Truc Viet Le Siyuan Liu GDAT: Analyzing Group Decision ... · Sol. Framework: GDAT Principle Metrics Comparative Comp. Analytic Component Case Study Dataset Results Conclusion References

Analyzing GroupDecision-Making

Truc Viet LeSiyuan Liu

Hoong Chuin LauRamayya Krishnan

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GDAT: Analyzing Group Decision-MakingBehaviors from Trajectories



Singapore Management UniversityCarnegie Mellon University

August 5, 2013

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Columbia Disaster (2003)

I At 9 AM (EST) on Feb. 1, 2003, the Columbia spaceshuttle disintegrated over the skies of Texas andLouisiana upon entering the Earth’s atmosphere

I Resulted in the death of all 7 crew members

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Columbia Disaster (2003)

I At 9 AM (EST) on Feb. 1, 2003, the Columbia spaceshuttle disintegrated over the skies of Texas andLouisiana upon entering the Earth’s atmosphere

I Resulted in the death of all 7 crew members

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Columbia Disaster (cont’d)I Cause: Piece of foam insulation broke off from the

external tank of the shuttle

I Columbia Accident Investigation Board was immediatelyset up to investigate the incident

I One of its memorable conclusions was that “faultydecision was endemic in NASA’s culture” and“groupthink pervaded decision-making at NASA”[Ferraris and Carveth, 2003]

I The mission was executed by the Mission ManagementTeam (MMT) headed by Linda Ham

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Columbia Disaster (cont’d)

I MMT met daily during the 16-day mission

I The group’s decisions were highly cohesivethroughout and dissenting opinions “not welcomed”

I There was significant unwillingness to discuss problemsand people were reluctant to raise flags

I Meetings were overpowered by Ham’s optimisticopinions and engineers later regretted not having raisedthe issues of foam debris observed during the launch

I “Everybody assumed that someone else would do [raisethe issue] it. No one wants to be the first.”[Ferraris and Carveth, 2003]

I Ham was in turn under great economic and politicalpressures to lead a successful mission

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Simple 2-person Group

Dyadic Dominance Structure of Group 퐶

Preference Lists:퐴: {푗 > 푚 > 푖 > 푙 > 푘}퐵: {푚 > 푛 > 푗 > 푖 > 푘}

퐶 = 퐴 ∪ 퐵: {푗 > 푚 > 푖 > 푛 > 푙}

I Two agents A and B. C is a group C = A ∪ B

I Each has a preference list (partially ordered list) over afinite set of abstract items

I Preference list of C is the joint decision of A & B overthe same ranking task → Whose preference dominatesin the joint decision?

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Simple 2-person Group

Dyadic Dominance Structure of Group 퐶

Preference Lists:퐴: {푗 > 푚 > 푖 > 푙 > 푘}퐵: {푚 > 푛 > 푗 > 푖 > 푘}

퐶 = 퐴 ∪ 퐵: {푗 > 푚 > 푖 > 푛 > 푙}

I Two agents A and B. C is a group C = A ∪ B

I Each has a preference list (partially ordered list) over afinite set of abstract items

I Preference list of C is the joint decision of A & B overthe same ranking task → Whose preference dominatesin the joint decision?

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Other Examples

I Decision-making in household consisting of 2 adultdecision-makers (husband & wife) over vacations[Cosenza, 1985, Lee and Beatty, 2002]

I Analyzing structural shift in U.S. politics throughSenate voting records from 1989 to 2009 (20-year)[Herlau et al., 2013]

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Other Examples

I Decision-making in household consisting of 2 adultdecision-makers (husband & wife) over vacations[Cosenza, 1985, Lee and Beatty, 2002]

I Analyzing structural shift in U.S. politics throughSenate voting records from 1989 to 2009 (20-year)[Herlau et al., 2013]

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What Are We Getting At?

I Analyze the dyadic dominance structure in groupdecision-making through a sequence of joint decisions

I Def. – “Pressure or appeal, both formal and informalas well as overt and covert, directed by one subset ofmembers to another.” [Cosenza, 1985]

I Dyadic → Bipartite structure in groups: male vs.female, left-wing vs. right-wing, etc.

