information aggregation in social networksaromuri/uploads/1/7/6/...chandrasekhar, larreguy, xandri...

Information Aggregation in Social Networks

A. Jadbabaie∗ P. Molavi∗∗ A. Tahbaz-Salehi†

∗Electrical and Systems Engineering, Computer and Information ScienceOperations and Information Management, UPenn

∗∗Electrical and Systems Engineering, UPenn

†Decision, Risk, and OperationsColumbia Graduate School of Business

Jadbabaie, Molavi, and Tahbaz-Salehi Information Aggregation in Social Networks 1 / 22

Motivating Example: Climate Change

General features:

There is a common underlying state of the world.

Agents’ optimal choices depend on the realization of the state.

Agents have heterogeneous information endowments.

Agents communicate their opinions with agents in their socialneighborhood.

Other examples:

adoption of microfinance (Banerjee et al. (2012))

agricultural methods (Hagerstrand (1969) and Rogers (1983))

medical procedures

General features:

Other examples:

medical procedures

General features:

Other examples:

medical procedures

General features:

Other examples:

medical procedures

General features:

Other examples:

medical procedures

The Problem

Research question

How do the network structure and agents’ information endowmentsdetermine the extent of information aggregation?

Our model: an extension of DeGroot’s learning model with

continuous flow of new information

heterogenous observations

asymptotic agreement with the Bayesian benchmark

DeGroot Learning Models

DeGroot based models are tractable:

DeGroot (1974)

Tsitsiklis (1984)

DeMarzo, Vayanos, Zwiebel (2003)

Jadbabaie, Lin, Morse (2003)

Acemoglu, Ozdaglar, Parandeh-Gheibi (2010)

Golub and Jackson (2010)

Jadbabaie, Molavi, Sandroni, Tahbaz-Salehi (2012)

There is empirical evidence in favor of DeGroot models:

Chandrasekhar, Larreguy, Xandri (2012)

DeGroot Learning Models

DeGroot based models are tractable:

DeGroot (1974)

Tsitsiklis (1984)

DeMarzo, Vayanos, Zwiebel (2003)

Jadbabaie, Lin, Morse (2003)

Acemoglu, Ozdaglar, Parandeh-Gheibi (2010)

Golub and Jackson (2010)

Jadbabaie, Molavi, Sandroni, Tahbaz-Salehi (2012)

There is empirical evidence in favor of DeGroot models:

Chandrasekhar, Larreguy, Xandri (2012)

Problem Setup

Finite set of agents N = {1, . . . , n}.

Agents are organized in a social network.

Ni ⊆ N is the set of neighbors of agent i .

Agent i can observe the beliefs of her neighbors, i.e., j ∈ Ni .

Finite set of conceivable states Θ.

Agents have beliefs over Θ and repeatedly update them.

Timing of the Events

At time 0:

A state θ∗ ∈ Θ is realized.

Agent i starts with the prior µi ,0 over Θ.

At time t:

Agent i observes {µj ,t−1}j∈Ni.

Agent i also privately observes signal ωi ,t .

Agents use the new information to update their beliefs:

µi ,t = fi (µi ,t−1; {µj ,t−1}j∈Ni, ωi ,t)

Timing of the Events

At time 0:

A state θ∗ ∈ Θ is realized.

Agent i starts with the prior µi ,0 over Θ.

At time t:

Agent i observes {µj ,t−1}j∈Ni.

Agent i also privately observes signal ωi ,t .

Agents use the new information to update their beliefs:

µi ,t = fi (µi ,t−1; {µj ,t−1}j∈Ni, ωi ,t)

Agents’ Private Observations

Signal profile ωt = (ω1,t , . . . , ωn,t) is generated at time t.

ω1, ω2, ω3, . . . are i.i.d.

ωt ∼ `(·|θ), given the realized state is θ ∈ Θ.

ωi ,t belongs to the finite set Si .

The marginal of `(·|θ) over Si is `i (·|θ).

Information Content of Agents’ Observations

Definition

Agent i ’s information for θ in favor of θ̂ when θ is realized:

Iθi (θ̂)def= DKL(`i (·|θ)‖`i (·|θ̂)) ≥ 0.

Definition

The realized state θ∗ is identifiable if I∗i (θ) > 0 for some i and allθ 6= θ∗.

Information Content of Agents’ Observations

Definition

Agent i ’s information for θ in favor of θ̂ when θ is realized:

Iθi (θ̂)def= DKL(`i (·|θ)‖`i (·|θ̂)) ≥ 0.

Definition

The realized state θ∗ is identifiable if I∗i (θ) > 0 for some i and allθ 6= θ∗.

The Model

Belief update

µi ,t = aiiBU(µi ,t−1;ωi ,t) +∑j∈Ni

aijµj ,t−1

Agents aggregate the beliefs of their neighbors linearly.

Agents process their private signals in a Bayesian way.

aii is the self-reliance of agent i .

aij is trust of agent i in agent j .

The Model

Belief update

µi ,t = aiiBU(µi ,t−1;ωi ,t) +∑j∈Ni

aijµj ,t−1

Agents aggregate the beliefs of their neighbors linearly.

