choice set optimization under discrete choice models of...
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Choice Set Optimization Under Discrete ChoiceModels of Group Decisions
Kiran Tomlinson and Austin R. Benson
Department of Computer Science, Cornell University
ICML 2020
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Discrete choice models
Goal
Model human choices
Given a set of items, produce probability distribution
Multinomial logit (MNL) model (McFadden, 1974)
Choice set
Utility 2 3 2 1 1
↓ softmax
Choice prob. 0.18 0.50 0.18 0.07 0.07
Pr(choose x from choice set C ) =exp(ux)∑y∈C exp(uy )
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Discrete choice models
Goal
Model human choicesGiven a set of items, produce probability distribution
Multinomial logit (MNL) model (McFadden, 1974)
Choice set
Utility 2 3 2 1 1
↓ softmax
Choice prob. 0.18 0.50 0.18 0.07 0.07
Pr(choose x from choice set C ) =exp(ux)∑y∈C exp(uy )
1 / 19
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Discrete choice models
Goal
Model human choicesGiven a set of items, produce probability distribution
Multinomial logit (MNL) model (McFadden, 1974)
Choice set
Utility 2 3 2 1 1
↓ softmax
Choice prob. 0.18 0.50 0.18 0.07 0.07
Pr(choose x from choice set C ) =exp(ux)∑y∈C exp(uy )
1 / 19
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Discrete choice models
Goal
Model human choicesGiven a set of items, produce probability distribution
Multinomial logit (MNL) model (McFadden, 1974)
Choice set
Utility 2 3 2 1 1
↓ softmax
Choice prob. 0.18 0.50 0.18 0.07 0.07
Pr(choose x from choice set C ) =exp(ux)∑y∈C exp(uy )
1 / 19
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Discrete choice models
Goal
Model human choicesGiven a set of items, produce probability distribution
Multinomial logit (MNL) model (McFadden, 1974)
Choice set
Utility 2 3 2 1 1
↓ softmax
Choice prob. 0.18 0.50 0.18 0.07 0.07
Pr(choose x from choice set C ) =exp(ux)∑y∈C exp(uy )
1 / 19
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The choice set influences preferences
e.g., preference for red fruit:
choice set 1
4 1
choice set 2
2 1 2
Not expressible with MNL
Context effects are common(Huber et al., 1982; Simonson & Tversky, 1992; Shafir et al., 1993; Trueblood et al., 2013)
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The choice set influences preferences
e.g., preference for red fruit:
choice set 1
4 1
choice set 2
2 1 2
Not expressible with MNL
Context effects are common(Huber et al., 1982; Simonson & Tversky, 1992; Shafir et al., 1993; Trueblood et al., 2013)
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The choice set influences preferences
e.g., preference for red fruit:
choice set 1
4 1
choice set 2
2 1 2
Not expressible with MNL
Context effects are common(Huber et al., 1982; Simonson & Tversky, 1992; Shafir et al., 1993; Trueblood et al., 2013)
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The choice set influences preferences
e.g., preference for red fruit:
choice set 1
4 1
choice set 2
2 1 2
Not expressible with MNL
Context effects are common(Huber et al., 1982; Simonson & Tversky, 1992; Shafir et al., 1993; Trueblood et al., 2013)
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Title breakdown
Choice Set Optimization UnderDiscrete Choice Models of Group Decisions
choice set
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Title breakdown
Choice Set Optimization UnderDiscrete Choice Models of Group Decisions
choice set
choosers
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Title breakdown
Choice Set Optimization UnderDiscrete Choice Models of Group Decisions
choice set
adults
1 3 2
1
children
1 2 1
1
(pretrained)
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Title breakdown
Choice Set Optimization UnderDiscrete Choice Models of Group Decisions
choice set
adult
1 3 2
1
child
1 2 1
1
(pretrained)
3 / 19
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Title breakdown
Choice Set Optimization UnderDiscrete Choice Models of Group Decisions
choice set
adult
1 2 1
1
child
2 1 3
1
(pretrained)
3 / 19
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Title breakdown
Choice Set Optimization UnderDiscrete Choice Models of Group Decisions
choice set
adult
1 2 1
1
child
2 1 3
1
(pretrained)
3 / 19
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Title breakdown
Choice Set Optimization UnderDiscrete Choice Models of Group Decisions
choice set
adult
1 3 2 1
child
1 2 1 1
(pretrained)
3 / 19
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Our contributions
Central algorithmic question
How can we influence the preferences of a group of decision-makers byintroducing new alternatives?
