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❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣

❲❛s❡❡♠ ❆s❧❛♠ ❙✉♣❡r✈✐s♦r✿ ❚r✐✈✐❦r❛♠ ❉♦❦❦❛

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✶ ✴ ✶✷

❈♦♥t❡♥ts

✶

▼♦t✐✈❛t✐♦♥

✷

❚❤❡ Pr♦❜❧❡♠

✸

❘♦❜✉st♥❡ss

✹

❘❡s✉❧ts

✺

❋✉rt❤❡r ❲♦r❦

✻

❘❡❢❡r❡♥❝❡s

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✷ ✴ ✶✷

▼♦t✐✈❛t✐♦♥

▼♦t✐✈❛t✐♦♥

Pr✐❝❡✲❞❡♠❛♥❞ r❡❧❛t✐♦♥s❤✐♣ ✉♥❦♥♦✇♥ t♦ s❡❧❧❡rs ❛t ♣r♦❞✉❝t ❧❛✉♥❝❤✳ ❊①♣❡r✐♠❡♥t❛t✐♦♥ ✇✐t❤ ♣r✐❝✐♥❣ ♥❡❝❡ss❛r② t♦ ❞❡t❡r♠✐♥❡ r❡✈❡♥✉❡✲♠❛①✐♠✐s✐♥❣ ♣r✐❝❡✳ ❆❞❥✉st✐♥❣ t❤❡ ♣r✐❝❡ t♦♦ ♦❢t❡♥ ❝❛♥ ❜❡ ❤❛r♠❢✉❧ t♦ s❛❧❡s✳ Pr✐❝❡ ❝❤❛♥❣✐♥❣ ❝❛♥ ❝♦♠❡ ✇✐t❤ ❤✉❣❡ ❛❞♠✐♥✐str❛t✐✈❡ ❛♥❞ ♦t❤❡r ❤✐❞❞❡♥ ❝♦sts✳ ▼✻ ❚♦❧❧ ❧♦s❡s ➾✷✺♠ ♣❡r ②❡❛r ♦♥ ❛✈❡r❛❣❡ ❛♥❞ ❤❛s ♥❡✈❡r ♠❛❞❡ ❛ ♣r♦✜t✳

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✸ ✴ ✶✷

❚❤❡ Pr♦❜❧❡♠

❚❤❡ Pr♦❜❧❡♠

wt,i rt

Pr✐❝❡s r ❛♥❞ w ❛r❡ s❡❧❡❝t❡❞ ❢r♦♠ Ω = p−, ..., p−✳ ❖♥ ❞❛② i ✐♥ t♦❧❧✐♥❣ ♣❡r✐♦❞ t✱ ✐❢ rt ≤ wt,i t❤❡ ❝✉st♦♠❡r ❝❤♦♦s❡s t❤❡ t♦❧❧ r♦❛❞✳ ❖♥❧② ✉s❛❣❡ ut ✐s ♦❜s❡r✈❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ t♦❧❧✐♥❣ ♣❡r✐♦❞✿ ut =

n−1

• i=0

✶{rt ≤ wt,i}✳ ❚❤❡ ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥ ♣❞❢ ❢♦r ♣✉❜❧✐❝ r♦❛❞ ♣r✐❝❡s ✐s ❤✐❞❞❡♥✱ ❜✉t t❤❡ ❝❞❢ ❝❛♥ ❜❡ s❛♠♣❧❡❞✳

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✹ ✴ ✶✷

❚❤❡ Pr♦❜❧❡♠

Pr✐❝❡ ❝❛♣ ❝♦♥str❛✐♥t✿ rt+1 ≤ ⌈rt × (1 + ς)⌉ ✱ ς > 0✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤❡ t♦❧❧ s❡tt❡r ✐s t♦ s❡❧❡❝t ❛ s❡q✉❡♥❝❡ ♦❢ ♣r✐❝❡s R = {rt : t = 0, ..., N − 1} s✉❝❤ t❤❛t✿ R = arg max

R

N−1

• t=0

ut × rt

• ❲❛s❡❡♠ ❆s❧❛♠

❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✺ ✴ ✶✷

❘♦❜✉st♥❡ss

❘♦❜✉st♥❡ss

❆ss✉♠❡ t❤❛t ♥❛t✉r❡ ✐s ♦♥ t❤❡ s✐❞❡ ♦❢ t❤❡ ❝✉st♦♠❡r ✐✳❡✳ ✬✇❛♥ts✬ t♦ ♠✐♥✐♠✐s❡ ❝✉st♦♠❡r ❝♦st✳ ❚❤❡ ❞✐str✐❜✉t✐♦♥ f ✐s ❡st✐♠❛t❡❞ ❛t ❡❛❝❤ t♦❧❧✐♥❣ ♣❡r✐♦❞ ✉s✐♥❣ ❤✐st♦r✐❝❛❧ ✉s❛❣❡ ✐♥❢♦r♠❛t✐♦♥✳ ◆❛t✉r❡ s❡ts f ✐s s❡t ❛❢t❡r t❤❡ t♦❧❧ s❡tt❡r s❡❧❡❝ts t❤❡ ♥❡①t t♦❧❧ ♣r✐❝❡ rt+1✳

