦♥ ✐s s✐♠✐❧❛r t♦ ❛ ❣r♦✉♣

♦❢ ❡①♣❡rts ✇❤♦ ♠✉st ❝❤♦♦s❡ t❤r♦✉❣❤ ❛ ♣♦❧❧✱ t♦ ❛♣♣r♦✈❡ ♦r r❡❥❡❝t ❛ ♣r♦❥❡❝t✳ ❚❤❡ ♠❛✐♥ ✐ss✉❡ ✐s t❤❛t t❤❡

tr✉t❤ ✐s ✉♥❦♥♦✇♥✳ ❇❡❢♦r❡ ♠❛❦✐♥❣ ❤❡r ❝❤♦✐❝❡✱ ❡❛❝❤ ❥✉r♦r ♦❜s❡r✈❡s ❛ s✐❣♥❛❧ t❤❛t ❣✐✈❡s ❛♥ ✐♥❞✐❝❛t✐♦♥ ❛❜♦✉t t❤❡ ✐♥♥♦❝❡♥❝❡ ♦r t❤❡ ❣✉✐❧t✐♥❡ss ♦❢ t❤❡ ❞❡❢❡♥❞❛♥t ♦r t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r♦❥❡❝t✳ ■♥ t❤❡ st❛t✐st✐❝❛❧ ❛♣♣r♦❛❝❤✱

❛s ✐♥ t❤❡ ❣❛♠❡ t❤❡♦r❡t✐❝ ♦♥❡✱ s✐❣♥❛❧s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡ ✉♥❞❡r❧②✐♥❣ ❤②♣♦t❤❡s✐s ✐s t❤❛t

❥✉r♦rs ✐♥t❡r♣r❡t ❡✈✐❞❡♥❝❡s ❞✐✛❡r❡♥t❧②✱ ❜❡❝❛✉s❡ ♦❢ t❤❡✐r ❞✐✛❡r❡♥t ❧✐❢❡ ❡①♣❡r✐❡♥❝❡s ❛♥❞ ❝♦♠♣❡t❡♥❝✐❡s✳ ❚❤✐s

❛ss✉♠♣t✐♦♥ ✐s ❝♦♥✈❡♥✐❡♥t ❜❡❝❛✉s❡ ✐t ♠❛❦❡s ❝♦♠♣✉t❛t✐♦♥ ❡❛s✐❡r✳ ❇✉t✱ ✐❢ ✇❡ ✐♥t❡r♣r❡t s✐❣♥❛❧s ❛s ❡✈✐❞❡♥❝❡s

♦r ❛s ❛ t❡❝❤♥✐❝❛❧ r❡♣♦rt✱ t❤❡r❡ ❛r❡ ❛t ❧❡❛st t✇♦ ♣♦ss✐❜❧❡ ✐♠♣❧✐❝❛t✐♦♥s✿ ❥✉r♦rs ❛r❡ ❛❧❧♦✇❡❞ t♦ ✐♥t❡r♣r❡t t❤❡♠

✐♥ ♦♣♣♦s✐t❡ ✇❛②s ♦r ❞✉r✐♥❣ t❤❡ tr✐❛❧ ♥♦ ❞❡❝✐s✐✈❡ ❡✈✐❞❡♥❝❡ ✐s ♣r♦❞✉❝❡❞ ❛t ❛❧❧✳ ❋♦r t❤✐s r❡❛s♦♥ ✐♥ ♦✉r ❛♥❛❧②s✐s

✇❡ ❛ss✉♠❡ t❤❛t✱ ✇✐t❤ ❛ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t② α ∈ (0,1) ❡✈✐❞❡♥❝❡s ❛r❡ s♦ str♦♥❣ t♦ ❧❡❛✈❡ ♥♦ ❝❤❛♥❝❡ t♦

✐♥t❡r♣r❡t❛t✐♦♥✳ ■♥ ❢❛❝t✱ ✐❢ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♠♦❞❡r♥ ✐♥✈❡st✐❣❛t✐♦♥ t❡❝❤♥✐q✉❡s✱ ✐✳❡✳ ✜♥❣❡r♣r✐♥ts✱ ❉◆❆ ♦r s♦♠❡

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

❞♦❡s ♥♦t s❡❡♠ r❡❛s♦♥❛❜❧❡✳ ❋♦❧❧♦✇✐♥❣ t❤❡ r❡s✉❧ts ✐♥ ❋❡❞❞❡rs❡♥ ❛♥❞ P❡s❡♥❞♦r❢❡r ✭✶✾✾✽✮ ✇❡ ✇✐❧❧ r❡str✐❝t ♦✉r

❛tt❡♥t✐♦♥ t♦ s✐♠♣❧❡ ♠❛❥♦r✐t② ❞❡❝✐s✐♦♥ r✉❧❡s ✇✐t❤ ♥♦ ❛❜st❡♥t✐♦♥✳ ❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ ❝❤❛♣t❡r ✐s ♦r❣❛♥✐③❡❞

❛s ❢♦❧❧♦✇s✿ ✐♥ s❡❝t✐♦♥ ✷✳✷ ✇❡ ♣r❡s❡♥t t❤❡ ♠♦❞❡❧✳ ■♥ s❡❝t✐♦♥ ✷✳✸ ✇❡ ❝♦♠♣✉t❡ t❤❡ s②♠♠❡tr✐❝ ❡q✉✐❧✐❜r✐❛✳ ■♥

s❡❝t✐♦♥ ✷✳✹ ✇❡ s✉♠♠❛r✐③❡ t❤❡ ❝❧❛ss✐❝❛❧ ♠♦❞❡❧ ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t s✐❣♥❛❧s ❛♥❞ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

s②♠♠❡tr✐❝ ❡q✉✐❧✐❜r✐❛✳ ■♥ s❡❝t✐♦♥ ✷✳✺ ✇❡ ❝♦♠♣❛r❡ t❤❡ t✇♦ ♠♦❞❡❧s tr②✐♥❣ t♦ ✜♥❞ ❛ ❜❡♥❝❤♠❛r❦✳ ❙❡❝t✐♦♥ ✷✳✻

❝♦♥❝❧✉❞❡s✳ ❆❧❧ t❤❡ ♣r♦♦❢s ❛r❡ ✐♥ t❤❡ ❆♣♣❡♥❞✐① ✭❙❡❝t✐♦♥ ✷✳✼✮✳

✷ ❚❤❡ ▼♦❞❡❧

▲❡tJ ❜❡ t❤❡ s❡t ♦❢ ❥✉r♦rs ✇✐t❤ ♦❞❞ ❝❛r❞✐♥❛❧✐t②✱ ❡❛❝❤j∈J ♠✉st ❝❤♦♦s❡ t♦ ❛❝q✉✐t ♦r ❝♦♥✈✐❝t t❤❡ ❞❡❢❡♥❞❛♥t✳

❚❤❡r❡ ✐s ♥♦ ❛❜st❡♥t✐♦♥ s♦ t❤❡ ❛❝t✐♦♥ s❡t ❢♦r ❡❛❝❤ ❥✉r♦r ✐sSj={a, c}✱ ♠♦r❡♦✈❡r ❥✉r♦rs s❤❛r❡ t❤❡ s❛♠❡ ♣❛②♦✛

❢✉♥❝t✐♦♥vj =v: Ω1×S→R✱ ❞❡✜♥❡❞ ❛s

v,u◦f

✇❤❡r❡ Ω1 ={■,●} ✐s t❤❡ s❡t ♦❢ st❛t❡s ♦❢ ◆❛t✉r❡✱ ✐♥♥♦❝❡♥t ♦r ❣✉✐❧t②✱ ❝♦♥❝❡r♥✐♥❣ t❤❡ ❞❡❢❡♥❞❛♥t✳ ❚❤❡ s❡t S=Sj×S−j ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ❛❝t✐♦♥ ♣r♦✜❧❡s✳ ❚❤❡ ❢✉♥❝t✐♦♥v✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❛❣❣r❡❣❛t✐♦♥

r✉❧❡f :Sj×S−j → {a, c} ♦❢ t❤❡ ❛❝t✐♦♥ ♣r♦✜❧❡s ✇✐t❤ t❤❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥u:{a, c} ×Ω1→R✱ ❞❡✜♥❡❞ ❛s u(a|ω1=■) =u(c|ω1=●) = 0

u(a|ω1=●) =−(1−q) u(c|ω1=■) =−q

✇✐t❤q∈(0,1)✳ ❚❤❡ ♣❛r❛♠❡t❡rq❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s ❛ t❤r❡s❤♦❧❞ ♦❢ r❡❛s♦♥❛❜❧❡ ❞♦✉❜t✳ ❚❤❡ tr✉❡ st❛t❡ ✐♥Ω1

✐s ✉♥❦♥♦✇♥ ❛♥❞ t❤❡ ❥✉r② ♠❡♠❜❡rs s❤❛r❡ t❤❡ s❛♠❡ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥πj =π= Pr(■)∈(0,1)✳

❇❡❢♦r❡ ❝❤♦♦s✐♥❣ ❛♥ ❛❝t✐♦♥ ✐♥Sj ❡❛❝❤ ❥✉r♦r j ∈J ♦❜s❡r✈❡s ❛ ♣r✐✈❛t❡ s✐❣♥❛❧tj t❤❛t ❝❛♥ ❛ss✉♠❡ ✈❛❧✉❡s ✐♥

Tj ={i, g} ✇✐t❤ st❛t❡ ❞❡♣❡♥❞❡♥t ❞✐str✐❜✉t✐♦♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t②α∈(0,1) ♥❛t✉r❡ r❡✈❡❛❧s t❤❡

tr✉❡ st❛t❡ ✐♥Ω1 ✇✐t❤ ❞❡❣❡♥❡r❛t❡ ✐♥❞❡♣❡♥❞❡♥t s✐❣♥❛❧s

Pr(tj=i|■,❘) = 1 ❛♥❞ Pr(tj =g|●,❘) = 1 Pr(tj=g|■,❘) = 0 ❛♥❞ Pr(tj=i|●,❘) = 0

✷

✇❤❡r❡ ❘ ✐s t❤❡ ❡✈❡♥t ✏❞❡❣❡♥❡r❛t❡ s✐❣♥❛❧s✑ ❛♥❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② (1−α) s✐❣♥❛❧s ❛r❡ st✐❧❧ ✐♥❞❡♣❡♥❞❡♥t ❜✉t

