Opinion dynamics and inﬂuence networks in...

Opinion dynamics and influence networksin economics

Agnieszka RUSINOWSKA

Paris School of Economics - CNRS

Universit Paris 1, Centre d’Economie de la Sorbonne

Agnieszka RUSINOWSKA c©2016 Opinion dynamics and influence networks in economics

Role of networks

◮ A social/physical/mathematical/graphical structure thatcaptures the relationships between a set of agents.

◮ Ubiquitous in nature and society: food-webs, internet, railwaynetworks, social networks, supply chains, ...

◮ Increasingly central role in representing (dynamic)socio-economic phenomena: transmission of information (e.g.,job opportunities); learning, influence, opinion formation;contagion, rumours; trade of goods and services; businessinteractions, financial networks; scientific collaboration;friendship, homophily; political interactions; criminal activities,...


Transmission of opinions

Figure: Twitter communities and sentiment around climate issues (Mozetic et

al. (2015), Project SIMPOL)


Influence network

Figure: Inferred network of influence from roll-call votes at the European

parliament (Karpf et al. (2015), Project SIMPOL)


Transaction network

Figure: Network of transactions on the EU-ETS market market (Karpf etal. (2015) “A network-based analysis of the European Emission Market”)


Production networks


Financial networks

Figure: Nodes represent financial institutions. Outgoing links represent the

estimated potential impact of an institution to another one (Battiston et

al. (2012) “Debtrank: too central to fail”)


Scientific collaboration

Figure: Network of scientific collaboration at the University of Florida (Raffaele

Vacca, “Studying and shaping networks of scientific collaboration at UF”)


Homophily

Figure: Friendship network among high-school students

Homophily (Lazarsfeld & Merton, 1954) → There is a bias in relationships

toward similar individuals.Agnieszka RUSINOWSKA c©2016 Opinion dynamics and influence networks in economics

Political interactions

Figure: Top 20 U.S. senators in the co-sponsorship network (Fowler (2006)

“Legislative Co-sponsorship Networks in the U.S. House and Senate”, Social

Networks 28 (4): 454-465)


Criminal activities

Figure: Duijn et al. (2014) “The relative ineffectiveness of criminal network

disruption”. Scientific reports, 4. Organization of a Deutch network of

cannabis cultivators


Objective of the talk

◮ Recent directions in network analysis used in economics?◮ What kind of questions/tools do we ask/use?◮ Focus on the opinion dynamics and influence networks (‘joint’

topic of interest shared with other disciplines).◮ Networks as very natural modeling tools!


Outline

◮ Bayesian and non-Bayesian models of opinion formation

◮ DeGroot model

◮ Targeting, key players and strategic influence in socialnetworks

◮ Model of influence based on aggregation functions

◮ Application to the Krackhardt’s advice network

◮ Anonymous influence

◮ Modeling contagion by aggregation functions (project)

◮ Bayesian models of information transmission


Dynamic models of opinion formation


Main components of opinion formation, implications

Components of opinion formation (Acemoglu & Ozdaglar, 2010):

◮ priors - views about a subject◮ sources of information - own experience, observing others’

actions and experiences, communication with others◮ method of information processing - Bayesian models (agents

use Bayes rule) vs. non-Bayesian models.

Implications of the form of learning, the sources of information andthe structure of social networks on the following questions:

◮ Consensus - Will social learning lead to consensus amongindividuals starting with different views?

◮ Asymptotic learning - Will social learning effectively aggregatedispersed information and guarantee that incorrect beliefs (notjustifiable by the data and evidence) disappear?

◮ Misinformation - How much room is there for beliefmanipulation and misinformation?


Non-Bayesian approaches

◮ Imitation - an agent may use a combination of beliefs of someothers he knows or has observed, e.g., DeGroot (1974),DeMarzo et al. (2003), Golub & Jackson (2009, 2010)

◮ imitation + information about the performance of differentalternatives - replicator dynamics inspired by evolutionarybiology (beliefs that have performed better are more likely toreplicate)

◮ Case-based decision theory (Gilboa & Schmeidler, 1995, 2001)- beliefs are formed according to empirical similarity (agentsform beliefs based on their experiences in similar situations)

◮ Models of interacting particle systems - motivation comesfrom statistical mechanism; opinions represented by finitelymany discrete values (e.g. Ising model of Glauber (1963) andthe voter model of Clifford & Sudbury (1973) and Holley &Liggett (1975)) or continuous values (e.g., DeGroot (1974),Krause (2002), Hegselmann & Krause (2002), etc.)


Questions and tools

Questions:

◮ When do we reach consensus?