1 Objectives: optimize agreement, promote an itemChoice models: MNL, context effect models (NL, CDM, EBA)
2 Optimizing agreement is NP-hard in all models (two people!)
3 Promoting an item is NP-hard with context effects
4 Restrictions can make promotion easy but leave agreement hard
5 Poly-time ε-additive approximation for small groups
6 Fast MIBLP for MNL agreement in larger groups
∗
∗See paper
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Our contributions
Central algorithmic question
How can we influence the preferences of a group of decision-makers byintroducing new alternatives?
1 Objectives: optimize agreement, promote an itemChoice models: MNL, context effect models (NL, CDM, EBA)
2 Optimizing agreement is NP-hard in all models (two people!)
3 Promoting an item is NP-hard with context effects
4 Restrictions can make promotion easy but leave agreement hard
5 Poly-time ε-additive approximation for small groups
6 Fast MIBLP for MNL agreement in larger groups
∗
∗See paper
4 / 19
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Our contributions
Central algorithmic question
How can we influence the preferences of a group of decision-makers byintroducing new alternatives?
1 Objectives: optimize agreement, promote an itemChoice models: MNL, context effect models (NL, CDM, EBA)
2 Optimizing agreement is NP-hard in all models (two people!)
3 Promoting an item is NP-hard with context effects
4 Restrictions can make promotion easy but leave agreement hard
5 Poly-time ε-additive approximation for small groups
6 Fast MIBLP for MNL agreement in larger groups
∗
∗See paper
4 / 19
![Page 21: Choice Set Optimization Under Discrete Choice Models of ...kt/publication/2020-tomlinson-optimizin… · Discrete choice models Goal Model human choices Given a set of items, produce](https://reader034.vdocuments.net/reader034/viewer/2022050112/5f49cbdf7e9af15440006fea/html5/thumbnails/21.jpg)
Our contributions
Central algorithmic question
How can we influence the preferences of a group of decision-makers byintroducing new alternatives?
1 Objectives: optimize agreement, promote an itemChoice models: MNL, context effect models (NL, CDM, EBA)
2 Optimizing agreement is NP-hard in all models (two people!)
3 Promoting an item is NP-hard with context effects
4 Restrictions can make promotion easy but leave agreement hard
5 Poly-time ε-additive approximation for small groups
6 Fast MIBLP for MNL agreement in larger groups
∗
∗See paper
4 / 19
![Page 22: Choice Set Optimization Under Discrete Choice Models of ...kt/publication/2020-tomlinson-optimizin… · Discrete choice models Goal Model human choices Given a set of items, produce](https://reader034.vdocuments.net/reader034/viewer/2022050112/5f49cbdf7e9af15440006fea/html5/thumbnails/22.jpg)
Our contributions
Central algorithmic question
How can we influence the preferences of a group of decision-makers byintroducing new alternatives?
1 Objectives: optimize agreement, promote an itemChoice models: MNL, context effect models (NL, CDM, EBA)
2 Optimizing agreement is NP-hard in all models (two people!)
3 Promoting an item is NP-hard with context effects
4 Restrictions can make promotion easy but leave agreement hard
5 Poly-time ε-additive approximation for small groups
6 Fast MIBLP for MNL agreement in larger groups
∗
∗See paper
4 / 19
![Page 23: Choice Set Optimization Under Discrete Choice Models of ...kt/publication/2020-tomlinson-optimizin… · Discrete choice models Goal Model human choices Given a set of items, produce](https://reader034.vdocuments.net/reader034/viewer/2022050112/5f49cbdf7e9af15440006fea/html5/thumbnails/23.jpg)
Our contributions
Central algorithmic question
How can we influence the preferences of a group of decision-makers byintroducing new alternatives?
