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■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

❆♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦ ▼✳ ❍❡✐♥③❡r✶ ❊✳ Pr♦❢✉♠♦✶

✶❉❡♣❛r♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s

❊❚❍ ❩ür✐❝❤

❙❡♠✐♥❛r ✐♥ ❙t❛t✐st✐❝s✿ ▲❡❛r♥✐♥❣ ❇❧❛❝❦❥❛❝❦✱ ❆♣r✐❧ ✷✵✶✻

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

• ♦❛❧s

◮ ❆s ❜❧❛❝❦❥❛❝❦ ♣❧❛②❡rs ✇❡ ✇❛♥t t♦ ✜♥❞ ❛♥ ♦♣t✐♠❛❧ str❛t❡❣②

t❤❛t ✐s ❛ ✇❛② ♦❢ s❡❧❡❝t✐♥❣ t❤❡ ❜❡st ❛❝t✐♦♥ ✐♥ ❡❛❝❤ st❛t❡ ♦❢ t❤❡ ❣❛♠❡✱ t❤❡ ♦♥❡ ✇❤✐❝❤ ✇✐❧❧ ♠❛①✐♠✐③❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ✉s ✇✐♥♥✐♥❣✳

◮ ❲❡ ✇✐❧❧ tr② t♦ ❧❡❛r♥ t❤❡ ♦♣t✐♠❛❧ str❛t❡❣② ♣r❡s❡♥t❡❞ ✐♥ ❛♥ ❡❛r❧✐❡r

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

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼❛t❤❡♠❛t✐❝❛❧ s❡tt✐♥❣

❲❡ ♣✐❝❦ ❛ t❤❡♦r❡t✐❝❛❧ ❢r❛♠❡✇♦r❦ ❢r♦♠ ❞❡❝✐s✐♦♥ t❤❡♦r②✳ ❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛ ❣❛♠❡ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ ✜♥✐t❡ ▼❛r❦♦✈ ❉❡❝✐s✐♦♥ Pr♦❝❡ss ✭▼❉P✮✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼❛t❤❡♠❛t✐❝❛❧ s❡tt✐♥❣

◆♦t❛t✐♦♥ ✿

◮ t ∈ {✶, ✷, ...Ti} ❞❡s❝r✐❜❡s t❤❡ ❞✐✛❡r❡♥t st❡♣s ♦❢ t❤❡ ❡♣✐s♦❞❡ i

✭✇❡ ✇✐❧❧ ❞r♦♣ i ❢♦r ❝❧❛r✐t②✮✳

◮ (St)t∈{✶,✷,...T} t❤❡ ♣r♦❝❡ss ♦❢ ❞✐✛❡r❡♥t st❛t❡s ♦❢ t❤❡ ❣❛♠❡✳ ◮ (St, at)t∈{✶,✷,...T} t❤❡ st❛t❡✲❛❝t✐♦♥ ♣❛✐rs✳ ❚❤❡ ❛❝t✐♦♥s ✇❤✐❝❤

❝❛♥ ❜❡ t❛❦❡♥ ❞❡♣❡♥❞ ♦♥ t❤❡ ❝✉rr❡♥t st❛t❡

◮ (Rt)t∈{✶,✷,...T} t❤❡ ♣r♦❝❡ss ♦❢ r❡✇❛r❞s ❢♦❧❧♦✇✐♥❣ ❛ tr✐♣❧❡

✭st❛t❡✱❛❝t✐♦♥✱r❡s✉❧t✐♥❣ st❛t❡✮✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼❛t❤❡♠❛t✐❝❛❧ s❡tt✐♥❣

❆t ❡❛❝❤ t✐♠❡ st❡♣ t t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❝❡ss ❢♦❧❧♦✇s ✶✳ ❲❡ ❛r❡ ✐♥ s♦♠❡ st❛t❡ St = st ✷✳ ❚❤❡ ♣❧❛②❡r t❛❦❡s ❛♥ ❛❝t✐♦♥ at ❢r♦♠ St✳ ✸✳ t❤❡ st❛t❡✲❛❝t✐♦♥ ♣❛✐r (St = st, at) ❧❡❛❞s t♦ ❛ r❛♥❞♦♠ ❢♦❧❧♦✇✐♥❣ st❛t❡ St+✶|st, at✳ ✹✳ t❤❡ tr✐♣❧❡ (st, at, st+✶) ❧❡❛❞s t♦ ❛ r❡✇❛r❞ Rt+✶✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❚❤❡ ❛❣❡♥t ❛♥❞ t❤❡ ❡♥✈✐r♦♥♠❡♥t