◮ Speed of convergence (How quickly beliefs reach their limit)?

◮ How does each agent influence the limiting belief (socialinfluence)?

◮ Can informations be correctly aggregated on a network(wisdom of crowds)?

◮ When do subgroups of different opinion form?

◮ Whom should I target in a network? Who are the key players?

Tools:

◮ Usual: Markov chain theory, weighted averages, establishedcentrality measures, ...

◮ New: aggregation functions, OWA, fuzzy quantifiers,hypergraphs, new centrality measures, ...


The DeGroot model


The DeGroot model

◮ Model of belief and consensus formation in social networks

◮ N = {1, . . . , n}: set of agents, each having some opinionrepresented by a number in [0, 1]

◮ Each agent updates his opinion by a weighted average of theopinion of the agents

◮ W = [wij ]: (row-stochastic) matrix of weights, with wij :weight (trust) that agent i places on the opinion of agent j .It follows that

x(t) = Wx(t − 1) = W tx(0)

where x(t) = (x1(t), . . . , xn(t))T is the opinion vector at time

step t

◮ We associate to W a directed graph Γ on N such that there isan arc (i , j) from i to j (meaning that i listens to j) iff wij > 0.


Convergence in the DeGroot model

◮ The asymptotic theory of the DeGroot model is identical tothis of Markov chains. Namely, convergence to a consensusoccurs if W is irreducible (strongly connected) and aperiodic.

◮ Irreducible means that the network is strongly connected, i.e.,there is a path from any node to any other node.

◮ Aperiodic means that the greatest common divisor of thelengths of all cycles is 1.


Social influence in the DeGroot model

◮ How does each agent influence the limiting belief?

◮ Let x(0) be an arbitrary starting belief vector andx(∞) = (x∞, ..., x∞) be the vector of limiting consensusbeliefs. We search for an influence vector s ∈ [0, 1]n such that∑

i si = 1 and

x∞ = s · x(0) =∑

i

sixi (0)

If it exists, then the limiting beliefs are weighted averages ofthe initial beliefs with the weights being the influence weightsof the agents, and s is the left-hand unit eigenvector of W ,i.e., sW = s.


The wisdom of crowds

◮ Consider sequences of societies of size n, each described bystrongly connected and aperiodic W n. Let n grows.

◮ There is a true state of nature µ ∈ [0, 1].

◮ Each agent i in network n observes a signal (which forms hisinitial opinion) according to a random variable with mean µ

and variance σ2.

◮ Each society converges to a consensus xn(∞).

◮ The crowd is wise when the limiting beliefs converge jointly inprobability to the true state µ as n grows. Conditions:

◮ “Balance” - no family below a certain size limit can be gettinginfinitely more weight from the remaining agents than it givesto the remaining agents.

◮ “Dispersion” - any large enough finite family must give at leastsome minimal weight to any family which makes up almost allof the society.


Generalizations of the DeGroot model

◮ The updating can vary with time and circumstances.◮ DeMarzo et al. (2003) (Time-Varying Weight on Own Beliefs)

The updating rule is

x(t) = [(1− λt)I + λtW ]x(t − 1)

I = identity matrix, λt ∈ (0, 1] = adjustment factor, ifconstant → DeGroot model, W = stochastic matrix

◮ Krause (2000) (Only Weighting Those with Similar Beliefs)An agent places equal weight on all opinions that are withinsome distance of his own current opinion, and zero otherwise.

◮ Despite many extensions, usually the aggregation of opinionsthrough a convex combination is assumed.

◮ Different approach to aggregating:◮ Model of opinion formation with arbitrary aggregation

functions (Grabisch and Rusinowska, 2013)◮ Model with ordered weighted averages – modeling anonymous

social influence (Foerster et al., 2013)


Targeting, key players and strategic influence

in social networks


Introduction - Whom should I target?

◮ Identifying optimal targets is a crucial problem for achievingsocial impact (marketing, lobbying, political campaigning).

◮ Complex systems (Albert et al., 2000), computer science(Kempe et al., 2003, 2005), mathematics (Bollobas andRiordan, 2004), economics (Ballester et al., 2006, ...) ...

◮ Computer science: the optimal diffusion (e.g., adoption of anew product or innovation) from an algorithmic perspective,maximizing the expected spread (targeting as an optimizationproblem) - Domingos and Richardson (2001, 2002), Kempe etal. (2003, 2005), ...


Strategic aspects, centrality measures for the key players

The key player characterized by some established or new centralitymeasures.

