Price Fluctuations: To Buy or to Rent
Price Fluctuations: To Buy or to Rent 1 ´ Marcin Bienkowski 1 Institute
of Computer Science, University of Wrocław
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
1 / 24
Outline
1
Ski Rental
2
Our extension to the model
3
Results Upper Bound for Known Game End Upper Bound for Stochastic Game End Lower Bound for Known Game End
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
2 / 24
Ski Rental
Outline
1
Ski Rental
2
Our extension to the model
3
Results Upper Bound for Known Game End Upper Bound for Stochastic Game End Lower Bound for Known Game End
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
3 / 24
Ski Rental
Ski rental problem (SRP)
“A toy example illustrating rent-or-buy paradigms” A skier At the beginning of each day may either rent skis for p, or buy them for s · p.
At the end of the day, may break legs and skis Input: the leg-breaking day Goal: find a cheapest strategy
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
4 / 24
Ski Rental
Ski rental problem (SRP)
“A toy example illustrating rent-or-buy paradigms” A skier At the beginning of each day may either rent skis for p, or buy them for s · p.
At the end of the day, may break legs and skis Input: the leg-breaking day Goal: find a cheapest strategy
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
4 / 24
Ski Rental
Ski Rental Problem, cont.
Offline setting is trivial Online setting: Best deterministic online algorithm: “rent for s − 1 days and buy on day s”. This strategy is (2 − 1s )-competitive (Karlin, Manasse, Sleator, FOCS 86) (Karp, IFIP 92)
Best randomized online algorithm (chooses a random day from e {1, . . . , s} and then buys on this day) is e−1 ≈ 1.58-competitive (Karlin, Manasse, McGeoch, Owicki, SODA 90)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
5 / 24
Ski Rental
Variations
Many problems exhibit the same “rent-or-buy” structure as SRP, e.g. Bahncard problem
(Fleischer, COCOON 98) (Karlin, Kenyon, Randall, STOC 01)
TCP Acknowledgement problem
(Dooly, Goldman, Scott, JACM 01) (Karlin, Kenyon, Randall, STOC 01)
Page replication problem
(Black, Sleator, CMU TR 89) (Albers, Koga, SWAT 94)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
6 / 24
Our extension to the model
Outline
1
Ski Rental
2
Our extension to the model
3
Results Upper Bound for Known Game End Upper Bound for Stochastic Game End Lower Bound for Known Game End
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
7 / 24
Our extension to the model
Adding dynamics
What if rental price p may change each day? (BUT: the purchase price = s × rental price)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
8 / 24
Our extension to the model
Arbitrary changes of rental price
At day 1, the skier sees the price p = 1. What should she/he do? 1
Skier rents at day 1. What may happen? The prices in next days are equal to k The skier never breaks legs CSKIER ≥ s · k , COPT = s.
2
Skier buys at day 1. What may happen? The prices in next days are equal to 1/k The skier never breaks legs CSKIER = s, COPT = 1 + s/k .
For arbitrary changes, if k = s, the competitive ratio is Ω(s) (large!)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
9 / 24
Our extension to the model
Arbitrary changes of rental price
At day 1, the skier sees the price p = 1. What should she/he do? 1
Skier rents at day 1. What may happen? The prices in next days are equal to k The skier never breaks legs CSKIER ≥ s · k , COPT = s.
2
Skier buys at day 1. What may happen? The prices in next days are equal to 1/k The skier never breaks legs CSKIER = s, COPT = 1 + s/k .
For arbitrary changes, if k = s, the competitive ratio is Ω(s) (large!)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
9 / 24
Our extension to the model
Arbitrary changes of rental price
At day 1, the skier sees the price p = 1. What should she/he do? 1
Skier rents at day 1. What may happen? The prices in next days are equal to k The skier never breaks legs CSKIER ≥ s · k , COPT = s.
2
Skier buys at day 1. What may happen? The prices in next days are equal to 1/k The skier never breaks legs CSKIER = s, COPT = 1 + s/k .
For arbitrary changes, if k = s, the competitive ratio is Ω(s) (large!)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
9 / 24
Our extension to the model
Arbitrary changes of rental price
At day 1, the skier sees the price p = 1. What should she/he do? 1
Skier rents at day 1. What may happen? The prices in next days are equal to k The skier never breaks legs CSKIER ≥ s · k , COPT = s.
2
Skier buys at day 1. What may happen? The prices in next days are equal to 1/k The skier never breaks legs CSKIER = s, COPT = 1 + s/k .
For arbitrary changes, if k = s, the competitive ratio is Ω(s) (large!)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
9 / 24
Our extension to the model
This paper
Additive model of price changes: rental price p is always at least 1 in two consecutive days prices differ by at most 1.
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
10 / 24
Results
Outline
1
Ski Rental
2
Our extension to the model
3
Results Upper Bound for Known Game End Upper Bound for Stochastic Game End Lower Bound for Known Game End
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
11 / 24
Results
Our Contribution We characterize the competitive ratio R in three scenarios (both for deterministic and randomized algorithms): 1
√ Unknown Game End (UGE): R = Θ( s).