1 Objectives: optimize agreement, promote an itemChoice models: MNL, context effect models (NL, CDM, EBA)
2 Optimizing agreement is NP-hard in all models (two people!)
3 Promoting an item is NP-hard with context effects
4 Restrictions can make promotion easy but leave agreement hard
5 Poly-time ε-additive approximation for small groups
6 Fast MIBLP for MNL agreement in larger groups
∗
∗See paper
4 / 19
![Page 24: Choice Set Optimization Under Discrete Choice Models of ...kt/publication/2020-tomlinson-optimizin… · Discrete choice models Goal Model human choices Given a set of items, produce](https://reader034.vdocuments.net/reader034/viewer/2022050112/5f49cbdf7e9af15440006fea/html5/thumbnails/24.jpg)
Our contributions
Central algorithmic question
How can we influence the preferences of a group of decision-makers byintroducing new alternatives?
1 Objectives: optimize agreement, promote an itemChoice models: MNL, context effect models (NL, CDM, EBA)
2 Optimizing agreement is NP-hard in all models (two people!)
3 Promoting an item is NP-hard with context effects
4 Restrictions can make promotion easy but leave agreement hard
5 Poly-time ε-additive approximation for small groups
6 Fast MIBLP for MNL agreement in larger groups
∗
∗See paper
4 / 19
![Page 25: Choice Set Optimization Under Discrete Choice Models of ...kt/publication/2020-tomlinson-optimizin… · Discrete choice models Goal Model human choices Given a set of items, produce](https://reader034.vdocuments.net/reader034/viewer/2022050112/5f49cbdf7e9af15440006fea/html5/thumbnails/25.jpg)
Our contributions
Central algorithmic question
How can we influence the preferences of a group of decision-makers byintroducing new alternatives?
1 Objectives: optimize agreement, promote an itemChoice models: MNL, context effect models (NL, CDM, EBA)
2 Optimizing agreement is NP-hard in all models (two people!)
3 Promoting an item is NP-hard with context effects
4 Restrictions can make promotion easy but leave agreement hard
5 Poly-time ε-additive approximation for small groups
6 Fast MIBLP for MNL agreement in larger groups∗
∗See paper
4 / 19
![Page 26: Choice Set Optimization Under Discrete Choice Models of ...kt/publication/2020-tomlinson-optimizin… · Discrete choice models Goal Model human choices Given a set of items, produce](https://reader034.vdocuments.net/reader034/viewer/2022050112/5f49cbdf7e9af15440006fea/html5/thumbnails/26.jpg)
Three models accounting for context effects
Nested logit (NL) (McFadden, 1978)
Context-dependent random utility model (CDM) (Seshadri et al., 2019)
Elimination-by-aspects (EBA) (Tversky, 1972)
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Three models accounting for context effects
Nested logit (NL) (McFadden, 1978)
start
red fruit
11 4
11
1
repeated softmaxover node children
Context-dependent random utility model (CDM) (Seshadri et al., 2019)
Elimination-by-aspects (EBA) (Tversky, 1972)
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Three models accounting for context effects
Nested logit (NL) (McFadden, 1978)
Context-dependent random utility model (CDM) (Seshadri et al., 2019)
pxy
0 −1 0 −1
0 0 0 0
−1 0 0 −1
0 0 0 0
−1 0 −1 0
softmax overpull-adjusted utilities:
ux +∑z∈C
pzx
Elimination-by-aspects (EBA) (Tversky, 1972)
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Three models accounting for context effects
Nested logit (NL) (McFadden, 1978)
Context-dependent random utility model (CDM) (Seshadri et al., 2019)
Elimination-by-aspects (EBA) (Tversky, 1972)
item aspects
{berry, red, sweet}
{berry, purple, sweet}
{red, crunchy}
{citrus, yellow, sour}
{red, sweet}
utility for each aspect
repeatedly choose an aspect,eliminate items without it