❲❡ ❤❛✈❡ ❛♥ ❛❣❡♥t ✇❤✐❝❤ ❝❤♦♦s❡s ❛❝t✐♦♥s ✐♥ ❛♥ ❡♥✈✐r♦♥♠❡♥t✳ ✶✳ ❆♥ ❛❣❡♥t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❞❡❝✐s✐♦♥ ♣r♦❝❡ss ✇❤✐❝❤ ❝❤♦♦s❡s ❛♥ ❛❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝✉rr❡♥t st❛t❡✱ ✇❡ ❝❛❧❧ ✐t ❛ ♣♦❧✐❝② ❢✉♥❝t✐♦♥ π✳ ■t ❝❛♥ ❜❡ ❞❡t❡r♠✐♥✐st✐❝✱ ❧✐❦❡ ❛ ❣r❡❡❞② ♣♦❧✐❝② ♦r st♦❝❤❛st✐❝✳ π(a|s) ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❛❦✐♥❣ ❛❝t✐♦♥ a ❜❡✐♥❣ ✐♥ st❛t❡ s✳ ✷✳ ❆❝t✐♦♥s ❛r❡ ♠♦t✐✈❛t❡❞ ❜② r❡✇❛r❞s✳ ❖✉r ✐♥t❡r❡st ✐s t♦ ♠❛①✐♠✐③❡ t❤❡ ❡①♣❡❝t❡❞ ❛❣❣r❡❣❛t❡❞ r❡✇❛r❞ E ∞

• t=✵

Rt

• ▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦

❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❇❡st ♣♦❧✐❝②✱ st❛t❡ ❛♥❞ ❛❝t✐♦♥✲st❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✳

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

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❇❡st ♣♦❧✐❝②✱ st❛t❡ ❛♥❞ ❛❝t✐♦♥✲st❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✳

❙t❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✿ vπ(s) = Eπ ∞

• i=✵

Rt+i+✶

• St = s
• ❑♥♦✇✐♥❣ t❤❛t ■ ❛♠ ✐♥ st❛t❡ s ✇❤✐❝❤ r❡✇❛r❞ ❝❛♥ ■ ❡①♣❡❝t ❢♦❧❧♦✇✐♥❣

♣♦❧✐❝② π✳

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

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

◮ ❇❡st ♣♦❧✐❝✐❡s s❤❛r❡ t❤❡ s❛♠❡ ♦♣t✐♠❛❧ st❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥

v⋆(s) = ♠❛①

π

vπ(s) ∀s ∈ S✱ ✇❡ ❞❡♥♦t❡ t❤❡♠ ❜② π⋆✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❇❡st ♣♦❧✐❝②✱ st❛t❡ ❛♥❞ ❛❝t✐♦♥✲st❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✳

■♥ ♦r❞❡r t♦ ❝♦♥str✉❝t ♣♦❧✐❝✐❡s ✇✐t❤ ❜❡tt❡r st❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✱ ✇❡ ♥❡❡❞ ❛ t♦♦❧ t❤❛t t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t ❛❝t✐♦♥s✳ ❲❡ ❝❛❧❧ ✐t ❆❝t✐♦♥✲❙t❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✿ qπ(s, a) = Eπ ∞

• i=✵

Rt+i+✶

• St = s, At = a
• ❑♥♦✇✐♥❣ t❤❛t ■ ❛♠ ✐♥ st❛t❡ s✱ ✇❤✐❝❤ r❡✇❛r❞ ❝❛♥ ■ ❡①♣❡❝t ❜② t❛❦✐♥❣

❛❝t✐♦♥ a ✉♥❞❡r ♣♦❧✐❝② π✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

• ❡♥❡r❛❧ ✐❞❡❛

❙♦ ❢❛r ✇❡ ❤❛✈❡ s❡❡♥ ❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣✱ ✇❤✐❝❤ ✇♦r❦❡❞ ❛s ❢♦❧❧♦✇s ✿

◮ ❈♦♠♣✉t❡ t❤❡ ✈❛❧✉❡ ♦❢ ❡❛❝❤ st❛t❡ ❢r♦♠ t❤❡ ❝♦♠♣❧❡t❡ ❦♥♦✇❧❡❞❣❡

♦❢ t❤❡ ❡♥✈✐r♦♥♠❡♥t✳ ❲❡ ❤❛❞ t♦ s♦❧✈❡ ❡q✉❛t✐♦♥s ❧✐❦❡ t❤✐s ✿ vπ(s) =

• a

π(a|s)

• s′,r

p(s′, r|s, a)[r + γvπ(s′)] ◆♦✇ ✇❡ ✇✐❧❧ s❡❡ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞s✱ ✇❤✐❝❤ ❧❡❛r♥ t❤❡ ✈❛❧✉❡ ♦❢ ❡❛❝❤ st❛t❡ ❢r♦♠ s❛♠♣❧❡ r❡t✉r♥s✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

• ❡♥❡r❛❧ ✐❞❡❛

❲❤❛t ✐s t❤❡ str✉❝t✉r❡ ♦❢ ❛ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❄

◮ ❋✐rst❧② ✇❡ ❣❡♥❡r❛t❡ ❡①♣❡r✐❡♥❝❡✳ ●♦ t❤r♦✉❣❤ ❛ s❡q✉❡♥❝❡ ♦❢

❛❝t✐♦♥s ❛♥❞ st❛t❡s ✉♥t✐❧ ✇❡ ❛rr✐✈❡ ❛t ❛ t❡r♠✐♥❛❧ st❛t❡✳ ❲❡ ❝❛❧❧ t❤✐s s❡q✉❡♥❝❡ ❛♥ ❡♣✐s♦❞❡✱ ✐♥ t❤✐s t❛❧❦ ♦♥❧② ✜♥✐t❡ ❡♣✐s♦❞❡s ✇✐❧❧ ❜❡ ❝♦♥s✐❞❡r❡❞✳ ❊①❛♠♣❧❡ ✿ ❚❤❡ r❡✇❛r❞ ✐s ♦♥❧② ❣✐✈❡♥ ❛t t❤❡ ❡♥❞ ♦❢ ❛♥ ❡♣✐s♦❞❡✳ ❚❤❡ ✈❛❧✉❡ ♦❢ ❡❛❝❤ st❛t❡ ✇❡ ❤❛✈❡ ❣♦♥❡ t❤r♦✉❣❤ ✇✐❧❧ ❜❡ ❛❞❛♣t❡❞ ❛❝❝♦r❞✐♥❣❧②✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