2
Known Game End (KGE): For game duration γ = sy , R = Θ(sL(y ) )
3
Stochastic Game End (SGE): Probability of breaking leg = 1/Γ, expected duration = Γ = sy , E[R] = O(sL(y ) · (log s)7/9 ) This scenario, but without price fluctuations, considered by Fujiwara and Iwama (ISAAC 02)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
12 / 24
Results
Our Contribution We characterize the competitive ratio R in three scenarios (both for deterministic and randomized algorithms): 1 2
√ Unknown Game End (UGE): R = Θ( s). Known Game End (KGE): For game duration γ = sy , R = Θ(sL(y ) ) L(y) 2/5 1/3
1/2 2/3 4/5
3
1
y
Stochastic Game End (SGE): Probability of breaking leg = 1/Γ, expected duration = Γ = sy , E[R] = O(sL(y ) · (log s)7/9 ) This scenario, but without price fluctuations, considered by Fujiwara and Iwama (ISAAC 02)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
12 / 24
Results
Our Contribution We characterize the competitive ratio R in three scenarios (both for deterministic and randomized algorithms): 1 2
√ Unknown Game End (UGE): R = Θ( s). Known Game End (KGE): For game duration γ = sy , R = Θ(sL(y ) ) L(y) 2/5 1/3
1/2 2/3 4/5
3
1
y
Stochastic Game End (SGE): Probability of breaking leg = 1/Γ, expected duration = Γ = sy , E[R] = O(sL(y ) · (log s)7/9 ) This scenario, but without price fluctuations, considered by Fujiwara and Iwama (ISAAC 02)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
12 / 24
Results
Our Contribution We characterize the competitive ratio R in three scenarios (both for deterministic and randomized algorithms): 1 2
√ Unknown Game End (UGE): R = Θ( s). Known Game End (KGE): This talk (upper + lower for γ = s) For game duration γ = sy , R = Θ(sL(y ) ) L(y) 2/5 1/3
1/2 2/3 4/5
3
1
y
Stochastic Game End (SGE): This talk Probability of breaking leg = 1/Γ, expected duration = Γ = sy , E[R] = O(sL(y ) · (log s)7/9 ) This scenario, but without price fluctuations, considered by Fujiwara and Iwama (ISAAC 02)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
12 / 24
Results
Upper Bound for Known Game End
Ideas for algorithm for KGE What should trigger ski purchase? In real life, there are two types of people: Those, who buy shares because their price is rising Those, who buy shares because their price is falling down
Apparently, we may learn from both types!
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
13 / 24
Results
Upper Bound for Known Game End
Ideas for algorithm for KGE What should trigger ski purchase? In real life, there are two types of people: Those, who buy shares because their price is rising Those, who buy shares because their price is falling down
Apparently, we may learn from both types!
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
13 / 24
Results
Upper Bound for Known Game End
Ideas for algorithm for KGE, cont.
What should trigger ski purchase? 1
Price drops below some level. A good oportunity for A LG. The downside: the price may go down even further and O PT may pay even less.
2
Price exceeds some threshold. Example: One should eventually buy!
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
14 / 24
Results
Upper Bound for Known Game End
Ideas for algorithm for KGE, cont.
What should trigger ski purchase? 1
Price drops below some level. A good oportunity for A LG. The downside: the price may go down even further and O PT may pay even less.
2
Price exceeds some threshold. Example: rental price
B
Alg = Θ(B 2 ) A
Opt buys here for sA time
One should eventually buy! M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
14 / 24
Results
Upper Bound for Known Game End
Designing an optimal online algorithm for KGE, cont. Algorithm P ROT Has two thresholds A < B. It buys if the rental price falls outside [A, B]. If the rental price remains within [A, B], then it buys after s days. P ROT run on a sequence of length γ = sy , comp. ratio R = O(sL(y ) )
For γ = s, P ROT chooses A = s1/3 and B = s2/3 . M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
15 / 24
Results
Upper Bound for Known Game End
Designing an optimal online algorithm for KGE, cont. Algorithm P ROT Has two thresholds A < B. It buys if the rental price falls outside [A, B]. If the rental price remains within [A, B], then it buys after s days. P ROT run on a sequence of length γ = sy , comp. ratio R = O(sL(y ) ) L(y) 2/5 1/3
2/3 3/5
logs B
1/3
logs A
1/5 1/2 2/3 4/5
1
y
1/2 2/3 4/5
1
y
For γ = s, P ROT chooses A = s1/3 and B = s2/3 . M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
15 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s
Theorem For Known End Scenario with γ = s, P ROT is O(s1/3 )-competitive. (Recall that in this case A = s1/3 , B = s2/3 ).
Observation If the rental price remains in [A, B] for s steps, then the competitive ratio is O(B/A) = O(s1/3 ). (Almost normal ski-rental process). Lemma If the price stays first within [A, B] and drops below A at some point, then P ROT is O(s1/3 )-competitive. (Proof omitted).
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
16 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s
Theorem For Known End Scenario with γ = s, P ROT is O(s1/3 )-competitive. (Recall that in this case A = s1/3 , B = s2/3 ).
Observation If the rental price remains in [A, B] for s steps, then the competitive ratio is O(B/A) = O(s1/3 ). (Almost normal ski-rental process). Lemma If the price stays first within [A, B] and drops below A at some point, then P ROT is O(s1/3 )-competitive. (Proof omitted).
