Online Resource Allocation Problems, with Applications to ... · PDF fileOnline Resource...

Online Resource Allocation Problems, with Applicationsto Revenue Management

David Simchi-Levi (MIT)

David Simchi-Levi (MIT) Online Resource Allocation 0 / 22

Introduction

Papers

1 Tight Weight-dependent Competitive Ratios for Online MatchingProblems

joint work with Will Ma

2 Dynamic Recommendation at Checkout under Inventory Constraint

joint work with Xi Chen, Will Ma, and Linwei Xin

Online Resource Allocation

Papers

1 Tight Weight-dependent Competitive Ratios for OnlineMatching Problems

joint work with Will Ma

2 Dynamic Recommendation at Checkout under Inventory Constraint

joint work with Xi Chen, Will Ma, and Linwei Xin

Online Resource Allocation Problem Definition

A General Online Resource Allocation Problem

Known at start:number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r

(1)i < . . . < r

(mi )i

For t = 1, 2, . . .1 Customer t Arrives:

observe p(j)t,i , the probability of customer t buying item i at price j , for

all i and j (can be based on customer features or classes)2 Decision:

offer an item it which has not stocked out, at a price jt , to customer t3 Realization:

customer makes purchase with probability p(jt)t,it

if she purchases, then earn revenue r(jt)it

and decrement inventory of itExtensions:

can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption

Known at start:

number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r

(1)i < . . . < r

(mi )i

Known at start:number of items (resources), n

unreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r

(1)i < . . . < r

(mi )i

Known at start:number of items (resources), nunreplenishable starting inventory of each item i , ki

mi potential prices of each item i , satisfying r(1)i < . . . < r

(mi )i

(1)i < . . . < r

(mi )i

(1)i < . . . < r

(mi )i

For t = 1, 2, . . .

1 Customer t Arrives:observe p

(j)t,i , the probability of customer t buying item i at price j , for

(1)i < . . . < r

(mi )i

all i and j (can be based on customer features or classes)

2 Decision:offer an item it which has not stocked out, at a price jt , to customer t

3 Realization:customer makes purchase with probability p

(jt)t,it

(1)i < . . . < r

(mi )i

offer an item it which has not stocked out, at a price jt , to customer t

3 Realization:customer makes purchase with probability p

(jt)t,it

(1)i < . . . < r

(mi )i

and decrement inventory of it

Extensions:can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption

(1)i < . . . < r

(mi )i

Competitive Ratio

compare algorithm’s performance vs. optimum which knows all arrivalinformation at the start

develop algorithms whose competitive ratio

infinstance I (incl. arrivals)

E[ALG(I)]

OPT(I)

is bounded by a constant independent of the number ofitems/customers

algorithm makes no assumption on arrival sequence

Competitive Ratio

E[ALG(I)]

OPT(I)

Competitive Ratio

E[ALG(I)]

OPT(I)

Competitive Ratio

E[ALG(I)]

OPT(I)

Online Resource Allocation Intuition

Key Challenges

1 How do we prioritize between different items to be sold?

Price of : $150

Price of : $200

2 When do we reserve the inventory of an item for customers willing topay higher prices?

Potential prices of : $150, $450

Key Challenges

Price of : $150

Price of : $200

Key Challenges

Price of : $150

Price of : $200

Key Challenges

Price of : $150

Price of : $200

Key Challenges

Price of : $150

Price of : $200

Challenge 1: Multiple Items

Price of : $150

Price of : $200

Solution: “Inventory Balancing”

balance the depletion of different items, relative to their prices

idea has been used in the:online b-matching problem (Kalyanasundaram/Pruhs ’00)Adwords problem (Mehta et al. ’05, Buchbinder/Jain/Naor ’07)personalized assortment problem(Golrezaei/Nazerzadeh/Rusmevichientong ’14)