• ❡♥❡r❛❧ ✐❞❡❛

❲❤❛t ✐s t❤❡ str✉❝t✉r❡ ♦❢ ❛ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❄

◮ ❙❡❝♦♥❞❧② ✇❡ ✉♣❞❛t❡ t❤❡ ♣♦❧✐❝② ❛❝❝♦r❞✐♥❣ t♦ ♦✉r ❡①♣❡r✐❡♥❝❡ ❢♦r

❡❛❝❤ st❛t❡✳ ■t ✇✐❧❧ ♦♥❧② ❜❡ ✉♣❞❛t❡❞ ❛❢t❡r t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ ♦♥❡ ♦r ♠♦r❡ ❡♣✐s♦❞❡s✳ ❊①❛♠♣❧❡ ✿ ❙t❛t❡ ✲ ❆❝t✐♦♥ ✷ ✺ ✻ ✼ ✽ ✾ ✲✶✵✶✲✶✵✵✵✵✵ ✲✶ ✲✶ ✲✶ ✵✳✹ ✲✶ ✲✶ ❖✉r ♣♦❧✐❝② ✇♦✉❧❞ ❜❡ ✿ ❲❤❡♥ ✐♥ st❛t❡ ✲✶✵✶✲✶✵✵✵✵✵ s❡❧❡❝t ❛❝t✐♦♥ ✼✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ✿ ❊st✐♠❛t✐♥❣ ✈❛❧✉❡s

❚❤❡ ✜rst st❡♣ ✐s t♦ ❡st✐♠❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ ❡❛❝❤ st❛t❡ ❣✐✈❡♥ ❛ ♣♦❧✐❝② π✳ ❍♦✇ ❞♦ ✇❡ ❝❛❧❝✉❧❛t❡ vπ(s) ❄ ❚✇♦ ♣♦ss✐❜✐❧✐t✐❡s ✿

◮ ❋✐rst✲❱✐s✐t ▼❈ ◮ ❊✈❡r②✲❱✐s✐t ▼❈

❊①❛♠♣❧❡ ✿ ❋♦r♠ ❤❡r❡ ♦♥ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ❋✐rst✲❱✐s✐t ▼❈

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ✿ ❊st✐♠❛t✐♥❣ ✈❛❧✉❡s

❋✐rst✲✈✐s✐t ▼❈ ❛✈❡r❛❣❡s t❤❡ r❡t✉r♥s ❢♦❧❧♦✇✐♥❣ ❛ ✈✐s✐t t♦ ❛ st❛t❡ s ✐♥ ❛❧❧ ❡♣✐s♦❞❡s✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ n ❡♣✐s♦❞❡s ❛♥❞ ❧❡t N(s) ❜❡ t❤❡ ❡♥✉♠❡r❛t✐♦♥ ♦❢ ❡♣✐s♦❞❡s ✇❤✐❝❤ ✈✐s✐t❡❞ s✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ N(s) ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s ♣r♦❝❡ss ❝♦✉❧❞ ❜❡ ✿ N(A) = {✶, ✷, ✸}✱ N(B) = {✶, ✸}

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ✿ ❊st✐♠❛t✐♥❣ ✈❛❧✉❡s

❋✐rst✲✈✐s✐t ▼❈ ❛✈❡r❛❣❡s t❤❡ r❡t✉r♥s ❢♦❧❧♦✇✐♥❣ ❛ ✈✐s✐t t♦ ❛ st❛t❡ s ✐♥ ❛❧❧ ❡♣✐s♦❞❡s✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ n ❡♣✐s♦❞❡s ❛♥❞ ❧❡t N(s) ❜❡ t❤❡ ❡♥✉♠❡r❛t✐♦♥ ♦❢ ❡♣✐s♦❞❡s ❞✉r✐♥❣ ✇❤✐❝❤ s ✇❛s ✈✐s✐t❡❞✳ ❚❤❡♥ ✇❡ ❞❡✜♥❡ ˆ vπ(s) ❛s ❢♦❧❧♦✇s ✿ ˆ vπ(s) = ✶ |N(s)|

n

• i=✶

RiIi∈N(s) ❊❛❝❤ r❡t✉r♥ ✐s ❛♥ ✐✳✐✳❞✳ ❡st✐♠❛t❡ ♦❢ t❤❡ tr✉❡ ✈❛❧✉❡ ♦❢ vπ(s)✳❚❤❡ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡r ❣✐✈❡s ✉s ❝♦♥✈❡r❣❡♥❝❡ t♦ t❤❡ ❡①♣❡❝t❛t✐♦♥✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ✿ ❊st✐♠❛t✐♥❣ ✈❛❧✉❡s

❙✉♣♣♦s❡ ✇❡ ❣✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡✇❛r❞s ❢♦r ❚✐❝ ❚❛❝ ❚♦❡ ✿