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
16 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s Lemma If the price stays first within [A, B] and exceeds B at time k , then P ROT is O(s1/3 )-competitive. rental price
B B/2
A
k
γ=s
time
Case 1. O PT buys skis till day k + B/2. P ROT k ·B+s·B ≤ = O(B/A) = O(s1/3 ) O PT s·A M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
17 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s Lemma If the price stays first within [A, B] and exceeds B at time k , then P ROT is O(s1/3 )-competitive. rental price
B B/2
X A
k − B/2
k
k + B/2
γ=s
time
Case 1. O PT buys skis till day k + B/2. P ROT k ·B+s·B ≤ = O(B/A) = O(s1/3 ) O PT s·A M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
17 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s Lemma If the price stays first within [A, B] and exceeds B at time k , then P ROT is O(s1/3 )-competitive. rental price
B B/2
X A
k − B/2
k
k + B/2
γ=s
time
Case 2. O PT rents skis till day k + B/2. P ROT X +s·B ≤ = O(s/B) = O(s1/3 ) O PT max{X , (B/2)2 } M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
18 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s Lemma If the price stays first within [A, B] and exceeds B at time k , then P ROT is O(s1/3 )-competitive. rental price
B B/2
X A
k − B/2
k
k + B/2
γ=s
time
Case 2. O PT rents skis till day k + B/2. P ROT X +s·B ≤ = O(s/B) = O(s1/3 ) O PT max{X , (B/2)2 } M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
18 / 24
Results
Upper Bound for Stochastic Game End
Known vs. Stochastic Game End Known Game End: On a sequence of length s, P ROT with thresholds A = s1/3 and B = s2/3 is O(s1/3 )-competitive. What if we run P ROT with these thresholds on sequence of arbitrary, unknown length γ? Then P ROT is Rγ = O(s1/3 · (1 + s/γ))-competitive! Stochastic Game End Scenario: We do not know γ, but we know that γ is geometricaly distributed and E[γ] = Γ. Solution: run P ROT with A and B suited for Γ. For Γ = s, we get Eγ [Rγ ] = O(s1/3 · s · logΓ Γ ) = O(s1/3 · log s). (The same (log s)-approximation result for different Γ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
19 / 24
Results
Upper Bound for Stochastic Game End
Known vs. Stochastic Game End Known Game End: On a sequence of length s, P ROT with thresholds A = s1/3 and B = s2/3 is O(s1/3 )-competitive. What if we run P ROT with these thresholds on sequence of arbitrary, unknown length γ? Then P ROT is Rγ = O(s1/3 · (1 + s/γ))-competitive! Stochastic Game End Scenario: We do not know γ, but we know that γ is geometricaly distributed and E[γ] = Γ. Solution: run P ROT with A and B suited for Γ. For Γ = s, we get Eγ [Rγ ] = O(s1/3 · s · logΓ Γ ) = O(s1/3 · log s). (The same (log s)-approximation result for different Γ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
19 / 24
Results
Upper Bound for Stochastic Game End
Known vs. Stochastic Game End Known Game End: On a sequence of length s, P ROT with thresholds A = s1/3 and B = s2/3 is O(s1/3 )-competitive. What if we run P ROT with these thresholds on sequence of arbitrary, unknown length γ? Then P ROT is Rγ = O(s1/3 · (1 + s/γ))-competitive! Stochastic Game End Scenario: We do not know γ, but we know that γ is geometricaly distributed and E[γ] = Γ. Solution: run P ROT with A and B suited for Γ. For Γ = s, we get Eγ [Rγ ] = O(s1/3 · s · logΓ Γ ) = O(s1/3 · log s). (The same (log s)-approximation result for different Γ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
19 / 24
Results
Upper Bound for Stochastic Game End
Known vs. Stochastic Game End Known Game End: On a sequence of length s, P ROT with thresholds A = s1/3 and B = s2/3 is O(s1/3 )-competitive. What if we run P ROT with these thresholds on sequence of arbitrary, unknown length γ? Then P ROT is Rγ = O(s1/3 · (1 + s/γ))-competitive! Stochastic Game End Scenario: We do not know γ, but we know that γ is geometricaly distributed and E[γ] = Γ. Solution: run P ROT with A and B suited for Γ. For Γ = s, we get Eγ [Rγ ] = O(s1/3 · s · logΓ Γ ) = O(s1/3 · log s). (The same (log s)-approximation result for different Γ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
19 / 24
Results
Upper Bound for Stochastic Game End
Stochastic Game End
In general, in SGE with parameter Γ, E[R] = RPROT(Γ) · log s. Can we improve this result? Yes! Problem with geometric (and with uniform) distribution: it is very likely that the game ends much earlier than Γ. Solution: assume the game ends at Γ/(log s)1/3 and run P ROT with appropriate thresholds A and B. Result: E[R] = RPROT(Γ) · (log s)7/9 .
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
20 / 24
Results
Upper Bound for Stochastic Game End
Stochastic Game End
In general, in SGE with parameter Γ, E[R] = RPROT(Γ) · log s. Can we improve this result? Yes! Problem with geometric (and with uniform) distribution: it is very likely that the game ends much earlier than Γ. Solution: assume the game ends at Γ/(log s)1/3 and run P ROT with appropriate thresholds A and B. Result: E[R] = RPROT(Γ) · (log s)7/9 .
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
20 / 24
Results
Upper Bound for Stochastic Game End
Stochastic Game End
In general, in SGE with parameter Γ, E[R] = RPROT(Γ) · log s. Can we improve this result? Yes! Problem with geometric (and with uniform) distribution: it is very likely that the game ends much earlier than Γ. Solution: assume the game ends at Γ/(log s)1/3 and run P ROT with appropriate thresholds A and B. Result: E[R] = RPROT(Γ) · (log s)7/9 .
