Introduction Benchmark model Belief-based model Empirical analysis Summary. Riot Networks. Lachlan Deer Michael D. König Fernando Vega-Redondo

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1 Riot Networks Lachlan Deer Michael D. König Fernando Vega-Redondo University of Zurich University of Zurich Bocconi University June 7, 2018 Deer & König &Vega-Redondo Riot Networks June 7, / 23

2 Motivation The emergence of collective action ( riots ), an important phenomenon Our approach emphasizes the role of social networks: They underlie payoffs, by peer interaction They channel crucial information, e.g. on the extent of support They 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 Deer & König &Vega-Redondo Riot Networks June 7, / 23

3 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), König 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), Barberà & 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. Deer & König &Vega-Redondo Riot Networks June 7, / 23

4 Outline 1 The benchmark model: complete (global) information The game-theoretic setup The law of motion: action and link adjustment The potential: characterization of the invariant distribution Stochastically stable states 2 The belief-based model: learning and incomplete (local) information The revised law of motion Stochastic stability under belief formation On beliefs and beachheads External manipulation of beliefs 3 Empirical application (ongoing): Arab Spring on Twitter 4 Summary and conclusions Deer & König &Vega-Redondo Riot Networks June 7, / 23

5 Benchmark model: the game-theoretic setup Players: (large) population i N = {1,..., n}. Network: G = (g ij ) n i,j=1, with g ij {1(i & j connected), 0 (i & j not connected)}, actions: s = (s 1,..., s n ), s i { 1(safe), +1(risky)} payoffs: π i (s, G) = (1 θ ρ)γ i s i + θ n j=1 g ijs i s j + ρ n j=1 s js i κs i ζd i where idiosyncratic characteristic of each agent i: γ i { 1, +1} coordination: local/peer effects θ ; global/population effects ρ costs: κ on risky action; ζ on linking (d i : i s degree) Deer & König &Vega-Redondo Riot Networks June 7, / 23

6 The law of motion Time is continuous. The dynamics induces a path of states {ω t } t 0 Ω where 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 a rate χ > 0 to revise his current action s it, in which case he chooses the new action s i { 1, +1} as a noisy best response to the prevailing state. Formally, for infinitesimal t, we posit: P ( ω t+ t = (s i, s it, G t ) ω t = (s it, s it, G t ) ) = χ P ( π i (s i, s it, G t ) π i (s it, s it, G t ) + ε it > 0 ) t + o( t). for i.i.d. shocks ε it, assumed logistically distributed with parameter η. Deer & König &Vega-Redondo Riot Networks June 7, / 23

7 The law of motion (cont.) (2) Link creation: At every t, each pair of unconnected agents i, j N is selected at a rate λ > 0 to create a link. Then, they do so iff they both find it profitable given some noisy perceptions of the entailed payoffs. Formally, for infinitesimal t + t, we have: P [ω t+ t = (s t, G t + ij) ω t 1 = (s, G t )] = λ P [{π i (s t, G t + ij) π i (s t, G t ) + ε ij,t > 0} {π j (s t, G t + ij) π j (s t, G t ) + ε ij,t > 0}] t + o( t) for i.i.d. shocks ε it, assumed logistically distributed with parameter η. Deer & König &Vega-Redondo Riot Networks June 7, / 23

8 The law of motion (cont.) (3) Link removal: At every t, each pair of unconnected agents i, j N is selected at a rate ξ > 0 to remove a link. Then, they do so iff at least one of them finds it profitable, given some noisy perceptions of the entailed payoffs. Formally, for infinitesimal t + t, we postulate: P [ω t+ t = (s t, G t ij) ω t 1 = (s, G t )] = λ P [{π i (s t, G t ij) π i (s t, G t ) + ε ij,t > 0} {π j (s t, G t ij) π j (s t, G t ) + ε ij,t > 0}] t + o( t) for i.i.d. shocks ε it, assumed logistically distributed with parameter η. Deer & König &Vega-Redondo Riot Networks June 7, / 23

9 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 γ i s i + θ 2 i=1 n i=1 j=1 n a ij s i s j + ρ 2 n n n s i s j κ s i mζ, i=1 j=1 i=1 where m i>j a ij stands for the number of links in G. Deer & König &Vega-Redondo Riot Networks June 7, / 23

10 Potential and the invariant distribution (cont.) For η <, the process is ergodic. The potential function then leads to the following Gibbs-measure characterizing its (unique) invariant distribution. Proposition The unique stationary distribution µ η defined on the measurable space (Ω, F) such that lim t P(ω t = (s, G) ω 0 = (s 0, G 0 )) = µ η (s, G). The probability measure µ η is given by µ η (s, G) = e ηφ(s,g) G G n s { 1,+1} n eηφ(s,g ) where Φ( ) is the potential. Deer & König &Vega-Redondo Riot Networks June 7, / 23

11 d d Checking the theory numerically ζ θ Actual & predicted avge degree for range of ζ and θ, n = 10, and equal number of γ = ±1 Deer & König &Vega-Redondo Riot Networks June 7, / 23

12 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. lim η µ η (ω) > 0}. Proposition Assume ζ < θ and let n + #({i N : γ i = +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 +, with agents in them choosing s i = γ i = +1 and s i = γ i = 1, respectively. Instead, if θ > θ, all states in Ω involve a complete network K n where all agents i N choose either s i = +1 if n + > n 2, or s i = 1 if n + < n 2. Observation: If pop. size n, global coordination effect are large, ρ, θ 0. Deer & König &Vega-Redondo Riot Networks June 7, / 23

