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Rent, Lease or Buy:

Randomized Algorithms for Multislope Ski Rental

Zvi LotkerBen-Gurion University

Boaz Patt-Shamir Dror RawitzTel Aviv University

Rent or Buy Dilemma

⊲ Rent or Buy

Ski Rental

Rent, Lease or Buy

Multislope SkiRental

Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples


Strategies & Profiles

Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Sleeping Baby Problem:

� You finally managed to put baby to sleep

Rent or Buy Dilemma

⊲ Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion




� Baby will wake up at some unknown timein the future

Rent or Buy Dilemma

⊲ Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion




� Baby will wake up at some unknown timein the future

� Should parent stay awake or go to sleep?When should the parent go to sleep?

– Going to sleep −→ fixed effort– Staying awake −→ effort per time unit

Rent or Buy Dilemma

⊲ Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Question:How do we measurethe quality of our solution?

Rent or Buy Dilemma

⊲ Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion



Competitive Analysis:

� Worst case analysis:Baby tries to make parent’s lifeas hard as possible

Rent or Buy Dilemma

⊲ Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion



Competitive Analysis:

� Worst case analysis:Baby tries to make parent’s lifeas hard as possible

� Our solution is comparedto best solution whenfuture is known in advance

Ski Rental

Rent or Buy

⊲ Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Discrete Version:

� Ski Rental Problem:

– Vacation at ski resort– End of vacation is unknown– Cost of skis is ¤B, rent is ¤1/day– Should we rent or buy the skis?– When should we buy?

Ski Rental

Rent or Buy

⊲ Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Discrete Version:

� Ski Rental Problem:

– Vacation at ski resort– End of vacation is unknown– Cost of skis is ¤B, rent is ¤1/day– Should we rent or buy the skis?– When should we buy?

� Optimal Offline Cost:

– Rent, if vacation isshorter than B days

– Buy at t = 0, otherwise

timeB

cost

B

Ski Rental

Rent or Buy

⊲ Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Optimal Online Strategies:

� 2-competitive deterministic strategy[Karlin et al. 88]

– Buy after B days timeB

cost

B

Ski Rental

Rent or Buy

⊲ Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion





cost

B

�e

e−1 -competitive randomized strategy [Karlin et al. 94]

– Buy at time t ≤ B chosenaccording to some distribution

Ski Rental

Rent or Buy

⊲ Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion





cost

B

�e



– Oblivious adversary

Ski Rental

Rent or Buy

⊲ Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion





cost

B

�e



– Oblivious adversary

– There exists a distribution,for which the best competitive factor is e

e−1

Rent, Lease or Buy

Rent or Buy

Ski Rental

⊲Rent, Lease orBuy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Sleeping Problem:

� Price of sleep increases with its depth� Sleep rent decreases with its depth

awake closing eyes naping asleep

Rent, Lease or Buy

Rent or Buy

Ski Rental

⊲Rent, Lease orBuy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Sleeping Problem:

� Price of sleep increases with its depth� Sleep rent decreases with its depth

awake closing eyes naping asleep

Extended Ski Rental:

� Mixed rent and buy options� Pure buy or pure rent may not exist

Multislope Ski Rental

Rent or Buy

Ski Rental

Rent, Lease or Buy

⊲Multislope SkiRental

Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Problem Definition:

� Several states/slopes

– Slope i: bi + ri · t

– bi+1 > bi for all i

– ri+1 < ri for all i

time

cost

Slope 0 Slope 1

b1

Slope 2

b2

Slope 3

b3

� End time is unknown� Which slope should we buy? When?


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Problem Definition:





time

cost

Slope 0 Slope 1

b1

Slope 2

b2

Slope 3

b3


� Online Buying Costs:Say we are in slope i, how much do we pay for slope j?


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Problem Definition:





time

cost

Slope 0 Slope 1

b1

Slope 2

b2

Slope 3

b3


� Online Buying Costs:Say we are in slope i, how much do we pay for slope j?