Measures of centrality can be categorized into the following maingroups (Jackson (2008)):

(1) Degree centrality - how connected a node is

(2) Closeness centrality - how easily a node can reach other nodes

(3) Betweenness centrality - how important a node is in terms ofconnecting other nodes

(4) Prestige- and eigenvector-related centrality - how important,central, or influential a node’s neighbors are.


Related literature in economics (1/2)

◮ Different types of agents: Acemoglu et al. (2010, 2013),Galeotti and Goyal (2009)

◮ The key player identified by a certain centrality measure:◮ Ballester et al. (2006; “intercentrality measure”) -

noncooperative game interpreted as a network game, whereplayers decide how much effort to exert; the Nash equilibriumaction of a player is proportional to his Bonacich centrality.

◮ Banerjee et al. (2013, 2014; “diffusion centrality”) - howextensively the information spreads from a given player; itnests degree centrality (if there is one time period ofcommunication), eigenvector centrality and Katz-Bonacichcentrality (if there are unlimited periods of communication)

◮ Bimpikis et al. (2015) - competition between firms whichtarget their marketing budgets to agents embedded in a socialnetwork; the targeted budget is an increasing function of theagent’s centrality.


Related literature in economics (2/2)

◮ Literature on strategic spending in political competitions:Lever (2010) - two persuaders exert effort in changing thevoters’ initial opinion

◮ Tsakas (2014a,b) - targeting agents in the diffusion of apreferred action / new product with the uncertain quality.


Strategic influence (Grabisch et al., 2015) (1/2)

◮ Most of the literature considers that a single external agent(or a group of non-competing actors) is targeting the network.

◮ Many situations where multiple actors are likely to competefor prominence over the network.

◮ What do we add to the DeGroot model and its extensions:◮ different types of agents (some agents influence others without

being influenced themselves)◮ targeting specific (central) agents in a social network according

to a new centrality measure◮ impact of the strategic aspects on the characterization of the

key target.


Strategic influence (Grabisch et al., 2015) (2/2)

◮ A network of (non-strategic) agents that update their opinionsas in the DeGroot model

◮ Two strategic players with extreme opinions competing forinfluence over the network

◮ Each strategic agent chooses one non-strategic agent totarget (forms a link with) in order to influence the averageopinion in the network

◮ A zero-sum game with players = strategic agents andstrategies = non-strategic agents

◮ Our analysis:◮ convergence and consensus reaching◮ the key target as pure Nash equilibrium strategy (existence

depends on the network structure)◮ characterization of pure strategy equilibria - 2 features of

target agents: (influenceability) and centrality/influence (newmeasure - intermediacy, differs from other centrality measures).


The proposed model (1/3)

◮ N = {1, . . . , n} set of non-strategic agents, each having aninitial opinion represented by a number xi (0) ∈ [0, 1].

◮ Agents update their opinions as in the DeGroot model (1974).

◮ Two strategic agents a1, a2 with fixed opinions 1 and 0respectively, who try to influence the non-strategic agents

◮ Each ai forms exactly one link with a non-strategic agent, i.e.,each ai chooses a strategy si in N.

◮ We assume that a1, a2 have an impact of λ and µ respectively(λ, µ > 0) on the network.

◮ When a1 (resp. a2) targets the non-strategic agent s1 (resp.s2), the n × n matrix of influence W is extended to a(n + 2)× (n + 2) matrix Pλ,µ(s).



One gets a new influence matrix

Pλ,µ(s) =

1 0 00 1 0

∆λ,µ(s)Eλ,µ(s) ∆λ,µ(s)W

where

◮ the weight renormalization matrix ∆λ,µ(s) is a diagonalmatrix with elements d1

d1+λδ1,s1+µδ1,s2, . . . , dn

dn+λδn,s1+µδn,s2di = degree of i (number of direct connections of i)

◮ the external influence matrix is Eλ,µ(s) = [ λds1

es1µds2

es2 ],

◮ ei is the unit vector with coordinate 1 at i , and δ theKronecker symbol, i.e., δi ,sj = 1 if i = sj and 0 otherwise.



The evolution law of the opinion vector x(t) is given by

x(t + 1) = Pλ,µ(s)x(t)

Let the opinion vector be x = [xa1 xa2 xN ]′. Then xa1(t), xa2(t)

are constant and equal to 1, 0 respectively and the evolution lawfor the opinions of the non-strategic agents is given by:

xN(t + 1) = ∆λ,µ(s)Eλ,µ(s)

[10

]+∆λ,µ(s)WxN(t).

In what follows we assume that W is irreducible (i.e., for every

i , j ∈ N there exists an integer m(i , j) such that w(m(i ,j))ij > 0).


