A Game-theoretic Analysis of
Catalog Optimization
JOEL OREN, UNIVERSITY OF TORONTO.
JOINT WORK WITH: NINA NARODYTSKA, AND CRAIG BOUTILIER
1
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: a set (assortment) of offered
items.
• Best-response: Optimizing one’s catalog
may be tricky –
≻
1
50
≻ ⋯
Vendor 1
$10
≻
≻
50
$5
$4
$15
$8
Catalog 1
$10
1
Vendor 2
Catalog 1
$15
$8
≻ ⋯ X 100
• 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)?
$10
≻
≻
≻
≻
3
The Formal Model
•The Catalog Selection game: 𝐺 = 𝐶 1 , … , 𝐶 𝑘 , 𝒑, 𝑁, 𝝅 .
•𝑘 strategic vendors (agents):
• Sets of items (𝐶 1 , … , 𝐶 𝑘 ) (not necessarily disjoint). Total # of items is 𝑚 =
𝑖
𝐶
. Each item has unlimited number of copies.
𝑖
• An exogenous price vector 𝑝 ∈ 𝑅+𝑚 .
• Vendor 𝑖’s strategy: a catalog 𝑅 ⊆ 𝐶 𝑖 . Strategy profile 𝐒 = (𝑆1 , … , 𝑆 𝑘 ).
• Each vendor’s goal is to maximize revenue, 𝑟𝑖 (𝑆𝑖 , 𝑺−𝒊 ).
•Set N of unit-demand consumers with rankings 𝜋 = (𝜋1 , … , 𝜋𝑛 ) over
• Each consumer 𝑖 buys her most preferred item 𝑐𝑖 in
𝑖.
𝑆
𝑖
𝑖.
𝐶
𝑖
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Selecting the Best Response – the Full
Information Setting
•The Full Information setting: the consumers’ preference profile 𝝅 is commonly
known.
•Given 𝑺−𝒊 , how should vendor 𝑖 select 𝑆 𝑖 ?
◦ Cheap items in 𝐶 𝑖 may be commonly preferred over expensive items in 𝐶 𝑖 .
◦ Not adding certain items in 𝐶 𝑖 -- 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 𝑖: they’re of length 𝐿, and they are single-peaked.
•Result: we can optimize vendor 𝑖’s best-response using a dynamicprogramming approach.
•See paper for details.
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Partial Information Setting
•Consumer rankings are unknown in
advance. Instead, they are drawn
from a commonly known
distribution 𝐷.
•Best response:
𝑎𝑟𝑔𝑚𝑎𝑥𝑆⊆𝑅𝑖 𝐸𝝅∼𝐷 [𝑟𝑖 𝑆, 𝑹−𝒋 ] .
•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
$10
Vendor 2
$5
$4
$15
Catalog 1
$10
≻
Catalog 1
$1
5
≻
$8
$8
≻
≻
𝐷
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Partial Information Setting
•Consumer preferences are unknown in advance.
Instead, they are drawn from a commonly known
distribution 𝐷.
•Best response: 𝑎𝑟𝑔𝑚𝑎𝑥𝑆⊆𝑅𝑖 𝐸𝝅∼𝐷 [𝑟𝑖
𝑆, 𝑹−𝒋
].
•Mallows 𝜑 −distribution: each 𝜋𝑖 ∼ 𝐷(𝜋, 𝜑), where
Pr 𝜋 = 𝜑 𝑑𝐾 (𝜋,𝜋) /𝑍𝑚 , and 𝑑𝐾 (⋅,⋅) is the Kendall’s 𝜏distance.
• 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
[ 𝑝1 , 𝐷1 , 𝑝2 , 𝐷2 , … , 𝑝𝑟 , 𝐷𝑟 ].
Vendor 2
Vendor 1
$10
$4
$5
$8
$15
Catalog 1
Catalog 1
$15
$10
≻
$8
≻
≻
≻
𝐷
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Equilibria and Stability of the Game
•The Catalog Selection game: 𝐺 =
𝐶 1 , … , 𝐶 𝑘 , 𝒑, 𝑁, 𝝅
All vendors’
Mutual
sets are
sets
disjoint
•Does this game admit PNE? If so, what are
the guarantees on them?
•The social outcome: total profit.
Full
information
•Price of Anarchy (PoA): ratio of the OPT social
Partial
to the worst-case PNE.
•Price of Stability (PoS): ratio of the OPT social
to the best PNE.
information - IC
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Full Information, Disjoint Sets
•𝝅 = (𝜋1 , … , 𝜋𝑛 ), 𝐶 𝑖 ∩ 𝐶𝑗 = ∅ for all 𝑖 ≠ 𝑗.
•There exist instances of the game w/o PNE.
𝑥+𝜖
≻
𝑥+𝜖
≻
2𝑥
≻
2𝑥
𝑥+𝜖
𝑥+𝜖
𝑥+𝜖
≻
≻
2𝑥
2𝑥
≻
≻
≻
𝑥+𝜖
2𝑥
≻
2𝑥
𝑥+𝜖
(2𝑥 + 2𝜖, 𝑥 + 𝜖)
(2𝑥 + 2𝜖, 2𝑥)
(2𝑥, 2𝑥 + 2𝜖)
(4𝑥, 2𝑥 + 𝜖)
2𝑥
𝑥+𝜖
2𝑥
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Partial Information, Disjoint Sets
•𝜋𝑖 ∼ 𝐷 𝜋, 𝜑 . If 𝜑 = 0, all preferences equal 𝜋 -- trivial PNE. So assume 𝜑 = 1
– an Impartial Culture (IC) – preferences are drawn u.i.d.
•Result: there always exists a PNE.
• Intuition:
1. Given 𝑹−𝒋 , best set of size 𝑡 is the set of 𝑡 most expensive items in 𝐶𝑗 .
2. If number of other vendors’ item increased – vendor 𝑗 can only bestrespond by adding items, never removing items.
•Result: The POS is Ω(𝑚) – the social welfare of the best PNE can be a
1
−fraction of the optimal welfare.
𝑚
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Full Information, Mutual Sets
•The preference profile 𝝅, is fully known. 𝐶 1 = ⋯ = 𝐶 𝑘 .
•If an item is offered by 𝑟 vendors, payment for it is split among them evenly.
•Observation: there is always a PNE: (𝐶 1 , … , 𝐶 𝑘 ) – no vendor has an incentive
to remove any items.
•Result: The PoS is Ω(2𝑚 ).
• Use a price vector that constitutes an “approximately” geometric series:
1
4
1
(1,
2
+
𝜖, + 2𝜖, … ). 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 Ω(𝑚). Idea: one item of value 1, 𝑚 − 1 items of value 𝜖. If
1
everyone selects all items, first item is picked with probability .
𝑚
𝑚⋅𝑘
𝑂(
)
log 𝑚
•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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