Wang ke mining revenue-maximizing bundling configuration
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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.
Wang ke mining revenue-maximizing bundling configuration
- 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.
- 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
Editor's Notes
- #31: + 834,251 users + 385,425 items + 1,743,804 ratings (remove some duplicated ratings)