I Accurate modeling of group dynamics anddecision-making mechanism – e.g., battle of the sexes,sequential bargaining games [Siegel et al., 1961,Manser and Brown, 1980, Dwyer and Walker Jr, 1981]

I Prediction of future decision outcomes, e.g., Nashequilibria of the game

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Terms & Concepts

I Bipartite group – 2 agent types A & B of differingpreferences and interests → Group is denoted as C

I Members of each type have homogeneous preferences

I Two dynamics of dominance structure:

1. Balanced dynamics – balanced power structurebetween two types in decision-making

2. Dominant dynamics – decisions dominated by one ofthe types (the dominant party)

I Trajectory – temporally ordered sequence of decisionsmade by A, B, or C that reflect both the spatial andtemporal dimensions of decision-making:

I “what” action to take, “where” to goI “when” to do that, for “how long”

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Two Decision-making Schemes

I 1. Consensus – everyone agrees to a given course ofaction → avoids having “winners” and “losers” →time-consuming due to deliberation

I 2. Voting – majority wins → reflects majority/minoritystructure of the group → takes much less time thanconsensus scheme

I Two dominance dynamics:

1. Balanced – more likely to arise from a consensusscheme

2. Dominant – more likely to arise from a voting scheme

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Problem Statement

I Given the decision data: independent trajectories of Aand B as well as the trajectories of C (joint decisions)over a certain activity

I Learn the differences between A and B:I Extract the set of spatio-temporal criteria in which A

and B might have “revealed comparative differences” intheir preferences

I Infer the (dyadic) dominance structure of the group CI Determine which dynamics it most likely admits:

balanced or dominance?I If it is the latter, which type is the dominant party?

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Problem Statement






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Problem Statement






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The Principle

I GDAT: Group Dominance Analysis via Trajectories – ageneric statistical solution framework

I Systematically compare each party’s influence on thegroup’s decisions based based on a flexible array ofspatio-temporal criteria

I Criteria used for comparison should reflect thefundamental differences between A and B, i.e., theirrevealed comparative differences – typically throughstatistical tests of difference

I Results of these comparisons are aggregated into a finalscore to compare the group’s dominance structure

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The Principle





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The Principle





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The Solution Framework

Spatial Metrics

• Transitional Probability• Transitional PMI• Positional Probability• Positional PMI

Temporal Metrics

• Time to Event (Mean/Median)

• Event Choice (Mean/Median)

Survival Analysis

Correlation Analysis

Test of Significance

Comparative Analysis

Fisher ݖ-Transform

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1

A B C

Output: Dominance Structure

Input: Trajectories

Parameters

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The Solution Framework (cont’d)

I Three sequential components / phases:

1. Metric component – proposes and calculates an arrayof spatio-temporal metrics that reflect the essentialaspects of decision-making → flexible: how many andexactly what metrics are not important and shouldrather be adaptive to the problem domain

2. Comparative component – elicits the differences ininfluence between the two agent types

3. Analytic component – tests the statistical significanceof the differences between each type’s influence on thegroup’s decisions & aggregates test results into a finalscore to make inference

I Parameters: θρ – threshold of correlation and θd –threshold of determination → to be flexibly tuned toadapt to the given problem

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Spatial Metrics

I Suppose the task is to select a subset s ⊆ S of abstract“locations” to visit over a finite time-frame T

I S is also finite and let i , j ∈ S be two specific locations

I Transitional probability: Pr(i → j |i)I Transitional Pointwise Mutual Information (PMI):

PMI(i → j) = log Pr(i→j)Pr(i) Pr(j)

I PMI – information-theoretic measure of the associationbetween two discrete random variables

I Pr(i → j) = # sequences containing i→j# all sequences

I Positional probability: Pr(i , k) = Pr(i : k |i) Pr(i), wherek, 1 ≤ k ≤ l , is a “position” and l = |s|

I Positional PMI: PMI(i ; k) = log Pr(i ,k)Pr(i) Pr(k≤l)

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Spatial Metrics



I Transitional probability: Pr(i → j |i)

I Transitional Pointwise Mutual Information (PMI):






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Spatial Metrics









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Temporal Metrics

I “event i”: a visit to location i

I τ̄i and τ̃i : mean and median time to event i

I Call the position k , 1 ≤ k ≤ l , of location i ∈ s the“event choice k” of x

I γ̄i and γ̃i : mean and median choice

I T : total duration of the activity – i.e., time taken tovisit all locations in s ⊆ S

I Survival function: S(t) = Pr(T ≥ t) = 1− F (t),where F (t) = Pr(T < t) and t is the time to finish

I Logrank test: non-parametric test of differencebetween two survival functions SA(t) & SB(t)

I If pS(A,B) ≤ 0.05 → A and B have a revealedcomparative difference w.r.t. their total duration

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Temporal Metrics









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Temporal Metrics









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I For each of the proposed metric (except the logranktest), calculate the Pearson correlation between thepairs A & B, A & C , and B & C : ρk(A,B), ρk(A,C ),and ρk(B,C ) respectively

I If ρk(A,B) < θρ (threshold of correlation) → A and Bhave a revealed comparative difference w.r.t. metric k

I Then we ask: Are ρk(A,C ) and ρk(B,C ) significantlydifferent from each other? And in which direction?

