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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♦❜❧❡♠ Pr✐❝❡s r ❛♥❞ w ❛r❡ s❡❧❡❝t❡❞ ❢r♦♠ Ω = p − , ..., p − ✳ ❖♥ ❞❛② i ✐♥ t♦❧❧✐♥❣ ♣❡r✐♦❞ t ✱ ✐❢ r t ≤ w t,i t❤❡ ❝✉st♦♠❡r ❝❤♦♦s❡s t❤❡ t♦❧❧ r♦❛❞✳ r t w t,i ❖♥❧② ✉s❛❣❡ u t ✐s ♦❜s❡r✈❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ n − 1 � t♦❧❧✐♥❣ ♣❡r✐♦❞✿ u t = ✶ { r t ≤ w t,i } ✳ i =0 ❚❤❡ ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥ ♣❞❢ ❢♦r ♣✉❜❧✐❝ r♦❛❞ ♣r✐❝❡s ✐s ❤✐❞❞❡♥✱ ❜✉t t❤❡ ❝❞❢ ❝❛♥ ❜❡ s❛♠♣❧❡❞✳ ❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✹ ✴ ✶✷

❚❤❡ Pr♦❜❧❡♠ Pr✐❝❡ ❝❛♣ ❝♦♥str❛✐♥t✿ r t +1 ≤ ⌈ r t × (1 + ς ) ⌉ ✱ ς > 0 ✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤❡ t♦❧❧ s❡tt❡r ✐s t♦ s❡❧❡❝t ❛ s❡q✉❡♥❝❡ ♦❢ ♣r✐❝❡s R = { r t : t = 0 , ..., N − 1 } s✉❝❤ t❤❛t✿ � N − 1 � � R = arg max u t × r t R t =0 ❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✺ ✴ ✶✷

❘♦❜✉st♥❡ss ❘♦❜✉st♥❡ss ❆ss✉♠❡ t❤❛t ♥❛t✉r❡ ✐s ♦♥ t❤❡ s✐❞❡ ♦❢ t❤❡ ❝✉st♦♠❡r ✐✳❡✳ ✬✇❛♥ts✬ f t♦ ♠✐♥✐♠✐s❡ t♦ ♠✐♥✐♠✐s❡ ❝✉st♦♠❡r ❝♦st✳ � � ( r t +1 · f ( i )) + ( f ( i ) · i ) i ∈ Ω i ∈ Ω ❚❤❡ ❞✐str✐❜✉t✐♦♥ f ✐s ❡st✐♠❛t❡❞ i ≥ r t +1 i<r t +1 ❛t ❡❛❝❤ t♦❧❧✐♥❣ ♣❡r✐♦❞ ✉s✐♥❣ ❤✐st♦r✐❝❛❧ ✉s❛❣❡ ✐♥❢♦r♠❛t✐♦♥✳ s✉❜❥❡❝t t♦ � f ( i ) = γ r t ◆❛t✉r❡ s❡ts f ✐s s❡t ❛❢t❡r t❤❡ t♦❧❧ i ∈ Ω s❡tt❡r s❡❧❡❝ts t❤❡ ♥❡①t t♦❧❧ ♣r✐❝❡ i ≥ r t r t +1 ✳ ❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✻ ✴ ✶✷

❘♦❜✉st♥❡ss ❘♦❜✉st ❚✇♦ P♦✐♥t Pr✐❝✐♥❣ 1 . 00 Estimate ❋♦r ❡❛❝❤ ♣♦ss✐❜❧❡ r t +1 ✜♥❞ ❛❧❧ f 1 0 . 80 ❢❡❛s✐❜❧❡ t✇♦✲♣♦✐♥t ❞✐str✐❜✉t✐♦♥s✳ f 2 0 . 60 f ( w ) ❋✐t t❤❡ t✇♦✲♣♦✐♥t ❞✐str✐❜✉t✐♦♥ 0 . 40 t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ✉s❡r✬s ❝♦st ✐❢ r t +1 ✇❡r❡ t♦ ❜❡ s❡t✳ 0 . 20 0 . 00 0 5 10 15 20 ❚❤❡♥ s❡❧❡❝t t❤❡ r t +1 t❤❛t w ♠❛①✐♠✐s❡s r❡✈❡♥✉❡✳ ❋✐❣✉r❡✿ ❋✐tt✐♥❣ t✇♦✲♣♦✐♥t ❞✐str✐❜✉t✐♦♥s ❢♦r s♣❡❝✐✜❝ r t +1 ✳ ❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✼ ✴ ✶✷

❘❡s✉❧ts ❘❡s✉❧ts ❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤❡s [1] ✳ 130 TollRoad Optimal 120 BZ1 BZ2 110 Prices and Optimal Prices 100 90 80 70 60 0 10 20 30 40 50 Time Period ❋✐❣✉r❡✿ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r U (40 , 130) ❞❛t❛ ❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✽ ✴ ✶✷

❘❡s✉❧ts ❖✈❡r❛❧❧ ♣❡r❢♦r♠❛♥❝❡✳ Toll capon 100 BZ1 capon BZ2 capon Toll capoff BZ1 capoff BZ2 capoff 80 Current Total Regret 60 40 20 0 10 20 30 40 50 Time Period ❋✐❣✉r❡✿ ❘❡❣r❡t ♦✈❡r ❚✐♠❡ ❲❛s❡❡♠ ❆s❧❛♠ ❘♦❜✉st ❉②♥❛♠✐❝ ❚♦❧❧ Pr✐❝✐♥❣ ✾ ✴ ✶✷

❋✉rt❤❡r ❲♦r❦ ❙❡❛s♦♥❛❧✐t② ❙✐❣♥✐✜❝❛♥t ❛❞✈❛♥t❛❣❡ ♦✈❡r ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤❡s ✇❤❡♥ w t,i s❛♠♣❧❡❞ ❢r♦♠ ❞✐✛❡r❡♥t ❞✐str✐❜✉t✐♦♥s s❡❛s♦♥❛❧❧②✳ 180 Toll Road Optimal 160 140 Prices and Optimal Prices 120 100 80 60 40 20 0 10 20 30 40 50 Time Period ❋✐❣✉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✐❝✐♥❣ ✶✷ ✴ ✶✷