♥♦t s♦ ❛❝❝✉r❛t❡

Pr(tj=i|■,¬❘) =z ❛♥❞ Pr(tj =g|●,¬❘) =z Pr(tj =g|■,¬❘) = 1−z ❛♥❞ Pr(tj=i|●,¬❘) = 1−z

✇✐t❤z∈(¹₂,1)✳ ❚♦ s✉♠♠❛r✐③❡ t❤❡ s❡t ♦❢ st❛t❡s ♦❢ ♥❛t✉r❡ ✐s ❞❡✜♥❡❞ ❛s Ω = Ω1×Ω2 ✇❤❡r❡Ω2={❘,¬❘}

❝♦♥t❛✐♥s t❤❡ ❡✈❡♥t ✏❞❡❣❡♥❡r❛t❡ s✐❣♥❛❧s✑ ❛♥❞ ✐ts ❝♦♠♣❧❡♠❡♥t✳ ❚❤❡ ❡✈❡♥t ❘ ✐s ♥♦t ❞✐r❡❝t❧② ♦❜s❡r✈❛❜❧❡ ❛♥❞ s♦

Pr(tj =g|●) =α+ (1−α)z Pr(tj=i|●) = (1−α)(1−z)

Pr(tj=i|■) =α+ (1−α)z Pr(tj=g|■) = (1−α)(1−z)

❆❢t❡r ❣❡tt✐♥❣ ❤❡r s✐❣♥❛❧✱ ❡❛❝❤ ❥✉r♦rj∈J ✉♣❞❛t❡s ❤❡r ❜❡❧✐❡❢ ✉s✐♥❣ ❇❛②❡s✬ r✉❧❡ ❛♥❞ t❤❡♥

Pr(■|tj =i) = απ+ (1−α)πz

απ+ (1−α)[πz+ (1−π)(1−z)]

Pr(■|tj =g) = (1−α)π(1−z)

α(1−π) + (1−α)[π(1−z) + (1−π)z]

Pr(●|tj =i) = (1−α)(1−π)(1−z) απ+ (1−α)[πz+ (1−π)(1−z)]

Pr(●|tj =g) = α(1−π) + (1−α)(1−π)z α(1−π) + (1−α)[π(1−z) + (1−π)z]

t❤❡ s❡ts(Sj, Tj)j∈J❛♥❞ t❤❡ ✈❛❧✉❡s ♦❢|J|, α, z, π, q❛r❡ ❝♦♠♠♦♥ ❦♥♦✇❧❡❞❣❡✳ ❊❛❝❤ ❥✉r♦r ✈♦t❡s s✐♠✉❧t❛♥❡♦✉s❧②

❛♥❞ ❜❡❢♦r❡ t❤❡ ♣♦❧❧ t❤❡② ❝❛♥♥♦t ❝♦♠♠✉♥✐❝❛t❡✳ ❚❤✐s ❛ss✉♠♣t✐♦♥ ❝♦✉❧❞ s❡❡♠ t♦♦ str♦♥❣ ❜✉t t❤❡ ❛♥❛❧②s✐s

♦❢ t❤❡ ❞❡❧✐❜❡r❛t✐♦♥ ♠❡❝❤❛♥✐s♠ ✐s ❜❡②♦♥❞ t❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ✇♦r❦✳ ▲❡t 0≤k(s)≤ |J|❜❡ t❤❡ ♥✉♠❜❡r ♦❢

❛❝q✉✐t ✈♦t❡s ✐♥ ❛♥ ❛❝t✐♦♥ ♣r♦✜❧❡s∈ S ❛♥❞ ❧❡t mˆ ❜❡ ❡q✉❛❧ t♦ mˆ = (|J| −1)/2^✶✳ ❚❤❡ ❛❣❣r❡❣❛t✐♥❣ r✉❧❡f

✉s❡❞ ✐s ❞❡✜♥❡❞ ❛s

f(s),

( a ✐❢ k(s)>mˆ c ✐❢ k(s)≤mˆ

❛♥❞ ✐t ❝♦rr❡s♣♦♥❞s t♦ s✐♠♣❧❡ ♠❛❥♦r✐t②✳ ❚❤✐s t②♣❡ ♦❢ ❢✉♥❝t✐♦♥ ❤❛s t✇♦ ✐♠♣♦rt❛♥t ♣r♦♣❡rt✐❡s✱ ✐t ✐s ❛♥♦♥②♠♦✉s

❛♥❞ ♠♦♥♦t♦♥✐❝✿ ✐t tr❡❛ts ❛❧❧ ✈♦t❡s ❡q✉❛❧❧② ❛♥❞ ✐❢ ❛ ❞❡❝✐s✐♦♥ ✐s t❛❦❡♥ ✇✐t❤n ✈♦t❡s✱ ✐t ✇✐❧❧ ♥♦t ❝❤❛♥❣❡ ✇✐t❤

n+ 1✈♦t❡s✳ ❚❤❡ ♣r❡s❡♥❝❡ ♦❢ ♣r✐✈❛t❡ s✐❣♥❛❧s ❛♥❞ t❤❡ ✐♥t❡r❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ♣❛②♦✛ ❢✉♥❝t✐♦♥s✱ ✐♥❞✉❝❡ ❡❛❝❤

❥✉r♦r t♦ ✈♦t❡ str❛t❡❣✐❝❛❧❧②✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳ ❆ ✈♦t✐♥❣ str❛t❡❣② ❢♦r ❥✉r♦rj∈J ✐s ❛ ♠❛♣

σj :Tj→∆(Sj)

✇❤❡r❡∆(Sj)⊃Sj ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ♣✉r❡ ❛♥❞ ♠✐①❡❞ ❛❝t✐♦♥s✳

❋♦❧❧♦✇✐♥❣ t❤❡ t❡r♠✐♥♦❧♦❣② ✐♥ ❆✉st❡♥✲❙♠✐t❤ ❛♥❞ ❇❛♥❦s ✭✶✾✾✻✮ ❡❛❝❤ ❥✉r♦r ❝❛♥ ❜❡❤❛✈❡ ✐♥ t❤r❡❡ ❞✐✛❡r❡♥t ✇❛②s

❉❡✜♥✐t✐♦♥ ✷✳✷✳ ❆ ✈♦t✐♥❣ str❛t❡❣②σj :Tj→∆(Sj)✐s

✐✳ ✐♥❢♦r♠❛t✐✈❡✱ ✐❢ σj(sj|tj=i) =a❛♥❞σj(sj|tj =g) =c✳

✐✐✳ s✐♥❝❡r❡✱ ✐❢ ❣✐✈❡♥ t❤❡ ♦❜s❡r✈❡❞ s✐❣♥❛❧✱ ✐t ♠❛①✐♠✐③❡s t❤❡ ❡①♣❡❝t❡❞ ✉t✐❧✐t② ♦❢ ❥✉r♦rj∈J✳

▲❡tσ= (σ1, σ2, . . . , σ2 ˆm+1)❜❡ t❤❡ ✈♦t✐♥❣ str❛t❡❣② ♣r♦✜❧❡ ♦❢ t❤❡ ❥✉r②✱ t❤❡♥

✶❙✐♥❝❡|J|✐s ❛ss✉♠❡❞ t♦ ❜❡ ❛♥ ♦❞❞ ♥✉♠❜❡r✱mˆ ✐s ❛❧✇❛②s ❛♥ ✐♥t❡❣❡r✳

✸

❉❡✜♥✐t✐♦♥ ✷✳✸✳ ❆ ✈♦t✐♥❣ str❛t❡❣② ♣r♦✜❧❡ σ : {i, g}^{2 ˆm+1} → {∆(Sj)}^{2 ˆ}_j=1^m+1 ✐s r❛t✐♦♥❛❧✱ ✐❢ ✐t ✐s ❛ ❜❛②❡s✐❛♥

◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤❡ ❣❛♠❡Γ^b =hJ,(Tj, Sj)j∈J,(α, z, π), vi✳

❉❡✜♥✐t✐♦♥ ✷✳✹✳ ❚❤❡ ✈♦t✐♥❣ str❛t❡❣② ♣r♦✜❧❡ σ^∗ = (σ^∗₁, . . . , σ^∗_{2 ˆm+1}) ✐s ❛ ❜❛②❡s✐❛♥ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤❡

❣❛♠❡Γ^b=hJ,(Tj, Sj)j∈J,(α, z, π), vi✐❢ ❢♦r ❛❧❧ j∈J✱tj∈Tj ❛♥❞σj∈∆(Sj) X

t_−j∈T−j

v(σj^∗|σ^∗_−j, tj, t−j) Pr(t−j|tj)≥ X

t_−j∈T−j

v(σj|σ^∗_−j, tj, t−j) Pr(t−j|tj)

❈♦r♦❧❧❛r② ✷✳✺✳ ❲❤❡♥❡✈❡r ❛ ✈♦t✐♥❣ str❛t❡❣② ♣r♦✜❧❡σ^∗ ✐s r❛t✐♦♥❛❧✱ ✐t ✐s ❛❧s♦ s✐♥❝❡r❡✳

■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ❛ r❛t✐♦♥❛❧ ❥✉r♦r ✐s ❝♦♥❝❡r♥❡❞ ♦♥❧② ✇❤❡♥ ❤❡r ✈♦t❡ ✐s ♣✐✈♦t❛❧✳ ■♥ ♣❛rt✐❝✉❧❛r ❢♦r t❤✐s

♠♦❞❡❧✱ ❛❢t❡r ♦❜s❡r✈✐♥❣ tj = g ❥✉r♦r j ∈ J ♣r❡❢❡rs t♦ ❛❝q✉✐t ✐❢ q > Pr(●|piv, tj = g) ♦r ❝♦♥✈✐❝t ✐❢ t❤❡

