This document presents algorithms for mining revenue-maximizing bundling configurations from consumer preference data. It discusses how willingness to pay for items can be estimated from online ratings data. The bundle configuration problem of grouping items into bundles to maximize total revenue is formulated and shown to be NP-hard for bundles of size 3 or more. Heuristic algorithms based on graph matching and greedy approaches are proposed to solve the problem approximately. The algorithms are evaluated on a real dataset of Amazon book ratings, demonstrating increased revenue from bundling over selling items individually.

1. Mining Revenue-Maximizing
Bundling Configuration
Loc Do*, Hady Lauw*, Ke Wang+
*Singapore Management University
+Simon Fraser University

2. Bundling
• Selling two or more items at one price
• Beneficial to consumers by lowering the
overall price (compared to buying the items
separately)
• Beneficial to sellers by gaining higher overall
revenue (from selling more items to more
consumers)

5. 0
2
4
6
8
10
12
14
1 2 3
Price(dollar)
No of consumers willing to pay the price or higher
The lower the price, the more consumers
willing to pay at least that price
Demand curve plots the number of consumers
willing to pay at least a certain price

6. 0
2
4
6
8
10
12
14
1 2 3
Price(dollar)
No of consumers willing to pay the price or higher
Seller
revenue
Consumer
surplus
Deadweight
loss
Suppose we now fix the price at $8

7. Willingness to Pay for Items
Consumer Willingness
to Pay
Buy A at
price = 8?
Seller
revenue
Consumer
surplus
Deadweigh
t loss
u1 $12 ✔ $8 $4 0
u2 $8 ✔ $8 0 0
u3 $5 ✖ 0 0 $5
Total 16 4 5

8. Willingness to Pay for Bundles
• A function of willingness to pay for items
– may incorporate discount factors for
complementary vs. substitute products
– assumed to be linear sum in this talk for simplicity
Consume
r
Item A Item B Bundle AB
u1 $12 $4 $16
u2 $8 $2 $10
u3 $5 $11 $16

9. Individual Items – Revenue: $27
Consu
mer
WA WB Pricing A
at $8
Pricing B
at $11
u1 $12 $4 ✓
u2 $8 $2 ✓
u3 $5 $11 ✓

10. Consu
mer
WA WB Pricing A
at $8
Pricing B
at $11
u1 $12 $4 ✓
u2 $8 $2 ✓
u3 $5 $11 ✓
Consu
mer
WA WB WAB Pricing AB at $16
u1 $12 $4 $16 ✓ ✓
u2 $8 $2 $10
u3 $5 $11 $16 ✓ ✓
Components – Revenue: $27
Pure Bundling – Revenue: $32

11. Challenges
• It works on some cases, but not for others
• Challenge #1 (Data):
– Dependent on willingness to pay profile of
consumers in the population
• Challenge #2 (Computation):
– Combination of items in a potential bundle
– For N items, up to 2N-1 potential bundles
– Many possible configurations of bundles

12. Traditional Marketing Approach to
Willingness to Pay
• Relying on explicit solicitation from consumers
– e.g., market research, surveys
– unscalable
• Conjoint analysis:
– decompose consumer preference for items to
preferences for specific features
• Discrete choice theory:
– give consumer a choice of two products with
different attributes, and observe the selection

13. Key Idea
• Leverage on the wealth of consumer
preference data

14. Estimating Willingness to Pay from Ratings
• Willingness to pay for an item is a function of:
– intrinsic value of the item (sales price)
– extrinsic value of the item to the user (rating)
• A simple model: linear function of price and rating
• Richer models: may consider features or regression
Rating Willingness to Pay
5 $12.50
4 $10.00
3 $7.50
2 $5.00
1 $2.50
Item sales price = $10
Rating base = 4
Using this mapping, we find that revenue-maximizing pricing is close
to Amazon’s actual pricing.

15. Modeling More Realistic Adoption Behaviors
15
Willingness to Pay = $10

16. Determination of “Optimal” Price
• Find the price that optimizes the expected
revenue of an item or a bundle
• Discretized price levels
– computation through bucketing strategies

17. Bundle Configuration Problem
• Given:
– a set of M users
– a set of N items
– M x N matrix of willingness to pay for items
• willingness to pay for bundles implicitly modeled by a function
– a positive integer k (maximum size of a bundle)
• Find:
– a partitioning of the N items
– each sub-partition is of size no larger than k
– resulting in the highest total revenue
17

18. Assumptions
• Revenue maximization objective
– Profit maximization if marginal cost is known
– Incorporate other factors e.g., consumer surplus
• Single unit demand
– can be relaxed if demanded quantity is known
• No budget constraint
• No supply constraint

19. Matching (graph theory)
• A set of edges without common vertices
– maximum: most number of edges
– maximum weight: highest total edge weights

20. Optimal Solution for k = 2 with Graph Matching
Consum
er
Item A
pA = 10
Item B
pB = 10
Item C
pC = 10
Bundle AB
pAB = 14.5
Bundle AC
pAC = 12
Bundle BC
pBC = 12
u1 $10.0 $10.0 $2.0 $20.0 $12.0 $12.0
u2 $10.0 $4.5 $10.0 $14.5 $20.0 $14.5
u3 $4.5 $10.0 $10.0 $14.5 $14.5 $20.0
A B
C
20 20
20
36 36
43.5
(a) Input Graph (b) Maximum Weight Matching
A B
C
20
43.5

