A Game-theoretic Analysis of Catalog Optimization

1. A Game-theoretic Analysis of Catalog Optimization Joel oren, university of Toronto. Joint work with: ninanarodytska, and craigboutilier

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.

3. Vendor 1 Vendor2 • Catalog: a set (assortment) of offered items. • Best-response: Optimizing one’s catalog may be tricky – • What are the convergence properties of these dynamics? • Do pure Nash eq. (PNE) exist? • What is the Price of Anarchy/Stability, (PoA/PoS)? $10 $5 $4 $15 $8 Catalog 1 50 Catalog 1 1 $8 $15 $10 50 1 X 100 $10

4. The Formal Model • The Catalog Selection game: • strategicvendors(agents): • Sets of items (not necessarily disjoint). Total # of items is . Each item has unlimited number of copies. • An exogenous price vector . • Vendor ’s strategy: a catalog . Strategy profile . • Each vendor’s goal is to maximize revenue, . • Set of unit-demand consumers with rankingsover . • Each consumer buys her most preferred item in .

5. Selecting the Best Response – the Full Information Setting • The Full Information setting: the consumers’ preference profile is commonly known. • Given , how should vendor select ? • Cheap items in may be commonly preferred over expensive items in . • Not adding certain items in -- may lose consumers due to competition. 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.

6. A Special Case: Single-Peaked Truncated Preference • Single-peaked, L-truncated preferences: if looking at the prefixes composed of only vendor : they’re of length and they are single-peaked. • Result: we can optimize vendor ’s best-response using a dynamic-programming approach. • See paper for details.

7. Partial Information Setting • Consumer rankings are unknown in advance. Instead, they are drawn from a commonly known distribution . • Best response: . • Warmup: preferences are drawn u.i.d. from the complete set of preferences. • Idea:greedily add items until expected revenue starts to decline. Vendor 1 Vendor2 $10 $8 $5 $4 $15 Catalog 1 Catalog 1 $15 $10 $8

8. Partial Information Setting • Consumer preferences are unknown in advance. Instead, they are drawn from a commonly known distribution . • Best response: . • Mallows distribution: each , where , and is the Kendall’s -distance. • Result: There is a polytime DP algorithm for optimizing the best response. • Algorithm can be generalized to handle mixtures of Mallows distributions; i.e., lotteries . Vendor 1 Vendor2 $10 $8 $5 $4 $15 Catalog 1 Catalog 1 $15 $8 $10

9. Equilibria and Stability of the Game • The Catalog Selection game: • Does this game admit PNE? If so, what are the guarantees on them? • The social outcome:total profit. • Price of Anarchy (PoA): ratio of the OPT social to the worst-case PNE. • Price of Stability (PoS): ratio of the OPT social to the best PNE. All vendors’ sets are disjoint Mutual sets Full information Partial information - IC

10. Full Information, Disjoint Sets • , for all . • There exist instances of the game w/o PNE.

11. Partial Information, Disjoint Sets • . If , all preferences equal -- trivial PNE. So assume – an Impartial Culture (IC) – preferences are drawn u.i.d. • Result: there always exists a PNE. • Intuition: • Given, best set of size is the set of most expensive items in . • If number of other vendors’ item increased – vendor can only best-respond by adding items, never removing items. • Result: The POS is – the social welfare of the best PNE can be a fraction of the optimal welfare.

12. Full Information, Mutual Sets • The preference profile, is fully known. . • If an item is offered by vendors, payment for it is split among them evenly. • Observation: there is always a PNE: – no vendor has an incentive to remove any items. • Result: The PoS is . • Use a price vector that constitutes an “approximately” geometric series: . A single consumer who ranks items in increasing order of price. • Cheapest item will always be offered, and bought.

13. Partial Information, Mutual Sets • Preferences uphold the Impartial Culture assumption. • A PNE always exists. • PoA: at least . Idea: one item of value , items of value . If everyone selects all items, first item is picked with probability . • Price of Stability: – ratio of the optimal pure Nash eq. to the optimal social welfare. • Idea: constructively find a Nash eq., by adding items in decreasing order prices; lower-bound value upon termination.

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.