✐♥❡q✉❛❧✐t② ✐s r❡✈❡rs❡❞✳ ■♥ t❤❡ s❛♠❡ ✇❛②✱ ❛❢t❡r ♦❜s❡r✈✐♥❣tj =is❤❡ ♣r❡❢❡rs t♦ ❛❝q✉✐t ✐❢q >Pr(●|piv, tj =i)

♦r ❝♦♥✈✐❝t ✐❢ t❤❡ ✐♥❡q✉❛❧✐t② ✐s r❡✈❡rs❡❞✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ❞❡❢❡♥❞❛♥t ✐s ❣✉✐❧t②✱ ❣✐✈❡♥ t❤❛t ❥✉r♦rj ∈J

✐s ♣✐✈♦t❛❧ ❛♥❞ ❣✐✈❡♥tj=g ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s

Pr(●|piv, tj=g) = Pr(●|tj=g) Pr(piv|●, tj =g)

Pr(●|tj =g) Pr(piv|●, tj=g) + Pr(■|tj=g) Pr(piv|■, tj=g)

✇❤❡r❡

Pr(piv|●, tj=g) = X

t_−j∈T−j

Pr(t−j|●, tj=g) Pr(piv|●, t−j, tj=g) Pr(piv|■, tj=g) = X

t−j∈T−j

Pr(t−j|■, tj =g) Pr(piv|■, t−j, tj =g)

❛♥❞ ✐t ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ s✐❣♥❛❧s ♣r♦✜❧❡ ❛♥❞ ♦♥ t❤❡ str❛t❡❣✐❡s ❛❞♦♣t❡❞ ❜② t❤❡ ❥✉r♦rs✳ ❚❤❡

♣r♦❜❛❜✐❧✐t✐❡sPr(●|piv, tj =i)✱Pr(■|piv, tj=g)❛♥❞Pr(■|piv, tj =i)❛r❡ s✐♠✐❧❛r❧② ❞❡✜♥❡❞✳

✸ ❊q✉✐❧✐❜r✐❛

■♥ t❤✐s ♣❛♣❡r ✇❡ ✇✐❧❧ ❛♥❛❧②③❡ ♦♥❧② t❤❡ s②♠♠❡tr✐❝ ❜❛②❡s✐❛♥ ◆❛s❤ ❡q✉✐❧✐❜r✐❛ ✐♥ ✇❤✐❝❤ ♣❧❛②❡rs ❞♦ ♥♦t ✉s❡

✇❡❛❦❧② ❞♦♠✐♥❛t❡❞ str❛t❡❣✐❡s✳ ❋r♦♠ ♥♦✇ ♦♥ ❛ss✉♠❡|J|= 3✳ ❚♦ ❛✈♦✐❞ tr✐✈✐❛❧ ❝❛s❡s t❤❡ ♣❛r❛♠❡t❡rq∈(0,1)

♠✉st ❜❡ r❡str✐❝t❡❞✳ ■♥ ❢❛❝t✱ ❧❡t ✉s ❛ss✉♠❡ t❤❛t ❛ s✐♥❣❧❡ ❥✉r♦r ❝❛♥ ♦❜s❡r✈❡ ❛❧❧ s✐❣♥❛❧s ❛♥❞ s❤❡ ♦❜s❡r✈❡s t❤❡

s✐❣♥❛❧s ♣r♦✜❧❡t = (g, g, g)✳ ■♥ t❤✐s ❝❛s❡✱ ✐❢ s❤❡ ❝❤♦♦s❡s t♦ ❛❝q✉✐t ✐rr❡s♣❡❝t✐✈❡❧② ♦❢ t❤❡ ♦❜s❡r✈❡❞ ♣r♦✜❧❡✱ ✐t

♠❡❛♥s t❤❛t s❤❡ ✐s ♥♦t r❡s♣♦♥s✐✈❡ ❛s ✐❢ s❤❡ ❝❤♦♦s❡s t♦ ❝♦♥✈✐❝t ❛❢t❡r ♦❜s❡r✈✐♥❣ t❤❡ s✐❣♥❛❧s ♣r♦✜❧❡t= (i, i, i)✳

❉❡✜♥✐t✐♦♥ ✸✳✶✳ ❆ ❥✉r♦rj∈J ✐s s❛✐❞ t♦ ❜❡ r❡s♣♦♥s✐✈❡✱ ✐❢ t❤❡r❡ ❡①✐st ❛t ❧❡❛st t✇♦ s✐❣♥❛❧s ♣r♦✜❧❡st^′, t^′′∈T s✉❝❤ t❤❛tσj(sj|t=t^′)6=σj(sj|t=t^′′)❢♦r s♦♠❡ sj∈Sj✱ ✇❤❡r❡T ={Tj}^{2 ˆ}_j=1^m+1✳

❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ✐♥ ❋❡❞❞❡rs❡♥ ❛♥❞ P❡s❡♥❞♦r❢❡r ✭✶✾✾✽✮✳ ❇❡❢♦r❡ ♦❜s❡r✈✐♥❣ t❤❡ s✐❣♥❛❧✱

❡❛❝❤ ❥✉r♦r ❤❛s ❛ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ π∈(0,1) ❛♥❞ ✐t r❡♣r❡s❡♥ts t❤❡ ❝♦♠♠♦♥ s❡♥t✐♠❡♥t ❛❜♦✉t t❤❡ ✐♥♥♦❝❡♥❝❡ ♦r t❤❡ ❣✉✐❧t✐♥❡ss ♦❢ t❤❡ ❞❡❢❡♥❞❛♥t✳ ■t s❡❡♠s ♥❛t✉r❛❧ t♦ st✉❞② t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❡q✉✐❧✐❜r❛t❡❞

❥✉r♦rs ✇❤♦ ❤❛✈❡ ♥♦t ❛ ❜✐❛s❡❞ ♦♣✐♥✐♦♥✳

❉❡✜♥✐t✐♦♥ ✸✳✷✳ ❆ ❥✉r♦rj∈J ✐s s❛✐❞ t♦ ❜❡ ✉♥❜✐❛s❡❞✱ ✐❢ π=¹₂✳

❚❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥ ❝❤❛r❛❝t❡r✐③❡s ❛♥ ✐♠♣❛rt✐❛❧ ❥✉r♦r✳

Pr♦♣♦s✐t✐♦♥ ✸✳✸✳ ❋♦r ❣✐✈❡♥α∈(0,1)✱ z∈(¹₂,1) ❛♥❞|T|= 3✱ ❛♥ ✉♥❜✐❛s❡❞ ❥✉r♦r j∈J ✐s r❡s♣♦♥s✐✈❡✱ ✐❢

❛♥❞ ♦♥❧② ✐❢

q∈ q_α^min, q^max_α

=

(1−α)(1−z)³

α+ (1−α)[z³+ (1−z)³], α+ (1−α)z³ α+ (1−α)[z³+ (1−z)³]

✹

❚❤❡ ❛❜♦✈❡ ♣r♦♣♦s✐t✐♦♥ ❣✐✈❡s t❤❡ ❜♦✉♥❞❛r✐❡s ✇✐t❤✐♥ t❤❡ ♣❛r❛♠❡t❡rq♠✉st ❧✐❡✱ ✇✐t❤ ❛ ❧✐tt❧❡ ❛❜✉s❡ ♦❢ ♥♦t❛t✐♦♥

✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝♦♥s✐❞❡rσj(sj|tj=g)❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝q✉✐t ❣✐✈❡♥ t❤❡ s✐❣♥❛❧tj=g❛♥❞σj(sj|tj =i)

❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝q✉✐t ❣✐✈❡♥ t❤❡ s✐❣♥❛❧tj=i✳ ❆s ✐t ✇✐❧❧ ❜❡ ❝❧❡❛r ❜❡❧♦✇✱ t❤❡ ✈❛❧✉❡ ♦❢q❞❡t❡r♠✐♥❡s t❤❡

❢❡❛s✐❜❧❡ str❛t❡❣✐❡s ❛♥❞ t❤❡ ❡q✉✐❧✐❜r✐❛✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ❞❡s❝r✐❜❡s ✇❤❡♥ t❤❡ ✐♥❢♦r♠❛t✐✈❡ str❛t❡❣②

✐s ❛❧s♦ r❛t✐♦♥❛❧✳

Pr♦♣♦s✐t✐♦♥ ✸✳✹✳ ▲❡tJ ❜❡ ❛ s❡t ♦❢ ✉♥❜✐❛s❡❞ ❛♥❞ r❡s♣♦♥s✐✈❡ ❥✉r♦rs ✇✐t❤ mˆ ∈N<∞❢♦r ❣✐✈❡♥ α∈(0,1)

❛♥❞z∈(¹₂,1)✱ ✐❢ q∈(1−z, z)t❤❡♥ t❤❡ ✐♥❢♦r♠❛t✐✈❡ ✈♦t✐♥❣ str❛t❡❣② ✭♣r♦✜❧❡✮ ✐s r❛t✐♦♥❛❧✳ ❚❤❛t ✐s✱ ❢♦r ❛♥② q∈(1−z, z) t❤❡ ✐♥❢♦r♠❛t✐✈❡ ✈♦t✐♥❣ str❛t❡❣② ♣r♦✜❧❡σ^∗ ✐s ❛ ❜❛②❡s✐❛♥ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♦❢ Γ^b✳

◆♦t✐❝❡ t❤❛t t❤❡ ❛❜♦✈❡ ♣r♦♣♦s✐t✐♦♥ ❤♦❧❞s ❢♦r ❛♥② ✜♥✐t❡m✳ ❲❤❡♥ˆ q /∈(1−z, z)t❤❡ t❤r❡s❤♦❧❞ ♦❢ r❡❛s♦♥❛❜❧❡

❞♦✉❜t ✐s s❤✐❢t❡❞ t♦✇❛r❞ ❛ ❜❡❤❛✈✐♦r ♠♦r❡ ❝♦♥❝❡r♥❡❞ ❛❜♦✉t t❤❡ r✐s❦ ♦❢ ♠❛❦❡ ❛ ♠✐st❛❦❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r q∈(q^min_α ,1−z)❥✉r♦rs ❛r❡ ♠♦r❡ ❛❢r❛✐❞ ♦❢ ❛❝q✉✐t ❛ ❣✉✐❧t②✱ r❛t❤❡r t❤❛♥ ❝♦♥✈✐❝t ❛♥ ✐♥♥♦❝❡♥t ❞❡❢❡♥❞❛♥t✳ ❚❤❡

❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ❞❡s❝r✐❜❡s t❤❡ s②♠♠❡tr✐❝ ❜❛②❡s✐❛♥ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♣❧❛②❡❞ ❜② t❤✐s t②♣❡ ♦❢ ❥✉r♦rs✳

Pr♦♣♦s✐t✐♦♥ ✸✳✺✳ ▲❡t J ❜❡ ❛ s❡t ♦❢ ✉♥❜✐❛s❡❞ ❛♥❞ r❡s♣♦♥s✐✈❡ ❥✉r♦rs ✇✐t❤ |J| = 3✱ ❢♦r ❣✐✈❡♥ α ∈ (0,1)✱

z∈(¹₂,1) ❛♥❞q∈(ˆqα^min,1−z)t❤❡ ✈♦t✐♥❣ str❛t❡❣②

Pr(a|tj=g) = 0

Pr(a|tj=i) = q[α+ (1−α)[z²+ (1−z)²]]−(1−α)(1−z)² q[α+ (1−α)[z³+ (1−z)³]]−(1−α)(1−z)³

✇❤❡r❡

ˆ

q^minα = (1−α)(1−z)²

α+ (1−α)[z²+ (1−z)²] > q^minα

✐s r❛t✐♦♥❛❧✳ ❚❤❛t ✐s✱ ❢♦r ❛♥②q∈(ˆq^minα ,1−z)t❤❡ ❛❜♦✈❡ str❛t❡❣② ✐s ❛ ❜❛②❡s✐❛♥ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♦❢ Γ^b✳

◆♦t✐❝❡ t❤❛t t❤❡ ❧♦✇❡r ❜♦✉♥❞ ✐♥ ♦r❞❡r t♦ ❤❛✈❡ ❛♥ ✉♥❜✐❛s❡❞✱ r❡s♣♦♥s✐✈❡ ❛♥❞ r❛t✐♦♥❛❧ ❥✉r♦r ✐s ❤✐❣❤❡r t❤❛♥

t❤❡ ♦♥❡ ✇❤❡♥ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❞❡❝✐s✐♦♥ ♠❛❦❡r✳ ■♥ ❛ ❝❡rt❛✐♥ s❡♥s❡✱ ✐t ✐s ❛s ✐❢ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♦t❤❡r ❥✉r♦rs

♥❛rr♦✇s t❤❡ r❡s♣♦♥s✐✈❡♥❡ss ✐♥t❡r✈❛❧✳ ▼♦r❡♦✈❡r✱ ❥✉r♦rs ❝❤♦♦s❡ t♦ ✏❛❝❝❡♣t✑ t❤❡ ♦❜s❡r✈❡❞ s✐❣♥❛❧ ♦♥❧② ✇❤❡♥ ✐t

✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡✐r ❛♣t✐t✉❞❡✱ ❛♥❞ s♦ t❤❡② tr✉t❤❢✉❧❧② r❡✈❡❛❧ ✐t t❤r♦✉❣❤ t❤❡ ✈♦t❡✳ ■♥ t❤❡ ♦t❤❡r ❝❛s❡ t❤❡②

♣r❡❢❡r t♦ r❛♥❞♦♠✐③❡ t❤❡✐r ✈♦t❡✳ ❲❤❡♥q∈(z, q^max_α )t❤❡ s✐t✉❛t✐♦♥ ✐s t❤❡ ♦♣♣♦s✐t❡✱ ❥✉r♦rs ❛r❡ ♠♦r❡ ❛❢r❛✐❞ ♦❢

❝♦♥✈✐❝t ❛ ♣♦ss✐❜❧❡ ✐♥♥♦❝❡♥t r❛t❤❡r t❤❛♥ t♦ ❛❝q✉✐t ❛ ❣✉✐❧t② ❞❡❢❡♥❞❛♥t✳

Pr♦♣♦s✐t✐♦♥ ✸✳✻✳ ▲❡t J ❜❡ ❛ s❡t ♦❢ ✉♥❜✐❛s❡❞ ❛♥❞ r❡s♣♦♥s✐✈❡ ❥✉r♦rs ✇✐t❤ |J| = 3✱ ❢♦r ❣✐✈❡♥ α ∈ (0,1)✱

z∈(¹₂,1) ❛♥❞q∈(z,qˆα^max)t❤❡ ✈♦t✐♥❣ str❛t❡❣②

Pr(a|tj =g) = q(1−α)z(1−z)−(1−α)z²(1−z) α+ (1−α)z³−q[α+ (1−α)[z³+ (1−z)³]]

Pr(a|tj=i) = 1

✇❤❡r❡

ˆ

q^maxα = α+ (1−α)z²

α+ (1−α)[z²+ (1−z)²] < q^maxα

✐s r❛t✐♦♥❛❧✳ ❚❤❛t ✐s✱ ❢♦r ❛♥②q∈(z,qˆ^maxα )t❤❡ ❛❜♦✈❡ str❛t❡❣② ✐s ❛ ❜❛②❡s✐❛♥ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♦❢ Γ^b✳

❆s ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣r♦♣♦s✐t✐♦♥✱ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❝❤❛♥❣❡s qˆα^max < q^maxα ❛♥❞ ❥✉r♦rs ✏❛❝❝❡♣t✑ t♦ r❡✈❡❛❧

tr✉t❤❢✉❧❧② t❤❡ s✐❣♥❛❧ ♦♥❧② ✇❤❡♥ ✐t ✐s ❝♦❤❡r❡♥t ✇✐t❤ t❤❡✐r ❝♦♥❝❡r♥✱ ♦t❤❡r✇✐s❡ t❤❡② ❝❤♦♦s❡ t♦ r❛♥❞♦♠✐③❡✳

✺

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦r

❚❤❡ ❈♦♥❞♦r❝❡t✬s ❏✉r② ❚❤❡♦r❡♠ st❛t❡s t❤❛t ♠❛❥♦r✐t② ❛r❡ ♠♦r❡ ❧✐❦❡❧② t♦ s❡❧❡❝t t❤❡ ❝♦rr❡❝t ❛❧t❡r♥❛t✐✈❡ ✇✐t❤

r❡s♣❡❝t t♦ t❤❡ s✐♥❣❧❡ ♦♥❡✳ ◆♦t ♦♥❧②✱ ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❥✉r♦rs ❣r♦✇s✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♠❛❦✐♥❣ t❤❡ ❝♦rr❡❝t

❝❤♦✐❝❡ ✐♥❝r❡❛s❡s✳ ❯♥❞❡r t❤❡ ✐♥❢♦r♠❛t✐✈❡ str❛t❡❣② ❛ ❞❡❢❡♥❞❛♥t ✐s ❛❝q✉✐tt❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢k(t)>m✱ ✇❤❡r❡ˆ k(t)✐s t❤❡ ♥✉♠❜❡r ♦❢is✐❣♥❛❧s ✐♥ t❤❡ ♣r♦✜❧❡ t∈T✳ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ k(t)❝♦♥❞✐t✐♦♥❡❞ t♦ ♦♥❡ ♦❢ t❤❡ t✇♦

st❛t❡s ✐♥Ω1 ✐s

Pr(k(t) = 2 ˆm+ 1|ω1=■) = α+ (1−α)z^{2 ˆm+1} Pr(k(t) =x|ω1=■) =

2 ˆm+ 1 x

(1−α)z^x(1−z)^{(2 ˆm+1)−x}

❢♦r0≤x <|J|= 2 ˆm+ 1 ❛♥❞

Pr(k(t) = 0|ω1=●) = α+ (1−α)z^{2 ˆm+1} Pr(k(t) =x|ω1=●) =

2 ˆm+ 1 x

(1−α)z^{(2 ˆm+1)−x}(1−z)^x

❢♦r0< x≤ |J|= 2 ˆm+ 1✳ ❙♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❝♦♥✈✐❝t ❛♥ ✐♥♥♦❝❡♥t ✐s Pr(conv|■) = (1−α)

ˆ

Xm

x=0

2 ˆm+ 1 x

z^x(1−z)^{(2 ˆm+1)−x}

❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝q✉✐t ❛ ❣✉✐❧t② ✐s Pr(acq|●) = (1−α)

ˆ

Xm

x=0

2 ˆm+ 1 x

z^x(1−z)^{(2 ˆm+1)−x}

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ s❤♦✇s t❤❛t ✇✐t❤ t❤❡ ✐♥❢♦r♠❛t✐✈❡ str❛t❡❣② t❤❡ ❈♦♥❞♦r❝❡t✬s ❚❤❡♦r❡♠ st✐❧❧ ❤♦❧❞s✳

Pr♦♣♦s✐t✐♦♥ ✸✳✼✳ ▲❡t J ❜❡ ❛ s❡t ♦❢ ✉♥❜✐❛s❡❞✱ r❛t✐♦♥❛❧ ❛♥❞ r❡s♣♦♥s✐✈❡ ❥✉r♦rs ✇✐t❤ mˆ ∈ N✱ ❢♦r ❣✐✈❡♥

α∈(0,1)✱z∈(¹₂,1) ❛♥❞q∈(1−z, z)t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❝♦♥✈✐❝t ❛♥ ✐♥♥♦❝❡♥t ❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝q✉✐t

❛ ❣✉✐❧t② ❣♦ t♦ ③❡r♦ ❛s t❤❡ ❥✉r② s✐③❡ ❣♦❡s t♦ ✐♥✜♥✐t②✳

❙✐♥❣❧❡ ❏✉❞❣❡ ✈s✳ ▼✉❧t✐♣❧❡ ❏✉r♦rs

■♥ t❤✐s ♣❛rt ✇❡ ❛♥❛❧②③❡ t❤❡ s✐t✉❛t✐♦♥ ✐♥ ✇❤✐❝❤ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❥✉❞❣❡✱ ✇❤♦ ♠✉st ❝❤♦♦s❡ t♦ ❛❝q✉✐t ♦r ❝♦♥✈✐❝t t❤❡ ❞❡❢❡♥❞❛♥t✳ ❚❤❡ ❜❛s✐❝ ❛ss✉♠♣t✐♦♥ ♠❛❞❡ ✐s t❤❛t✱ s✐♥❝❡ ❛ ❥✉❞❣❡ ❝❛♥ ✐♥t❡r❛❝t ✇✐t❤ t❤❡ ✐♥✈♦❧✈❡❞ ♣❛rt✐❡s

