ISSN 1471-0498

DEPARTMENT OF ECONOMICS

DISCUSSION PAPER SERIES

Coordinated Adoption of Social Innovations

Dominik Karos

Number 797

July, 2016

Manor Road Building, Oxford OX1 3UQ

Coordinated Adoption of Social Innovations∗

Dominik Karos†

July 4, 2016

Abstract The members of a society are faced with the decision whetheror not to participate in an anti-government protest. Their utilities depend ontheir own decision but also on those of their neighbors in an underlying socialnetwork. They randomly observe other people’s decisions, gather informationon who is already active, and base their decision on their information. Themodel uses a Markov process (that depends on the underlying social network)to analyze who will become active over time. Two new features are essen-tial: first, only very mild assumptions about the underlying social networkare made, in particular agents can be entirely heterogeneous. Second, indi-viduals are allowed to coordinate their decision if they mutually observe eachother. The government can use political violence in order to change people’sutility from being active. The probability of a revolution can thereby be re-duced in the short run, but not in the long run. Under political repressionprotests do not increase gradually, but suddenly; and the conditional probabil-ity of a quick revolution given a protest increases if the regime turns violentlyagainst the protesters. Since large jumps in the number of activists dependon their capability to coordinate, the repression of political activism is moreeffective in countries where social media are not easily accessible. The findingsare illustrated by data on the number of protests and revolutions world-widedepending on a country’s number on the Political Terror Scale.

Keywords: Social Networks, Coordination, Strong Nash Equilibrium, Inno-vation Diffusion, Unanticipated Revolutions, Political Repression

JEL Classification: C72, D85, O33

∗I would like to thank Margareth Meyer, Arno Riedl, Larry Samuelson, Berno Buchel, andthe participants of the Economic Theory Workshop at the University of Oxford and of theEconomic Seminar at Maastricht University for their valuable comments.†St Edmund Hall and The Department of Economics in the University of Oxford

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1 Introduction

Around 8:00pm I posted on Facebook: “Come on guys, let’s beserious. If you really want to do something, don’t just ‘like’ this post.Write that you are ready, and we can try to start something.” Withinan hour there were more than 600 comments. I posted again: “Letsmeet at 10:30pm near the monument to independence in the middleof the Maidan.” When I arrived, maybe 50 people had gathered.Soon the crowd had swelled to more than 1,000.

— Mustafa Nayem1

Coordinated behavior is essential when modeling social uprising or the adoptionof an innovation that involves high cost (or risk) of participation and that re-wards participants only if a critical mass is reached. While many models on thedynamics of revolutions exist, their consideration of unilateral decisions ratherthan group decision is a drawback that must be overcome on our way to un-derstand the events during the first half of the current decade. The model Ipropose in this article shall contribute to the theory of innovation diffusion insocial networks and provide a theoretical foundation for stylized facts on theemergence of revolutions.According to Kuran (1989) revolutions under repressive regimes are quick andunanticipated. The reason is that a privately hated regime may enjoy widespreadpublic support because of people’s reluctance to take the lead in publicizing theiropposition. A sudden change in living standards can then be a coordinationdevice that causes sudden mass protest. Similarly, Olson (1971) argues thatfree-riders hinder the emergence of a revolution: people prefer a regime changethat is caused by the protest of others, so that they do not have to risk theirown life fighting for it.Given this coordination problem a regime can influence the probability of a rev-olution by repressing their people: if protesting is more costly to the individual,fewer people will openly challenge the system without being backed up by manyothers. Hence, under a repressive regime an anti-government protest will occuronly if chances are high that it will cause the government to step back. Figure 1depicts2 how the anti-government protests and revolutions world-wide between1976 and 2014 are distributed over the five levels on the Political Terror Scale.3

It can be seen that the probability that an anti-government protest turns intoa revolution is 0 in PTS-1-countries and gradually increases to its maximum ofabout 1% in PTS-5-countries.Figure 1 suggests that political repression actually increases the chance thata protest turns into a revolution. In order to understand these dynamics one

1Mustafa Nayem is one of the initiators of the Maidan protests in Kiev 2013/2014 andnow member of the Ukrainian parliament. This quote is from an article he wrote forwww.opensocietyfoundations.org on April 4, 2014

2Data are based on Banks and Wilson (2015), Gibney et al. (2015), and Marshall andMarshall (2015), for details see Appendix A

3The Political Terror Scale has five levels where level 1 contains countries under a securerule of law and level 5 consists of those where terror has expanded to the whole population.