Convergence and consensus reaching

Proposition 1For any initial vector of opinions x(0) := [xa1(0) xa2(0) xN(0)]

′

with xa1(0) = 1, xa2(0) = 0 and xN(0) ∈ [0, 1]n, we have:

limt→+∞

(Pλ,µ(s))t [xa1(0) xa2(0) xN(0)]

′ = [xa1 xa2 xN ]′

where xa1 = 1, xa2 = 0 and

xN = (I −∆λ,µ(s)W )−1 λ

ds1∆λ,µ(s)es1

In particular, if the two strategic agents choose the same target,one obtains convergence towards a consensus obtained in N.

Proposition 2If s1 = s2, then agents in N reach a consensus α, given byα = λ

λ+µ .


Payoffs of the strategic agents (1/3)

◮ Consider the constant-sum game Gλ,µ with players a1, a2,their impacts λ, µ > 0 and their set of strategies N.

◮ The payoff of each strategic agent depends on the asymptoticaverage belief, i.e., the payoff πλ,µ(s) of player a1 is

πλ,µ(s) = 1′ · (I −∆λ,µ(s)W )−1 λ

ds1∆λ,µ(s)es1

and of player a2 is n − πλ,µ(s).

◮ Gλ,µ being a zero-sum game with finite strategy space has atleast an equilibrium in mixed strategies.

◮ Conditions for equilibria in pure strategies?

◮ Derive an explicit expression of the payoff function πλ,µ(s).



◮ For any walk p = (i1, . . . , im):w(p) = “weight” of p measured according to W

w(p) :=

m−1∏

j=1

wij ,ij+1

◮ For i , j ∈ N, i 6= j :c ji = sum of weights of cycles around i that pass through j(probability for i to be reached by the influence of j before hereceives the “echo” of his own opinion)

bji = sum of weights of walks to i that pass through j (sum ofthe probabilities for the agents other than i to be reached bythe influence of j before this of i)

c ji :=∑

p∈Cji

w(p), bji :=∑

p∈Bji

w(p)



TheoremThe payoff of player a1 given the strategy s = (s1, s2) is obtainedas follows:

1. If s1 = s2 = i , i.e., if both strategic agents target the sameagent i ,

πλ,µ(i , i) =nλ

λ+ µ

2. If s1 = i and s2 = j 6= i , i.e., the strategic agents targetdifferent agents i , j ,

πλ,µ(i , j) =λ[µbij + djc

ij n]

µdicji + λdjc

ij + λµ


Influence and influenceability

◮ djcij influenceability of j given that i is the agent targeted by

the other strategic agent - the larger djcij , the less

influenceable j is:◮ the larger dj is, the more opinions j takes into account and the

slower he can be influenced by an additional opinion◮ cycles around j account for the self-feedback (echo) that j

receives of his opinion: the larger c ij is, the more i interfereswith this self reinforcement process and the lesser the influencethat can be exerted on j by a strategic agent

◮ bij influence (intermediacy) of i relatively to j : extent to whichi can interpose himself between j and other non-strategicagents, extent to which the influence of i reaches the networkbefore this of j .


Characterizing equilibria - equal influence (λ = µ) (1/2)

TheoremA pair (i , i) is an equilibrium of Gλ if for all j ∈ N \ {i}:

bij − bji ≥n

λ

[dic

ji − djc

ij

]

◮ There are networks for which there exist equilibria in purestrategies (e.g., perfectly symmetric networks) as well asnetworks for which there exists no equilibrium in purestrategies (e.g., the circle).

◮ (i , i) is an equilibrium if for all j ∈ N, the excessinfluence/intermediacy of i over j is greater than the excess

influenceability of j over i , scaled by the factorn

λ.

◮ The relative importance of influence vis-a-vis influenceabilityincreases with the level of external influence λ.

◮ Conversely, as the size of the network increases, the relativeimportance of influence vis-a-vis influenceability decreases,i.e., highly influenceable agents become optimal targets.Agnieszka RUSINOWSKA c©2016 Opinion dynamics and influence networks in economics

Characterizing equilibria - equal influence (λ = µ) (2/2)

Proposition

(i) For distinct i , j ∈ N, limλ→0 πλ(i , j) =djc

ij n

dicji + djc ij

, so that

(i , i) is an equilibrium of Gλ as λ → 0 iff for all j ∈ N:

djcij ≥ dic

ji

(ii) For distinct i , j ∈ N, limλ→+∞ πλ(i , j) = n − bji , so that (i , i)is an equilibrium of Gλ as λ → +∞ if and only if for all j ∈ N:

bij ≥ bji .