Price of : $150

Price of : $200

Price of : $150

Price of : $200

Price of : $150

Price of : $200

Challenge 2: Multiple Prices

Solution: “Booking Limits”

stop offering the item at the lower price after some fraction has beensold

idea has been used in the:

one-way-trading problem (Lavi/Nisan ’00, El-Yaniv et al. ’01)single-leg booking problem (Ball/Queyranne ’09)online dynamic pricing problem (Ma/S.-L./Teo ’17)

Online Resource Allocation Example

Combining the Challenges

Potential prices for : $150, $450

Potential prices for : $200

p = .2

Bid-price Control Policy: for each customer t, offer the item i andprice j maximizing the expected “profit”

p(j)t,i (r

(j)i − λi ),

where λi is the “cost”, or bid price, for one unit of inventory of i

Key Question: what is the value of λi?

p = .2

p(j)t,i (r

(j)i − λi ),

p = .2

p(j)t,i (r

(j)i − λi ),

p = .2

p(j)t,i (r

(j)i − λi ),

The Value of Inventory

Bid-price Control Policy:

maxi ,j

p(j)t,i (r

(j)i − λi )

Typical definition of λi :

solve a deterministic LP based on the forecasted demand over theremaining time horizon

λi is the “reduced cost” of the inventory constraint on item i

Our definition of λi :

let wi be the fraction of item i ’s starting inventory already sold

λi = Φi (wi ), where Φi is an increasing function

maxi ,j

p(j)t,i (r

(j)i − λi )

maxi ,j

p(j)t,i (r

(j)i − λi )

maxi ,j

p(j)t,i (r

(j)i − λi )

maxi ,j

p(j)t,i (r

(j)i − λi )

maxi ,j

p(j)t,i (r

(j)i − λi )

maxi ,j

p(j)t,i (r

(j)i − λi )

Returning to the Example

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p = .2

p(j)t,i (r

(j)i − λi )

= .2(450− 50)= 80

p(j)t,i (r

(j)i − λi )

= 1(200− 150)= 50

value of is low: λi = Φi (15 ) ≈ 50 (most units still remaining)

value of is high: λi = Φi (34 ) ≈ 150 (most units already sold)

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p = .2

p(j)t,i (r

(j)i − λi )

= .2(450− 50)= 80

p(j)t,i (r

(j)i − λi )

= 1(200− 150)= 50

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p = .2

p(j)t,i (r

(j)i − λi )

= .2(450− 50)= 80

p(j)t,i (r

(j)i − λi )

= 1(200− 150)= 50

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p = .2

p(j)t,i (r

(j)i − λi )

= .2(450− 50)= 80

p(j)t,i (r

(j)i − λi )

= 1(200− 150)= 50

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p = .2

p(j)t,i (r

(j)i − λi )

= .2(450− 50)= 80

p(j)t,i (r

(j)i − λi )

= 1(200− 150)= 50

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p = .2

p(j)t,i (r

(j)i − λi )

= .2(450− 50)= 80

p(j)t,i (r

(j)i − λi )

= 1(200− 150)= 50

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p = .2

p(j)t,i (r

(j)i − λi )

= .2(450− 50)= 80

p(j)t,i (r

(j)i − λi )

= 1(200− 150)= 50

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p(j)t,i (r

(j)i − λi )

= 1(150− 50)= 100

p = .2

p(j)t,i (r

(j)i − λi )

= .2(450− 50)= 80

p(j)t,i (r

(j)i − λi )

= 1(200− 150)= 50

Exact Value Function Φi for i =

Fraction Sold, wi

Φi (wi )

Φi increases from 0 to themaximum price of $450 over [0, 1]

Φi is piecewise-convex

the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price

the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α

α = ln 2(r−1)√1+4r(r−1)/e−1

, where r is the

ratio of high price to low price

different than the optimal single-itembooking limit σ = r

2r−1of

Ball/Queyranne1 5

σ = r2r−1

α = ln2(r−1)√

1+4r(r−1)/e−1

Fraction Sold, wi

Φi (wi )