– Additive Model: bj − bi

– From-scratch Model: bj

– Non-Additive Model: bij

Offline Strategy

Rent or Buy

Ski Rental

Rent, Lease or Buy


⊲ Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Offline Strategy:

� Knows end time in advance� Buys a slope at time 0 according to end time

time

cost

Slope 0 Slope 1

b1

Slope 2

b2

Slope 3

b3

Offline Strategy

Rent or Buy

Ski Rental

Rent, Lease or Buy


⊲ Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Offline Strategy:

� Knows end time in advance� Buys a slope at time 0 according to end time

time

cost

Slope 0 Slope 1

b1

Slope 2

b2

Slope 3

b3

opt(t)

s1 s2 s3

Competitive Analysis:The online strategy is compared to the offline strategy

Previous Results & Applications

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

⊲ Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Non-additive/from-scratch models:

� Online Capital Investments:

– Deter. 6.83-competitive strategy(slopes may arrive over time)[Azar et al. 99]

– Deter. lower bound 3.618Random. 2.88-competitive strategy[Damaschke 03]


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

⊲ Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Non-additive/from-scratch models:

� Online Capital Investments:

– Deter. 6.83-competitive strategy(slopes may arrive over time)[Azar et al. 99]

– Deter. lower bound 3.618Random. 2.88-competitive strategy[Damaschke 03]

� Rerouting in ATM networks:

– 4-competitive deterministic strategy(slopes may be concave)[BCN 00]


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

⊲ Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Energy Saving:

� Slopes are hibernation modes


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

⊲ Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Energy Saving:

� Slopes are hibernation modes

� Additive model [IGS 02]:

– Deterministic 2-competitive strategy

� Non-additive model [AIS 04]:

– Algorithm that computesbest deterministic strategy

– Algorithm that computes beststrategy given distribution of idle period length(complexity depends on #bins in histogram)

Our Results

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

⊲ Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Two slopes without pure buy:

� Deterministic Strategy:

– Buy at s where r0s = b1 + r1s

– Competitive ratio is 1 + r0−r1r0

time

cost

b1

s

r0t

b1 + r1t

Our Results

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

⊲ Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Two slopes without pure buy:

� Deterministic Strategy:

– Buy at s where r0s = b1 + r1s

– Competitive ratio is 1 + r0−r1r0

time

cost

b1

s

r0t

b1 + r1t

� Our Results [LPR08a]:

– Randomized ee−1+

r1r0

-competitive strategy

– Matching lower bound (using Yao’s Lemma)

Our Results

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

⊲ Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Multislope Ski Rental [LPR08b]:

� Additive Model:

– Randomized e−rk/r0

e−1 -competitive strategy(Decomposition into k classical ski rental instances)

– Main Result:Algorithm that computesbest randomized strategy

time

cost

Slope 0 Slope 1

b1

Slope 2

b2

Slope 3

b3

opt(t)

s1s2 s3

Our Results

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

⊲ Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Multislope Ski Rental [LPR08b]:

� Additive Model:


e−1 -competitive strategy(Decomposition into k classical ski rental instances)

– Main Result:Algorithm that computesbest randomized strategy

time

cost

Slope 0 Slope 1

b1

Slope 2

b2

Slope 3

b3

opt(t)

s1s2 s3

� Non-additive Model

– Randomized e-competitive strategy(Repeated doubling)

Two Slopes without Pure Buy

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

⊲ Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Randomized strategy:

� “Distribution” on buying time

time

cost

b1

s

r0t

b1 + r1t


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

⊲ Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion




time

cost

b1

s

r0t

b1 + r1t

� p1(t) — prob. of buying slope 1 until t


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

⊲ Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion




time

cost

b1

s

r0t

b1 + r1t


� Expected rent at time t:

r0 · (1 − p1(t)) + r1 · p1(t) = r0 + (r1 − r0)p1(t)


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

⊲ Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion




time

cost

b1

s

r0t

b1 + r1t



r0 · (1 − p1(t)) + r1 · p1(t) = r0 + (r1 − r0)p1(t)

� We spend as much as possible on buying for t ≤ s:

r0 + (r1 − r0)p1(t) + b1 ·ddtp1(t) = c · r0


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

⊲ Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion




time

cost

b1

s

r0t

b1 + r1t



r0 · (1 − p1(t)) + r1 · p1(t) = r0 + (r1 − r0)p1(t)