Agents process their private signals in a Bayesian way.

aii is the self-reliance of agent i .

aij is trust of agent i in agent j .

Sufficient Conditions for Learning

Proposition

Assume

(a) the network is strongly connected;

(b) agents’ priors have full support;

(c) all the states are identifiable;

(d) agents have strictly positive self-reliance; i.e., aii > 0 for all i .

Then all agents learn the realized state asymptotically almostsurely; i.e.,

µi ,t(θ∗) −→ 1 ∀i ∈ N .

Rate of Learning (1/2)

Definition

Eigenvector centrality of agent i is the normalized fixed point of

vi =n∑

aijvj .

Proposition

et : the total expected error at time t.

limt→+∞1

tlog et = −r ,

wherer ≈ min

θminθ̂ 6=θ

∑i∈N

viaiiIθi (θ̂).

Definition

Eigenvector centrality of agent i is the normalized fixed point of

vi =n∑

aijvj .

Proposition

et : the total expected error at time t.

limt→+∞1

tlog et = −r ,

wherer ≈ min

θminθ̂ 6=θ

∑i∈N

viaiiIθi (θ̂).

r ≈ minθ

minθ̂ 6=θ

∑i∈N

viaiiIθi (θ̂)

minθ minθ̂ 6=θ Iθi (θ̂) is the rate of learning for a Bayesian agent.

1− aii is the reduction in the rate as a result of agent’s bias.

vi captures the effective attention that agent i receives.

Information Endowments

Definition

Ii � Ij if

Iθi (θ̂) ≥ Iθj (θ̂) for all θ, θ̂ ∈ Θ.

Ii � Ij if Ij � Ik and

Iθi (θ̂) > Iθj (θ̂) for some θ, θ̂ ∈ Θ.

� captures a notion of dominance.

Ii provides more information than Ij ≺ Ii .

This is regardless of which state is realized.

Information Allocation (1/4)

Definition

An allocation rule is a bijective mapping

σ : {1, 2, . . . , n} → {1, 2, . . . , n}.

Question

Given signals with information {Ij}j∈N , which allocation rulemaximizes the rate of learning?

Proposition

Assume that {Ij}j∈N are fully ordered with respect to �. Then forany optimal allocation σ,

aiivi ≥ ajjvj ⇔ Iσ(i) � Iσ(j),aiivi > ajjvj ⇔ Iσ(i) � Iσ(j).

This corresponds to positive assortative matching.

Corollary

Assume that aii = a for all i and for some m ≤ n,

{I for j = 1, . . . ,m,

0 for j = m + 1, . . . , n.

Then in any optimal allocation the m most central agents makethe informative observations.

Corollary

If |Θ| = 2, in any optimal allocation the most central agentsobserve the most informative signals.

These are consistent with the empirical observation of Banerjee,Chandrasekhar, Duflo, & Jackson (2012).

Corollary

{I for j = 1, . . . ,m,

0 for j = m + 1, . . . , n.

Corollary

{I for j = 1, . . . ,m,

0 for j = m + 1, . . . , n.

Corollary

The information ordering which leads to positive assortativematching is very strong (dominance).

Other orderings could lead to dramatically different results.

In fact, the optimal allocation could correspond to the least centralagent making the “best” observations.

An Example (1/4)

aii = a

Θ = {θ0, θ1, . . . , θn}

Si = {H,T}

`j(·|·) :

θ0 1− αj αj

θ1 1− αj αj...

...θj αj 1− αj

......

θn 1− αj αj

An Example (2/4)

r ∝ mink 6=0

n∑i=1

viI0i (θk)

I0j (·) :

θ1 θ2 θn

j = 1 I1 0 . . . 0j = 2 0 I2 . . . 0

......

. . ....

j = n 0 0 . . . In

Agent with signal j is “expert” in state j and knows nothingabout other states.

Nobody is expert in state 0.

An Example (3/4)

Given the allocation rule σ, the rate of learning is proportional to

vi Iσ(i).

The rate of learning is determined by the agent with the smallestcentrality × expertise.

An Example (4/4)

Proposition

For any optimal allocation σ,

vi ≥ vj ⇔ Iσ(i) ≤ Iσ(j),vi > vj ⇔ Iσ(i) < Iσ(j).

In any optimal allocation, the least central agent has thehighest expertise.

There needs to be a mismatch between the attention andinformation bottlenecks.

This corresponds to negative assortative matching.

An Example (4/4)

Proposition

For any optimal allocation σ,

vi ≥ vj ⇔ Iσ(i) ≤ Iσ(j),vi > vj ⇔ Iσ(i) < Iσ(j).

In any optimal allocation, the least central agent has thehighest expertise.

There needs to be a mismatch between the attention andinformation bottlenecks.

This corresponds to negative assortative matching.

Conclusion

The interplay of information and network structures determinethe extent of information aggregation.

When the information content of signals are fully orderedaccording to a notion of dominance, optimal allocation ofsignals entails central agents to receive better signals.

Using different notions of information ordering can result insignificantly different conclusion.

In the optimal allocation, the least central agents could makethe best observations.

Future research:

a more complete characterization of the rate of learning

information aggregation in social networksaromuri/uploads/1/7/6/...chandrasekhar, larreguy, xandri...

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