I If yes: ρk(A,C ) < ρk(B,C ) → B exerts a greaterinfluence than A in the group decision-making w.r.t.metric k → carry on to the third phase [Similarly forρk(A,C ) > ρk(B,C )]

I If no: k will not be considered further in the analysis

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Fisher z-transform

I To quantify the significance of the difference betweentwo correlation coefficients

I z-transform of ρ: z = 12 ln 1+ρ

1−ρ

I For each k , ρk(A,B) < θρ, → z-transform ρk(A,C )and ρk(B,C ): zk(A,C ) and zk(B,C )

I Two-tailed z-test → Test of difference betweenzk(A,C ) and zk(B,C ): Zk = zk (A,C)−zk (B,C)√

(2/(n−3))

I Zk ∼ N(0, 1) → if Zk > +/− 1.96: zk(A,C ) andzk(B,C ) are significantly different at the 5% level

I If so, check the sign of Zk (too see which agent type ismore influential w.r.t. k)

I Otherwise, neither A nor B has an influence w.r.t. k

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Fisher z-transform



1−ρ



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Fisher z-transform



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I We propose a simple comparative procedure toaggregate all the tests of significance in the previousphase to determine the dominant type, if any

I Given a threshold of determination θd > 1/2, denoteΣA and ΣB the domination “score” of A and B

I Idea: for each metric k in which A and B have RCD, ifZk > 0 (significantly): ΣA ← ΣA + 1, else:ΣB ← ΣB + 1 → “voting” determination scheme

I Return +1 if ΣA ≥ θd |K | → A is the dominant type or−1 if ΣB ≥ θd |K | → B is the dominant type (|K | is the# metrics considered)

I Return 0 otherwise → balanced dynamics

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Dataset Description

I Real-world trajectory dataset collected from visitor’smovements in Setosa (Aug – Dec, 2012)

I Each trajectory is a sequence of visits to at most 17attractions in the theme park recorded by RFID systemswith exact location and time of the visit

I Agent types: A (Adult), B (Child), and C (Family)I Child: < 12 y.o., Adult: ≥ 13 y.o.I Groups C : ≥ 2 visitors of different types each

I Question: Given the trajectories of A, B, and C , whatis the dominance structure of C?

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Dataset Description (cont’d)

Table : Summary of the Case Study Dataset

Child (B) Adult (A) Fam. (C ) |C | %B

n 151 942 459 NA NAMin. 1 1 1 2 13%

1st Qu. 5 6 7 3 33%Median 8 8 9 3 50%

Mean 7.834 8.274 8.492 3.769 43%3rd Qu. 10 11 10 4 50%

Max. 14 15 15 16 80%ρ(T , l) 0.828 0.737 0.764 NA NA

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Results – Illustration of a Spatial Metric

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−8 −6 −4 −2 0 2 4

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02

4

Positional PMI: Adult vs. Child

Adult

Chi

ld

ρ = 0.753

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02

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Positional PMI: Adult vs. Family

Adult

Fam

ily

ρ = 0.864

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−8 −6 −4 −2 0 2 4

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02

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Positional PMI: Child vs. Family

Child

Fam

ily

ρ = 0.793

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Principle

Metrics

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Analytic Component

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References

Results – Illustration of a Temporal Metric

0 100 200 300 400 500 600

0.0

0.4

0.8

Survivorship Comparison: Adult vs. Child

Minutes

Pro

babi

lity

AdultChild

p = 2e−05

0 100 200 300 400 500 600

0.0

0.4

0.8

Survivorship Comparison: Adult vs. Family

Minutes

Pro

babi

lity

AdultFamily

p = 0.3564

0 100 200 300 400 500 600

0.0

0.4

0.8

Survivorship Comparison: Child vs. Family

Minutes

Pro

babi

lity

ChildFamily

p = 0.0035

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Sol. Framework:GDAT

Principle

Metrics

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Results – Conclusion

Table : z-scores & p-values of the Correlation Metrics

Zk pkPr(i → j |i) 6.98 < 0.001PMI(i → j) 3.29 0.001

Pr(i , k) 5.11 < 0.001PMI(i ; k) 2.74 0.0061

τ̄i 1.56 0.1188τ̃i 2.56 0.0105γ̃i 2.02 0.0434

I Conclusion: Adult is the dominant decision-maker inthe Family groups

I Foregone conclusion?

