Riot Networks - UZHIntroduction Benchmark model Belief-based model Empirical analysis Summary Riot Networks Lachlan Deer Michael D. K onig Fernando Vega-Redondo University of Zurich

Introduction Benchmark model Belief-based model Empirical analysis Summary

Riot Networks

Lachlan Deer Michael D. Konig Fernando Vega-Redondo

University of Zurich University of Zurich Bocconi University

June 7, 2018

Deer & Konig &Vega-Redondo Riot Networks June 7, 2018 1 / 23


Motivation

The emergence of collective action ( “riots”), an important phenomenon

Our approach emphasizes the role of social networks:

They underlie payoffs, by peer interactionThey channel crucial information, e.g. on the extent of supportThey are endogenous, i.e. co-evolve with actions

Our model sheds (theoretical) light on the following issues:

How does a large local population coordinate on collective action?How do expectations form and adapt along the process?What is the role of individual heterogeneity (e.g. in preferences)?

Ensuing key objective:

Collection of “big” Twitter data on rioting events (Arab Spring)Structural estimation of our model parameters



Related literature

A wide number of related literature strands

Coordination games in networks:

Fixed networks: Blume (1993), Brock & Durlauf (2001), Morris (2000)

Co-evolving networks: Jackson & Watts (2002), Goyal & V-R (2005),Konig et al. (2014), Marsili & V-R (2017)

Learning:DeMarzo et al. (2003), Golub & Jackson (2010), Acemoglu et al. (2014)

Collective action & threshold behavior:Granovetter (1978), Chwe (2000), Barbera & Jackson (2016)

Our model integrates above features into a single framework leading to:(i) a closed-form characterization of equilib. paths & full comparative analysis

(ii) a likelihood to be used in structural estimation of the model

Badev (2013) has model in similar vein, but equilibrium not characterized— hence full-fledged theoretical analysis or structural estimation not possible.



Outline

1 The benchmark model: complete (global) information

The game-theoretic setupThe law of motion: action and link adjustmentThe potential: characterization of the invariant distributionStochastically stable states

2 The belief-based model: learning and incomplete (local) information

The revised law of motionStochastic stability under belief formationOn beliefs and beachheadsExternal manipulation of beliefs

3 Empirical application (ongoing): Arab Spring on Twitter

4 Summary and conclusions



Benchmark model: the game-theoretic setup

Players: (large) population i ∈ N = 1, . . . , n.

Network: G = (gij)ni ,j=1, with

gij ∈ 1(i & j connected), 0 (i & j not connected),

actions: s = (s1, ..., sn), si ∈ −1(safe),+1(risky)

payoffs:

πi (s,G ) = (1− θ − ρ)γi si + θ∑n

j=1 gijsi sj + ρ∑n

j=1 sjsi − κsi − ζdiwhere

idiosyncratic characteristic of each agent i : γi ∈ −1,+1coordination: local/peer effects θ ; global/population effects ρcosts: κ on risky action; ζ on linking (di : i ’s degree)



The law of motion

Time is continuous. The dynamics induces a path of states ωtt≥0 ⊂ Ωwhere every ω = (s,G ) ∈ Ω specifies an action profile and a network.

The dynamics involves three components:

(1) Action adjustment: At every t, each agent i ∈ N is selected at arate χ > 0 to revise his current action sit , in which case he choosesthe new action s ′i ∈ −1,+1 as a noisy best response to theprevailing state. Formally, for infinitesimal ∆t, we posit:

P(ωt+∆t = (s ′i , s−it ,Gt)|ωt = (sit , s−it ,Gt)

)=

χ P(πi (s

′i , s−it ,Gt)− πi (sit , s−it ,Gt) + εit > 0

)∆t + o(∆t).

for i.i.d. shocks εit , assumed logistically distributed with parameter η.



The law of motion (cont.)

(2) Link creation: At every t, each pair of unconnected agents i , j ∈ Nis selected at a rate λ > 0 to create a link. Then, they do so iff theyboth find it profitable given some noisy perceptions of the entailedpayoffs. Formally, for infinitesimal t + ∆t, we have:

P [ωt+∆t = (st ,Gt + ij)|ωt−1 = (s,Gt)] =

λ P [πi (st ,Gt + ij)− πi (st ,Gt) + εij ,t > 0 ∩πj(st ,Gt + ij)− πj(st ,Gt) + εij ,t > 0] ∆t + o(∆t)




The law of motion (cont.)