◮ ✶ ✐❢ ✇❡ ✇✐♥ ◮ ✵ ❢♦r ❛ ❞r❛✇ ◮ −✶ ✐❢ ✇❡ ❧♦s❡

■♥ ♦✉r t♦② ❡①❛♠♣❧❡ ❚✐❝ ❚❛❝ ❚♦❡ ✇❡ ❝♦✉❧❞ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣❡r✐❡♥❝❡ ✿ ❆❢t❡r t❤♦s❡ ❡①♣❡r✐❡♥❝❡s ✇❡ ❤❛✈❡ ˆ vπ(s) = − ✶

✹

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ✿ ❊st✐♠❛t✐♥❣ ✈❛❧✉❡s

❖t❤❡r r❡♠❛r❦s ✿

◮ ❆❧❧ ♦❢ t❤✐s ✐s ❞♦♥❡ ❢♦r ❛ ❣✐✈❡♥ ♣♦❧✐❝② ◮ ❊st✐♠❛t❡s ❢♦r ❡❛❝❤ st❛t❡ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✇❡

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

◮ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✈❛❧✉❡ ♦❢ ♦♥❡ st❛t❡ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡

♥✉♠❜❡r ♦❢ st❛t❡s

◮ ❆ss✉♠♣t✐♦♥ ♦❢ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❡♣✐s♦❞❡s ✦

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ✿ ❊st✐♠❛t✐♥❣ ✈❛❧✉❡s

❚❤❡ ✇❤♦❧❡ ❛♣♣r♦❛❝❤ ❝❛♥ ❛❧s♦ ❜❡ ✉s❡❞ t♦ ❛♣♣r♦①✐♠❛t❡ ❛❝t✐♦♥ ✈❛❧✉❡s✳ ❲❡ ❥✉st ❝♦♥s✐❞❡r st❛t❡ ❛❝t✐♦♥ ♣❛✐rs ❛♥❞ ❡st✐♠❛t❡ t❤❡✐r ✈❛❧✉❡s✳ ❚❤✐s ✐s ♥❡❡❞❡❞ t♦ ✐♠♣r♦✈❡ t❤❡ ♣♦❧✐❝②✳ ❋♦r♠❛❧❧② ✿ ˆ qπ(s, a) = ✶ |N(s, a)|

n

• i=✶

RiIi∈N(s,a)

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ✿ ❊st✐♠❛t✐♥❣ ✈❛❧✉❡s

❲❡ ♥❡❡❞ ❛♥♦t❤❡r ❛ss✉♠♣t✐♦♥ ✐♥ ♦r❞❡r t♦ ❡st✐♠❛t❡ ❛❧❧ ♣❛✐rs ♦❢ st❛t❡✲❛❝t✐♦♥ ✈❛❧✉❡s✱ ❛s ♠❛♥② ❝♦♠❜✐♥❛t✐♦♥s ♠❛② ♥❡✈❡r ❜❡ ✈✐s✐t❡❞✳ Pr♦❜❧❡♠ ♦❢ ♠❛✐♥t❛✐♥✐♥❣ ❡①♣❧♦r❛t✐♦♥✱ ❛s ✇✐t❤ n✲❛r♠❡❞ ❜❛♥❞✐ts✳ ❚✇♦ ♣♦ss✐❜✐❧✐t✐❡s ✿

◮ ❊✈❡r② ♣❛✐r ❤❛s ♥♦♥✲③❡r♦ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ s❡❧❡❝t❡❞ ❛s st❛rt✳

❲❡ ❝❛❧❧ t❤✐s ❊①♣❧♦r✐♥❣ ❙t❛rts✳

◮ ❯s❡ st♦❝❤❛st✐❝ ♣♦❧✐❝✐❡s ✇❤✐❝❤ ❤❛✈❡ ❛ ♥♦♥✲③❡r♦ ♣r♦❜❛❜✐❧✐t② ♦❢

s❡❧❡❝t✐♥❣ ❛❧❧ ❛✈❛✐❧❛❜❧❡ ❛❝t✐♦♥s ✐♥ ❡❛❝❤ st❛t❡✳ ◗ ✿ ❉♦ ✇❡ ❤❛✈❡ ❡①♣❧♦r✐♥❣ st❛rts ✐♥ ❚✐❝ ❚❛❝ ❚♦❡ ❄ ◗ ✿ ❉♦ ✇❡ ❤❛✈❡ ❡①♣❧♦r✐♥❣ st❛rts ✐♥ ❇❧❛❝❦❥❛❝❦ ❄

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❙♦ ❢❛r ✇❡ ❤❛✈❡ ♦♥❧② ❝♦♥s✐❞❡r❡❞ ❤♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✳ ❇✉t ✇❤❛t ❛❜♦✉t t❤❡ ♦t❤❡r ❞✐r❡❝t✐♦♥ ❄

