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▼✐s❛❧❧♦❝❛t✐♦♥ ❈♦sts ♦❢ ❉✐❣❣✐♥❣ ❉❡❡♣❡r ✐♥t♦ t❤❡ ❈❡♥tr❛❧ ❇❛♥❦ ❚♦♦❧❦✐t ❘♦❜❡rt ❑✉rt③♠❛♥ 1 ❛♥❞ ❉❛✈✐❞ ❩❡❦❡ 2 1 ❋❡❞❡r❛❧ ❘❡s❡r✈❡ ❇♦❛r❞ ♦❢ ●♦✈❡r♥♦rs 2 ❯♥✐✈❡rs✐t② ♦❢ ❙♦✉t❤❡r♥ ❈❛❧✐❢♦r♥✐❛ ◆♦✈❡♠❜❡r✱ ✷✵✶✼ ❉■❙❈▲❆■▼❊❘✿ ❚❤❡ ✈✐❡✇s ❡①♣r❡ss❡❞ ❛r❡ s♦❧❡❧② t❤❡ r❡s♣♦♥s✐❜✐❧✐t② ♦❢ t❤❡ ❛✉t❤♦rs ❛♥❞ s❤♦✉❧❞ ♥♦t ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s r❡✢❡❝t✐♥❣ t❤❡ ✈✐❡✇s ♦❢ t❤❡ ❇♦❛r❞ ♦❢ ●♦✈❡r♥♦rs ♦❢ t❤❡ ❋❡❞❡r❛❧ ❘❡s❡r✈❡ ❙②st❡♠ ♦r ♦❢ ❛♥②♦♥❡ ❡❧s❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❋❡❞❡r❛❧ ❘❡s❡r✈❡ ❙②st❡♠✳

▼♦t✐✈❛t✐♦♥ ▲❛r❣❡✲s❝❛❧❡ ❛ss❡t ♣✉r❝❤❛s❡s ✭▲❙❆Ps✮ ❜② ❋❡❞✱ ❊❈❇✱ ❡t❝✳ ◮ ●♦✈❡r♥♠❡♥t ❞❡❜t ✭▲❙●❇❇✮✱ ✐♥✈❡st♠❡♥t✲❣r❛❞❡ ❝♦r♣♦r❛t❡ ❞❡❜t ✭ ▲❙❈❇❇ ✮✱ ❊❚❋s✱ ▼❇❙✱ ♦t❤❡r s❡❝✉r✐t✐❡s ◮ ❆✈❡r❛❣❡ ❡✛❡❝ts✿ ▲❛r❣❡ ❡✛❡❝t ♦♥ s♣r❡❛❞s✱ ✐ss✉❛♥❝❡s ◮ ❘❡❧❛t✐✈❡ ❡✛❡❝ts✿ ❇✐❣❣❡r ❡✛❡❝ts ♦♥ ❡❧✐❣✐❜❧❡ ❛ss❡ts ❊①❛♠♣❧❡✿ ❊❈❇ ❈♦r♣♦r❛t❡ ❙❡❝✉r✐t② P✉r❝❤❛s❡ Pr♦❣r❛♠ ◮ P✉r❝❤❛s❡ ♦❢ ❊❯✲❛r❡❛ ✐♥✈❡st♠❡♥t ❣r❛❞❡ ❝♦r♣✳ ❜♦♥❞s ◮ ❙♣r❡❛❞s ♦❢ ♣✉r❝❤❛s❡❞ ❜♦♥❞s ❢❡❧❧ r❡❧❛t✐✈❡❧② ♠♦r❡ ❍❡t❡r♦❣❡♥❡♦✉s ❡✛❡❝ts ♦♥ s♣r❡❛❞s ◮ ❍❡t❡r♦❣❡♥❡♦✉s ❡✛❡❝t ♦♥ ❝♦st ♦❢ ❝❛♣✐t❛❧✱ ✐♥✈❡st♠❡♥t ◮ ■♠♣❧✐❝❛t✐♦♥s ❢♦r ❝❛♣✐t❛❧ ✭♠✐s✮❛❧❧♦❝❛t✐♦♥✦ ✶ ✴ ✷✻