I Validating ground truth for GDAT

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Zk pkPr(i → j |i) 6.98 < 0.001PMI(i → j) 3.29 0.001

Pr(i , k) 5.11 < 0.001PMI(i ; k) 2.74 0.0061

τ̄i 1.56 0.1188τ̃i 2.56 0.0105γ̃i 2.02 0.0434




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Principle

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Zk pkPr(i → j |i) 6.98 < 0.001PMI(i → j) 3.29 0.001

Pr(i , k) 5.11 < 0.001PMI(i ; k) 2.74 0.0061

τ̄i 1.56 0.1188τ̃i 2.56 0.0105γ̃i 2.02 0.0434




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Zk pkPr(i → j |i) 6.98 < 0.001PMI(i → j) 3.29 0.001

Pr(i , k) 5.11 < 0.001PMI(i ; k) 2.74 0.0061

τ̄i 1.56 0.1188τ̃i 2.56 0.0105γ̃i 2.02 0.0434




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Sol. Framework:GDAT

Principle

Metrics

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References

Conclusion

I Generic statistical solution framework to a problem ofinterdisciplinary interest – group decision-making

I Problem proposed in: Business & OR[Arora and Allenby, 1999, Saaty and Peniwati, 2008],Economics & Poli-Sci [Black, 1948, Siegel et al., 1961],Biology & Sociology[Wilson, 2007, King and Cowlishaw, 2009], & much inPsychology[Montgomery, 1983, Stasser and Stewart, 1992]

I Our approach: large-scale spatio-temporal data analysis

I Generic: set of metrics are not fixed or strictly definedbut rather adaptive to the problem domain and thenature of the available data

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Conclusion





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Conclusion





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Conclusion





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References

References I

Arora, N. and Allenby, G. M. (1999).Measuring the influence of individual preferencestructures in group decision making.Journal of Marketing Research, pages 476–487.

Black, D. (1948).On the rationale of group decision-making.The Journal of Political Economy, 56(1):23–34.

Cosenza, R. M. (1985).Family decision making decision dominance structureanalysis – An extension.Journal of the Academy of Marketing Science,13(1-2):91–103.

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References

References II

Dwyer, F. R. and Walker Jr, O. C. (1981).Bargaining in an asymmetrical power structure.The Journal of Marketing, pages 104–115.

Ferraris, C. and Carveth, R. (2003).Nasa and the columbia disaster: Decision-making bygroupthink?In Proceedings of the 2003 Association for BusinessCommunication Annual Convention, page 12.

Herlau, T., Mørup, M., and Schmidt, M. N. (2013).Modeling temporal evolution and multiscale structure innetworks.In Proceedings of the 30th International Conference onMachine Learning.

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References

References III

King, A. J. and Cowlishaw, G. (2009).Leaders, followers, and group decision-making.Communicative & integrative biology, 2(2):147–150.

Lee, C. K. and Beatty, S. E. (2002).Family structure and influence in family decisionmaking.Journal of consumer marketing, 19(1):24–41.

Manser, M. and Brown, M. (1980).Marriage and household decision-making: A bargaininganalysis.International Economic Review, 21(1):31–44.

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References IV

Montgomery, H. (1983).Decision rules and the search for a dominance structure:Towards a process model of decision making.Analysing and aiding decision processes, pages 343–369.

Saaty, T. and Peniwati, K. (2008).Group decision making: Drawing out and reconcilingdifferences.RWS Pub.: Pittsburgh, PA.

Siegel, S., Fouraker, L. E., and Ellsberg, D. (1961).Bargaining and group decision making.

Stasser, G. and Stewart, D. (1992).Discovery of hidden profiles by decision-making groups:Solving a problem versus making a judgment.Journal of personality and social psychology, 63(3):426.

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References V

Wilson, R. (2007).Simulating the effect of social influence ondecision-making in small, task-oriented, groups.Journal of Artificial Societies and Social Simulation,10(4):4.

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Documents

Truc Viet Le Siyuan Liu GDAT: Analyzing Group Decision ... · Sol. Framework: GDAT Principle Metrics Comparative Comp. Analytic Component Case Study Dataset Results Conclusion References