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Figure 1: Mass protests and revolutions between 1976 and 2014.

has to ask: When does an individual decide to participate in a protest? Afterher personal threshold has been crossed? (Granovetter, 1978) or: When herpersonal threshold will be crossed because others will join her? (Chwe, 1999)or: When she is sufficiently influential to make others join after her? (Lohmann,1994a,b) or: When her friends are already participating? (Granovetter, 1973;McAdam and Paulson, 1993)The model I propose here is sufficiently general to capture all these cases: anindividual joins a protest if she meets a group of individuals, so that for eachmember of this group it is optimal to join the protest if all others join. Thismeans they can simply take each other by the hand and go, and do not have totrust that others will show up. By optimal I mean they prefer being active overbeing inactive. The utility functions that I use are more general than thresholdmodels and allow in particular for free riders and opportunists.The decision an individual takes is based on her information, that is her knowl-edge about the behavior of people she cares about. Hence, there are two networkstructures: On the one hand side the utility functions determine who is impor-tant for an individual, on the other hand side an individual might observe manymore people than those she cares about. This distinction is losely related tothe concept of strong and weak ties in social networks Granovetter (1973), butI shall give a very specific meaning to both networks. The utility functionsimplement a network of influence and control, usually represented by a controlstructure. An observation is a directed graph that is randomly drawn from theset of all directed graphs with vertice set N ; it is closely related to the stochasticneighborhood defined by Acemoglu et al. (2011). There are other ways to model

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the information that is available to an individual, for instance assuming thateach individual’s neighbors behave in the same way as the whole society (seefor instance Jackson and Rogers, 2007; Lopez-Pintado, 2008). However, in thecontext of revolutions where people often segregate into supporters and enemiesof the regime this assumption seems very unrealistic.Endowing individuals with utility functions and the two strategies to be activeor inactive defines a network game (Galeotti et al., 2010), and the point whereno individual joins or leaves the protest is an equilibrium. In most models, fromSchelling (1971) to Morris (2000) to Young (2009), only unilateral deviations areconsidered. But: suppose there is a couple, both prefer both being active overnobody being active over being active without their partner. If only unilateraldecisions are allowed, this couple will never join a protest. It seems thereforethat the Nash equilibrium is not strong enough to capture that reasoning.The strong (or coalitional) Nash equilibrium (Aumann, 1967) is stable withrespect to all coalitional deviations. But this seems unjustified as well: Whyshould a group of people that never meet deviate? In the model presented hereonly cliques in the observation graph can coordinate and jointly take an action.The equilibrium of interest is, hence, an equilibrium that is stable with respectto all coalitional deviations by cliques that can appear with positive probability.(The case where only pairs of players can communicate has been investigatedby Aliprantis et al., 2007).After having set up the model one can start analyzing the effects of policychanges or new technologies to the process. The separation of influence fromobservation allows to independently investigate the effects of political instru-ments (see for instance Ginkel and Smith, 1999) and the availability of newmedia (for instance Edmond, 2013). It turns out that easier communication canfasten the process but not necessarily increase the number of active individualsin the long run. Repression has a negative effect on the long run outcome onlyif communication is sufficiently difficult. In particular, it can increase the con-ditional probability of a successful revolution given a protest as suggested byFigure 1.Section 2 formally introduces control structures, observations, and the informa-tion players gather during the process. In Section 3 the the adoption cascadeis described, and some properties are derived. Section 4 defines a continuousdiffusion process based on the adoption cascade. Convergence properties are de-rived, the steady states are described, and robustness properties of the processwith respect to small perturbations are investigated. Section 5 returns to thedynamics of revolutions. In particular, it is shown how political repression canincrease the probability that anti-government protests turn into revolutions asseen in Figure 1. Section 6 contains some concluding remarks. All proofs andsome technical lemmas are postponed to the appendix.