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Three models accounting for context effects
Nested logit (NL) (McFadden, 1978)
Context-dependent random utility model (CDM) (Seshadri et al., 2019)
Elimination-by-aspects (EBA) (Tversky, 1972)
item aspects
{berry, red, sweet}
{berry, purple, sweet}
{red, crunchy}
{citrus, yellow, sour}
{red, sweet}
utility for each aspect
repeatedly choose an aspect,eliminate items without it
5 / 19
![Page 31: Choice Set Optimization Under Discrete Choice Models of ...kt/publication/2020-tomlinson-optimizin… · Discrete choice models Goal Model human choices Given a set of items, produce](https://reader034.vdocuments.net/reader034/viewer/2022050112/5f49cbdf7e9af15440006fea/html5/thumbnails/31.jpg)
Three models accounting for context effects
Nested logit (NL) (McFadden, 1978)
Context-dependent random utility model (CDM) (Seshadri et al., 2019)
Elimination-by-aspects (EBA) (Tversky, 1972)
Notes
1 NL, CDM, and EBA all subsume MNL2 These are all random utility models (RUMs) (Block & Marschak, 1960)
3 Can learn utilities from choice data (SGD on NLL)
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Three models accounting for context effects
Nested logit (NL) (McFadden, 1978)
Context-dependent random utility model (CDM) (Seshadri et al., 2019)
Elimination-by-aspects (EBA) (Tversky, 1972)
Notes
1 NL, CDM, and EBA all subsume MNL
2 These are all random utility models (RUMs) (Block & Marschak, 1960)
3 Can learn utilities from choice data (SGD on NLL)
5 / 19
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Three models accounting for context effects
Nested logit (NL) (McFadden, 1978)
Context-dependent random utility model (CDM) (Seshadri et al., 2019)
Elimination-by-aspects (EBA) (Tversky, 1972)
Notes
1 NL, CDM, and EBA all subsume MNL2 These are all random utility models (RUMs) (Block & Marschak, 1960)
3 Can learn utilities from choice data (SGD on NLL)
5 / 19
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Three models accounting for context effects
Nested logit (NL) (McFadden, 1978)
Context-dependent random utility model (CDM) (Seshadri et al., 2019)
Elimination-by-aspects (EBA) (Tversky, 1972)
Notes
1 NL, CDM, and EBA all subsume MNL2 These are all random utility models (RUMs) (Block & Marschak, 1960)
3 Can learn utilities from choice data (SGD on NLL)
5 / 19
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Outline
1 Overview
2 Agreement, Disagreement, and Promotion
3 Hardness Results
4 Approximation Algorithm
5 Experimental Results
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Problem setup
A
UC C Z
set of individuals making choices A
universe of items Uinitial choice set C ⊆ Upossible new alternatives C = U \ Cset of alternatives Z ⊆ C we add to C
choice probabilities Pr(a← x | C ∪ Z ) for each person a and item x
Choice set optimization
Find Z ⊆ C that optimizes some function of Pr(a← x | C ∪ Z )
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Problem setup
A U
C C Z
set of individuals making choices A
universe of items U
initial choice set C ⊆ Upossible new alternatives C = U \ Cset of alternatives Z ⊆ C we add to C
choice probabilities Pr(a← x | C ∪ Z ) for each person a and item x
Choice set optimization
Find Z ⊆ C that optimizes some function of Pr(a← x | C ∪ Z )
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Problem setup
A UC
C Z
set of individuals making choices A
universe of items Uinitial choice set C ⊆ U
possible new alternatives C = U \ Cset of alternatives Z ⊆ C we add to C
choice probabilities Pr(a← x | C ∪ Z ) for each person a and item x
Choice set optimization
Find Z ⊆ C that optimizes some function of Pr(a← x | C ∪ Z )