� We spend as much as possible on buying for t ≤ s:

r0 + (r1 − r0)p1(t) + b1 ·ddtp1(t) = c · r0

� Solution with boundary condition p1(0) = 0:

p1(t) =r0(c − 1)

r0 − r1(e

r0−r1b1

t− 1)


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

⊲ Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


� Guess: strategy stops buying at time s

time

cost

b1

s

r0t

b1 + r1t

� For t ≥ s = b1r0−r1

:

p1(t) = p1(s) = r0(c−1)r0−r1

(e − 1)

c · r1 = r0 + (r1 − r0)p1(s)


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

⊲ Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion



time

cost

b1

s

r0t

b1 + r1t

� For t ≥ s = b1r0−r1

:

p1(t) = p1(s) = r0(c−1)r0−r1

(e − 1)

c · r1 = r0 + (r1 − r0)p1(s)

⇓

c =e

e − 1 + r1r0


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

⊲ Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion



time

cost

b1

s

r0t

b1 + r1t

� For t ≥ s = b1r0−r1

:

p1(t) = p1(s) = r0(c−1)r0−r1

(e − 1)

c · r1 = r0 + (r1 − r0)p1(s)

⇓

c =e

e − 1 + r1r0

⇓

p1(t) =

et·

r0−r1b1 −1

e−1+r1r0

t ≤ s

e−1e−1+

r1r0

t > s

Two Slopes — Strategy & Examples

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

⊲Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


� Probability of beingat slope 1 at time t: p1(t) =

et·

r0−r1b1 −1

e−1+r1r0

t ≤ s

e−1e−1+

r1r0

t > s

� Strategy: choose t at random according to p1

Two Slopes — Strategy & Examples

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

⊲Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


� Probability of beingat slope 1 at time t: p1(t) =

et·

r0−r1b1 −1

e−1+r1r0

t ≤ s

e−1e−1+

r1r0

t > s

� Strategy: choose t at random according to p1

� Examples:

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

r1/r0 = 0, c = e

e−1≈ 1.582 r1/r0 = 0.5, c = e

e−0.5≈ 1.2254

Classical ski rental Pr[No buy] = 12e−1


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


� Multiple Slopes:

time

cost

Slope 0 Slope 1

b1

Slope 2

b2

Slope 3

b3

opt(t)

s1 s2 s3

� Optimal Solution:Buys slope at time 0 and stays on it


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


� Multiple Slopes:

time

cost

Slope 0 Slope 1

b1

Slope 2

b2

Slope 3

b3

opt(t)

s1 s2 s3

� Optimal Solution:Buys slope at time 0 and stays on it

� Additive Model:

– We pay bj − bi for moving from i to j

– W.l.o.g. we always go from slope i to slope i + 1


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples


⊲Strategies &Profiles

Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


� Randomized Strategy:

– Should we move to the next slope?– When do we move to the next slope?– Probability distribution over deterministic strategies


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion




� Randomized Profile:

– pi(t) — probability of beingin slope i at time t

–∑

i pi(t) = 1 for all t ≥ 0


Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion




� Randomized Profile:

– pi(t) — probability of beingin slope i at time t

–∑

i pi(t) = 1 for all t ≥ 0

� Plan:

– Every strategy induces a profile– Find best profile– Construct strategy

Profiles Costs

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



⊲ Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


� Rent:

– Expected rent at time t: Rp(t)△

=∑

i

pi(t) · ri

– Expected rental cost at time t:

∫ t

0Rp(z)dz

Profiles Costs

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



⊲ Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


� Rent:


=∑

i

pi(t) · ri


∫ t

0Rp(z)dz

� Buy:

– Expected buying cost at time t: Bp(t)△

=∑

i

pi(t) · bi

Profiles Costs

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



⊲ Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


� Rent:


=∑

i

pi(t) · ri


∫ t

0Rp(z)dz

� Buy:

– Expected buying cost at time t: Bp(t)△

=∑

i

pi(t) · bi

� Total:

– Expected cost at time t: Bp(t) +

∫ t

0Rp(z)dz

Profiles

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

⊲ Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


Chain of Transformations:

� Profile

⇛ Continuous profileContinuous in t

⇛ Prudent profileOnly one or two consecutive active slopes

⇛ Tight profileMoves to the next slope as soon as possible

Profiles

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

⊲ Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


cont.