◮ (i , i) is a Nash equilibrium for a vanishingly small level ofinfluence λ only if i is more influenceable than any other j .

◮ (i , i) is a Nash equilibrium of Gλ for an arbitrarily large level ofinfluence λ only if the relative influence (intermediacy) of i isgreater than this of any other agent j .


Characterizing equilibria - unequal influence (λ > µ)

PropositionA pair of strategies (i , i) is an equilibrium of the game Gλ,µ iff

(µ+ λ)bij − λn ≥ n[dicji − djc

ij ]

ConclusionIf one of the strategic agents has a much larger level of influencethan the other one, then the low influence agent should not choosethe same target as the high influence one because his influencewould get washed away.

◮ The low influence player a2 shall choose a target which has a highinfluenceability and which lowers the intermediacy of his opponent.

◮ The lower his weight µ is, the more important the influenceability of histarget is.

◮ The high influence player shall choose a target which has a highintermediacy and which lowers the influenceability of his opponent’starget.


A model of influence

based on aggregation functions


A Markovian model of influence (1/2)

◮ N := {1, . . . , n} set of agents making a yes-no decision.

◮ Each agent has an initial opinion to say ‘yes’ or ‘no’ whichmay change due to influence of others.

◮ Let bS ,T denote the probability that the set S of ‘yes’-votersbecomes T after one step of influence.

◮ Assuming that the process of influence may iterate, we obtaina stochastic influence process, depicting the evolution of thecoalition of ‘yes’-players in time.

◮ We assume that the process is Markovian (bS ,T depends on Sand T but not on the whole history) and stationary (bS ,T isconstant over time).

◮ States of this Markovian chain are all S ⊆ N, representing theset of ‘yes’-agents; its transition matrix B := [bS ,T ]S ,T⊆N is a2n × 2n row-stochastic matrix.


A Markovian model of influence (2/2)

◮ A class (strongly connected component) is◮ either transient (there is an outgoing arc)◮ or terminal (no outgoing arc).

◮ For the qualitative description of the convergence of theprocess (terminal classes, ignoring their probabilities), it issufficient to know the reduced matrix B given by

bS ,T =

{1, if bS ,T > 0

0, otherwise.

and equivalently represented by the transition graphΓ = (2N ,E ), where E is the set of arcs, its vertices are allpossible coalitions, and an arc (S ,T ) from state S to state Texists if and only if bS ,T = 1.


Aggregation functions

◮ Idea: each agent modifies his opinion by aggregating thecurrent opinion of all agents, possibly including himself.

◮ DefinitionAn n-place aggregation function is any mapping A : [0, 1]n → [0, 1]satisfying

1. A(0, . . . , 0) = 0, A(1, . . . , 1) = 1 (boundary conditions)2. If x ≤ x′ then A(x) ≤ A(x′) (nondecreasingness).

◮ We associate an aggregation function Ai to every agent i ∈ N(it specifies the way i modifies his opinion).

◮ Let A := (A1, . . . ,An) denote the vector of aggregationfunctions. We compute A(1S) = (A1(1S), . . . ,An(1S)), where1S is the characteristic vector of S andAi (1S) - the probability for agent i to say “yes” at next timestep when S is the current set of yes-agents.


Influential coalitions

DefinitionLet Ai be the aggregation function of agent i . A nonemptycoalition S ⊆ N is yes-influential for i if

1. Ai (1S) > 0

2. For all S ′ ⊂ S , Ai (1S ′) = 0.

Similarly, a coalition S is no-influential for i if

1. Ai (1N\S) < 1

2. For all S ′ ⊂ S , Ai (1N\S ′) = 1.

◮ We call Cyes

i and Cno

i the collections of yes- and no-influentialcoalitions for i .

◮ Cyes

i and Cno

i are nonempty antichains, i.e., any two sets inCyes

i or Cno

i are incomparable w.r.t. set inclusion.


Influential players and (hyper)graphs of influence

◮ If an influential coalition is formed by only one player, then wecall it influential player.

◮ The graph of yes-influence is a directed graph Gyes

A = (N,E )whose set of nodes is N, and there is an arc (j , i) from j to iif j is yes-influential in Ai (idem for the graph of no-influenceGno

A ).

◮ The hypergraph of yes-influence Hyes

A is a directed hypergraphwhere the set of vertices is N, and there is an hyperarc(C , {i}) for each C ∈ Cyes

i (similarly for Hno

A ).

Consider an influence process B based on aggregation functionsA1, . . . ,An. Then B can be reconstructed from the collections Cyes

i

and Cno

i , i ∈ N.Conclusion: it is not necessary to know the aggregation functionsfor a qualitative description of convergence.