7 / 19
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Problem setup
A UC C
Z
set of individuals making choices A
universe of items Uinitial choice set C ⊆ Upossible new alternatives C = U \ C
set of alternatives Z ⊆ C we add to C
choice probabilities Pr(a← x | C ∪ Z ) for each person a and item x
Choice set optimization
Find Z ⊆ C that optimizes some function of Pr(a← x | C ∪ Z )
7 / 19
![Page 40: Choice Set Optimization Under Discrete Choice Models of ...kt/publication/2020-tomlinson-optimizin… · Discrete choice models Goal Model human choices Given a set of items, produce](https://reader034.vdocuments.net/reader034/viewer/2022050112/5f49cbdf7e9af15440006fea/html5/thumbnails/40.jpg)
Problem setup
A UC C Z
set of individuals making choices A
universe of items Uinitial choice set C ⊆ Upossible new alternatives C = U \ Cset of alternatives Z ⊆ C we add to C
choice probabilities Pr(a← x | C ∪ Z ) for each person a and item x
Choice set optimization
Find Z ⊆ C that optimizes some function of Pr(a← x | C ∪ Z )
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Problem setup
A UC C Z
set of individuals making choices A
universe of items Uinitial choice set C ⊆ Upossible new alternatives C = U \ Cset of alternatives Z ⊆ C we add to C
choice probabilities Pr(a← x | C ∪ Z ) for each person a and item x
Choice set optimization
Find Z ⊆ C that optimizes some function of Pr(a← x | C ∪ Z )
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Problem setup
A UC C Z
set of individuals making choices A
universe of items Uinitial choice set C ⊆ Upossible new alternatives C = U \ Cset of alternatives Z ⊆ C we add to C
choice probabilities Pr(a← x | C ∪ Z ) for each person a and item x
Choice set optimization
Find Z ⊆ C that optimizes some function of Pr(a← x | C ∪ Z )
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Three choice set optimization problems
Disagreement induced by Z
D(Z ) =∑{a,b}⊆Ax∈C
|Pr(a← x | C ∪ Z )− Pr(b ← x | C ∪ Z )|
Agreement
Find Z that minimizes D(Z )
Disagreement
Find Z that maximizes D(Z )
Promotion
Find Z that maximizes number of people whose favorite item in C is x∗
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Three choice set optimization problems
Disagreement induced by Z
D(Z ) =∑{a,b}⊆Ax∈C
|Pr(a← x | C ∪ Z )− Pr(b ← x | C ∪ Z )|
Agreement
Find Z that minimizes D(Z )
Disagreement
Find Z that maximizes D(Z )
Promotion
Find Z that maximizes number of people whose favorite item in C is x∗
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Three choice set optimization problems
Disagreement induced by Z
D(Z ) =∑{a,b}⊆Ax∈C
|Pr(a← x | C ∪ Z )− Pr(b ← x | C ∪ Z )|
Agreement
Find Z that minimizes D(Z )
Disagreement
Find Z that maximizes D(Z )
Promotion
Find Z that maximizes number of people whose favorite item in C is x∗
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Three choice set optimization problems
Disagreement induced by Z
D(Z ) =∑{a,b}⊆Ax∈C
|Pr(a← x | C ∪ Z )− Pr(b ← x | C ∪ Z )|
Agreement
Find Z that minimizes D(Z )
Disagreement
Find Z that maximizes D(Z )
Promotion
Find Z that maximizes number of people whose favorite item in C is x∗
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Outline
1 Overview
2 Agreement, Disagreement, and Promotion
3 Hardness Results
4 Approximation Algorithm
5 Experimental Results
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Making even two people agree (or disagree) is hard
Theorem
MNL Agreement is NP-hard, even when |A| = 2 and the twoindividuals have identical utilities on items in C .
Theorem
MNL Disagreement is similarly NP-hard.
Corollary
NL, CDM, and EBA Agreement/Disagreement are NP-hard.