f t♦ ♠✐♥✐♠✐s❡

• i∈Ω

i≥rt+1

(rt+1 · f(i)) +

• i∈Ω

i<rt+1

(f(i) · i) s✉❜❥❡❝t t♦

• i∈Ω

i≥rt

f(i) = γrt

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✻ ✴ ✶✷

❘♦❜✉st♥❡ss

❘♦❜✉st ❚✇♦ P♦✐♥t Pr✐❝✐♥❣

5 10 15 20 0.00 0.20 0.40 0.60 0.80 1.00 w f(w) Estimate f1 f2

❋✐❣✉r❡✿ ❋✐tt✐♥❣ t✇♦✲♣♦✐♥t ❞✐str✐❜✉t✐♦♥s ❢♦r s♣❡❝✐✜❝ rt+1✳

❋♦r ❡❛❝❤ ♣♦ss✐❜❧❡ rt+1 ✜♥❞ ❛❧❧ ❢❡❛s✐❜❧❡ t✇♦✲♣♦✐♥t ❞✐str✐❜✉t✐♦♥s✳ ❋✐t t❤❡ t✇♦✲♣♦✐♥t ❞✐str✐❜✉t✐♦♥ t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ✉s❡r✬s ❝♦st ✐❢ rt+1 ✇❡r❡ t♦ ❜❡ s❡t✳ ❚❤❡♥ s❡❧❡❝t t❤❡ rt+1 t❤❛t ♠❛①✐♠✐s❡s r❡✈❡♥✉❡✳

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✼ ✴ ✶✷

❘❡s✉❧ts

❘❡s✉❧ts

❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤❡s[1]✳

10 20 30 40 50 Time Period 60 70 80 90 100 110 120 130 Prices and Optimal Prices TollRoad Optimal BZ1 BZ2

❋✐❣✉r❡✿ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r U(40, 130) ❞❛t❛

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✽ ✴ ✶✷

❘❡s✉❧ts

❖✈❡r❛❧❧ ♣❡r❢♦r♠❛♥❝❡✳

10 20 30 40 50 Time Period 20 40 60 80 100 Current Total Regret Toll capon BZ1 capon BZ2 capon Toll capoff BZ1 capoff BZ2 capoff

❋✐❣✉r❡✿ ❘❡❣r❡t ♦✈❡r ❚✐♠❡

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✾ ✴ ✶✷

❋✉rt❤❡r ❲♦r❦

❙❡❛s♦♥❛❧✐t②

❙✐❣♥✐✜❝❛♥t ❛❞✈❛♥t❛❣❡ ♦✈❡r ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤❡s ✇❤❡♥ wt,i s❛♠♣❧❡❞ ❢r♦♠ ❞✐✛❡r❡♥t ❞✐str✐❜✉t✐♦♥s s❡❛s♦♥❛❧❧②✳

10 20 30 40 50 Time Period 20 40 60 80 100 120 140 160 180 Prices and Optimal Prices Toll Road Optimal

❋✐❣✉r❡✿ ❆❧❣♦r✐t❤♠ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ s❡❛s♦♥❛❧ ❞❛t❛

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✶✵ ✴ ✶✷

❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s

❬✶❪✳ ❖✳ ❇❡s❜❡s ❛♥❞ ❆✳ ❩❡❡✈✐✱ ✭✷✵✵✾✮✳ ❉②♥❛♠✐❝ ♣r✐❝✐♥❣ ✇✐t❤♦✉t ❦♥♦✇✐♥❣ t❤❡ ❞❡♠❛♥❞ ❢✉♥❝t✐♦♥✿ ❘✐s❦ ❜♦✉♥❞s ❛♥❞ ♥❡❛r✲♦♣t✐♠❛❧ ❛❧❣♦r✐t❤♠s✳ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✳ ❱♦❧ ✺✼✭✻✮ ✳ ✶✹✵✼✲✶✹✷✵ ❬✷❪✳ ❚✳ ❉♦❦❦❛✱ P✳ ❏❛❝❦♦ ❛♥❞ ❲✳ ❆s❧❛♠✱ ✭✷✵✶✼✮✳ ❘♦❜✉st ❉②♥❛♠✐❝ Pr✐❝✐♥❣ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♣r✐❝❡ ❝♦♥tr♦❧s✳ ▲❯▼❙ ✇♦r❦✐♥❣ ♣❛♣❡r✳ ❱♦❧ ✶✳ ✶✲✷✵

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✶✶ ✴ ✶✷

❚❤❛♥❦ ②♦✉ ❛❧❧ ❢♦r ❧✐st❡♥✐♥❣✦

❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✶✷ ✴ ✶✷