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2 Model Ingredients

2.1 Control Structures

Throughout the paper let N be a finite set of players. An individual i ∈ Nmay be convinced by a coalition S ⊆ N of active members to join a protest.But similarly, i might have a partner j such that i likes the idea of following S,but would not do so without j – and of course j may feel the same way withrespect to i. In this case S could convince i and j together, but not separately.A control structure captures this idea: a control structure is a correspondenceC : S 7→ C(S) ⊆ 2N , and if T ∈ C(S), I shall say that S is convincing forT . Throughout this paper the following two assumptions on control structuresshall be made:

1. C is monotonic, i.e. C(S) ⊆ C(T ) whenever S ⊆ T ,

2. C is additive, i.e. T1 ∪ T2 ∈ C(S) whenever T1, T2 ∈ C(S).

Monotonicity excludes “congestion effects”, i.e. the adoption of an innovation orjoining a protest does not become less attractive if more individuals are alreadyactive. Additivity is similar in the sense that cases were two “arch enemies”would join a coalition S separately but not together are excluded.

Both assumptions may be challenged in various contexts: When talkingfor instance about fashion, it seems that many people are interested in a newproduct only as long as it is not used by too many others. However, the mainfocus of this paper shall lie on the emergence of mass protests, where bothassumptions are reasonable. Similarly in applications like innovation diffusionwith positive externalities both assumptions will hardly be challenged.

Example 2.1. One way to construct a control structure is by starting fromindividual utility functions:

Suppose that the members of N have monotonic utility function ui : 2N →R, i.e. utility functions with

ui (S ∪ T )− ui (S) ≥ 0 and ui (S ∪ T )− ui (S) ≥ ui (S′ ∪ T )− ui (S′)

for all S ⊆ N with i ∈ S, S′ ⊆ S \ i, and T ⊆ N \ i. That is an activeplayer i prefers more players being active and derives greater marginal utilityfrom additional players adopting than an inactive individual. Then the jointbest response of a coalition T given that all members of S are active is to jointlyadopt if ui ((S ∪ T )) ≥ ui (S \ i) for all i ∈ T . Hence, the correspondence Cu,defined by

Cu(S) = T ⊆ N : ui ((S ∪ T )) ≥ ui (S \ i) for all i ∈ T (1)

for all S ⊆ N is a monotonic and additive control structure with 2S ⊆ C(S) forall S ⊆ N .

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While control structures with coordination are a new concepts, similar objectswithout coordination have been investigated before under the names commandfunctions (Hu and Shapley, 2003), follower functions (Grabisch and Rusinowska,2011), or mutual control structures (Karos and Peters, 2015). In all three casesC is a monotonic map from 2N to 2N , defining for each coalition S the set ofplayers S is convincing for.

2.2 Invariant Control Structures

When investigating the spread of protest or innovations, one essential questionis: How far will it spread? Let C be a control structure, let S, T,R ⊆ N , andsuppose that T ∈ C(S) and R ∈ C(T ). Then S indirectly convinces R via T : ifall members of S have adopted, the members of T will (jointly) do so as well,and after them, the members of R will do so. I shall call C invariant if R ∈ C(S)for all S, T,R ⊆ N with T ∈ C(S) and R ∈ C(T ). Equivalently, C(R) ⊆ C(S)for all R ∈ C(S). An extension of C is a control structure D with C(S) ⊆ D(S)for all S ⊆ N . If D is, additionally, invariant, it is an invariant extension ofC; and D is called minimal if D(S) ⊆ D′(S) for all invariant extensions D′

of C. It is easy to see that for any two invariant extensions D1 and D2 of Ctheir intersection4 D1 ∩D2 is an invariant extension of C as well (Lemma B.1in the appendix). Hence, the minimal invariant extension of C is unique; in thefollowing it shall be denoted by C∗. The following lemma describes an intuitiveway to find it, its proof is very similar to the one in Karos and Peters (2015)for mutual control structure and is therefore postponed to the appendix.

Lemma 2.2. Let C be a mutual control structure, let C1 = C, and recursivelydefine Ck by

Ck(S) = Ck−1(S) ∪⋃

T∈Ck−1(S)

C (T )

Ck(S) =

T : there is C ⊆ Ck(S) with T =

⋃T ′∈C

T ′

for all S ⊆ N . Then there is r ≥ 0 such that Cr = Ck for all k ≥ r, andCr = C∗.

The algorithm in Lemma 2.2 is neither the only, nor the most efficient, way tofind C∗, but it describes how an adoption cascade would work, if individualscould permanently observe each other. For control structures without corre-lation it corresponds to discrete informational cascade models as proposed forinstance by Schelling (1978) or analyzed more recently by Lim et al. (2015).