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❙♦ ❢❛r ✇❡ ❤❛✈❡ ♦♥❧② ❝♦♥s✐❞❡r❡❞ ❤♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✳ ◆♦✇ ✇❡ ✇❛♥t t♦ ❛♣♣r♦①✐♠❛t❡ ♦♣t✐♠❛❧ ♣♦❧✐❝✐❡s ❛s ✇❡❧❧✳ ❚❤❡ ✐❞❡❛ ✐s t❤❡ s❛♠❡ ❛s ✐♥ ●❡♥❡r❛❧✐③❡❞ P♦❧✐❝② ■t❡r❛t✐♦♥ ✭●P■✮ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣r❡s❡♥t❛t✐♦♥ ❛❜♦✉t ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❙♦ ❢❛r ✇❡ ❤❛✈❡ ♦♥❧② ❝♦♥s✐❞❡r❡❞ ❤♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✳ ❲❡ st❛rt ✇✐t❤ ❛♥ ❛r❜✐tr❛r② ♣♦❧✐❝② π✵✳ ❆t ❡❛❝❤ st❡♣ ✇❡ ❡✈❛❧✉❛t❡ t❤❡ st❛t❡✲❛❝t✐♦♥ ❢✉♥❝t✐♦♥ qπi✱ ❛♥❞ s❡❧❡❝t ❛s ♥❡✇ ♣♦❧✐❝② πi+✶ t❤❡ ❣r❡❡❞② ♣♦❧✐❝② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ qπi ✿ πi+✶(s) = ❛r❣ ♠❛①

a

qi(s, a)

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

◗ ✿ ❉♦ ✇❡ ❛❝t✉❛❧❧② ❝♦♥✈❡r❣❡ t♦ t❤❡ ♦♣t✐♠❛❧ ♣♦❧✐❝② ❄ ❆ ✿ ❨❡s ✦ ✭❇❡ ❛✇❛r❡ ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥s✮

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❈♦♥✈❡r❣❡♥❝❡ ✐s ❣✉❛r❛♥t❡❡❞ ❜② t❤❡ ♣♦❧✐❝② ✐♠♣r♦✈❡♠❡♥t t❤❡♦r❡♠✳ ❲❡ ♦♥❧② ♥❡❡❞ t♦ ✈❡r✐❢② t❤❛t t❤❡ ♥❡✇ ♣♦❧✐❝② ✐s ✉♥✐❢♦r♠❧② ❜❡tt❡r✳ ❲❡ ❤❛✈❡ ❢♦r ❛❧❧ πk✱ πk+✶ ❛♥❞ s ∈ S ✿ qπi(s, πi+✶(s)) = qπi(s, ❛r❣ ♠❛①

a

qπi(s, a)) = ♠❛①

a

qπi(s, a) ≥ qπi(s, πi(s)) = vπi(s) ❚❤❡r❡❢♦r❡ ❜② t❤❡ ♣♦❧✐❝② ✐♠♣r♦✈❡♠❡♥t t❤❡♦r❡♠ ✿ qπi+✶ ≥ qπi ◗ ✿ ❉♦ ✇❡ ❤❛✈❡ s♦♠❡t❤✐♥❣ ✉s❡❢✉❧ ❄

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❖✉r ❛ss✉♠♣t✐♦♥s ♦❢ ❡①♣❧♦r✐♥❣ st❛rts ❛♥❞ ✐♥✜♥✐t❡❧② ♠❛♥② ❡♣✐s♦❞❡s ❤❛✈❡ t♦ ❜❡ r❡♠♦✈❡❞✳ ❚❤❡ ❧❛tt❡r ✐s ❡❛s② t♦ r❡♠♦✈❡✳ ❲❡ ❝❛♥ ✿

◮ ❈♦❧❧❡❝t ❡♥♦✉❣❤ ❡♣✐s♦❞❡s ✉♥t✐❧ t❤❡ ♠❛r❣✐♥ ♦❢ ❡rr♦r ✐s s♠❛❧❧

❡♥♦✉❣❤✱ ❝❛♥ st✐❧❧ ❜❡ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✐♥t❡♥s✐✈❡

◮ ❯♣❞❛t❡ t❤❡ ♣♦❧✐❝② ❛❢t❡r ❡❛❝❤ ❡♣✐s♦❞❡✱

❲❡ ♠♦✈❡ ♦✉r ✈❛❧✉❡ ❢✉♥❝t✐♦♥ t♦✇❛r❞s t❤❡ r❡❛❧ ♦♥❡ ✿ ◗ ✿ ❉♦❡s ✐t st✐❧❧ ❝♦♥✈❡r❣❡ ❄ ❖♣❡♥ ♣r♦❜❧❡♠ ✦

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❍♦✇ ❞♦ ✇❡ t❛❦❡ ❛✇❛② t❤❡ ❊①♣❧♦r✐♥❣ ❙t❛rts ❛ss✉♠♣t✐♦♥ ❄ ❚✇♦ ♠❛✐♥ ✐❞❡❛s ✿

◮ ❖♥✲♣♦❧✐❝② ♠❡t❤♦❞s ✿

❙❡❧❡❝t t❤❡ ♣♦❧✐❝② ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t π(a|s) > ✵ ∀s ∈ S, a ∈ A(s)✳ ❲❡ ❝❛❧❧ t❤✐s ❛ s♦❢t ♣♦❧✐❝②✳ ❊①❛♠♣❧❡ ✿ ǫ✲❣r❡❡❞②

◮ ❖✛✲♣♦❧✐❝② ♠❡t❤♦❞s ✿

▼❛✐♥t❛✐♥ t✇♦ ♣♦❧✐❝✐❡s✱ ♦♥❡ ❢♦r ❡①♣❧♦r✐♥❣ ❛♥❞ ❛♥♦t❤❡r ❢♦r ♦♣t✐♠✐③❛t✐♦♥✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧ ✲ ♦♥ ♣♦❧✐❝②