(3) Link removal: At every t, each pair of unconnected agents i , j ∈ Nis selected at a rate ξ > 0 to remove a link. Then, they do so iff atleast one of them finds it profitable, given some noisy perceptions ofthe entailed payoffs. Formally, for infinitesimal t + ∆t, we postulate:

P [ωt+∆t = (st ,Gt − ij)|ωt−1 = (s,Gt)] =

λ P [πi (st ,Gt − ij)− πi (st ,Gt) + εij ,t > 0 ∪πj(st ,Gt − ij)− πj(st ,Gt) + εij ,t > 0] ∆t + o(∆t)




Potential and the invariant distribution

A first important result is that the game is a potential game(and hence noiseless best-response adjustment converges to some NE).

Proposition

The payoffs of the game admit a potential Φ : Ω→ R defined by

Φ(s,G ) = (1−θ−ρ)n∑

i=1

γi si+θ

2

n∑i=1

n∑j=1

aijsi sj+ρ

2

n∑i=1

n∑j=1

si sj−κn∑

i=1

si−mζ,

where m ≡∑

i>j aij stands for the number of links in G .



Potential and the invariant distribution (cont.)

For η <∞, the process is ergodic. The potential function then leads to thefollowing Gibbs-measure characterizing its (unique) invariant distribution.

Proposition

The unique stationary distribution µη defined on the measurable space(Ω,F) such that limt→∞ P(ωt = (s,G )|ω0 = (s0,G0)) = µη(s,G ).The probability measure µη is given by

µη(s,G ) =eηΦ(s,G)∑

G ′∈Gn∑

s′∈−1,+1n eηΦ(s′,G ′)

where Φ(·) is the potential.



Checking the theory numerically

0 2 4 6 8 100

1

2

3

4

5

6

ζ

d

0 0.2 0.4 0.6 0.8 1

2.5

2.6

2.7

2.8

2.9

3

3.1

θ

d

Actual & predicted avge degree for range of ζ and θ, n = 10, and equal number of γ = ±1



Stochastically stable states

Identify the states visited with positive prob. in the long run for vanishing noise,i.e. the stochastically stable states Ω∗ ≡ ω ∈ Ω s.t. limη→∞ µη(ω) > 0.

Proposition

Assume ζ < θ and let n+ ≡ #(i ∈ N : γi = +1). If

θ < θ∗ =(n − n+)(ζ − 2ρ) + 2(1− κ− ρ)

2 + n − n+,

all states in Ω∗ involve a network with two cliques of sizes n+ and n − n+, withagents in them choosing si = γi = +1 and si = γi = −1, respectively. Instead, ifθ > θ∗, all states in Ω∗ involve a complete network Kn where all agents i ∈ Nchoose either si = +1 if n+ > n

2 , or si = −1 if n+ < n2 .

Observation: If pop. size n, global coordination effect are large, ρ, θ∗ ≤ 0.



The belief-based model: revised law of motion

Here, each agent i holds expectations pi ∈ [0, 1] on the fraction of agentsj ∈ N in the overall population choosing sj = +1.

Then, the agent guides his behavior according to the expected payoff

Ei (πi |s,p,G ) = (1− θ − ρ)γi si + θ

n∑j=1

aijsi sj + ρnpi si − κsi − ζdi ,

in terms of which we posit direct counterparts of prior (1)-(3):

(1′) Action adjustment

(2′) Link creation

(3′) Link removal



Law of motion for beliefs under incomplete information

Agents revise beliefs by combining the following sources of local info.:

- the action frequencies of current partners, extrapolated globally;

- the beliefs of current partners, integrated a la DeGroot.

(4) Belief adjustment: At every t, each agent i ∈ N is selected at a rate τ > 0to revise his current belief pi . In that case, he carries out a convexcombination of the action frequencies and average beliefs of his currentpartners. Formally, for weight ϕ ∈ (0, 1], and

fi (st ,pt ,Gt) = ϕ1

dit

n∑j=1

aij,tsjt + (1− ϕ)1

dit

n∑j=1

aij,tpjt



Potential & invariant distribution: incomplete information

Again, we find that the game has a potential (a different one!)and hence the usual best-response adjustment converges to some NE.

Proposition

The payoffs of the game admit a potential Φ : Ω→ R defined by

Φi(s,G ) = (1−θ−ρ)n∑

i=1

γi si+θ

2

n∑i=1

n∑j=1

aijsi sj+ρpi

n∑i=1

pi si−κn∑

i=1

si−mζ,



Potential and invariant distribution: incomplete info. (cont.)

Again, for η <∞, the process is ergodic. Thus we obtain an identicalcharacterization of its unique invariant distribution – in terms of the newpotential Φ(·), instead of the former Φ(·).

Proposition

The unique stationary distribution µη defined on the measurable space(Ω,F) such that limt→∞ P(ωt = (s,G )|ω0 = (s0,G0)) = µη(s,G ).The probability measure µη is given by

µη(s,G ) =eηΦ(s,G)∑

G ′∈Gn∑

s′∈−1,+1n eηΦ(s′,G ′)

where Φ(·) is the potential.