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

✐♥❢♦r♠❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❛ s✐♥❣❧❡ ❥✉r♦r✳ ❆♥♦t❤❡r ♣♦ss✐❜❧❡ ❥✉st✐✜❝❛t✐♦♥ t♦ t❤✐s ❛ss✉♠♣t✐♦♥ ✐s t❤❛t ❥✉❞❣❡s

✉s✉❛❧❧② ❛r❡ ♠♦r❡ ❝♦♠♣❡t❡♥t✱ t❤❛♥ ❛ s✐♥❣❧❡ ❥✉r♦r ❞r❛✇♥ ❢r♦♠ ❛ ❧✐st ♦❢ ❝♦♠♠♦♥ ♣❡♦♣❧❡✳ ■♥ ♦r❞❡r t♦ ❦❡❡♣

s♦♠❡ ❛♥❛❧♦❣✐❡s ✇✐t❤ t❤❡ ❝❛s❡ ♦❢ t❤r❡❡ ❥✉r♦rs✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ♦❜s❡r✈❡❞ s✐❣♥❛❧s ✐s ❡q✉❛❧ t♦ ✸✳

❲❡ ❝♦♠♣❛r❡ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛❦✐♥❣ t❤❡ ❝♦rr❡❝t ❝❤♦✐❝❡ ❢♦r t❤❡ s❛♠❡ ✈❛❧✉❡ ♦❢q✳ ❲❡ ♠❛✐♥t❛✐♥ t❤❡ s❛♠❡

✉t✐❧✐t② ❢✉♥❝t✐♦♥u✱ ❜✉t ✐♥ t❤✐s ❝❛s❡ t❤❡ ❛❣❣r❡❣❛t✐♦♥ r✉❧❡f ✐s s✐♠♣❧② t❤❡ ✐❞❡♥t✐t② ❢r♦♠ t❤❡ ❥✉❞❣❡✬s ❞❡❝✐s✐♦♥

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

❥✉❞❣❡✬s ❜❡❤❛✈✐♦r ✐s t❤❡ s❛♠❡ ❛s t❤❡ s✐♥❣❧❡ ❥✉r♦r✳ ❍❡ ❝❤♦♦s❡s t♦ ❛❝q✉✐t ✐❢ ❛♥❞ ♦♥❧② ✐❢ q >Pr(●|ˆt)✱ ✇❤❡r❡

ˆt∈ T ✐s t❤❡ ♦❜s❡r✈❡❞ s✐❣♥❛❧s ♣r♦✜❧❡✱ ♦t❤❡r✇✐s❡ ❤❡ ❝❤♦♦s❡s t♦ ❝♦♥✈✐❝t t❤❡ ❞❡❢❡♥❞❛♥t✳ ❆ss✉♠✐♥❣ t❤❛t t❤❡

❥✉❞❣❡ ✐s ♥♦t ❜✐❛s❡❞✱ ❧❡t ✉s ❝♦♠♣✉t❡ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❜❡✐♥❣ ❣✉✐❧t② ❣✐✈❡♥ t❤❡ ❞✐✛❡r❡♥t s✐❣♥❛❧s ♣r♦✜❧❡s Pr(●|g, g, g) = α+ (1−α)z³

α+ (1−α)[z³+ (1−z)³] =θ1 Pr(●|i, g, g) =z=θ2

Pr(●|i, i, g) = 1−z=θ3 Pr(●|i, i, i) = (1−α)(1−z)³

α+ (1−α)[z³+ (1−z)³] =θ4

✻

♥♦t✐❝❡ t❤❛tθ2❛♥❞θ3 ❛r❡ t❤❡ s❛♠❡ ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ t❤❡ ♣❡r♠✉t❛t✐♦♥ ♦❢ t❤❡ s✐❣♥❛❧s✳ Pr♦♣♦s✐t✐♦♥ ✸✳✸ ❣✐✈❡s

✉s t❤❡ ✐♥t❡r✈❛❧ ♦❢ r❡s♣♦♥s✐✈❡♥❡ss ❢♦r ❛ s✐♥❣❧❡ ❞❡❝✐s✐♦♥ ♠❛❦❡r✱ ✇❡ ❝❛♥ s❡❡ t❤❛tθ1=qα^max❛♥❞θ4=q^minα ✳ ▲❡t Pr(●|conv)❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ❣✉✐❧t② ✐s ❝♦♥✈✐❝t❡❞ ❜② ❛ ❥✉❞❣❡ ❛♥❞ ❧❡t Pr(●|conv)t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

♣r♦❜❛❜✐❧✐t② ✇❤❡♥ t❤❡ ❞❡❧✐❜❡r❛t✐♦♥ ✐s ♠❛❞❡ ❜② ❛ ❣r♦✉♣ ♦❢ ❥✉r♦rs✳ ❲❤❡♥q∈(0, q^min_α )❜♦t❤ ❥✉❞❣❡ ❛♥❞ ❥✉r♦rs

❛r❡ ♦✉ts✐❞❡ t❤❡ r❡s♣♦♥s✐✈❡♥❡ss ✐♥t❡r✈❛❧✱ ❛♥❞ t❤❡② ❛❧✇❛②s ❝❤♦♦s❡ t♦ ❝♦♥✈✐❝t t❤❡ ❞❡❢❡♥❞❛♥t✳

Pr(●|conv) = 1

2 =Pr(●|conv)

■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ✐♥ t❤✐s ❝❛s❡✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝q✉✐tt✐♥❣ ❛♥ ✐♥♥♦❝❡♥t ❞❡❢❡♥❞❛♥t ✐sPr(■|acq) = 0 = Pr(■|acq)✳ ❋♦r q∈(q^minα ,qˆα^min) ❥✉r♦rs ❛❧✇❛②s ❝❤♦♦s❡ t♦ ❝♦♥✈✐❝t t❤❡ ❞❡❢❡♥❞❛♥t✱ ✇❤✐❧❡ t❤❡ s✐♥❣❧❡ ❥✉❞❣❡ ✇✐❧❧

❛❝q✉✐t t❤❡ ❞❡❢❡♥❞❛♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❤❡ ♦❜s❡r✈❡rs t❤❡ s✐❣♥❛❧ ♣r♦✜❧❡t= (i, i, i)✳ ❍❡♥❝❡

Pr(●|conv) =α+ (1−α)[z³+ 3z(1−z)]

1 + 3(1−α)z(1−z) >1

2 =Pr(●|conv) s✐♥❝❡α+ (1−α)[z³−(1−z)³] >0 ❤♦❧❞s ❢♦r ❛♥② α >0 ❛♥❞ z ∈ ¹₂,1

t❤❡ ❛❜♦✈❡ ✐♥❡q✉❛❧✐t② ✐s ❛❧✇❛②s s❛t✐s✜❡❞✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝q✉✐tt✐♥❣ ❛♥ ✐♥♥♦❝❡♥t ❞❡❢❡♥❞❛♥t✱ ✐♥ t❤✐s ❝❛s❡ ✐s

Pr(■|acq) = α+ (1−α)z³

α+ (1−α)[z³+ (1−z)³] >0 =Pr(■|acq)

✇❤❡♥q∈(1−z, z)t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❝♦♥✈✐❝t ❛ ❣✉✐❧t② ❞❡❢❡♥❞❛♥t ❛r❡ t❤❡ s❛♠❡

Pr(●|conv) =α+ (1−α)[z³+ 3z²(1−z)] =Pr(●|conv)

❛s t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❛❝q✉✐tt✐♥❣ ❛♥ ✐♥♥♦❝❡♥t ♦♥❡

Pr(■|acq) =α+ (1−α)[z³+ 3z²(1−z)] =Pr(■|acq)

❋♦rq∈(ˆq_α^max, q_α^max) ❥✉r② ♠❡♠❜❡rs ❛❧✇❛②s ❝❤♦♦s❡ t♦ ❛❝q✉✐t ❛ ❞❡❢❡♥❞❛♥t✱ ✇❤✐❧❡ ❛ s✐♥❣❧❡ ❥✉r♦r ✇✐❧❧ ❝♦♥✈✐❝t

✇❤❡♥ ❤❡ ♦❜s❡r✈❡s t❤❡ s✐❣♥❛❧ ♣r♦✜❧❡t= (g, g, g)✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡

Pr(●|conv) = α+ (1−α)z³

α+ (1−alpha)[z³+ (1−z)³ >0 =Pr(●|conv)

✇❤✐❧❡ ❢♦r t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❛❝q✉✐tt✐♥❣ ❛♥ ✐♥♥♦❝❡♥t ❞❡❢❡♥❞❛♥t t❤❡ r❡s✉❧t ✐s s✐♠✐❧❛r ❛s t❤❡ ✜rst ❝❛s❡ ❛♥❛❧②③❡❞

Pr(■|acq) =α+ (1−α)[z³+ 3z(1−z)]

1 + 3(1−α)z(1−z) >1

2 =Pr(■|acq)

❋♦rq∈(q^max_α ,1) ❜♦t❤ ❥✉❞❣❡ ❛♥❞ ❥✉r♦rs ❝❤♦♦s❡ t♦ ❛❝q✉✐t t❤❡ ❞❡❢❡♥❞❛♥t✱ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❝♦♥✈✐❝t✐♥❣ ❛

❣✉✐❧t② ❞❡❢❡♥❞❛♥t ❛r❡ ③❡r♦ ❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❛❝q✉✐tt✐♥❣ ❛♥ ✐♥♥♦❝❡♥t ♦♥❡ ❛r❡ ❡q✉❛❧ t♦ ♦♥❡ ❤❛❧❢✳