❲❡ ✇✐❧❧ ❝❤❡❝❦ ✐❢ ♦♥ ♣♦❧✐❝② ❧❡❛r♥✐♥❣ st✐❧❧ ❢✉❧✜❧❧s t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ ♣♦❧✐❝② ✐♠♣r♦✈❡♠❡♥t t❤❡♦r❡♠✳ ❋✐rst ❧❡t ✉s ❝♦♥s✐❞❡r ❛♥ ǫ✲❣r❡❡❞② ♣♦❧✐❝② ❛s ❢♦❧❧♦✇s ✿ π(a|s) =

• ǫ

|A(s)|

❢♦r t❤❡ ♥♦♥✲❣r❡❡❞② ❛❝t✐♦♥ ✶ − ǫ +

ǫ |A(s)|

❢♦r t❤❡ ❣r❡❡❞② ❛❝t✐♦♥

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧ ✲ ♦♥ ♣♦❧✐❝②

❲❡ ❤❛✈❡ ❢♦r ❛♥② s ∈ S ✿ qπ(s, π′(s)) =

• a

π′(a|s)qπ(s, a) = ǫ |A(s)|

• a

qπ(s, a) + (✶ − ǫ) ♠❛①

a

qπ(s, a) ≥ ǫ |A(s)|

• a

qπ(s, a) + (✶ − ǫ)

• a

π(a|s) −

ǫ |A(s)|

✶ − ǫ qπ(s, a) = ǫ |A(s)|

a

qπ(s, a) −

• a

qπ(s, a)

• +
• a

π(a|s)qπ(s, a) = vπ(v) ❚❤✉s π′ ≥ π ❲❡ ❤❛✈❡ t❤✉s s❤♦✇♥ t❤❛t ♣♦❧✐❝② ✐t❡r❛t✐♦♥ ✇♦r❦s ❢♦r ǫ✲s♦❢t ♣♦❧✐❝✐❡s✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧ ✲ ♦✛ ♣♦❧✐❝②

❙✉♣♣♦s❡ ✇❡ ✇❛♥t t♦ ❡st✐♠❛t❡ vπ ♦r qπ ♦❢ ❛ ♣♦❧✐❝② π ❜✉t ✇❡ ❝❛♥✬t t❡st ✐t ❞✐r❡❝t❧②✳ ❍♦✇ ❞♦ ✇❡ ❣❛t❤❡r ❡①♣❡r✐❡♥❝❡ ❄ ❆♥ ❡①❛♠♣❧❡ ✿ ❙✉♣♣♦s❡ ②♦✉ ✇❛♥t t♦ t❡st ②♦✉r ♥❡✇ ❇❧❛❝❦ ❏❛❝❦ str❛t❡❣②✱ ❜✉t ❛s ♣♦♦r st✉❞❡♥t ②♦✉ ❞♦♥✬t ❤❛✈❡ ❡♥♦✉❣❤ ♠♦♥❡② t♦ ♣❧❛② ❛t t❤❡ ❝❛s✐♥♦✳ ❖✛ ♣♦❧✐❝② ♠❡t❤♦❞s ❛❧❧♦✇ ②♦✉ t♦ ❛ss❡ss ②♦✉r str❛t❡❣② ♦❜s❡r✈✐♥❣ ♦t❤❡r ♣❧❛②❡rs✳ ❋♦r♠❛❧❧② ✇❡ ✉s❡ ❛♥♦t❤❡r ♣♦❧✐❝② µ t♦ ❣❡♥❡r❛t❡ ❞❛t❛✱ ❛♥❞ ❡st✐♠❛t❡ vπ ♦r qπ✳ ❲❡ ❝❛❧❧

◮ π t❤❡ t❛r❣❡t ♣♦❧✐❝② ◮ µ t❤❡ ❜❡❤❛✈✐♦r ♣♦❧✐❝②

▼♦r❡♦✈❡r ✇❡ ❞♦ ♥♦t ♥❡❡❞ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❡①♣❧♦r✐♥❣ st❛rts✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

❍♦✇❡✈❡r ✇❡ ♥❡❡❞ µ t♦ s❛t✐s❢② ❛ ❝♦♥❞✐t✐♦♥ ✿ π(a, s) > ✵ ⇒ µ(a, s) > ✵ ❊✈❡r② ❛❝t✐♦♥ ✇❤✐❝❤ ✐s t❛❦❡♥ ✉♥❞❡r ♣♦❧✐❝② π ♠✉st ❤❛✈❡ ❛ ♥♦♥✲③❡r♦ ♣r♦❜❛❜✐❧✐t② t♦ ❜❡ t❛❦❡♥ ❛s ✇❡❧❧ ✉♥❞❡r ♣♦❧✐❝② µ✳ ❲❡ ❝❛❧❧ t❤✐s t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❝♦✈❡r❛❣❡✳ ❚②♣✐❝❛❧❧② t❤❡ t❛r❣❡t ♣♦❧✐❝② π ✇♦✉❧❞ ❜❡ ❛ ❣r❡❡❞② ♣♦❧✐❝② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝✉rr❡♥t ❛❝t✐♦♥✲✈❛❧✉❡ ❢✉♥❝t✐♦♥✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧ ✲ ♦✛ ♣♦❧✐❝②