� Profile




Profiles

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

⊲ Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


prud.


� Profile




Profiles

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

⊲ Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


tight


� Profile




Prudent Profiles

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

⊲ Prudent Profiles

Tight Profile

Algorithm

Examples

Conclusion


prud.

cont.

Continuous to Prudent:

� ∀t two (or one) consecutive slopes are determinedsuch that buying cost is preserved

⇛ Rent may only decrease

Prudent Profiles

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles


Tight Profile

Algorithm

Examples

Conclusion


tight

prud.




Prudent to Tight:

� Buy next slope as soon as possible⇛ Rent may only decrease

Prudent Profiles

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles


Tight Profile

Algorithm

Examples

Conclusion


tight




Prudent to Tight:

� Buy next slope as soon as possible⇛ Rent may only decrease

Theorem: There exists a tight optimal profile

Computing Tight Profile

Rent or Buy

Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

⊲ Tight Profile

Algorithm

Examples

Conclusion


Computing tight profile for a given c:

� Solve series of diff. equations:

ddtBp(t) + Rp(t) = c · d

dtopt(t)

⇛ For tight profiles

ddtpi(t) + α · pi(t) = β

Computing Tight Profile

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Ski Rental

Rent, Lease or Buy


Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

⊲ Tight Profile

Algorithm

Examples

Conclusion


Computing tight profile for a given c:

� Solve series of diff. equations:

ddtBp(t) + Rp(t) = c · d

dtopt(t)

⇛ For tight profiles

ddtpi(t) + α · pi(t) = β

� Solve equation using boundary conditions:

– Initially: p0(0) = 1 and pi(0) = 0 for i > 0– Next boundary condition determined by events:

⊲ pi(t) = 1 for some slope i

⊲ddtopt(t) changes

– O(k) events

Algorithm

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Offline Strategy

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Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

⊲ Algorithm

Examples

Conclusion


Algorithm:Binary search for best competitive ratio c ∈ [1, e

e−1]

Algorithm

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Two Slopes

Strategy &Examples



Profile Costs

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Prudent Profiles

Tight Profile

⊲ Algorithm

Examples

Conclusion



e−1]

Computing a Strategy:

� Let p be a tight profile� X ∼ U(0, 1)� Move from slope i − 1 to slope i at t s.t. pi(t) = X

Algorithm

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Two Slopes

Strategy &Examples



Profile Costs

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Prudent Profiles

Tight Profile

⊲ Algorithm

Examples

Conclusion



e−1]

Computing a Strategy:

� Let p be a tight profile� X ∼ U(0, 1)� Move from slope i − 1 to slope i at t s.t. pi(t) = X

Theorem:A randomized (c + ε)-competitive strategycan be found in O(k log 1

ε) time,

where c is the best possible competitive ratio

Three Slopes — Examples

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Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

⊲ Examples

Conclusion


0.2 0.4 0.6 0.8 1.0t

0.2

0.4

0.6

0.8

1.0

Prob

0.2 0.4 0.6 0.8 1.0t

0.2

0.4

0.6

0.8

1.0

Prob

0.2 0.4 0.6 0.8 1.0t

0.2

0.4

0.6

0.8

1.0

Prob

p0 p1 p2

Conclusion

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Offline Strategy

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Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

⊲ Conclusion


Summary of Results:

� Two Slopes without Pure Rent:

– ee−1+r1/r0


– Matching lower bound

� Additive Model:


e−1-competitive strategy

– Algorithm that computes best randomized strategy

� Non-additive Model:

– Randomized e-competitive strategy

Conclusion

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Offline Strategy

Prev. Results

Our Results

Two Slopes

Strategy &Examples



Profile Costs

Profiles

Prudent Profiles

Tight Profile

Algorithm

Examples

⊲ Conclusion


Summary of Results:

� Two Slopes without Pure Rent:

– ee−1+r1/r0


– Matching lower bound

� Additive Model:


e−1-competitive strategy

– Algorithm that computes best randomized strategy

� Non-additive Model:

– Randomized e-competitive strategy

Open Problems:

� Compute best randomized strategy for non-additive model� Improve general upper bound for additive model

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