21. How about for k = 3 and beyond?
Maximum weight hypergraph matching is NP-hard
A B
C
20 20
20
36 36
43.5
66
Co
nsu
me
r
Item A
pA =
10
Item B
pB = 10
Item C
pC =
10
Bundle AB
pAB = 14.5
Bundle AC
pAC = 12
Bundle BC
pBC = 12
Bundle ABC
pABC = 22
u1 $10.0 $10.0 $2.0 $20.0 $12.0 $12.0 $22.0
u2 $10.0 $4.5 $10.0 $14.5 $20.0 $14.5 $24.5
u3 $4.5 $10.0 $10.0 $14.5 $14.5 $20.0 $24.5
A B
C
66

22. 3-uniform hypergraph
A B
C
Maximum matching on 3-uniform hypergraph is
known to be NP-hard
C D
E

23. Bundle Configuration is NP-hard for k ≥ 3
A B
C
3-uniform hypergraph matching is an instance of
3-sized bundle configuration problem
A B
C
1 1
1
2 2
2
3 + e

24. How about for k = 3 and beyond?
Maximum weight hypergraph matching is NP-hard
A B
C
20 20
20
36 36
43.5
66
Co
nsu
me
r
Item A
pA =
10
Item B
pB = 10
Item C
pC =
10
Bundle AB
pAB = 14.5
Bundle AC
pAC = 12
Bundle BC
pBC = 12
Bundle ABC
pABC = 22
u1 $10.0 $10.0 $2.0 $20.0 $12.0 $12.0 $22.0
u2 $10.0 $4.5 $10.0 $14.5 $20.0 $14.5 $24.5
u3 $4.5 $10.0 $10.0 $14.5 $14.5 $20.0 $24.5
A B
C
66

25. Heuristic#1: Iterative Graph Matching (k=2)
A B
C
20 20
20
36 36
43.5
(1a) Input Graph (1b) Output Matching
Iteration #1
A B
C
20
43.5
C
AB
20
43.5
66 ABC66
(2a) Input Graph (2b) Output Matching
Iteration #2

26. Heuristic #2: Greedy Algorithm
• Begin with a configuration of individual items
(i.e., bundles of size 1)
• Iterate till no more revenue gain:
– For each pair of bundles
• Compute the expected revenue gain
– Merge the pair with highest gain into a new bundle

27. Heuristic #1:
Iterative Matching
Heuristic #2:
Greedy Algorithm
In each iteration, only
one new bundle is
formed.
In each iteration, up to
N/2 new bundles are
formed.
O(MN2) for revenue
computation.
O(N2.5) for matching.
O(MN2) for revenue
computation.
O(N2log N) for greedy.

28. Bundling Strategies
A ✓
B
AB
✓
✓
✓
✓
✓
Individual
Components
Pure
Bundling
Mixed
Bundling

29. Consu
mer
WA WB WAB Pricing AB at $16
u1 $12 $4 $16 ✓ ✓
u2 $8 $2 $10
u3 $5 $11 $16 ✓ ✓
Pure Bundling – Revenue: $32
Consu
mer
WA WB WAB Pricing A at $8
Pricing AB at $16
u1 $12 $4 $16 ✓ ✓
u2 $8 $2 $10 ✓
u3 $5 $11 $16 ✓ ✓
Mixed Bundling – Revenue: $40

30. Data
• User preferences as indicated by ratings
• Amazon ratings on books (UIC)
• 4,449 users, and 5,028 items
30
Rating Frequency Percentage
5 52,811 49%
4 31,499 29%
3 14,356 13%
2 5,841 5%
1 3,784 3%
Total 108,291 100%
Item Price Frequency Percentage
<= $5 57 1%
$5.01 - $10 2,464 49%
$10.01 - $15 1,280 25%
$15.01 - $20 994 20%
$20.01 - $25 140 3%
> $25 93 2%
Total 5,028 100%

31. Metrics
• Revenue coverage
– total possible revenue = sum all willingness to pay
– percentage of total possible revenue that is captured
– up to 100%
• Revenue gain
– compare to revenue from selling items (no bundles)
– percentage gain of bundling over non-bundling

32. Methods
• Mixed bundling:
– MixedMatching
– MixedGreedy
– MixedFreqItemset (baseline)
• Pure bundling:
– PureMatching
– PureGreedy
– PureFreqItemset (baseline)

34. Revenue Gain vs. Time

35. Scalability

36. Comparison to Optimal Solution (ILP)
Method N = 10 N = 15 N = 20 N = 25
PureMatching 78.1 77.8 77.9 77.2
PureGreedy 78.1 77.8 77.9 77.2
Optimal (ILP) 78.1 77.8 77.9 -
Revenue Coverage (percent)
Method N = 10 N = 15 N = 20 N = 25
PureMatching 0.01 0.01 0.01 0.02
PureGreedy 0.07 0.10 0.13 0.16
Optimal (ILP) 0.20 4.60 235.38 -
Running Time (seconds)

37. Conclusion
• Willingness to pay:
– mining consumer preferences from ratings data
• Bundle configuration:
– NP-hard for k ≥ 3
– Graph matching and greedy algorithms
• Future work:
– Generalization to non-monetary utility functions