❚❤❡ t♦♦❧ ✇❡ ✉s❡ ❢♦r ❡st✐♠❛t✐♦♥ ✐s ❝❛❧❧❡❞ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣✳ ■t ✐s ❛ ❣❡♥❡r❛❧ t❡❝❤♥✐q✉❡ ❢♦r ❡st✐♠❛t✐♥❣ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ✉♥❞❡r ♦♥❡ ❞✐str✐❜✉t✐♦♥ ❣✐✈❡♥ s❛♠♣❧❡s ❢r♦♠ ❛♥♦t❤❡r✳

• ✐✈❡♥ ❛ st❛t❡ St✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ s✉❜s❡q✉❡♥t st❛t❡✲❛❝t✐♦♥

tr❛❥❡❝t♦r② At, St+✶, At+✶, . . . , ST ♦❝❝✉rr✐♥❣ ✉♥❞❡r ♣♦❧✐❝② π ✐s Pπ({{St, At}, . . . , {ST}}) =

T−✶

• k=t

π(Ak|Sk)p(Sk+✶|Sk, Ak) ✇❤❡r❡ p ✐s t❤❡ st❛t❡✲tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t②✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧ ✲ ♦✛ ♣♦❧✐❝②

❚❤❡ r❡❧❛t✐✈❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ tr❛❥❡❝t♦r② ✉♥❞❡r t❤❡ t❛r❣❡t ❛♥❞ ❜❡❤❛✈✐♦r ♣♦❧✐❝✐❡s✱ ♦r t❤❡ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣✱ ✐s ✿ ρT

t :=

T−✶

k=t π(Ak|Sk)p(Sk+✶|Sk, Ak)

T−✶

k=t µ(Ak|Sk)p(Sk+✶|Sk, Ak)

=

T−✶

• k=t

π(Ak|Sk) µ(Ak|Sk) ❚❤❡ tr❛❥❡❝t♦r② ♣r♦❜❛❜✐❧✐t✐❡s ❞❡♣❡♥❞ ♦♥ t❤❡ ▼❉P✱ ✇❤✐❝❤ ❛r❡ ❣❡♥❡r❛❧❧② ✉♥❦♥♦✇♥✱ ❜✉t ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r ♦✉t✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧ ✲ ♦✛ ♣♦❧✐❝②

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❣❛t❤❡r❡❞ ❡①♣❡r✐❡♥❝❡ ✐♥ t❤❡ ❢♦r♠ ♦❢ n ❡♣✐s♦❞❡s✳ ▲❡t N(s) ❜❡ t❤❡ ❡♥✉♠❡r❛t✐♦♥ ♦❢ ❡♣✐s♦❞❡s ✇❤✐❝❤ ✈✐s✐t❡❞ st❛t❡ s✳ ▲❡t T(s, k) ❜❡ t❤❡ ✜rst t✐♠❡ ✇❤❡♥ st❛t❡ s ✐s ✈✐s✐t❡❞ ✐♥ ❡♣✐s♦❞❡ k✳ ❚❤❡ t✐♠❡ ♦❢ t❤❡ t❡r♠✐♥❛❧ st❛t❡ ♦❢ ❡♣✐s♦❞❡ k ✐s ❞❡♥♦t❡❞ ❛s T(k) ✳ ❆♥ ❡①❛♠♣❧❡ ❝♦✉❧❞ ❧♦♦❦ ❧✐❦❡ t❤✐s ✿ ❲❡ ✇♦✉❧❞ ❤❛✈❡ ✿ N(E) = {✶, ✷}✱ T(B, ✶) = ✷✱ T(✶) = ✹ ❛♥❞ G✷ = ✽✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧ ✲ ♦✛ ♣♦❧✐❝②

❖♥❡ ✇❛② t♦ ❡st✐♠❛t❡ vπ(s) ✐s t♦ s❝❛❧❡ t❤❡ r❡t✉r♥s ❜② t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s ✇❡ ✈✐s✐t❡❞ st❛t❡ s ✿ ˆ vπ(s) =

• i∈N(s) ρT(i)

T(s,i)Gi

|N(s)| ❚❤✐s ✐s ✇❤❛t ✇❡ ❝❛❧❧ ♦r❞✐♥❛r② ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣✳ ◗ ✿ ❲❤❛t ♣r♦❜❧❡♠ ❞♦❡s t❤✐s ❡st✐♠❛t♦r ❤❛✈❡ ❄ ❚❤❡ ✈❛r✐❛♥❝❡ ✐s ✉♥❜♦✉♥❞❡❞ ✐♥ ❣❡♥❡r❛❧ ✦ ❊①❛♠♣❧❡ ✿ ❖♥❡ ✈✐s✐t ♦❢ s ✇✐t❤ r❛t✐♦ ✶✵✵ ❛♥❞ r❡t✉r♥ ✶✱ ˆ vπ(s) = ✶✵✵✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧ ✲ ♦✛ ♣♦❧✐❝②

❆♥ ❛❧t❡r♥❛t✐✈❡ ✐s ✇❡✐❣❤t❡❞ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣✱ ✇❤✐❝❤ ✐s ✉s✐♥❣ ❛♥ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ✿ ˆ vπ(s) =

