1. A Game-theoretic Analysis of Catalog Optimization JOEL OREN, UNIVERSITY OF TORONTO. JOINT WORK WITH: NINA NARODYTSKA, AND CRAIG BOUTILIER 1

2. Motivating Story: Competitive Adjustment of Offerings A large retail chain opens a new store. Multiple competitors. Multiple potential customers: Typically doesn’t buy too many items – say, just one item. Buy their most preferred item, given what is offered in total – over all stores. Exogenous (fixed) prices. How should they choose what to offer, so as to maximize their profits? A form of assortment optimization. 2

3. Catalog 1 3 Vendor 1 Vendor 2 $10 $5 $15 $4 $8 Catalog 1 $10 $15 $8 $10 Catalog: a set (assortment) of offered items. Best-response: Optimizing one’s catalog may be tricky – What are the convergence properties of these dynamics? 1.Do pure Nash eq. (PNE) exist? 2.What is the Price of Anarchy/Stability, (PoA/PoS)? 1 50 1 X 100

4. The Formal Model 4

5. Selecting the Best Response – the Full Information Setting Theorem: Computing a best response is Max-SNP hard. Implication: there is a constant, such that approximating the maximal profit beyond this constant is NP-hard. 5

6. A Special Case: Single-Peaked Truncated Preference 6

7. Partial Information Setting 7 Catalog 1 Vendor 1 Vendor 2 $10 $5 $15 $4 $8 Catalog 1 $10 $1 5 $8

8. Partial Information Setting 8 Catalog 1 Vendor 1 Vendor 2 $10 $5 $15 $4 $8 Catalog 1 $10 $15 $8

9. Equilibria and Stability of the Game 9 Full information Partial information - IC Mutual sets All vendors’ sets are disjoint

10. Full Information, Disjoint Sets 10

11. Partial Information, Disjoint Sets 11

12. Full Information, Mutual Sets 12

13. Partial Information, Mutual Sets 13

14. Conclusions and Future Directions Best response: Max-SNP hard in general, easier under some assumptions. Question: can we design an approximation algorithm for the full-info. case? Game theoretic analysis: PoA/PoS under complete VS partial information, disjoint VS mutual sets. Question: can we show that there is always a PNE under a general Mallows dist.? Additional directions: Other classes of preferences. Study the game when prices are set endogenously. 14