Stochastically stable states: incomplete information

Identify the states visited with positive prob. in the long run for vanishing noise,i.e. the stochastically stable states Ω∗ ≡ ω ∈ Ω s.t. limη→∞ µη(ω) > 0.

Proposition

Assume ζ < θ and let n+ ≡ #(i ∈ N : γi = +1). If

θ < θ =(n − n+)ζ + 2(1− κ− ρ)

2 + n − n+,

all states in Ω∗ involve a network with two cliques of sizes n+ and n − n+, withagents in them choosing si = γi = +1 and si = γi = −1, respectively. Instead, ifθ > θ∗, all states in Ω∗ involve a complete network Kn where all agents i ∈ Nchoose either si = −1 if n+ > n

2 , or si = −1 if n+ < n2 .



On peer pressure, segregation, and information

Comparing segregation thresholds under complete & incomplete info.:

(n − n+)(ζ − 2ρ) + 2(1− κ− ρ)

2 + n − n+= θ∗ < θ =

(n − n+)ζ + 2(1− κ− ρ)

2 + n − n+

segregation

conformityθ* (beliefs)

θ* (compl. info.)

0.00 0.05 0.10 0.150.0

0.1

0.2

0.3

0.4

0.5

ρ

θ

For large n and ρ, θ∗ = 0 while θ almost constant



On segregation and beliefs (cont.)

Why is the comparison of thresholds θ∗ and θ important?

It bears on whether a relatively small group of n− ≡ n − n+

revolutionaries can gain a stable beachhead.

Thereafter, through “drift” (or gradual change in socio-politicalconditions), it can take over.

Explicit modeling of beliefs also allows one to study belief manipulation(by, say, a government), modifying the belief formation rule as

fi (st ,pt ,Gt) = νg︸︷︷︸propaganda

+ϕ1

dit

n∑j=1

aij ,tsjt + (1− ϕ− ν)1

dit

n∑j=1

aij ,tpjt

ν ∈ (0, 1] parametrizes manipulation strength, g = −1 is preferred action

Naturally, manipulation seen to reduce the “segregation parameter region”



Empirical application: the Egyptian Arab Spring

Events: The counter-revolutionary backlash on Morsi’s government

Focus: the unfolding and aftermath of Mohamed Morsi’s toppling(July 3, 2013), part of what has been called the “Arab Winter”

Time span: June 21-September 3, bracketing Morsi’s fall– around 2 1

2 years after first protests against Mubarak erupted (Jan.2011) and a year after Morsi’s access to power (June 2012)

Data: Twitter information, downloaded through its public API

700K individuals, 5M ArabSpring-related tweets

Network constructed by relying on @mentions and retweets

Content (NLP) analysis to impute beliefs and behavior

Objectives: Large panel data set on the evolution of collective actionthat can be operationalized in terms of our model.To be used for:

testing the modelunderstanding the process in terms of the estimated parameters



Illustration: sample network, 8K nodes from 10% sample



Structural estimation of the model: the essential idea

Our explicit (closed-form) solution of the model underlies the econometricmaximum-likelihood strategy, applied either statically or dynamically:

Static approach: For any observed strategy profile and network (s,G ), weuse as likelihood the invariant distribution:

µη(s,G ;$) =eηΦ(s,G)∑

G ′∈Gn∑

s′∈−1,+1n eηΦ(s′,G ′)

where $ = (γ, θ, ρ, κ, ζ) are the parameters to be used as likelihood maximizers.

Dynamic approach: For any observed sequence of networks (Gt)t2t=t1

andaction profiles (st)

t2t=t1

we use as likelihood:

P((Gt , st)t2t=t1+1|st1 ,Gt1 ;$) =

t2∏t=t1+1

P((Gt , st)t2t=t1+1|Gt−1, st−1;$).



Summary

The rise of collective action in large populations is not wellunderstood – cannot be modeled as a standard coordination game

We propose a rich model where actions and links co-evolve, asdictated by (myopic) payoff considerations

Motivated by the large-population context, incomplete informationand locally-formed expectations alternative to benchmark setup

The model can be fully solved analytically, which permits itsstructural estimation, both statically and dynamically

This approach can be applied to a wide range of collective-actionproblems, for many of which extensive data are becoming available.An example is provided: turning point in Egyptian Arab Spring


Documents

Riot Networks - UZHIntroduction Benchmark model Belief-based model Empirical analysis Summary Riot Networks Lachlan Deer Michael D. K onig Fernando Vega-Redondo University of Zurich