❚❤✐s P❛♣❡r ▼✐s❛❧❧♦❝❛t✐♦♥ ✭♦r ✉♥❞♦✐♥❣✮ ❛ r❡s✉❧t ♦❢ ▲❙❈❇❇ ◮ ■♠♣❧✐❡s ❛ ♥♦✈❡❧ ▲❙❆P ♣♦❧✐❝② tr❛❞❡✲♦✛ ◮ ✏❉✐str✐❜✉t✐♦♥❛❧ ❡✛❡❝ts✑ ♦❢ ▼P ♦♥ ✜r♠s ◮ ❈♦st ♦❢ ❞✐❣❣✐♥❣ ❞❡❡♣❡r ✐♥t♦ ❝❡♥tr❛❧ ❜❛♥❦ t♦♦❧❦✐t ▼♦❞❡❧ ♦❢ ▲❙❆Ps ✇✐t❤ ❤❡t❡r♦❣❡♥❡♦✉s ✜r♠s ◮ ❋✐r♠s ❤❡t❡r♦❣❡♥❡♦✉s ✐♥ ♣r♦❞✉❝t✐♦♥ ✫ ✜♥❛♥❝✐♥❣ ◮ ■♥❝♦♠♣❧❡t❡ ✫ s❡❣♠❡♥t❡❞ ♠❛r❦❡ts ◮ ❈♦♠♣❛r❡ ▲❙❈❇❇ ✇✐t❤ ▲❙●❇❇ ❚✇♦ ❧❡✈❡❧s ♦❢ ❆♥❛❧②s✐s ◮ ❙t❛t✐❝ ♠♦❞❡❧ t♦ ❞❡♠♦♥str❛t❡ ♠❡❝❤❛♥✐s♠ ◮ ❋✉❧❧ ◆❑ ♠♦❞❡❧ ✷ ✴ ✷✻

❘❡s✉❧ts ❙t❛t✐❝ ♠♦❞❡❧ t♦ ❞❡♠♦♥str❛t❡ ♠❡❝❤❛♥✐s♠ ◮ ▲❙●❇❇ ❛✛❡❝ts s♣r❡❛❞s ❛♣♣r♦①✐♠❛t❡❧② ❡✈❡♥❧② ◮ ▲❙❈❇❇ ❛✛❡❝ts s♣r❡❛❞s ❤❡t❡r♦❣❡♥❡♦✉s❧② ◮ ❚❤✉s✱ ❞✐✛❡r❡♥t ✐♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ✭♠✐s✮❛❧❧♦❝❛t✐♦♥ ♦❢ ❝❛♣✐t❛❧ ◗✉❛♥t✐t❛t✐✈❡ ▼♦❞❡❧✿ ◆❡✇ ❑❡②♥❡s✐❛♥ ❉❙●❊ ♠♦❞❡❧ ◮ ❊♠❜❡❞ ✐♥ ◆❑ ♠♦❞❡❧✱ ❛❧❛ ●❡rt❧❡r ❛♥❞ ❑❛r❛❞✐ ✭✷✵✶✸✮ ◮ ❆❧❧ ▲❙❆Ps ❧♦✇❡r ❛✈❡r❛❣❡ s♣r❡❛❞s✱ r❛✐s❡ ✐♥✈❡st♠❡♥t ◮ ▲❙❈❇❇s ✐♥❞✉❝❡ ❤❡t❡r♦❣❡♥❡♦✉s ♠♦✈❡♠❡♥ts ✐♥ s♣r❡❛❞s ❛♥❞ ✐♥❞✉❝❡ q✉❛♥t✐t❛t✐✈❡❧② s✐❣♥✐✜❝❛♥t ♠✐s❛❧❧♦❝❛t✐♦♥ ❡✛❡❝t ❩❡r♦ ▲♦✇❡r ❇♦✉♥❞ ◮ ▼✐s❛❧❧♦❝❛t✐♦♥ ♥♦t ❛♠♣❧✐✜❡❞ ❜② ❩▲❇ ✭t♦ ✜rst✲♦r❞❡r✮ ◮ ▼✐s❛❧❧♦❝✳ ❡✛❡❝t ❃ ♦✉t♣✉t ❣❛✐♥ ❛✇❛② ❢r♦♠ ❩▲❇ ◮ ▼✐s❛❧❧♦❝✳ ❡✛❡❝t ❁ ♦✉t♣✉t ❣❛✐♥ ❛t ❩▲❇ ✸ ✴ ✷✻