Some results on qualitative convergence

TheoremConsider an influence process B based on aggregation functionsA = (A1, . . .An). Then terminal classes are:

1. either singletons {S}, S ∈ 2N (terminal states)

2. or cycles of nonempty sets {S1, . . . , Sk} of any length2 ≤ k ≤

(n

⌊n/2⌋

)(and therefore they are periodic of period k)

with the condition that all sets are pairwise incomparable (byinclusion) (cyclic terminal classes)

3. or collections C = C1 ∪ · · · ∪ Cp of nonempty sets, where eachsubcollection Cj is a Boolean lattice [Sj , Sj ∪ Kj ], Sj 6= ∅,Sj ∪ Kj 6= N, and at least one Kj is nonempty (regularterminal classes, Boolean terminal classes).

[S , S ∪ K ] → no consensus is reached, but all agents in S agree tosay ‘yes’, all agents in N \ (S ∪ K ) agree to say ‘no’, the agents inK oscillate between ‘yes’ and ‘no’ without ending.


Example of nontrivial terminal states

∅

1 2 3

12 13 23

123

Then {12} and {3} are nontrivial terminal states.


Example of cyclic classes

∅

1 2 3

12 23 13

123

Then {1, 2, 3} and {12, 23, 13} are cyclic classes.


Example of regular terminal class

Example - {1, 12} is a regular terminal class.

∅

1 2 3

12 13 23

123


Some results on qualitative convergence

Consider an influence process B based on A = (A1, . . .An).

Proposition

1. If the graph (Gyes

A )∗ ∪ Gno

A is strongly connected, then there isno nontrivial terminal state, where (·)∗ indicates the graphwith inverted arcs.

2. If for all i ∈ N, i is yes- and no-influential for i , then there isno cyclic class.

For an agent i ∈ N, its closure in Gno

A , denoted by cl(i), is the setof agents which can reach i by a path in Gno

A . By convention,i ∈ cl(i).

TheoremThere is no normal regular terminal class if for each i ∈ N, everyagent outside cl(i) can be reached by a path from cl(i) in Gyes

A .


Symmetric decomposable models

◮ An influence model based on aggregation functions A isdecomposable if all influential coalitions are singletons (i.e., allcan be described by influential players / Gyes

A and Gno

A ).

◮ An influence model based on aggregation functions A issymmetric if any yes-influential coalition is also no-influentialand vice-versa.

◮ Consequence: a symmetric decomposable model is describedthrough a single graph of influence GA.


Application to the Krackhardt’s

advice network


Introduction

◮ In 1987, Krackhardt published in Social Networks an empiricalstudy based on data collected from managers of a smallmanufacturing firm in the US, to see who sought advice fromwhom.

◮ In 2008, Jackson analyzed these data with the help of theDeGroot’s opinion formation model, and concluded thatmanagers converge to the same opinion in [0, 1].

◮ We provide a different analysis, considering the advice networkof Krackhardt as giving information on the influence betweenmanagers.


The Krackhardt’s data

◮ 21 managers in a manufacturing firm of about 100 employees

◮ Each manager is asked if manager i asks advice to manager j

◮ Graph of the consensus structure: there is an arc frommanager i to manager j if there is a majority of people whothink that i is asking advice to j


The Krackhardt’s advice network

1

23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

1920

21


Assumptions

◮ There are only influential agents, no influential coalition.

◮ If agent i asks advice to agent j , we consider that j isinfluential for i .

◮ Each agent is self-influential.

◮ There is no way to distinguish betwen yes- and no-influence,hence we consider that they coincide.

◮ Consequence: the model is symmetric decomposable, and itsgraph of influence GA is the advice network with all arcsinverted + loops on each agents.


The influence graph

1

23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

1920

21


Qualitative analysis of convergence

From previous results:

◮ No nontrivial terminal state because G 0 (the undirectedgraph) is connected

◮ No cyclic class by the presence of loops on every node

◮ Can regular terminal class exist? Let us construct the closurecl(i) of each agent i , and see if one can find agents outsidecl(i) reachable from cl(i) in G

◮ We compute cl(2):

2 receives arcs from 4, 7, 14, 18 and 21. Thesenodes in turn receive arcs from 6, 8, 9, 10, 11, thenwe find 5, 12, 19, 20, and there is no new node anymore. Hence cl(2) ={2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 18, 19, 20, 21}.