✹ ❚❤❡ ❈❧❛ss✐❝❛❧ ▼♦❞❡❧

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ r❡♣❧✐❝❛t❡ t❤❡ ❛♥❛❧②s✐s ❛s ✐♥ ❙❡❝t✐♦♥ ✸ ✇❤❡♥ t❤❡ ❝❧❛ss✐❝❛❧ ❛ss✉♠♣t✐♦♥ ✐s ✉s❡❞✳ ❚♦ ❞♦

t❤✐s✱ ✐t ✐s ❡♥♦✉❣❤ t♦ ❝❤❛♥❣❡ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ s✐❣♥❛❧s ❞✐str✐❜✉t✐♦♥ ❛s Pr(tj=i|■) =p ❛♥❞ Pr(tj =g|●) =p Pr(tj=g|■) = 1−p ❛♥❞ Pr(tj=i|●) = 1−p

✇✐t❤ p∈ (¹₂,1) ❛♥❞ ✇❡ ❛ss✉♠❡ t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ s✐❣♥❛❧s✳ ❆s ✐♥ Pr♦♣♦s✐t✐♦♥ ✸✳✸ ✐t ✐s ♣♦ss✐❜❧❡ t♦

❞❡✜♥❡ t❤❡ ✐♥t❡r✈❛❧ ✇✐t❤✐♥ ❛ ❥✉r♦r ✐s r❡s♣♦♥s✐✈❡✱ ✇❤❡♥ s❤❡ ❝❛♥ ♦❜s❡r✈❡ ❛❧❧ t❤❡ s✐❣♥❛❧s ♣r♦✜❧❡

✼

Pr♦♣♦s✐t✐♦♥ ✹✳✶✳ ❋♦r ❣✐✈❡♥p∈(¹₂,1) ❛♥❞|T|= 3 ❛♥ ✉♥❜✐❛s❡❞ ❥✉r♦rj∈J ✐s r❡s♣♦♥s✐✈❡✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢

q∈ q^min₀ , q₀^max

=

(1−p)³

p³+ (1−p)³, p³ p³+ (1−p)³

◆♦t✐❝❡ t❤❛t ❧❡tt✐♥❣p❡q✉❛❧ t♦z✱ t❤✐s ✐♠♣❧✐❡s t❤❛t s✐❣♥❛❧s ❤❛✈❡ t❤❡ s❛♠❡ q✉❛❧✐t②✱ t❤❡ r❡s♣♦♥s✐✈❡♥❡ss ✐♥t❡r✈❛❧

✐♥ ❙❡❝t✐♦♥ ✸ ✐s ❛❧✇❛②s ✇✐❞❡r t❤❛♥ t❤✐s ♦♥❡ q₀^min, q^max₀

⊂ q_α^min, q^max_α

✳ ■t ✐s s✉✣❝✐❡♥t t♦ ❦♥♦✇ t❤❛t ✇✐t❤ ❛

♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✱ ♥♦ ♠❛tt❡r ♦❢ ✐ts ✈❛❧✉❡✱ ◆❛t✉r❡ r❡✈❡❛❧s t❤❡ tr✉t❤ t♦ ❝♦♥✈✐♥❝❡ ♠♦r❡ ❝♦♥❝❡r♥❡❞ ❥✉r♦rs t♦

r❡❛❝t t♦ t❤❡ ✐♥❢♦r♠❛t✐♦♥✳

Pr♦♣♦s✐t✐♦♥ ✹✳✷✳ ▲❡tJ ❜❡ ❛ s❡t ♦❢ ✉♥❜✐❛s❡❞ ❛♥❞ r❡s♣♦♥s✐✈❡ ❥✉r♦rs ✇✐t❤mˆ ∈N<∞❢♦r ❣✐✈❡♥ p∈(¹₂,1)✱

✐❢q∈(1−p, p)t❤❡♥ t❤❡ ✐♥❢♦r♠❛t✐✈❡ ✈♦t✐♥❣ str❛t❡❣② ✭♣r♦✜❧❡✮ ✐s r❛t✐♦♥❛❧✳ ❚❤❛t ✐s✱ ❢♦r ❛♥②q∈(1−p, p)t❤❡

✐♥❢♦r♠❛t✐✈❡ ✈♦t✐♥❣ str❛t❡❣② ♣r♦✜❧❡σ^∗ ✐s ❛ ❜❛②❡s✐❛♥ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♦❢Γ^b₀✳

✇❤❡r❡Γ^b₀ =hJ,(Tj, Sj)j∈J,(p, π), vi ✐s t❤❡ ♠♦❞✐✜❡❞ ❜❛②❡s✐❛♥ ❣❛♠❡✳ ❲❤❡♥q /∈(1−p, p)t❤❡ t❤r❡s❤♦❧❞ ♦❢

r❡❛s♦♥❛❜❧❡ ❞♦✉❜t ✐s s❤✐❢t❡❞ ❛♥❞ s♦

Pr♦♣♦s✐t✐♦♥ ✹✳✸✳ ▲❡tJ ❜❡ ❛ s❡t ♦❢ ✉♥❜✐❛s❡❞ ❛♥❞ r❡s♣♦♥s✐✈❡ ❥✉r♦rs ✇✐t❤|J|= 3✱ ❢♦r ❣✐✈❡♥p∈(¹₂,1) ❛♥❞

q∈(ˆq^min₀ ,1−p)t❤❡ ✈♦t✐♥❣ str❛t❡❣②

Pr(a|tj=g) = 0

Pr(a|tj=i) = q[p²+ (1−p)²]−(1−p)² q[p³+ (1−p)³]−(1−p)³

✇❤❡r❡

ˆ

q^min₀ = (1−p)²

p²+ (1−p)² > q^min₀

✐s r❛t✐♦♥❛❧✳ ❚❤❛t ✐s✱ ❢♦r ❛♥②q∈(ˆq^min₀ ,1−p)t❤❡ ❛❜♦✈❡ str❛t❡❣② ✐s ❛ ❜❛②❡s✐❛♥ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♦❢ Γ^b₀✳

❙✐♠✐❧❛r❧② t♦ t❤❡ Pr♦♣♦s✐t✐♦♥ ✸✳✺ t❤❡ ❧♦✇❡r ❜♦✉♥❞qˆ^min₀ ✐s ❤✐❣❤❡r t❤❛♥ q₀^min ❛♥❞ t❤❡ s✐❣♥❛❧ tj ✐s ✏r❡✈❡❛❧❡❞✑

♦♥❧② ✇❤❡♥ ✐t ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ❛♣t✐t✉❞❡ ♦❢ t❤❡ ❥✉r♦rs✳

Pr♦♣♦s✐t✐♦♥ ✹✳✹✳ ▲❡tJ ❜❡ ❛ s❡t ♦❢ ✉♥❜✐❛s❡❞ ❛♥❞ r❡s♣♦♥s✐✈❡ ❥✉r♦rs ✇✐t❤|J|= 3✱ ❢♦r ❣✐✈❡♥p∈(¹₂,1) ❛♥❞

q∈(p,qˆ^max₀ )t❤❡ ✈♦t✐♥❣ str❛t❡❣②

Pr(a|tj=g) =qp(1−p)−p²(1−p) p³−q[p³+ (1−p)³] Pr(a|tj=i) = 1

✇❤❡r❡

ˆ

q₀^max= p²

p²+ (1−p)² < q^max₀

✐s r❛t✐♦♥❛❧✳ ❚❤❛t ✐s✱ ❢♦r ❛♥②q∈(p,qˆ^max₀ )t❤❡ ❛❜♦✈❡ str❛t❡❣② ✐s ❛ ❜❛②❡s✐❛♥ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♦❢ Γ^b₀✳

❛s Pr♦♣♦s✐t✐♦♥ ✸✳✻ t❤❡ ✉♣♣❡r ❜♦✉♥❞qˆ^max₀ ✐s ❧♦✇❡r t❤❛♥q^max₀ ❛♥❞tj ✐s ✏r❡✈❡❛❧❡❞✑ ♦♥❧② ✇❤❡♥tj =i✳

✺ ❇❡♥❝❤♠❛r❦

❉❡♥♦t❡ ❛s α✲♠♦❞❡❧ t❤❡ ♦♥❡ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✷ ❛♥❞ ❛s 0✲♠♦❞❡❧ t❤❡ ❝❧❛ss✐❝ ♦♥❡ ✐♥ ❙❡❝t✐♦♥ ✹✳ ◆♦✇

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

✐♥❢♦r♠❛t✐✈❡ str❛t❡❣② ♣r♦✜❧❡✱ ❛r❡ t❤❡ s❛♠❡✳ ❚♦ ❞♦ t❤✐s ✐t ✐s ♥❡❝❡ss❛r② t♦ ✜♥❞ t❤❡ r❡❧❛t✐♦♥✱ ✐❢ ❡①✐sts✱ ❜❡t✇❡❡♥

t❤❡ ❞✐✛❡r❡♥t ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡❧s✳ ❇❡❢♦r❡ ♦❜s❡r✈✐♥❣ t❤❡ s✐❣♥❛❧

❊❯[v] =−1

2[Pr(conv|■)q+ Pr(acq|●)(1−q)]

✽

✇❤❡r❡ t❤❡ ❝♦♥❞✐t✐♦♥❡❞ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❛❝q✉✐t ❛♥❞ ❝♦♥✈✐❝t ❞❡♣❡♥❞ ❛❧s♦ ❜② t❤❡ ✈♦t✐♥❣ str❛t❡❣② ♣r♦✜❧❡

Pr(conv|■) = X

(t−i,t_i)∈T

Pr(conv|t−i, ti,■) Pr(t−i, ti|■) Pr(acq|●) = X

(t−i,t_i)∈T

Pr(acq|t−i, ti,●) Pr(t−i, ti|●)

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

❊❯α[v] =−1

2(1−α)[(1−z)³+ 3z(1−z)²]

❊❯0[v] =−1

2[(1−p)³+ 3p(1−p)²]

■♥ t❤✐s ❝❛s❡ ❊❯α[v]✐s ❡q✉❛❧ t♦ ❊❯0[v]✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜❡t✇❡❡♥α✱z ❛♥❞p❤♦❧❞s t❤✐s r❡❧❛t✐♦♥