Subset Sumreductions
Agreement
0 1 2sZ/t
0.16
0.18
0.20
0.22
D(Z
)
Disagreement
0 1 2 3 4 5sZ/t
0.00.10.20.30.4
D(Z
)
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Making even two people agree (or disagree) is hard
Theorem
MNL Agreement is NP-hard, even when |A| = 2 and the twoindividuals have identical utilities on items in C .
Theorem
MNL Disagreement is similarly NP-hard.
Corollary
NL, CDM, and EBA Agreement/Disagreement are NP-hard.
Subset Sumreductions
Agreement
0 1 2sZ/t
0.16
0.18
0.20
0.22
D(Z
)
Disagreement
0 1 2 3 4 5sZ/t
0.00.10.20.30.4
D(Z
)
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Making even two people agree (or disagree) is hard
Theorem
MNL Agreement is NP-hard, even when |A| = 2 and the twoindividuals have identical utilities on items in C .
Theorem
MNL Disagreement is similarly NP-hard.
Corollary
NL, CDM, and EBA Agreement/Disagreement are NP-hard.
Subset Sumreductions
Agreement
0 1 2sZ/t
0.16
0.18
0.20
0.22
D(Z
)
Disagreement
0 1 2 3 4 5sZ/t
0.00.10.20.30.4
D(Z
)
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Making even two people agree (or disagree) is hard
Theorem
MNL Agreement is NP-hard, even when |A| = 2 and the twoindividuals have identical utilities on items in C .
Theorem
MNL Disagreement is similarly NP-hard.
Corollary
NL, CDM, and EBA Agreement/Disagreement are NP-hard.
Subset Sumreductions
Agreement
0 1 2sZ/t
0.16
0.18
0.20
0.22
D(Z
)
Disagreement
0 1 2 3 4 5sZ/t
0.00.10.20.30.4
D(Z
)
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Making even two people agree (or disagree) is hard
Theorem
MNL Agreement is NP-hard, even when |A| = 2 and the twoindividuals have identical utilities on items in C .
Theorem
MNL Disagreement is similarly NP-hard.
Corollary
NL, CDM, and EBA Agreement/Disagreement are NP-hard.
Subset Sumreductions
Agreement
0 1 2sZ/t
0.16
0.18
0.20
0.22
D(Z
)
Disagreement
0 1 2 3 4 5sZ/t
0.00.10.20.30.4
D(Z
)
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Promoting an item is hard (with context effects)
Promotion is impossible with MNL
MNL preserves relative preferences across choice sets.
Theorem
Promotion is NP-hard under NL, CDM, and EBA.
a’s root
y r
x∗ z1 . . . zn
0 log 2
log(t + ε)log z1
log zn
b’s root
x∗ r
y z1 . . . zn
0 log 2
log(t − ε)log z1
log zn
Promotion is “easier” than Agreement
Model restrictions make Promotion easy, but leave Agreement hard.e.g., same-tree NL
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Promoting an item is hard (with context effects)
Promotion is impossible with MNL
MNL preserves relative preferences across choice sets.
Theorem
Promotion is NP-hard under NL, CDM, and EBA.
a’s root
y r
x∗ z1 . . . zn
0 log 2
log(t + ε)log z1
log zn
b’s root
x∗ r
y z1 . . . zn
0 log 2
log(t − ε)log z1
log zn
Promotion is “easier” than Agreement
Model restrictions make Promotion easy, but leave Agreement hard.e.g., same-tree NL
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Promoting an item is hard (with context effects)
Promotion is impossible with MNL
MNL preserves relative preferences across choice sets.
Theorem
Promotion is NP-hard under NL, CDM, and EBA.
a’s root
y r
x∗ z1 . . . zn
0 log 2
log(t + ε)log z1
log zn
b’s root
x∗ r
y z1 . . . zn
0 log 2
log(t − ε)log z1
log zn
Promotion is “easier” than Agreement
Model restrictions make Promotion easy, but leave Agreement hard.e.g., same-tree NL
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Promoting an item is hard (with context effects)
Promotion is impossible with MNL
MNL preserves relative preferences across choice sets.