Closure of agent 2 (in blue). Each remaining agent 1, 3, 13, 15,16, 17 receives an arc from cl(2)

S

1

23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

1920

21



◮ Since agent 2 influences agents 1, 3, 4, 5, 7, 9, 11, 13, 14, 15,16, 17, 18, 19, 20, 21, the closure of these agents containscl(2), hence the same conclusion is reached for these agents

◮ 6 is influenced by 7 who is influenced by 2, hence again cl(6)contains cl(2)

◮ 8, 10 and 12 receive indirectly an influence from 2, hence theirclosure contain cl(2)

◮ In summary, for each agent i , every agent outside cl(i)receives an arc from cl(i). Hence, we conclude that no normalregular class exists.


Anonymous social influence


Anonymous influence (1/2)

◮ Customers often can express their opinion on the product orthe hotel, e.g., customer reviews on amazon.com orbooking.com → People follow anonymous customers thathave expressed their positive/negative opinion on the product.

◮ Anonymous influence means: the change in opinion of anactor depends only on how many actors hold an opinion,instead of which actors do so.

◮ An n-place aggregation function A is anonymous if for allx ∈ {0, 1}n and any permutation σ : N → N,A(x1, . . . , xn) = A(xσ(1), . . . , xσ(n)).

◮ Example: the majority model: an actor changes his opinion ifthe proportion of actors with opposite opinion is greater thana threshold η ≥ 1

2 .


Anonymous influence (2/2)

◮ More generally, an aggregation function called orderedweighted average (OWA) models this kind of behavior:

OWAW (x) =n∑

i=1

wix(i) for all x ∈ [0, 1]n

where x(1) ≥ x(2) ≥ · · · ≥ x(n) are the ordered components ofx and wi ≥ 0 ∀i ,

∑i wi = 1.

◮ Example: Maj[m]i (x) := x(m) for all i ∈ N, i.e., wm = 1

◮ An aggregation function A is anonymous if and only if it is anordered weighted average.


Convergence in the anonymous model

A necessary and sufficient condition for convergence to consensus.

TheoremConsider the aggregation model Ai = OWAw i , i ∈ N. Then, thereare only the trivial terminal classes if and only if there existsk ∈ {1, . . . , n} s.t. both:

1. For all k = k , . . . , n − 1, there are k + 1 distinct actors s.t.coalitions of size k are “yes”-influential on each of them.

2. For all k = 1, . . . , k − 1, there are n− k +1 distinct actors s.t.coalitions of size n − k are “no”-influential on each of them.

◮ In other words, we have a cascade that leads either to the“yes”- (part (i)) or “no”-consensus (part (ii)).


Generalization

◮ We say that an aggregation function A isOWAw -decomposable if there exists λ ∈ ]0, 1] and anaggregation function A′ s.t.

A = λOWAw + (1− λ)A′.

◮ Such aggregation functions do exist since convex combinationsof aggregation functions are again aggregation functions.

◮ It turns out that the sufficiency part of our Theorem alsoholds if agents use such decomposable aggregation functions.


Example: Mass psychology (1/2)

The phenomenon of mass psychology (herding behavior):

◮ If at least a certain number of actors share the same opinion,then these actors attract others.

◮ Let n = 3 and whenever only 2 agents have the same opinion,the third changes her opinion with probability λ ∈ (0, 1):

Mass[2]i (x) = λx(2) + (1− λ)xi for all i ∈ N

◮ This function is OWAw -decomposable (convex combination ofthe majority model and a completely self-centered actor):

Mass[2]i (x) = λMaj

[2]i (x) + (1− λ)xi for all i ∈ N

and by the above Theorem (k = 2) the group eventuallyreaches a consensus.


Example: Mass psychology (2/2)

∅

1

2

3

12

13

23

N

λ

1− λ

λ 1− λ

λ

1− λ

1− λ

λ

1− λ λ

1− λ

λ

The aggregation functions are not anonymous since actors considertheir own opinion with weight 1− λ > 0.However, the model is anonymous (there is no differentiationbetween different coalitions of the same size).


Example: Important actors (1/2)

We consider n = 3 actors and that each actor follows to someextent λ ∈ (0, 1) the simple majority model. Moreover, for eachactor, the actor with the next higher index has a relativeimportance of 1− λ. This corresponds to the following importantactors aggregation model:

Imp[2;i+1]i (x) = λMaj

[2]i (x) + (1− λ)xi+1 for all i ∈ N.

Actor i + 1 is “yes”- and “no”-influential on actor i for all i ∈ Nand coalitions of size two or more are “yes”- and “no”-influentialon all actors.


Example: Important actors (2/2)

The model gives the following digraph of the Markov chain:

∅

1

2

3

12

13

23

N

λ

1− λλ

1− λ

λ 1− λ

λ1− λ

λ

1− λ λ

1− λ

The model is anonymous although the aggregation functions arenot.