▲✐t❡r❛t✉r❡ ❉❙●❊ ♠♦❞❡❧s ✇✐t❤ ✐♠♣❡r❢❡❝t ❛ss❡t✲s✉❜st✐t✉t❛❜✐❧✐t② ❛♥❞ ✉♥❝♦♥✈❡♥t✐♦♥❛❧ ♠♦♥❡t❛r② ♣♦❧✐❝② ◮ ●❡rt❧❡r ✫ ❑❛r❛❞✐ ✭✷✵✶✶✴✷✵✶✸✮❀ ❈✉r❞✐❛ ❛♥❞ ❲♦♦❞❢♦r❞ ✭✷✵✶✺✮❀ ❍❡ ❛♥❞ ❑r✐s❤♥❛♠✉rt❤② ✭✷✵✶✸✮ ▼✐s❛❧❧♦❝❛t✐♦♥✱ ✜♥❛♥❝✐❛❧ ❝♦♥str❛✐♥ts✴❜♦rr♦✇✐♥❣ ❝♦sts ◮ ▼✐❞r✐❣❛♥ ✫ ❳✉ ✭✷✵✶✹✮❀ ●✐❧❝❤r✐st✱ ❙✐♠✱ ❩❛❦r❛❥s❡❦ ✭✷✵✶✸✮❀ ❉❛✈✐❞✱ ❙❝❤♠✐❞✱ ❩❡❦❡ ✭✷✵✶✼✮ ❊♠♣✐r✐❝❛❧ ❡✈✐❞❡♥❝❡ ♦❢ ❤❡t❡r♦❣❡♥❡♦✉s ❡✛❡❝ts ♦❢ ▲❙❆Ps ◮ ❑r✐s❤♥❛♠✉rt❤② ❛♥❞ ❱✐ss✐♥❣✲❏♦r❣❡♥s❡♥ ✭✷✵✶✶✮❀ ❋♦❧❡②✲❋✐s❤❡r✱ ❘❛♠❝❤❛r❛♥✱ ❨✉ ✭✷✵✶✻✮❀ ❉❛r♠♦✉r✐ ✫ ❘♦❞♥②❛♥s❦② ✭✷✵✶✻✮❀ ❉✐ ▼❛❣❣✐♦✱ ❑❡r♠❛♥✐✱ ❛♥❞ P❛❧♠❡r ✭✷✵✶✻✮❀ ❑✉rt③♠❛♥✱ ▲✉❝❦✱ ❛♥❞ ❩✐♠♠❡r♠❛♥♥ ✭✷✵✶✼✮ ✹ ✴ ✷✻

❙t❛t✐❝ ▼♦❞❡❧ ❈♦♥s✉♠♣t✐♦♥ ❛♥❞ Pr♦❞✉❝t✐♦♥ � 1 �� ρ − � y ρ ♠❛① i di p i y i di y i � y i � ρ − 1 ❨✐❡❧❞s p i = Y ❋✐♥❛❧ ❣♦♦❞ ✭♥✉♠❡r❛✐r❡✮ ❝❛♥ ❜❡ ❝♦♥s✉♠❡❞ ♦r ✉s❡❞ ❢♦r ❝❛♣✐t❛❧ ◮ Pr♦❞✉❝✐♥❣ K ❝❛♣✐t❛❧ ✉s❡s h k K bk +1 ✜♥❛❧ ❣♦♦❞s b k +1 C = Y − h k K bk +1 b k +1 ❈♦♠♣❡t✐t✐✈❡ s❡❝t♦r ❝♦♥✈❡rts ✜♥❛❧ ❣♦♦❞ t♦ K K QK − h k K bk +1 ◮ ♠❛① b k +1 ⋆ Q = h k K b k ✺ ✴ ✷✻