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
20 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s. Input construction s
rental price
α
α-peaky sequence
s1/3 1 γ=s
time
α is chosen randomly with distribution π: with prob. 1/3, α = s1/2 with prob. 1/3, α = s with prob. 1/3, α chosen uniformly from [s1/2 , 2s2/3 ]. We show that for any det. alg. D ET, Eπ [D ET/O PT] = Ω(s1/3 ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
21 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s. Input construction s
rental price
α
α-peaky sequence
s1/3 1 time
γ=s
α is chosen randomly with distribution π: with prob. 1/3, α = s1/2 with prob. 1/3, α = s with prob. 1/3, α chosen uniformly from [s1/2 , 2s2/3 ].
s
rental price
2 · s2/3 s1/2 s1/3 1 γ=s
time
We show that for any det. alg. D ET, Eπ [D ET/O PT] = Ω(s1/3 ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
21 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s. Input construction s
rental price
α
α-peaky sequence
s1/3 1 time
γ=s
α is chosen randomly with distribution π: with prob. 1/3, α = s1/2 with prob. 1/3, α = s with prob. 1/3, α chosen uniformly from [s1/2 , 2s2/3 ].
s
rental price
2 · s2/3 α s1/2 s1/3 1 γ=s
time
We show that for any det. alg. D ET, Eπ [D ET/O PT] = Ω(s1/3 ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
21 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. Only two reasonable types of deterministic algorithms: 1
D ET always rents
2
D ET chooses r ∈ [s1/3 , s]. If the price reaches r , D ET buys. (it does not make sense to buy after the peak)
What if D ET chooses “always rent” strategy or chooses r ≥ s2/3 ? With prob. 1/3 input is s-peaky: Eπ
D ET 1 s5/3 ≥ · 4/3 = Ω(s1/3 ) O PT 3 s
Similar argument for r < s1/2 . M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
22 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. Only two reasonable types of deterministic algorithms: 1
D ET always rents
2
D ET chooses r ∈ [s1/3 , s]. If the price reaches r , D ET buys. (it does not make sense to buy after the peak)
What if D ET chooses “always rent” strategy or chooses r ≥ s2/3 ? With prob. 1/3 input is s-peaky: Eπ
D ET 1 s5/3 ≥ · 4/3 = Ω(s1/3 ) O PT 3 s
Similar argument for r < s1/2 . M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
22 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. Only two reasonable types of deterministic algorithms: 1
D ET always rents
2
D ET chooses r ∈ [s1/3 , s]. If the price reaches r , D ET buys. (it does not make sense to buy after the peak)
What if D ET chooses “always rent” strategy or chooses r ≥ s2/3 ? With prob. 1/3 input is s-peaky: Eπ
D ET 1 s5/3 ≥ · 4/3 = Ω(s1/3 ) O PT 3 s
s 2 · s2/3 s1/2 s1/3 1
Similar argument for r < s1/2 . M. Bienkowski (University of Wrocław)
rental price
Price Fluctuations: To Buy or to Rent
Θ(s2 ) γ = s time
22 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. Only two reasonable types of deterministic algorithms: 1
D ET always rents
2
D ET chooses r ∈ [s1/3 , s]. If the price reaches r , D ET buys. (it does not make sense to buy after the peak)
What if D ET chooses “always rent” strategy or chooses r ≥ s2/3 ? With prob. 1/3 input is s-peaky: Eπ
D ET 1 s5/3 ≥ · 4/3 = Ω(s1/3 ) O PT 3 s
s 2 · s2/3 s1/2 s1/3 1
Similar argument for r < s1/2 . M. Bienkowski (University of Wrocław)
rental price
Price Fluctuations: To Buy or to Rent
Θ(s2 ) γ = s time
22 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. D ET chooses r ∈ [s1/2 , s2/3 ]. If the price reaches r , D ET buys. We consider only α-peaky sequences for α ∈R [r , 2s2/3 ] Probability density: µ(α) = 31 · 2s2/31−s1/2 = Θ(s−2/3 ).
Eπ
D ET O PT
2s2/3
s·r · µ(α) · dα O(α2 ) r s · r s·r =Ω − 2/3 · s−2/3 = Ω s1/3 | r {z2s }
≥
Z
≥s−s/2
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
23 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. D ET chooses r ∈ [s1/2 , s2/3 ]. If the price reaches r , D ET buys. We consider only α-peaky sequences for α ∈R [r , 2s2/3 ]
s 2 · s2/3
Probability density: µ(α) = 31 · 2s2/31−s1/2 = Θ(s−2/3 ).
Eπ
D ET O PT
rental price
r s1/2 s1/3 1
γ = s time
2s2/3
s·r · µ(α) · dα O(α2 ) r s · r s·r =Ω − 2/3 · s−2/3 = Ω s1/3 | r {z2s }
≥
Z
≥s−s/2
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
23 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. D ET chooses r ∈ [s1/2 , s2/3 ]. If the price reaches r , D ET buys. We consider only α-peaky sequences for α ∈R [r , 2s2/3 ]
s 2 · s2/3 α
Probability density: µ(α) = 31 · 2s2/31−s1/2 = Θ(s−2/3 ).
Eπ
D ET O PT
rental price
r s1/2 s1/3 1
Θ(α2 ) γ = s time
2s2/3
s·r · µ(α) · dα O(α2 ) r s · r s·r =Ω − 2/3 · s−2/3 = Ω s1/3 | r {z2s }
≥
Z
≥s−s/2
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
23 / 24
Results
Lower Bound for Known Game End
Thank you for you attention!