Example 2.3. Let C be the minimal control structure with C(1) = 12 andC(2) = 23. Then C2(1) = C(1) ∪ C(12) = 12, 23, which is not additive,and C2(1) = 12, 23, 123. (Throughout the paper I use the notation ij as anabbreviation for i, j in order to avoid unnecessary brackets.)

4The intersection D1 ∩D2 is defined by (D1 ∩D2) (S) = D1(S) ∩D2(S) for all S ⊆ N .

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2.3 Observation and Information

One aim of this paper is to separate influence from observation. While theprevious sections considered the former, I shall now turn to the latter. Anobservation is a collection P = (P1, . . . ,PN ) with i ∈ Pi for all i ∈ N ; in thiscase i observes all members of Pi.5 The set of all observations shall be denotedby Ω and β shall denote a probability distribution over Ω.

Example 2.4. Let N = 1, 2, 3 and suppose that all observations are physicalmeetings, i.e. i ∈ Pj if and only if Pi = Pj , that the probability of two individ-uals’ meeting is 1

2 , and that all meetings are stochastically independent of eachother. Then there are 5 observations that can happen with positive probability,namely: nobody observes anybody, 1 and 2 meet, 1 and 3 meet, 2 and 3 meet,all players meet; and each of these observations happens with probability 1

5 .Hence,

supp (β) = (1, 2, 3) , (12, 12, 3) , (13, 2, 13) , (1, 23, 23) , (123, 123, 123) ⊆ Ω.

For i ∈ N let Si be the coalition of players of whom i thinks that they haveadopted the innovation. The vector S = (S1, . . . ,SN ) is called an informationstructure.

Example 2.5. Let N = 1, 2, 3. Suppose that all players know player 3 ownsa certain product, and that player 2 also owns this product but is the only onewho knows that. This situation can be represented by the information structureS = (3, 23, 3).

Let Ψ be the set of information structures. Clearly, Ω ⊆ Ψ. There is a natural(partial) order on Ψ (and hence on Ω), namely

S T if and only if Si ⊆ Ti for all i ∈ N.

That is S T if and only if each individual i who thinks that j is active in Tthinks the same about j in S.

An individual i’s decision about joining a protest depends on her informationSi rather than the true set of active individuals. Hence, a state shall consistof a coalition S of active individuals and an information structure S such thati ∈ Si if and only if i ∈ S, i.e. i knows whether or not she’s active. The setof states shall be denoted by Υ. Again, there is a natural (partial) order on Υ,namely

(S,S) (T, T ) if and only if S T and S ⊆ T. (2)

In a state (S,S) individual i is incorrectly informed if Si 6= S. There are twokinds of incorrect information.

5An observation could be described by a directed graph as well, but graph theoretic prop-erties are not interesting for what follows.

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1. If the information Si satisfies Si ( S, then individual i is underinformed.In this case i does not know all active agents, but there is no inactiveagent who i believes is active.

2. If there is j ∈ Si \ S then i is called misinformed. In this case i believesthat j is active, but this is not true.

In the remainder of this article I make the assumption that players don’t forgetwho they have seen, and they update their information about another individualonly if they observe this individual. In the model of Bala and Goyal (1998) itis crucial that players do not use their observation of same players to draw anyconclusions about other players they don’t see. In the present paper this aspectshall not be almost ignored: Player’s expected utility depends on the activeplayers she knows of. However, the model remains absolutely silent about howthis utility is derived. It could be the case that a player assigns some positiveprobability to somebody else’s being active after having seen a third person.The only assumption made here is that this probability is strictly less than 1.In this case we must consider Si as the set of players about whom i thinks thatthey are active with probability 1.

3 An Information Cascade

3.1 Innovation Adoption

In the following let C be a fixed control structure. Let (T, T ) be a state and sup-pose the observation P is made. Then each individual i updates her informationaccording to

Ri = (Ti \ Pi) ∪ (Pi ∩ T ) . (3)

That is agent i will not change her belief about players she does not observe,and she correctly updates her belief about those players she observes.After having updated their information, agents have to decide on whether or notto adopt the innovation, based on their new information. Agents who mutuallyobserve each other have the chance to coordinate their actions. A coalition Kis called a coordination unit if K ⊆ Pj ∪Rj and K ∈ C (Rj) for all j ∈ K. Thefirst condition ensures that each individual i ∈ K is observed by each individualj who does not think that i has switched yet. The second condition ensures thatall members of K agree to jointly adopt the innovation based on their individualinformation. Hence, coordination units are exactly those coalitions who agree tojointly adopt the innovation given an observation and an information structure.The set of coordination units shall be denoted by K (R,P), i.e.