■♥t❡r♠❡❞✐❛t❡ ●♦♦❞ ❋✐r♠ ❚❡❝❤♥♦❧♦❣② y i = A i k α i ▼❛①✐♠✐③❡s r❡✈❡♥✉❡s ❧❡ss s♣❡♥❞✐♥❣ ♦♥ ❝❛♣✐t❛❧ ◮ ▼✉st ✜♥❛♥❝❡ ❝❛♣✐t❛❧✱ ❣r♦ss ✐♥t❡r❡st r❛t❡ r i ◮ ❆❧s♦ ❛❧❧♦✇ ❢♦r ❛♥ ❡①♦❣❡♥♦✉s ✇❡❞❣❡✱ τ k i p i y i ◮ ♠❛① − r i Qk i τ k k i i ❋❖❈✿ � αρY 1 − ρ A ρ 1 � 1 − αρ ◮ k i = i τ k i r i Q ❚❤❡r❡ ❛r❡ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❣r♦✉♣s ♦❢ ✜r♠s✱ ✐♥❞❡①❡❞ ❜② j ◮ ❊✈❡r② ✜r♠ i ❜❡❧♦♥❣s ❡①❛❝t❧② t♦ ♦♥❡ ❣r♦✉♣ j ✻ ✴ ✷✻

❋✐♥❛♥❝✐❛❧ ■♥t❡r♠❡❞✐❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ✐♥t❡r♠❡❞✐❛r✐❡s ♦r ❝❡♥tr❛❧ ❜❛♥❦ ✜♥❛♥❝❡ ✜r♠s ◮ S b,i + S g,i = Qk i ●♦✈❡r♥♠❡♥t ❜♦♥❞s ✐♥ ♣♦s✐t✐✈❡ ♥❡t s✉♣♣❧②✿ B S = B b + B g ▼❛①✐♠✐③❡s ✈❛❧✉❡ s✉❜❥❡❝t t♦ r❡❣✉❧❛t♦r② ❝♦❧❧❛t❡r❛❧ ❝♦♥str❛✐♥t � � � � � � ♠❛① S b,i r i di + B b r b + N − S b,i di − B b r S b,i ,B b i ∈ j i ∈ j j j �� � ν j � s✳t✳ V ≥ θ ∆ j + θB b S b,i di i ∈ j j ✼ ✴ ✷✻

❊♥❞♦❣❡♥♦✉s ❙♣r❡❛❞s ❢r♦♠ ▲❙❆Ps λ ✐s ▲❛❣r❛♥❣✐❛♥ ▼✉❧t✐♣❧✐❡r λ r b − r = 1+ λ θ � S g,j = i ∈ j S g,i di �� � ν j − 1 λ r i − r = 1+ λ θ ∆ j ν j s ∈ j Qk s − S g,s ❊q✉✐❧✐❜r✐✉♠ ✽ ✴ ✷✻

▲❙●❇❇✴▲❙❈❇❇✿ ❉✐r❡❝t ❊✛❡❝ts ♦♥ ❙♣r❡❛❞s ❍♦❧❞✐♥❣ ✜r♠ ✐♥✈❡st♠❡♥t ❞❡❝✐s✐♦♥s✱ k i ✱ ✜①❡❞✿ ▲❙●❇❇✱ B g ↑ ◮ ▲♦♦s❡♥s ❝♦❧❧❛t❡r❛❧ ❝♦♥str❛✐♥t ❛♥❞ ♠✉❧t✐♣❧✐❡r λ 1+ λ ◮ Pr♦♣♦rt✐♦♥❛❧❧② ❞❡❝r❡❛s❡s ✜r♠ s♣r❡❛❞s ▲❙❈❇❇✱ S g,i ↑ λ ◮ ▲♦♦s❡♥s ❝♦❧❧❛t❡r❛❧ ❝♦♥str❛✐♥t ❛♥❞ ♠✉❧t✐♣❧✐❡r 1+ λ ◮ ❉✐r❡❝t❧② ❧♦✇❡rs s♣r❡❛❞ ❢♦r ✜r♠s i ∈ j ❜✉t ♥♦t i / ∈ j ✭✉♥❞❡r ❝♦♥❞✐t✐♦♥ t❤❛t ❜❛♥❦s ❞♦ ♥♦t ❤♦❧❞ t♦♦ ♠✉❝❤ ❣♦✈❡r♥♠❡♥t ❞❡❜t r❡❧❛t✐✈❡ t♦ t❤❡✐r ✇❡❛❧t❤✮ ✾ ✴ ✷✻