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
24 / 24
of Computer Science, University of Wrocław
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
1 / 24
Outline
1
Ski Rental
2
Our extension to the model
3
Results Upper Bound for Known Game End Upper Bound for Stochastic Game End Lower Bound for Known Game End
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
2 / 24
Ski Rental
Outline
1
Ski Rental
2
Our extension to the model
3
Results Upper Bound for Known Game End Upper Bound for Stochastic Game End Lower Bound for Known Game End
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
3 / 24
Ski Rental
Ski rental problem (SRP)
“A toy example illustrating rent-or-buy paradigms” A skier At the beginning of each day may either rent skis for p, or buy them for s · p.
At the end of the day, may break legs and skis Input: the leg-breaking day Goal: find a cheapest strategy
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
4 / 24
Ski Rental
Ski rental problem (SRP)
“A toy example illustrating rent-or-buy paradigms” A skier At the beginning of each day may either rent skis for p, or buy them for s · p.
At the end of the day, may break legs and skis Input: the leg-breaking day Goal: find a cheapest strategy
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
4 / 24
Ski Rental
Ski Rental Problem, cont.
Offline setting is trivial Online setting: Best deterministic online algorithm: “rent for s − 1 days and buy on day s”. This strategy is (2 − 1s )-competitive (Karlin, Manasse, Sleator, FOCS 86) (Karp, IFIP 92)
Best randomized online algorithm (chooses a random day from e {1, . . . , s} and then buys on this day) is e−1 ≈ 1.58-competitive (Karlin, Manasse, McGeoch, Owicki, SODA 90)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
5 / 24
Ski Rental
Variations
Many problems exhibit the same “rent-or-buy” structure as SRP, e.g. Bahncard problem
(Fleischer, COCOON 98) (Karlin, Kenyon, Randall, STOC 01)
TCP Acknowledgement problem
(Dooly, Goldman, Scott, JACM 01) (Karlin, Kenyon, Randall, STOC 01)
Page replication problem
(Black, Sleator, CMU TR 89) (Albers, Koga, SWAT 94)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
6 / 24
Our extension to the model
Outline
1
Ski Rental
2
Our extension to the model
3
Results Upper Bound for Known Game End Upper Bound for Stochastic Game End Lower Bound for Known Game End
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
7 / 24
Our extension to the model
Adding dynamics
What if rental price p may change each day? (BUT: the purchase price = s × rental price)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
8 / 24
Our extension to the model
Arbitrary changes of rental price
At day 1, the skier sees the price p = 1. What should she/he do? 1
Skier rents at day 1. What may happen? The prices in next days are equal to k The skier never breaks legs CSKIER ≥ s · k , COPT = s.
2
Skier buys at day 1. What may happen? The prices in next days are equal to 1/k The skier never breaks legs CSKIER = s, COPT = 1 + s/k .
For arbitrary changes, if k = s, the competitive ratio is Ω(s) (large!)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
9 / 24
Our extension to the model
Arbitrary changes of rental price
At day 1, the skier sees the price p = 1. What should she/he do? 1
Skier rents at day 1. What may happen? The prices in next days are equal to k The skier never breaks legs CSKIER ≥ s · k , COPT = s.
2
Skier buys at day 1. What may happen? The prices in next days are equal to 1/k The skier never breaks legs CSKIER = s, COPT = 1 + s/k .
For arbitrary changes, if k = s, the competitive ratio is Ω(s) (large!)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
9 / 24
Our extension to the model
Arbitrary changes of rental price
At day 1, the skier sees the price p = 1. What should she/he do? 1
Skier rents at day 1. What may happen? The prices in next days are equal to k The skier never breaks legs CSKIER ≥ s · k , COPT = s.
2
Skier buys at day 1. What may happen? The prices in next days are equal to 1/k The skier never breaks legs CSKIER = s, COPT = 1 + s/k .
For arbitrary changes, if k = s, the competitive ratio is Ω(s) (large!)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
9 / 24
Our extension to the model
Arbitrary changes of rental price
At day 1, the skier sees the price p = 1. What should she/he do? 1
Skier rents at day 1. What may happen? The prices in next days are equal to k The skier never breaks legs CSKIER ≥ s · k , COPT = s.
2
Skier buys at day 1. What may happen? The prices in next days are equal to 1/k The skier never breaks legs CSKIER = s, COPT = 1 + s/k .
For arbitrary changes, if k = s, the competitive ratio is Ω(s) (large!)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
9 / 24
Our extension to the model
This paper
Additive model of price changes: rental price p is always at least 1 in two consecutive days prices differ by at most 1.
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
10 / 24
Results
Outline
1
Ski Rental
2
Our extension to the model
3
Results Upper Bound for Known Game End Upper Bound for Stochastic Game End Lower Bound for Known Game End
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
11 / 24
Results
Our Contribution We characterize the competitive ratio R in three scenarios (both for deterministic and randomized algorithms): 1
√ Unknown Game End (UGE): R = Θ( s).