−1

2(1−α)[(1−z)³+ 3z(1−z)²] =−1

2[(1−p)³+ 3p(1−p)²] ✭⋆✮

❞❡r✐✈❡❞ ✐♥ t❤❡ ❆♣♣❡♥❞✐①✳ ❲❤❡♥α= 0 ❜♦t❤ ♠♦❞❡❧s ❝♦✐♥❝✐❞❡ ❛♥❞ p=z ❛s ✇❤❡♥ α= 1 ♦r z = 1✱ ✐♥ ❛❧❧

♦t❤❡r ❝❛s❡sp > z✳

❊①❛♠♣❧❡ ✺✳✶✳ ❋♦r α = 0.5 ❛♥❞ z = 0.65 t❤❡ ❡① ❛♥t❡ ❡①♣❡❝t❡❞ ✉t✐❧✐t② ✐s ❊❯α[v] = −0.0704 ❛♥❞ t❤❡

♣r♦❜❛❜✐❧✐t②pt❤❛t s❛t✐s✜❡s ❡q✉❛t✐♦♥ ✭⋆✮ ✐s0.7639✳

▲❡tqˆ^min = min{ˆq₀^min,qˆ_α^min} ❛♥❞qˆ^max = max{qˆ^max₀ ,qˆ_α^max}✳ ❲❤❡♥ q <qˆ^min ✐♥ ❜♦t❤ ♠♦❞❡❧s ❥✉r♦rs ❛❧✇❛②s

❝❤♦♦s❡s t♦ ❝♦♥✈✐❝t ❛♥❞ t❤❡ ❡① ❛♥t❡ ❡①♣❡❝t❡❞ ✉t✐❧✐t✐❡s ❛r❡

❊❯α[v] =❊❯0[v] =−1 2q

✇❤❡♥q >qˆ^max ✐♥ ❜♦t❤ ♠♦❞❡❧s ❥✉r♦rs ❛❧✇❛②s ❝❤♦♦s❡s t♦ ❛❝q✉✐t ❛♥❞ t❤❡ ❡① ❛♥t❡ ❡①♣❡❝t❡❞ ✉t✐❧✐t✐❡s ❛r❡

❊❯α[v] =❊❯0[v] =−1 2(1−q)

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

Pr♦♣♦s✐t✐♦♥ ✺✳✷✳ ❋♦r ❣✐✈❡♥α∈(¹₂,1)✱z∈(¹₂,1) ❛♥❞p∈(¹₂,1) s✉❝❤ t❤❛t ❡q✉❛t✐♦♥ ✭⋆✮ ❤♦❧❞s✱

ˆ

q^min= ˆq^min_α (ˆq^min= ˆq^min₀ ) ❛♥❞ qˆ^max= ˆq^max_α (ˆq^max= ˆq₀^max)

✐❢ ❛♥❞ ♦♥❧② ✐❢

α >(<) ˆα

✇✐t❤

ˆ

α= p²(1−z)²−(1−p)²z² (1−p)²+p²(1−z)²−(1−p)²z²

❙✐♥❝❡α∈(0,1) ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♦❜s❡r✈✐♥❣ ❛ str❛✐❣❤t❢♦r✇❛r❞ ❡✈✐❞❡♥❝❡ t❤❛t ❧❡❛✈❡s ♥♦

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

❥✉r♦rs✳ ◆♦✇ ✐t ✐s t✐♠❡ t♦ ❝♦♠♣❛r❡ t❤❡ ❡① ❛♥t❡ ❡①♣❡❝t❡❞ ✉t✐❧✐t✐❡s ❢♦r s♦♠❡ ✈❛❧✉❡s ♦❢q✳

Pr♦♣♦s✐t✐♦♥ ✺✳✸✳ ▲❡t J ❜❡ ❛ s❡t ♦❢ ✉♥❜✐❛s❡❞ ❛♥❞ r❡s♣♦♥s✐✈❡ ❥✉r♦rs ✇✐t❤ |J| = 3✱ ❢♦r ❣✐✈❡♥ α ∈ (0,1)✱

z∈(¹₂,1) ❛♥❞p∈(¹₂,1) s✉❝❤ t❤❛t ❡q✉❛t✐♦♥ ✭⋆✮ ❤♦❧❞s✱

✐✮ ✐❢q∈(0,qˆ^min)✐♥ ❜♦t❤ ♠♦❞❡❧s ❥✉r♦rs ❛❧✇❛②s ❝❤♦♦s❡ t♦ ❝♦♥✈✐❝t ❛♥❞ ❊❯α[v] =❊❯0[v]✳

✐✐✮ ✐❢ q ∈ (1−p,1−z) ✐♥ t❤❡ 0−♠♦❞❡❧ t❤❡ ✐♥❢♦r♠❛t✐✈❡ str❛t❡❣② ✐s r❛t✐♦♥❛❧✱ ✇❤✐❧❡ ✐♥ t❤❡ α✲♠♦❞❡❧ ✐s r❛t✐♦♥❛❧ t❤❡ str❛t❡❣② ❞❡s❝r✐❜❡❞ ✐♥ ♣r♦♣♦s✐t✐♦♥ ✸✳✺ ❛♥❞ ❊❯α[v]>❊❯0[v]✳

✐✐✐✮ ✐❢q∈(1−z, z) ✐♥ ❜♦t❤ ♠♦❞❡❧s t❤❡ ✐♥❢♦r♠❛t✐✈❡ str❛t❡❣② ✐s r❛t✐♦♥❛❧ ❛♥❞ ❊❯α[v] =❊❯0[v]✳

✐✈✮ ✐❢ q∈(z, p)✐♥ t❤❡0−♠♦❞❡❧ t❤❡ ✐♥❢♦r♠❛t✐✈❡ str❛t❡❣② ✐s r❛t✐♦♥❛❧✱ ✇❤✐❧❡ ✐♥ t❤❡α✲♠♦❞❡❧ ✐s r❛t✐♦♥❛❧ t❤❡

str❛t❡❣② ❞❡s❝r✐❜❡❞ ✐♥ ♣r♦♣♦s✐t✐♦♥ ✸✳✻ ❛♥❞ ❊❯α[v]>❊❯0[v]✳

✈✮ ✐❢q∈(ˆq^max,1) ✐♥ ❜♦t❤ ♠♦❞❡❧s ❥✉r♦rs ❛❧✇❛②s ❝❤♦♦s❡ t♦ ❛❝q✉✐t ❛♥❞ ❊❯α[v] =❊❯0[v]✳

✾

✻ ❈♦♥❝❧✉s✐♦♥s

❚❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t② ✇✐t❤ ✇❤✐❝❤ ◆❛t✉r❡ r❡✈❡❛❧s t❤❡ tr✉❡ st❛t❡ ♦❢ t❤❡ ✇♦r❧❞✱ ♠❛❦❡s t❤❡

r❡s♣♦♥s✐✈❡♥❡ss ✐♥t❡r✈❛❧ ✇✐❞❡r✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❥✉r♦rs ✐♥❝r❡❛s❡s t❤❡ ✈❛❧✉❡s ♦❢qα^min

❛♥❞q^min₀ ❛♣♣r♦❛❝❤ t♦ ③❡r♦ ❛♥❞ t❤❡ ✈❛❧✉❡s ♦❢ q^max_α ❛♥❞ q^max₀ ❛♣♣r♦❛❝❤ t♦ ♦♥❡✳ ❚❤❡ r❡❛s♦♥ ✐s ✈❡r② s✐♠♣❧❡✿

t❤❡ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❛ s✐♥❣❧❡ ❞❡❝✐s✐♦♥ ♠❛❦❡r ❝❛♥ ♦❜s❡r✈❡ ❛♥❞ t❤❡ ❤✐❣❤❡r ✐s ❤❡r ✐♥❝❧✐♥❛t✐♦♥ t♦ ❝❤❛♥❣❡

❤❡r ❥✉❞❣♠❡♥t ❛❜♦✉t t❤❡ ❞❡❢❡♥❞❛♥t✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❥✉r② ♠❡♠❜❡rs ✐♥❝r❡❛s❡s t❤❡

♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ ♣✐✈♦t❛❧ ❞❡❝r❡❛s❡s✳ ❚❤✐s ❡✛❡❝t s❤♦✉❧❞ ♥❛rr♦✇ t❤❡ ✐♥t❡r✈❛❧s(ˆq^min_α ,1−z)✱(ˆq^min₀ ,1−z)✱

(z,qˆ_α^max) ❛♥❞ (p,qˆ₀^max)✳ ❆ ♣♦ss✐❜❧❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✷ ✐s t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢

❛ ♣✉❜❧✐❝ ♦❜s❡r✈❛❜❧❡ s✐❣♥❛❧ ✇✐t❤ st❛t❡ ❞❡♣❡♥❞❡♥t ❞✐str✐❜✉t✐♦♥✳ ■t ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ r♦❧❡ ♣❧❛②❡❞ ❜②

✏♦♣✐♥✐♦♥ ❧❡❛❞❡rs✑ ♦r ❜② t❤❡ ♠❡❞✐❛✱ ✐✳❡✳ t❡❧❡✈✐s✐♦♥✱ ♥❡✇s♣❛♣❡rs✱ ■♥t❡r♥❡t ❢♦r✉♠s✱ ❡t❝✳ ■♥ t❤✐s ❝❛s❡✱ ✐t s❡❡♠s r❡❛s♦♥❛❜❧❡ t❤❛t t❤❡ q✉❛❧✐t② ♦❢ t❤❡ s✐❣♥❛❧ s❤♦✉❧❞ ❜❡ ❧♦✇❡r ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s✐❣♥❛❧s ♣r♦❞✉❝❡❞ ❞✉r✐♥❣ t❤❡

tr✐❛❧✳ ❆♥♦t❤❡r ♣♦ss✐❜❧❡ ❡①t❡♥s✐♦♥ ✐s t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❞✐✛❡r❡♥t t❤r❡s❤♦❧❞s ♦❢ r❡❛s♦♥❛❜❧❡ ❞♦✉❜t ✭❛s ✐♥

