A Game-theoretic Analysis of Catalog Optimization

1

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

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

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 ≻ 50 ⋯≻≻ X 100

4

The Formal Model•The Catalog Selection game: • strategic vendors (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 rankings over .• 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.

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.

7

Catalog 1

Vendor 1 Vendor 2

$10$5

$15$4

$8

≻ ≻ ≻ ≻

Catalog 1

$10$15

$8

𝐷

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 .

8

Catalog 1

Vendor 1 Vendor 2$10

$5$15

$4

$8

≻ ≻ ≻ ≻

Catalog 1

$10$15

$8

𝐷

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.

9

Full information

Partial information - IC

Mutual sets

All vendors’ sets are disjoint

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

2 𝑥 𝑥+𝜖

2 𝑥𝑥+𝜖

𝑥+𝜖

𝑥+𝜖

2 𝑥2 𝑥

𝑥+𝜖 2 𝑥 𝑥+𝜖

2 𝑥

𝑥+𝜖

2 𝑥

𝑥+𝜖 2 𝑥

10

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:

1. Given , best set of size is the set of most expensive items in .2. 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.

11

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.

12

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.

13

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