2
Known Game End (KGE): For game duration γ = sy , R = Θ(sL(y ) )
3
Stochastic Game End (SGE): Probability of breaking leg = 1/Γ, expected duration = Γ = sy , E[R] = O(sL(y ) · (log s)7/9 ) This scenario, but without price fluctuations, considered by Fujiwara and Iwama (ISAAC 02)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
12 / 24
Results
Our Contribution We characterize the competitive ratio R in three scenarios (both for deterministic and randomized algorithms): 1 2
√ Unknown Game End (UGE): R = Θ( s). Known Game End (KGE): For game duration γ = sy , R = Θ(sL(y ) ) L(y) 2/5 1/3
1/2 2/3 4/5
3
1
y
Stochastic Game End (SGE): Probability of breaking leg = 1/Γ, expected duration = Γ = sy , E[R] = O(sL(y ) · (log s)7/9 ) This scenario, but without price fluctuations, considered by Fujiwara and Iwama (ISAAC 02)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
12 / 24
Results
Our Contribution We characterize the competitive ratio R in three scenarios (both for deterministic and randomized algorithms): 1 2
√ Unknown Game End (UGE): R = Θ( s). Known Game End (KGE): For game duration γ = sy , R = Θ(sL(y ) ) L(y) 2/5 1/3
1/2 2/3 4/5
3
1
y
Stochastic Game End (SGE): Probability of breaking leg = 1/Γ, expected duration = Γ = sy , E[R] = O(sL(y ) · (log s)7/9 ) This scenario, but without price fluctuations, considered by Fujiwara and Iwama (ISAAC 02)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
12 / 24
Results
Our Contribution We characterize the competitive ratio R in three scenarios (both for deterministic and randomized algorithms): 1 2
√ Unknown Game End (UGE): R = Θ( s). Known Game End (KGE): This talk (upper + lower for γ = s) For game duration γ = sy , R = Θ(sL(y ) ) L(y) 2/5 1/3
1/2 2/3 4/5
3
1
y
Stochastic Game End (SGE): This talk Probability of breaking leg = 1/Γ, expected duration = Γ = sy , E[R] = O(sL(y ) · (log s)7/9 ) This scenario, but without price fluctuations, considered by Fujiwara and Iwama (ISAAC 02)
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
12 / 24
Results
Upper Bound for Known Game End
Ideas for algorithm for KGE What should trigger ski purchase? In real life, there are two types of people: Those, who buy shares because their price is rising Those, who buy shares because their price is falling down
Apparently, we may learn from both types!
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
13 / 24
Results
Upper Bound for Known Game End
Ideas for algorithm for KGE What should trigger ski purchase? In real life, there are two types of people: Those, who buy shares because their price is rising Those, who buy shares because their price is falling down
Apparently, we may learn from both types!
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
13 / 24
Results
Upper Bound for Known Game End
Ideas for algorithm for KGE, cont.
What should trigger ski purchase? 1
Price drops below some level. A good oportunity for A LG. The downside: the price may go down even further and O PT may pay even less.
2
Price exceeds some threshold. Example: One should eventually buy!
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
14 / 24
Results
Upper Bound for Known Game End
Ideas for algorithm for KGE, cont.
What should trigger ski purchase? 1
Price drops below some level. A good oportunity for A LG. The downside: the price may go down even further and O PT may pay even less.
2
Price exceeds some threshold. Example: rental price
B
Alg = Θ(B 2 ) A
Opt buys here for sA time
One should eventually buy! M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
14 / 24
Results
Upper Bound for Known Game End
Designing an optimal online algorithm for KGE, cont. Algorithm P ROT Has two thresholds A < B. It buys if the rental price falls outside [A, B]. If the rental price remains within [A, B], then it buys after s days. P ROT run on a sequence of length γ = sy , comp. ratio R = O(sL(y ) )
For γ = s, P ROT chooses A = s1/3 and B = s2/3 . M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
15 / 24
Results
Upper Bound for Known Game End
Designing an optimal online algorithm for KGE, cont. Algorithm P ROT Has two thresholds A < B. It buys if the rental price falls outside [A, B]. If the rental price remains within [A, B], then it buys after s days. P ROT run on a sequence of length γ = sy , comp. ratio R = O(sL(y ) ) L(y) 2/5 1/3
2/3 3/5
logs B
1/3
logs A
1/5 1/2 2/3 4/5
1
y
1/2 2/3 4/5
1
y
For γ = s, P ROT chooses A = s1/3 and B = s2/3 . M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
15 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s
Theorem For Known End Scenario with γ = s, P ROT is O(s1/3 )-competitive. (Recall that in this case A = s1/3 , B = s2/3 ).
Observation If the rental price remains in [A, B] for s steps, then the competitive ratio is O(B/A) = O(s1/3 ). (Almost normal ski-rental process). Lemma If the price stays first within [A, B] and drops below A at some point, then P ROT is O(s1/3 )-competitive. (Proof omitted).
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
16 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s
Theorem For Known End Scenario with γ = s, P ROT is O(s1/3 )-competitive. (Recall that in this case A = s1/3 , B = s2/3 ).
Observation If the rental price remains in [A, B] for s steps, then the competitive ratio is O(B/A) = O(s1/3 ). (Almost normal ski-rental process). Lemma If the price stays first within [A, B] and drops below A at some point, then P ROT is O(s1/3 )-competitive. (Proof omitted).