Theorem
Promotion is NP-hard under NL, CDM, and EBA.
a’s root
y r
x∗ z1 . . . zn
0 log 2
log(t + ε)log z1
log zn
b’s root
x∗ r
y z1 . . . zn
0 log 2
log(t − ε)log z1
log zn
Promotion is “easier” than Agreement
Model restrictions make Promotion easy, but leave Agreement hard.e.g., same-tree NL
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Promoting an item is hard (with context effects)
Promotion is impossible with MNL
MNL preserves relative preferences across choice sets.
Theorem
Promotion is NP-hard under NL, CDM, and EBA.
a’s root
y r
x∗ z1 . . . zn
0 log 2
log(t + ε)log z1
log zn
b’s root
x∗ r
y z1 . . . zn
0 log 2
log(t − ε)log z1
log zn
Promotion is “easier” than Agreement
Model restrictions make Promotion easy, but leave Agreement hard.
e.g., same-tree NL
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Promoting an item is hard (with context effects)
Promotion is impossible with MNL
MNL preserves relative preferences across choice sets.
Theorem
Promotion is NP-hard under NL, CDM, and EBA.
a’s root
y r
x∗ z1 . . . zn
0 log 2
log(t + ε)log z1
log zn
b’s root
x∗ r
y z1 . . . zn
0 log 2
log(t − ε)log z1
log zn
Promotion is “easier” than Agreement
Model restrictions make Promotion easy, but leave Agreement hard.e.g., same-tree NL
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Outline
1 Overview
2 Agreement, Disagreement, and Promotion
3 Hardness Results
4 Approximation Algorithm
5 Experimental Results
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Poly-time approximation for small group Agreement
Idea (inspired by Subset Sum FPTAS from CLRS)
Discretize possible utility sums of Z s
⇒ compute fewer sets than brute-force
Theorem
We can ε-additively approximate MNL Agreement in timeO(poly( 1
ε , |C |, |C |)).
∅ { }
{ , }{ }
a
b
c
can be adapted forCDM, NL,Disagreement,Promotion
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Poly-time approximation for small group Agreement
Idea (inspired by Subset Sum FPTAS from CLRS)
Discretize possible utility sums of Z s⇒ compute fewer sets than brute-force
Theorem
We can ε-additively approximate MNL Agreement in timeO(poly( 1
ε , |C |, |C |)).
∅ { }
{ , }{ }
a
b
c
can be adapted forCDM, NL,Disagreement,Promotion
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Poly-time approximation for small group Agreement
Idea (inspired by Subset Sum FPTAS from CLRS)
Discretize possible utility sums of Z s⇒ compute fewer sets than brute-force
Theorem
We can ε-additively approximate MNL Agreement in timeO(poly( 1
ε , |C |, |C |)).
∅ { }
{ , }{ }
a
b
c
can be adapted forCDM, NL,Disagreement,Promotion
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Poly-time approximation for small group Agreement
Idea (inspired by Subset Sum FPTAS from CLRS)
Discretize possible utility sums of Z s⇒ compute fewer sets than brute-force
Theorem
We can ε-additively approximate MNL Agreement in timeO(poly( 1
ε , |C |, |C |)).
∅ { }
{ , }{ }
a
b
c
can be adapted forCDM, NL,Disagreement,Promotion
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Poly-time approximation for small group Agreement
Idea (inspired by Subset Sum FPTAS from CLRS)
Discretize possible utility sums of Z s⇒ compute fewer sets than brute-force
Theorem
We can ε-additively approximate MNL Agreement in timeO(poly( 1
ε , |C |, |C |)).