α = ln 2(r−1)√1+4r(r−1)/e−1

, where r is the

2r−1of

Ball/Queyranne1 5

σ = r2r−1

α = ln2(r−1)√

1+4r(r−1)/e−1

Fraction Sold, wi

Φi (wi )

α = ln 2(r−1)√1+4r(r−1)/e−1

, where r is the

2r−1of

Ball/Queyranne1 5

σ = r2r−1

α = ln2(r−1)√

1+4r(r−1)/e−1

Fraction Sold, wi

Φi (wi )

α = ln 2(r−1)√1+4r(r−1)/e−1

, where r is the

2r−1of

Ball/Queyranne1 5

σ = r2r−1

α = ln2(r−1)√

1+4r(r−1)/e−1

Fraction Sold, wi

Φi (wi )

α = ln 2(r−1)√1+4r(r−1)/e−1

, where r is the

2r−1of

Ball/Queyranne1 5

σ = r2r−1

α = ln2(r−1)√

1+4r(r−1)/e−1

Fraction Sold, wi

Φi (wi )

α = ln 2(r−1)√1+4r(r−1)/e−1

, where r is the

2r−1of

Ball/Queyranne1 5

σ = r2r−1

α = ln2(r−1)√

1+4r(r−1)/e−1

Fraction Sold, wi

Φi (wi )

α = ln 2(r−1)√1+4r(r−1)/e−1

, where r is the

2r−1of

Ball/Queyranne1 5

σ = r2r−1

α = ln2(r−1)√

1+4r(r−1)/e−1

Fraction Sold, wi

Φi (wi )

α = ln 2(r−1)√1+4r(r−1)/e−1

, where r is the

2r−1of

Ball/Queyranne

1 5r0.5

σ = r2r−1

α = ln2(r−1)√

1+4r(r−1)/e−1

Fraction Sold, wi

Φi (wi )

α = ln 2(r−1)√1+4r(r−1)/e−1

, where r is the

2r−1of

Ball/Queyranne1 5

σ = r2r−1

α = ln2(r−1)√

1+4r(r−1)/e−1

Online Resource Allocation Algorithm and Analysis

General Value Function Φi for 2 Prices

when mi = 2, Φi is piecewise-convex with 2 pieces, separated by α:

ewi−1eα−1

r(1)i + (r

(2)i − r

(1)i ) ewi−α−1

e1−α−1

α is given by α(r(2)i /r

(1)i ), where α(r) = ln 2(r−1)√

1+4r(r−1)/e−1

the competitive ratio associated with Φi , CRi , is then 1− e−α

ewi−1eα−1

r(1)i + (r

(2)i − r

(1)i ) ewi−α−1

e1−α−1

(1)i ), where α(r) = ln 2(r−1)√

1+4r(r−1)/e−1

ewi−1eα−1

r(1)i + (r

(2)i − r

(1)i ) ewi−α−1

e1−α−1

(1)i ), where α(r) = ln 2(r−1)√

1+4r(r−1)/e−1

ewi−1eα−1

r(1)i + (r

(2)i − r

(1)i ) ewi−α−1

e1−α−1

(1)i ), where α(r) = ln 2(r−1)√

1+4r(r−1)/e−1

General Value Function Φi for Multiple Prices

for any number mi of prices, Φi is designed, based on r(1)i , . . . , r

(mi )i , to

maximize CRi

we derive Φi by solving a differential equation arising from a primal-dualanalysis

in general, Φi is piecewise-convex with mi pieces, separated by mi − 1

booking limits α(1)i , . . . , α

(mi−1)i

α(1)i , . . . , α

(mi−1)i correspond to roots of a degree-mi polynomial, which can

be computed using bisection search

(mi )i , to

maximize CRi

(mi−1)i

α(1)i , . . . , α

(mi )i , to

maximize CRi

(mi−1)i

α(1)i , . . . , α

(mi )i , to

maximize CRi

(mi−1)i

α(1)i , . . . , α

(mi )i , to

maximize CRi

(mi−1)i

α(1)i , . . . , α

Overall Algorithm

maxi ,j

p(j)t,i (r

(j)i − λi )