K (R,P) = K ⊆ N : K ⊆ Rj ∪ Pj and K ∈ C (Rj) for all j ∈ K . (4)

Example 3.1. Let N = 1, 2, 3, 4 and let C be the minimal control structureC with C(1) = C(2) = C(3) = ∅ and C(4) = 12, 4, 124, i.e. C(S) =

⋃i∈S C(i)

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for all other S ⊆ N . Suppose that 4 has adopted the innovation, and that theavailable information is (4, 4, 4, 4). Suppose further that the following observa-tion (14, 24, 34, 34) is made. The singleton 4 is a coordination unit. Although 1and 2 would jointly adopt the innovation, they do not form a coordination unitas neither of them has adopted the innovation yet, nor do they observe eachother.Suppose instead that the observation is (14, 124, 34, 34). Although player 2observes player 1, a coordination between 1 and 2 is impossible as 1 does notobserve 2. Hence, the only coordination unit is 4.Suppose that the observation is (124, 124, 34, 34). Then both 4 and 12 arecoordination units: both 1 and 2 observe 4, so they would adapt the innovationtogether, and they observe each other and therefore can coordinate.

For an observation P and an (updated) information structure R denote thecollection of coordination units by K (R,P), and let

S =⋃

K∈K(R,P)

K. (5)

Hence, given R and P the set of players who will adopt the innovation is S.Note that for the definition of K (R,P) the information agents possess accordingto R is not necessarily correct.After all individuals in S have adopted the innovation, there is a second roundof information updating, as individuals know whom they have coordinated with.For i ∈ N let

κi =⋃

K∈K(R,P):i∈K

K, (6)

and note that κi = ∅ if i /∈ S∗ (R,P). I make the assumption here that allcoordination units form and that each individual observes all coordination unitsshe is a member of.6 Hence, the new information structure is given by

Si = (Ri \ κi) ∪ (S∗ (R,P) ∩ κi) = (Ri \ i) ∪ κi (7)

for all i ∈ N . Note that this process does not assume that individuals whohave adopted the innovation cannot switch back. They rather reevaluate theirdecision after each observation they make.Let

ηPC (T, T ;S,S) =

1, if P, T, T , S,S satisfy Equations (3) - (7),

0, otherwise.

6Intuitively, this make sense as coordination units are not exclusive and individuals withmonotonic utility functions have an incentive to coordinate with as many others as possible.Technically, it would be sufficient to assume that each individual for which a coordinationunit exists adopts the innovation and updates her information about at least one coordinationunit.

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That is ηPC indicates whether or not there is a transition from (T, T ) to (S,S)given P. Note that by definition for each (T, T ) ∈ Υ and each P ∈ Ω there isa unique (S,S) ∈ Υ with ηPC (T, T ;S,S) = 1. This (S,S) shall be called thesuccessor of (T, T ) via P.

3.2 Self-sufficiency, Consistency, and Monotonicity

The process defined in the previous subsection allows for coordinated adoptionof a social innovation but only allows unilateral switching back. Although thisis a problem in general, it will be shown that this is no loss of generality fortypical starting conditions. The reason is that in these cases no group of playershas an incentive to switch back.

Let C be a control structure that is derived form utility functions u. A state(T, T ) ∈ Υ shall be called self-sufficient if for each nonempty T ′ ⊆ T there isi ∈ T ′ with ui (Ti) ≥ ui (Ti \ T ′). In this case there is no group of players thathas an incentive to switch back. Denote the collection of all self-sufficient statesin which no player is misinformed by Υ. In general, it is not clear whether anarbitrary state that is reached via the adoption process defined by Equations(3) - (7) is self-sufficient. However, provided no misinformation, a state that isreached from a self-sufficient state is self-sufficient as well.

Lemma 3.2. Let u be a monotonic utility function and let C = Cu. Let(T, T ) ∈ Υ, let P ∈ Ω, and let (S,S) be the successor of (T, T ) via P. Then(S,S) ∈ Υ and (S,S) (T, T ).