●❡r❛r❞✐✱ ✷✵✵✵✮✳ ■♥ t❤✐s ❢r❛♠❡✇♦r❦ ❞♦❡s ❛ ❤❡t❡r♦❣❡♥❡♦✉s ❥✉r② ♣❡r❢♦r♠s ❜❡tt❡r t❤❛♥ ❛ ❤♦♠♦❣❡♥❡♦✉s ♦♥❡❄

✼ ❆♣♣❡♥❞✐①

Pr♦❜❛❜✐❧✐st✐❝ ❙tr✉❝t✉r❡

▲❡tN =|J|✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ s✐❣♥❛❧s ♣r♦✜❧❡st= (t1, . . . , tN)∈ {i, g}^N ✐s ❞❡✜♥❡❞ ❛s f(t) =10(t)·p(0, N) +1k(t)·p(k, N) +1N(t)·p(N, N)

✇❤❡r❡10(t)✐s t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ③❡r♦ ✈❡❝t♦r✱

1k(t) =

( 1 ✐❢ PN

j=1tj=k∈ {1,2, . . . , N−1}

0 ♦t❤❡r✇✐s❡

❛♥❞ 1N(t) ✐s t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r ✇✐t❤ ❛❧❧ ❝♦♠♣♦♥❡♥ts ❡q✉❛❧ t♦ ✶✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

♣r♦❜❛❜✐❧✐t✐❡s ❛r❡

p(0, N) = (1−π)α+ (1−α)[π(1−z)^N + (1−π)z^N] p(k, N) = (1−α)[πz^k(1−z)^N−k+ (1−π)z^N−k(1−z)^k]

p(N, N) =απ+ (1−α)[πz^N+ (1−π)(1−z)^N]

❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ s✐❣♥❛❧s ❢♦rk∈ {1,2, . . . , N −1} ✐s ❞✐str✐❜✉✐t❡❞ ❛s g(k) =

N j

N−1 j=k

π(1−α)z^j(1−z)^N−j+ (1−π)(1−α)z^N−j(1−z)^j

❢♦rk= 0

g(0) =π(1−α)(1−z)^N + (1−π)[α+ (1−α)z^N]

❛♥❞ ❢♦rk=N

g(N) =π[α+ (1−α)z^N] + (1−π)(1−α)(1−z)^N

❊①♣❧✐❝✐t❛t✐♦♥ ♦❢ ❈♦♥❞✐t✐♦♥ ✭⋆✮

❈♦♥❞✐t✐♦♥s ✭⋆✮ ❤♦❧❞s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜❡t✇❡❡♥α∈(0,1)✱ z∈(¹₂,1) ❛♥❞p∈(¹₂,1) ❤♦❧❞s t❤✐s r❡❧❛t✐♦♥

p= 1 2−sin

arcsin(2(1−α)[(1−z)³+ 3z(1−z)²]−1) 3

∈ 1

2,1

✶✵

✇❤❡r❡

arcsin(·) : [−1,1]→h

−π 2,π

2 i

✐s t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ sin(·)✳ ❚❤✐s ✐s t❤❡ ♦♥❧② ❛❞♠✐ss✐❜❧❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ t❤✐r❞ ❞❡❣r❡❡ ❡q✉❛t✐♦♥

❞❡r✐✈❡❞ ❜② ❝♦♥❞✐t✐♦♥ ✭⋆✮✳ ❚❛❜❧❡ ✶ r❡♣♦rts s♦♠❡ ✈❛❧✉❡s ♦❢p❢♦r ❣✐✈❡♥α❛♥❞z✳

α

✵✳✵✵✵ ✵✳✶✵✵ ✵✳✸✵✵ ✵✳✺✵✵ ✵✳✼✵✵ ✵✳✾✵✵ ✶✳✵✵✵

③

✵✳✺✺✵ ✵✳✺✺✵ ✵✳✺✼✾ ✵✳✻✸✽ ✵✳✼✵✸ ✵✳✼✼✻ ✵✳✽✼✻ ✶✳✵✵✵

✵✳✻✵✵ ✵✳✻✵✵ ✵✳✻✷✺ ✵✳✻✼✻ ✵✳✼✸✸ ✵✳✼✾✽ ✵✳✽✽✼ ✶✳✵✵✵

✵✳✻✺✵ ✵✳✻✺✵ ✵✳✻✼✶ ✵✳✼✶✺ ✵✳✼✻✹ ✵✳✽✷✶ ✵✳✾✵✵ ✶✳✵✵✵

✵✳✼✵✵ ✵✳✼✵✵ ✵✳✼✶✼ ✵✳✼✺✺ ✵✳✼✾✻ ✵✳✽✹✺ ✵✳✾✶✸ ✶✳✵✵✵

✵✳✼✺✵ ✵✳✼✺✵ ✵✳✼✻✹ ✵✳✼✾✹ ✵✳✽✷✾ ✵✳✽✻✾ ✵✳✾✷✻ ✶✳✵✵✵

✵✳✽✵✵ ✵✳✽✵✵ ✵✳✽✶✶ ✵✳✽✸✺ ✵✳✽✻✷ ✵✳✽✾✹ ✵✳✾✹✵ ✶✳✵✵✵

✵✳✽✺✵ ✵✳✽✺✵ ✵✳✽✺✽ ✵✳✽✼✻ ✵✳✽✾✻ ✵✳✾✷✵ ✵✳✾✺✹ ✶✳✵✵✵

✵✳✾✵✵ ✵✳✾✵✵ ✵✳✾✵✺ ✵✳✾✶✼ ✵✳✾✸✵ ✵✳✾✹✻ ✵✳✾✻✾ ✶✳✵✵✵

✵✳✾✺✵ ✵✳✾✺✵ ✵✳✾✺✸ ✵✳✾✺✽ ✵✳✾✻✺ ✵✳✾✼✸ ✵✳✾✽✹ ✶✳✵✵✵

✶✳✵✵✵ ✶✳✵✵✵ ✶✳✵✵✵ ✶✳✵✵✵ ✶✳✵✵✵ ✶✳✵✵✵ ✶✳✵✵✵ ✶✳✵✵✵

❚❛❜❧❡ ✶✿ ❱❛❧✉❡s ♦❢p❢♦r ✜①❡❞z ❛♥❞α✳

Pr♦♦❢s

Pr♦♦❢ ♦❢ ❈♦r♦❧❧❛r② ✷✳✺

❚r✐✈✐❛❧✳

Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✸

✭✐❢ ♣❛rt✮ ❋✐① α ∈ (0,1)✱ z ∈ (¹₂,1) ❛♥❞ t❛❦❡ ❛♥ ✉♥❜✐❛s❡❞ ❥✉r♦r ✇✐t❤ q ∈ (q_α^min, q_α^max)✳ ❲✐t❤♦✉t ❧♦ss ♦❢

❣❡♥❡r❛❧✐t② t❛❦❡ t❤❡ s✐❣♥❛❧s ♣r♦✜❧❡s t1 = (g, g, g)❛♥❞ t2 = (i, i, i)✳ ❙✐♥❝❡ s❤❡ ♦❜s❡r✈❡s ❛❧❧ t❤❡ s✐❣♥❛❧s t❤❡

♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ ♣✐✈♦t❛❧ ✐s ♦♥❡ ❛♥❞ s♦ s❤❡ ❝♦♠♣❛r❡s ❤❡rq✇✐t❤ t❤❡s❡ t✇♦ ♣r♦❜❛❜✐❧✐t✐❡s Pr(●|t1= (g, g, g)) = α+ (1−α)z³

α+ (1−α)[z³+ (1−z)³] Pr(●|t2= (i, i, i)) = (1−α)(1−z)³

α+ (1−α)[z³+ (1−z)³]

✐t r❡s✉❧ts t❤❛tPr(●|t2)< q < Pr(●|t1) ❛♥❞ s♦ s❤❡ ❝❤♦♦s❡s t♦ ❝♦♥✈✐❝t t❤❡ ❞❡❢❡♥❞❛♥t ❛❢t❡r ♦❜s❡r✈✐♥❣t1

❛♥❞ ❛❝q✉✐t ❤✐♠ ❛❢t❡r ♦❜s❡r✈✐♥❣t2✳ ■❢ t❤❡r❡ ❡①✐st ❛ s✐❣♥❛❧ ♣r♦✜❧❡ˆt∈T s✉❝❤ t❤❛tPr(●|ˆt) =qt❤❡♥ t❤❡ ❥✉r♦r

❝❤♦♦s❡s t♦ r❛♥❞♦♠✐③❡ ❤❡r ❝❤♦✐❝❡✳

✭♦♥❧② ✐❢✮ ▲❡t✬s ❛ss✉♠❡ t❤❛tq < q^minα ❛♥❞ s✉♣♣♦s❡ t❤❛t t❤❡ ❥✉r♦r ✐s r❡s♣♦♥s✐✈❡✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ❢♦r ❛♥② s✐❣♥❛❧s ♣r♦✜❧❡st∈T t❤❡ ♣❛r❛♠❡t❡rq✐s s♠❛❧❧❡r t❤❛♥Pr(●|t)❛♥❞ s♦ t❤❡ ❥✉r♦r ❛❧✇❛②s ❝❤♦♦s❡s t♦ ❝♦♥✈✐❝t t❤❡

❞❡❢❡♥❞❛♥t✱ ✐♥❞❡♣❡♥❞❡♥t❧② ❜② t❤❡ s✐❣♥❛❧s ♣r♦✜❧❡ ❛♥❞ s♦ s❤❡ ✐s ♥♦t r❡s♣♦♥s✐✈❡✳ ❋♦rq > q^max_α t❤❡ r❡❛s♦♥✐♥❣

✐s s✐♠✐❧❛r✳

Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✹

❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥ ❛♥❞ ❛❢t❡r ♠❛❦✐♥❣ s✐♠♣❧❡ ❝❛❧❝✉❧❛t✐♦♥s ✐t r❡s✉❧ts t❤❛t Pr(●|piv, tj =g) =z ❛♥❞ Pr(●|piv, tj =i) = 1−z

✶✶