Having derived Φ1, . . . ,Φn, two different algorithms are possible:1 Algorithm 1:

λi = Φi (wi ), where wi is the fraction of item i ’s starting inventory soldwi is dynamically incremented as sales are realized

2 Algorithm 2:

more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i

Overall Algorithm

maxi ,j

p(j)t,i (r

(j)i − λi )

Having derived Φ1, . . . ,Φn, two different algorithms are possible:

1 Algorithm 1:

2 Algorithm 2:

Overall Algorithm

maxi ,j

p(j)t,i (r

(j)i − λi )

2 Algorithm 2:

Overall Algorithm

maxi ,j

p(j)t,i (r

(j)i − λi )

2 Algorithm 2:

Overall Algorithm

maxi ,j

p(j)t,i (r

(j)i − λi )

2 Algorithm 2:

more applicable when the starting inventories ki are small

assume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i

Overall Algorithm

maxi ,j

p(j)t,i (r

(j)i − λi )

2 Algorithm 2:

more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all i

then λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i

Overall Algorithm

maxi ,j

p(j)t,i (r

(j)i − λi )

2 Algorithm 2:

more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]

Wi is a random seed which is initialized independently for each i

Overall Algorithm

maxi ,j

p(j)t,i (r

(j)i − λi )

2 Algorithm 2:

The Competitive Ratio

largestarting

inventories

Algorithm 1achieves mini CRi

deterministiccustomer

purchasing

Algorithm 1= Algorithm 2

counterexample counterexample counterexample

We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi

largestarting

inventories

purchasing

largestarting

inventories

purchasing

largestarting

inventories

purchasing

largestarting

inventories

purchasing

largestarting

inventories

purchasing

largestarting

inventories

purchasing

The Optimal Competitive Ratio

largestarting

inventories

purchasing

largestarting

inventories

purchasing

counterexample

counterexample counterexample

largestarting

inventories

purchasing

counterexample

largestarting

inventories

purchasing

counterexample counterexample

counterexample

Illustration of Competitive Ratios with Two Prices per Item

r = maxir

General Stochastic Setting

1k ∞

k = mini ki

StochasticPurchasing

DeterministicPurchasing

Multiple Items

Single Item

tight results

non-tight results

increasingcompetitive ratios

1− 1e≈ .632

[KP00,MSVV05,BJN07,GNR14]

1− 1e≈ .632

[KVV90,AGKM11,DJK13]

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14][ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16][ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14][ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16][ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14][ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16][ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14][ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16][ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14][ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16][ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16][ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12][ 1

2,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16][ 1

4,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387][↙, e−1

2e−1≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14]

[ 12,≈ .621][MP12]

[ 12,≈ .621]

r2(2r−1)

[MS17]

[CMSX16]

[ 14,↗][CMSX16]

[ 14,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387]

[↙, e−12e−1

≈ .387]

r = maxir

1k ∞

k = mini ki

Multiple Items

Single Item

tight results

non-tight results

1− 1e≈ .632

1− e−α(r)[MS17]

1− 1√e≈ .393

[MP12,GNR14][ 12,≈ .621][MP12]

[ 12,≈ .621]

r2(2r−1)

[MS17]

[CMSX16][ 14,↗][CMSX16]

[ 14,↗]

1−1/e

(1+k)(1−e−1/k )[MS17]

1−e−α(r)

(1+k)(e1/k−1)[MS17]

r2r−1

[BQ09]

1− 1e

[MST17][↙, e−1

2e−1≈ .387]

[↙, e−12e−1

≈ .387]

Online Resource Allocation Simulations

Simulations on Hotel Data Set

publicly-accessible hotel data set from MSOM journal(Bodea/Ferguson/Garrow ’09)

multiple items (room categories) with multiple prices (rates)

Room Category Discounted Rate Rack Rate Number of Rooms