Fuzzy linguistic quantifiers

◮ Actors might adjust their opinion according to softmajorities/minorities, e.g., they could say “yes” if “most ofthe actors say ‘yes’.”

◮ Words like “most” or “many” are fuzzy linguistic quantifiers.

◮ We define a quantifier by a function of the actors’ proportionsaying “yes” to the degree the quantifier is satisfied.

◮ A fuzzy linguistic quantifier Q is defined by a nondecreasingfunction

µQ : [0, 1] → [0, 1] s.t. µQ(0) = 0 and µQ(1) = 1.

The quantifier is regular if the function is strictly increasing onsome interval (c, c) ⊆ [0, 1] and otherwise constant.

◮ For all quantifiers, there exists a corresponding orderedweighted average that represents it.


Examples

Example: Typical quantifiers

We define

(i) Qaa = “almost all” by

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1


Examples

(ii) Qmo = “most”by

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

(iii) Qma = “many”by

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

(iv) Qaf = “at leasta few” by

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1


Some results on convergence

If actors use similar quantifiers and not too many actors deviate,they will eventually reach a consensus.

Proposition

Consider an aggregation model with aggregation functionsAi = OWAQi , i ∈ N.

(i) If Qi is regular for all i ∈ N and ∩i∈N(ci , ci ) 6= ∅, then thereare no other terminal classes than the trivial terminal classes.

(ii) Suppose mini∈N ci > 0, then the result in (i) still holds if lessthan ⌈cdn⌉ actors deviate to a regular quantifier Qd suchthat cd < mini∈N ci .

(iii) Suppose maxi∈N ci < 1, then the result in (i) still holds if lessthan ⌈(1− cd)n⌉ actors deviate to a regular quantifier Qd

such that maxi∈N ci < cd .


Modeling contagion by aggregation functions(work in progress)


Introduction

◮ Contagion occurs if an action can spread from a finite set ofindividuals to the whole population.

◮ An important contribution to the analysis of contagion is in(Morris, 2000), where the author focuses on thecharacterization of the contagion threshold.

◮ The contagion model is a particular instance of the influencemodel based on aggregation functions (work in progress).


The contagion model of Morris

◮ countably infinite set X of players◮ Γ(x): neighborhood of player x of size γ; threshold 0 ≤ q ≤ 1◮ Rule of contagion: given a configuration X (t) at time t (set

of ‘yes’ players), next configuration X (t +1) is the set of play-ers having a proportion of neighbors in X (t) at least equal to q:

X (t + 1) = {x ∈ X ||Γ(x) ∩ X (t)|

γ≥ q}.

◮ Contagion threshold ξ is the largest q such that ‘yes’ spreadsover X from some finite group X (0)

◮ The contagion model is a particular influence model based onthe following aggregation function for each player x :

Ax(1X ) =

{1, if |X∩Γ(x)|

γ ≥ q

0, otherwise.◮ If q is below the contagion threshold, the terminal states are

the trivial states ∅ and X. Otherwise, other terminal classes(nontrivial terminal states, cycles) can occur.


Bayesian models of information transmission


Bayesian approach

In social learning models agents observe choices over time andupdate their beliefs accordingly (different from the models in whichthe choices depend on the influence of others).

◮ A state of the world θ equal to 0 or 1 with probability 0.5.

◮ A population of n agents, choosing an action xi ∈ {0, 1}, withutility 1 if xi = θ (0 otherwise).

◮ Each agent receives a private signal si ∈ {0, 1} withP(si = θ) = q > 0.5.

◮ Signals are private observation but actions are observed.

◮ Perfect Bayesian Equilibrium: agents update their beliefs in aBayesian fashion and choose the utility maximizing actiongiven beliefs.


Bayesian observation/communication learning

Bayesian observation learning

◮ “herd behavior” (Banerjee, 1992; Scharfstein & Stein, 1990),Acemoglu et al. (2009), Acemoglu (2010), etc.

◮ “informational cascades” (Bikhchandani et al., 1992;Anderson & Holt, 1997) - people get information by observingothers’ actions and are inclined to imitate those who aresupposed to be better informed. Informational cascades formas people ignore their internal signals and follow others.

◮ Bala & Goyal (1998, 2001) - Agents are connected in anundirected social network and in each period theysimultaneously choose among a finite set of actions. Thepayoffs are random and their distribution depends on anunknown state of nature. In each period, an agent alsoobserves choices and outcomes of the neighbors.

Bayesian communication learning - Acemoglu et al. (2009),Galeotti et al. (2010), Hagenbach & Koessler (2010), etc.


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