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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.

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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

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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 .

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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.

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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.

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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 .

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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.

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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

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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.

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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.

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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.

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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.

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