∅ { }
{ , }{ }
a
b
c
can be adapted forCDM, NL,Disagreement,Promotion
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Outline
1 Overview
2 Agreement, Disagreement, and Promotion
3 Hardness Results
4 Approximation Algorithm
5 Experimental Results
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Datasets and training procedure
SFWork: survey of San Fransisco transportation choicesGroups: live in city center, live in suburbs(Koppelman & Bhat, 2006)
Allstate: online insurance policy shoppingGroups: homeowners, non-homeowners(Kaggle, 2014)
Yoochoose: online retail shoppingGroups: first half, second half (by timestamp)(Ben-Shimon et al., 2015)
Model training
Optimize NLL using PyTorch’s Adam with amsgrad fix(Kingma & Ba, 2015; Reddi et al., 2018; Paszke et al., 2019)
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Datasets and training procedure
SFWork: survey of San Fransisco transportation choicesGroups: live in city center, live in suburbs(Koppelman & Bhat, 2006)
Allstate: online insurance policy shoppingGroups: homeowners, non-homeowners(Kaggle, 2014)
Yoochoose: online retail shoppingGroups: first half, second half (by timestamp)(Ben-Shimon et al., 2015)
Model training
Optimize NLL using PyTorch’s Adam with amsgrad fix(Kingma & Ba, 2015; Reddi et al., 2018; Paszke et al., 2019)
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Datasets and training procedure
SFWork: survey of San Fransisco transportation choicesGroups: live in city center, live in suburbs(Koppelman & Bhat, 2006)
Allstate: online insurance policy shoppingGroups: homeowners, non-homeowners(Kaggle, 2014)
Yoochoose: online retail shoppingGroups: first half, second half (by timestamp)(Ben-Shimon et al., 2015)
Model training
Optimize NLL using PyTorch’s Adam with amsgrad fix(Kingma & Ba, 2015; Reddi et al., 2018; Paszke et al., 2019)
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Datasets and training procedure
SFWork: survey of San Fransisco transportation choicesGroups: live in city center, live in suburbs(Koppelman & Bhat, 2006)
Allstate: online insurance policy shoppingGroups: homeowners, non-homeowners(Kaggle, 2014)
Yoochoose: online retail shoppingGroups: first half, second half (by timestamp)(Ben-Shimon et al., 2015)
Model training
Optimize NLL using PyTorch’s Adam with amsgrad fix(Kingma & Ba, 2015; Reddi et al., 2018; Paszke et al., 2019)
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Greedy algorithm fails in small examples
SFWork CDM Agreement
C = {drive alone, transit}
GreedyZ = {carpool}
OptimalZ = {bike, walk}
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Greedy algorithm fails in small examples
SFWork CDM Agreement
C = {drive alone, transit}
GreedyZ = {carpool}
OptimalZ = {bike, walk}
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Greedy algorithm fails in small examples
SFWork CDM Agreement
C = {drive alone, transit}
GreedyZ = {carpool}
OptimalZ = {bike, walk}
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Approximation outperforms greedy on 2-item choice sets
Allstate
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Approximation outperforms greedy on 2-item choice sets
Yoochoose
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Approximation outperforms theoretical guarantee
Allstate CDM Promotion on all 2-item choice sets
10 1 100 101 102 103
Approximation
0.00.20.40.60.81.0
Frac
. Ins
tanc
es S
olve
d
Algorithm 1GreedyBrute force
10 1 100 101 102 103
Approximation
102103104105106107
# Se
ts C
ompu
ted
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Approximation outperforms theoretical guarantee
Allstate CDM Promotion on all 2-item choice sets
10 1 100 101 102 103
Approximation
0.00.20.40.60.81.0
Frac
. Ins
tanc
es S
olve
d
Algorithm 1GreedyBrute force
10 1 100 101 102 103
Approximation
102103104105106107
# Se
ts C
ompu
ted
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Acknowledgment
This research was supported by NSF Award DMS-1830274, ARO AwardW911NF19-1-0057, and ARO MURI. We thank Johan Ugander forhelpful conversations.
Takeaways
1 Influence group preferencesby modifying the choice set
2 NP-hard to maximizeconsensus or promote items
3 Promotion is easier thanachieving consensus
4 Approximation algorithmthat works well in practice
Availability
Data and source code hosted athttps://github.com/
tomlinsonk/choice-set-opt.
19 / 19