• i∈N(s) ρT(i)

N(s,i)Gi

• i∈N(s) ρT(i)

N(s,i)

◗ ✿ ❲❤❛t ✐s t❤❡ ♣r♦❜❧❡♠ ✇✐t❤ t❤✐s ❡st✐♠❛t♦r ❄ ❚❤❡ ❡st✐♠❛t♦r ✐s ❜✐❛s❡❞ ✐♥ t❤❡ st❛t✐st✐❝❛❧ s❡♥s❡✱ ✐ts ❡①♣❡❝t❛t✐♦♥ ✐s vµ(s) ✐♥st❡❛❞ ♦❢ vπ(s)✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧ ✲ ♦✛ ♣♦❧✐❝②

■♥ ♣r❛❝t✐❝❡ t❤❡ ✇❡✐❣❤t❡❞ ❡st✐♠❛t♦r ❤❛s ❞r❛♠❛t✐❝❛❧❧② ❧♦✇❡r ✈❛r✐❛♥❝❡ ❛♥❞ ✐s t❤❡r❡❢♦r❡ str♦♥❣❧② ♣r❡❢❡rr❡❞✳ ❊①❛♠♣❧❡ ♦❢ ❛ ❜❧❛❝❦❥❛❝❦ st❛t❡ ✿

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ✲ ▲❡❛r♥✐♥❣ ❚✐❝ ❚❛❝ ❚♦❡

❲❡ ❧❡t ♦✉r ▼♦♥t❡ ❈❛r❧♦ ❧❡❛r♥❡r ♣❧❛② ❛❣❛✐♥st ❛ r❛♥❞♦♠ str❛t❡❣② ✿

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ✲ ▲❡❛r♥✐♥❣ ❚✐❝ ❚❛❝ ❚♦❡

❲❡ ❧❡t ♦✉r ▼♦♥t❡ ❈❛r❧♦ ❧❡❛r♥❡r ♣❧❛② ❛❣❛✐♥st t❤❡ ❑✐♥❣ ▲✉✐s ✿

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ✲ ▲❡❛r♥✐♥❣ ❚✐❝ ❚❛❝ ❚♦❡

❲❡ ❧❡t ♦✉r ▼♦♥t❡ ❈❛r❧♦ ❧❡❛r♥❡r ♣❧❛② ❛❣❛✐♥st t❤❡ ♦♣t✐♠❛❧ str❛t❡❣② ❢r♦♠ ❉P ✿

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ✲ ▲❡❛r♥✐♥❣ ❚✐❝ ❚❛❝ ❚♦❡

❲❡ ✉s❡❞ ✐t t♦ ♣❧❛② ❚✐❝ ❚❛❝ ❚♦❡ ❛❣❛✐♥st t❤❡ ❜❡st str❛t❡❣② ✇❡ ❝♦✉❧❞ ❝♦♠❡ ✉♣ ✇✐t❤ ✭■♠♣r♦✈❡❞ ▲❛t✐♥ P❧❛②❡rs✮ ✿

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ✲ ▲❡❛r♥✐♥❣ ❇❧❛❝❦ ❏❛❝❦

❲❡ ❧❡t ✐t ♣❧❛② ❇❧❛❝❦ ❏❛❝❦ ✇✐t❤ t❤❡ ❘❛♥❞♦♠ ❛♥❞ t❤❡ ❖♣t✐♠❛❧ P❧❛②❡r ✿ ❉✉❡ t♦ t❤❡ ǫ✲s♦❢t ♣♦❧✐❝② ✐t st✐❧❧ t❛❦❡s ❛ ❧♦t ♦❢ ✇r♦♥❣ ❞❡❝✐s✐♦♥s✳

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ✲ ▲❡❛r♥✐♥❣ ❇❧❛❝❦ ❏❛❝❦

❲❡ ✉♣❞❛t❡ ♦✉r str❛t❡❣② ❛❢t❡r ❡❛❝❤ ❣❛♠❡✱ ✉s✐♥❣ ❛ r✉♥♥✐♥❣ ❛✈❡r❛❣❡ ✿

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ✲ ▲❡❛r♥✐♥❣ ❇❧❛❝❦ ❏❛❝❦

❚❤❡ t❤r❡❡ ❝r✉❝✐❛❧ ♠❡t❤♦❞s ✇❤✐❝❤ ❣❡♥❡r❛t❡ ❛♥❞ s❡❧❡❝t ❛❝t✐♦♥s ✿

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s

■♥tr♦❞✉❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ Pr❡❞✐❝t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ❈♦♥tr♦❧

▼♦♥t❡ ❈❛r❧♦ ✲ ▲❡❛r♥✐♥❣ ❇❧❛❝❦ ❏❛❝❦

❆ ❡①❝❡r♣t ♦❢ t❤❡ str❛t❡❣② t❛❜❧❡✱ ✇✐t❤♦✉t ❛♥ ❛❝❡ ❛♥❞ s♣❧✐tt✐♥❣ ♣♦ss✐❜✐❧✐t② ✿

▼✐❝❤❛❡❧ ❍❡✐♥③❡r✱ ❊♠♠❛♥✉❡❧ Pr♦❢✉♠♦ ❘❡✐♥❢♦r❝❡♠❡♥t ▲❡❛r♥✐♥❣ ✲ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s