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
16 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s Lemma If the price stays first within [A, B] and exceeds B at time k , then P ROT is O(s1/3 )-competitive. rental price
B B/2
A
k
γ=s
time
Case 1. O PT buys skis till day k + B/2. P ROT k ·B+s·B ≤ = O(B/A) = O(s1/3 ) O PT s·A M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
17 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s Lemma If the price stays first within [A, B] and exceeds B at time k , then P ROT is O(s1/3 )-competitive. rental price
B B/2
X A
k − B/2
k
k + B/2
γ=s
time
Case 1. O PT buys skis till day k + B/2. P ROT k ·B+s·B ≤ = O(B/A) = O(s1/3 ) O PT s·A M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
17 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s Lemma If the price stays first within [A, B] and exceeds B at time k , then P ROT is O(s1/3 )-competitive. rental price
B B/2
X A
k − B/2
k
k + B/2
γ=s
time
Case 2. O PT rents skis till day k + B/2. P ROT X +s·B ≤ = O(s/B) = O(s1/3 ) O PT max{X , (B/2)2 } M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
18 / 24
Results
Upper Bound for Known Game End
Performance of P ROT for γ = s Lemma If the price stays first within [A, B] and exceeds B at time k , then P ROT is O(s1/3 )-competitive. rental price
B B/2
X A
k − B/2
k
k + B/2
γ=s
time
Case 2. O PT rents skis till day k + B/2. P ROT X +s·B ≤ = O(s/B) = O(s1/3 ) O PT max{X , (B/2)2 } M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
18 / 24
Results
Upper Bound for Stochastic Game End
Known vs. Stochastic Game End Known Game End: On a sequence of length s, P ROT with thresholds A = s1/3 and B = s2/3 is O(s1/3 )-competitive. What if we run P ROT with these thresholds on sequence of arbitrary, unknown length γ? Then P ROT is Rγ = O(s1/3 · (1 + s/γ))-competitive! Stochastic Game End Scenario: We do not know γ, but we know that γ is geometricaly distributed and E[γ] = Γ. Solution: run P ROT with A and B suited for Γ. For Γ = s, we get Eγ [Rγ ] = O(s1/3 · s · logΓ Γ ) = O(s1/3 · log s). (The same (log s)-approximation result for different Γ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
19 / 24
Results
Upper Bound for Stochastic Game End
Known vs. Stochastic Game End Known Game End: On a sequence of length s, P ROT with thresholds A = s1/3 and B = s2/3 is O(s1/3 )-competitive. What if we run P ROT with these thresholds on sequence of arbitrary, unknown length γ? Then P ROT is Rγ = O(s1/3 · (1 + s/γ))-competitive! Stochastic Game End Scenario: We do not know γ, but we know that γ is geometricaly distributed and E[γ] = Γ. Solution: run P ROT with A and B suited for Γ. For Γ = s, we get Eγ [Rγ ] = O(s1/3 · s · logΓ Γ ) = O(s1/3 · log s). (The same (log s)-approximation result for different Γ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
19 / 24
Results
Upper Bound for Stochastic Game End
Known vs. Stochastic Game End Known Game End: On a sequence of length s, P ROT with thresholds A = s1/3 and B = s2/3 is O(s1/3 )-competitive. What if we run P ROT with these thresholds on sequence of arbitrary, unknown length γ? Then P ROT is Rγ = O(s1/3 · (1 + s/γ))-competitive! Stochastic Game End Scenario: We do not know γ, but we know that γ is geometricaly distributed and E[γ] = Γ. Solution: run P ROT with A and B suited for Γ. For Γ = s, we get Eγ [Rγ ] = O(s1/3 · s · logΓ Γ ) = O(s1/3 · log s). (The same (log s)-approximation result for different Γ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
19 / 24
Results
Upper Bound for Stochastic Game End
Known vs. Stochastic Game End Known Game End: On a sequence of length s, P ROT with thresholds A = s1/3 and B = s2/3 is O(s1/3 )-competitive. What if we run P ROT with these thresholds on sequence of arbitrary, unknown length γ? Then P ROT is Rγ = O(s1/3 · (1 + s/γ))-competitive! Stochastic Game End Scenario: We do not know γ, but we know that γ is geometricaly distributed and E[γ] = Γ. Solution: run P ROT with A and B suited for Γ. For Γ = s, we get Eγ [Rγ ] = O(s1/3 · s · logΓ Γ ) = O(s1/3 · log s). (The same (log s)-approximation result for different Γ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
19 / 24
Results
Upper Bound for Stochastic Game End
Stochastic Game End
In general, in SGE with parameter Γ, E[R] = RPROT(Γ) · log s. Can we improve this result? Yes! Problem with geometric (and with uniform) distribution: it is very likely that the game ends much earlier than Γ. Solution: assume the game ends at Γ/(log s)1/3 and run P ROT with appropriate thresholds A and B. Result: E[R] = RPROT(Γ) · (log s)7/9 .
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
20 / 24
Results
Upper Bound for Stochastic Game End
Stochastic Game End
In general, in SGE with parameter Γ, E[R] = RPROT(Γ) · log s. Can we improve this result? Yes! Problem with geometric (and with uniform) distribution: it is very likely that the game ends much earlier than Γ. Solution: assume the game ends at Γ/(log s)1/3 and run P ROT with appropriate thresholds A and B. Result: E[R] = RPROT(Γ) · (log s)7/9 .
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
20 / 24
Results
Upper Bound for Stochastic Game End
Stochastic Game End
In general, in SGE with parameter Γ, E[R] = RPROT(Γ) · log s. Can we improve this result? Yes! Problem with geometric (and with uniform) distribution: it is very likely that the game ends much earlier than Γ. Solution: assume the game ends at Γ/(log s)1/3 and run P ROT with appropriate thresholds A and B. Result: E[R] = RPROT(Γ) · (log s)7/9 .