13 The belief-based model: revised law of motion Here, each agent i holds expectations p i [0, 1] on the fraction of agents j N in the overall population choosing s j = +1. Then, the agent guides his behavior according to the expected payoff E i (π i s, p, G) = (1 θ ρ)γ i s i + θ n a ij s i s j + ρnp i s i κs i ζd i, in terms of which we posit direct counterparts of prior (1)-(3): (1 ) Action adjustment (2 ) Link creation (3 ) Link removal j=1 Deer & König &Vega-Redondo Riot Networks June 7, / 23

14 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 à la DeGroot. (4) Belief adjustment: At every t, each agent i N is selected at a rate τ > 0 to revise his current belief p i. In that case, he carries out a convex combination of the action frequencies and average beliefs of his current partners. Formally, for weight ϕ (0, 1], and f i (s t, p t, G t ) = ϕ 1 d it n j=1 a ij,t s jt + (1 ϕ) 1 d it n a ij,t p jt j=1 Deer & König &Vega-Redondo Riot Networks June 7, / 23

15 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 γ i s i + θ 2 i=1 n n a ij s i s j +ρp i i=1 j=1 n i=1 n p i s i κ s i mζ, i=1 Deer & König &Vega-Redondo Riot Networks June 7, / 23

16 Potential and invariant distribution: incomplete info. (cont.) Again, for η <, the process is ergodic. Thus we obtain an identical characterization of its unique invariant distribution in terms of the new potential Φ( ), instead of the former Φ( ). Proposition The unique stationary distribution µ η defined on the measurable space (Ω, F) such that lim t P(ω t = (s, G) ω 0 = (s 0, G 0 )) = µ η (s, G). The probability measure µ η is given by µ η (s, G) = e η Φ(s,G) G G n s { 1,+1} n eη Φ(s,G ) where Φ( ) is the potential. Deer & König &Vega-Redondo Riot Networks June 7, / 23

17 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. lim η µ η (ω) > 0}. Proposition Assume ζ < θ and let n + #({i N : γ i = +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 +, with agents in them choosing s i = γ i = +1 and s i = γ i = 1, respectively. Instead, if θ > θ, all states in Ω involve a complete network K n where all agents i N choose either s i = 1 if n + > n 2, or s i = 1 if n + < n 2. Deer & König &Vega-Redondo Riot Networks June 7, / 23

18 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 conformity θ * (beliefs) θ 0.2 θ * (compl. info.) 0.1 segregation ρ For large n and ρ, θ = 0 while ˆθ almost constant Deer & König &Vega-Redondo Riot Networks June 7, / 23

19 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-political conditions), 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 f i (s t, p t, G t ) = νg +ϕ 1 n a }{{} ij,t s jt + (1 ϕ ν) 1 n a ij,t p jt d it d it j=1 j=1 propaganda ν (0, 1] parametrizes manipulation strength, g = 1 is preferred action Naturally, manipulation seen to reduce the segregation parameter region Deer & König &Vega-Redondo Riot Networks June 7, / 23

20 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 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 and retweets Content (NLP) analysis to impute beliefs and behavior Objectives: Large panel data set on the evolution of collective action that can be operationalized in terms of our model.to be used for: testing the model understanding the process in terms of the estimated parameters Deer & König &Vega-Redondo Riot Networks June 7, / 23

network, 8K nodes from 10% sample Deer & Ko nig

21 Introduction Benchmark model Belief-based model Empirical analysis Summary Illustration: sample network, 8K nodes from 10% sample Deer & Ko nig &Vega-Redondo Riot Networks June 7, / 23

22 Structural estimation of the model: the essential idea Our explicit (closed-form) solution of the model underlies the econometric maximum-likelihood strategy, applied either statically or dynamically: Static approach: For any observed strategy profile and network (s, G), we use as likelihood the invariant distribution: µ η (s, G; ϖ) = e η Φ(s,G) G G n s { 1,+1} n e η Φ(s,G ) where ϖ = (γ, θ, ρ, κ, ζ) are the parameters to be used as likelihood maximizers. Dynamic approach: For any observed sequence of networks (G t ) t2 t=t 1 action profiles (s t ) t2 t=t 1 we use as likelihood: P((G t, s t) t 2 t=t1 +1 st 1, G t1 ; ϖ) = t 2 t=t 1 +1 P((G t, s t) t 2 t=t1 +1 Gt 1, st 1; ϖ). and Deer & König &Vega-Redondo Riot Networks June 7, / 23

23 Summary The rise of collective action in large populations is not well understood cannot be modeled as a standard coordination game We propose a rich model where actions and links co-evolve, as dictated by (myopic) payoff considerations Motivated by the large-population context, incomplete information and locally-formed expectations alternative to benchmark setup The model can be fully solved analytically, which permits its structural estimation, both statically and dynamically This approach can be applied to a wide range of collective-action problems, for many of which extensive data are becoming available. An example is provided: turning point in Egyptian Arab Spring Deer & König &Vega-Redondo Riot Networks June 7, / 23

1. Introduction. 2. A Simple Model

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