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
20 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s. Input construction s
rental price
α
α-peaky sequence
s1/3 1 γ=s
time
α is chosen randomly with distribution π: with prob. 1/3, α = s1/2 with prob. 1/3, α = s with prob. 1/3, α chosen uniformly from [s1/2 , 2s2/3 ]. We show that for any det. alg. D ET, Eπ [D ET/O PT] = Ω(s1/3 ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
21 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s. Input construction s
rental price
α
α-peaky sequence
s1/3 1 time
γ=s
α is chosen randomly with distribution π: with prob. 1/3, α = s1/2 with prob. 1/3, α = s with prob. 1/3, α chosen uniformly from [s1/2 , 2s2/3 ].
s
rental price
2 · s2/3 s1/2 s1/3 1 γ=s
time
We show that for any det. alg. D ET, Eπ [D ET/O PT] = Ω(s1/3 ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
21 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s. Input construction s
rental price
α
α-peaky sequence
s1/3 1 time
γ=s
α is chosen randomly with distribution π: with prob. 1/3, α = s1/2 with prob. 1/3, α = s with prob. 1/3, α chosen uniformly from [s1/2 , 2s2/3 ].
s
rental price
2 · s2/3 α s1/2 s1/3 1 γ=s
time
We show that for any det. alg. D ET, Eπ [D ET/O PT] = Ω(s1/3 ). M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
21 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. Only two reasonable types of deterministic algorithms: 1
D ET always rents
2
D ET chooses r ∈ [s1/3 , s]. If the price reaches r , D ET buys. (it does not make sense to buy after the peak)
What if D ET chooses “always rent” strategy or chooses r ≥ s2/3 ? With prob. 1/3 input is s-peaky: Eπ
D ET 1 s5/3 ≥ · 4/3 = Ω(s1/3 ) O PT 3 s
Similar argument for r < s1/2 . M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
22 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. Only two reasonable types of deterministic algorithms: 1
D ET always rents
2
D ET chooses r ∈ [s1/3 , s]. If the price reaches r , D ET buys. (it does not make sense to buy after the peak)
What if D ET chooses “always rent” strategy or chooses r ≥ s2/3 ? With prob. 1/3 input is s-peaky: Eπ
D ET 1 s5/3 ≥ · 4/3 = Ω(s1/3 ) O PT 3 s
Similar argument for r < s1/2 . M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
22 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. Only two reasonable types of deterministic algorithms: 1
D ET always rents
2
D ET chooses r ∈ [s1/3 , s]. If the price reaches r , D ET buys. (it does not make sense to buy after the peak)
What if D ET chooses “always rent” strategy or chooses r ≥ s2/3 ? With prob. 1/3 input is s-peaky: Eπ
D ET 1 s5/3 ≥ · 4/3 = Ω(s1/3 ) O PT 3 s
s 2 · s2/3 s1/2 s1/3 1
Similar argument for r < s1/2 . M. Bienkowski (University of Wrocław)
rental price
Price Fluctuations: To Buy or to Rent
Θ(s2 ) γ = s time
22 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. Only two reasonable types of deterministic algorithms: 1
D ET always rents
2
D ET chooses r ∈ [s1/3 , s]. If the price reaches r , D ET buys. (it does not make sense to buy after the peak)
What if D ET chooses “always rent” strategy or chooses r ≥ s2/3 ? With prob. 1/3 input is s-peaky: Eπ
D ET 1 s5/3 ≥ · 4/3 = Ω(s1/3 ) O PT 3 s
s 2 · s2/3 s1/2 s1/3 1
Similar argument for r < s1/2 . M. Bienkowski (University of Wrocław)
rental price
Price Fluctuations: To Buy or to Rent
Θ(s2 ) γ = s time
22 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. D ET chooses r ∈ [s1/2 , s2/3 ]. If the price reaches r , D ET buys. We consider only α-peaky sequences for α ∈R [r , 2s2/3 ] Probability density: µ(α) = 31 · 2s2/31−s1/2 = Θ(s−2/3 ).
Eπ
D ET O PT
2s2/3
s·r · µ(α) · dα O(α2 ) r s · r s·r =Ω − 2/3 · s−2/3 = Ω s1/3 | r {z2s }
≥
Z
≥s−s/2
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
23 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. D ET chooses r ∈ [s1/2 , s2/3 ]. If the price reaches r , D ET buys. We consider only α-peaky sequences for α ∈R [r , 2s2/3 ]
s 2 · s2/3
Probability density: µ(α) = 31 · 2s2/31−s1/2 = Θ(s−2/3 ).
Eπ
D ET O PT
rental price
r s1/2 s1/3 1
γ = s time
2s2/3
s·r · µ(α) · dα O(α2 ) r s · r s·r =Ω − 2/3 · s−2/3 = Ω s1/3 | r {z2s }
≥
Z
≥s−s/2
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
23 / 24
Results
Lower Bound for Known Game End
Lower bound for KGE, γ = s, cont. D ET chooses r ∈ [s1/2 , s2/3 ]. If the price reaches r , D ET buys. We consider only α-peaky sequences for α ∈R [r , 2s2/3 ]
s 2 · s2/3 α
Probability density: µ(α) = 31 · 2s2/31−s1/2 = Θ(s−2/3 ).
Eπ
D ET O PT
rental price
r s1/2 s1/3 1
Θ(α2 ) γ = s time
2s2/3
s·r · µ(α) · dα O(α2 ) r s · r s·r =Ω − 2/3 · s−2/3 = Ω s1/3 | r {z2s }
≥
Z
≥s−s/2
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
23 / 24
Results
Lower Bound for Known Game End
Thank you for you attention!
M. Bienkowski (University of Wrocław)
Price Fluctuations: To Buy or to Rent
24 / 24