For more general control structure a similar concept can be defined: thestate (S,S) shall be called consistent if for each i ∈ S there is K ⊆ N withi ∈ K, K ⊆ Sj , and K ∈ C (Sj) for all j ∈ K. That is for each active playeri there is a group of active individuals (containing i) such that each j in thisgroup thinks it is a good idea that the group is active, given j’s information.Let Υ∗ denote the set of consistent states without misinformation. Similar toself-sufficiency, consistency is a property that remains true during the adoptionprocess, provided no misinformation.

Lemma 3.3. Let (T, T ) ∈ Υ∗, let P ∈ Ω, and let (S,S) be the successor of(T, T ) via P. Then (S,S) ∈ Υ∗ and (S,S) (T, T ).

Clearly, if β (P) > 0 for all P ∈ Ω, it is easy to see that misinformation will bedetected eventually: after the observation P = (N, . . . , N) any misinformationis eliminated. Moreover, the resulting state is consistent. However, one couldassume that it would be sufficient for this detection that for any i, j ∈ N thereis an observation P with j ∈ Pi and β (P) > 0. The following example showsthat this intuition is wrong.

Example 3.4. Let N = 12345 and let C be the minimal control structure withC ((k mod 5) + 1) = k for k ∈ N .7 Define

Σk = (S,S) ∈ Υ : S = k and ((k mod 5) + 1) ∈ Sk .7For k, n ∈ N the term n mod k is defined as n− k maxl∈N:lk≤n l.

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In each (S,S) ∈ Σk, individual k is misinformed about individual (k mod 5)+1,nevertheless all such states are consistent. Let P lk = k, ((k + l) mod 5) + 1for l = 0, . . . , 3 and k ∈ N . For all such P l all coordination units must besingletons. Suppose (S,S) ∈ Σ(k mod 5)+1 and that observation P l is made. Ifl 6= 0, no player observes a player they consider important, and nobody willchange their strategy. If l = 0 then player k observes that (k mod 5) + 1 isactive, and she becomes active as well; but (k mod 5) + 1 discovers that shehas been misinformed and is not contained in any coordination unit. Hence, thesuccessor of (S,S) lies in Σk. Therefore a state where nobody is misinformedcannot be reached.

3.3 Maximal Active Sets

An individual i ∈ N is called loyal if for each coalition T ⊆ N with i ∈ T andui (N) ≥ ui (N \ T ) there is j ∈ T with uj (N) < uj (N \ T ). Let L0 be theset of loyal individuals. They will never adopt the innovation, and this decisionmight influence other people: for k ≥ 1 define recursively Lk by i ∈ Lk if andonly if for each T ⊇ Lk−1 with i ∈ T \ Lk−1 and ui

(N \ Lk−1

)≥ ui (N \ T )

there is j ∈ T \ Lk−1 with uj(N \ Lk−1

)< uj (N \ T ). Then L1 contains all

those individuals that will never adopt as long as the members of L0 have notadopted, L2 contains all those individuals that will never adopt as long as themembers of L1 have not adopted, and so on. Clearly, L0 ⊆ L1 ⊆ . . ., and thereis r ≥ 0 such that Lr = Lk for each k ≥ r. Let L∗ = Lr for such r. Membersof L∗ shall be called skeptic. Intuitively, the set L∗ defines an upper bound forthe maximal set of individuals that can become active: As long as the set Sof active individuals does not contain any member of L∗, it has not chance toconvince any (group) of these individuals.

Lemma 3.5. Let ui be a monotonic utility functions, let C = Cu, and letS ⊆ N \ L∗. Then T ⊆ N \ L∗ for each T ∈ C∗ (S).

In general the upper bound from the previous lemma is not sharp. Supposethe individuals in N = 1, 2, 3, 4 have the following utility functions: u1(S) ≥u1 (S \ 1) for all S ⊆ N , i.e. 1 always wants to be active, ui(N) ≥ ui(1) >ui(1, 2, 3) ≥ ui(1, 2, 3 \ i for i = 2, 3, i.e. 2 and 3 are free-riders whoprefer 1’ being active over their joint being active over leaving the other alonewith player 1. For player 4 let his utility function satisfy u4 (S) > u4 (S ∪ 4)for all S ⊆ N \ 4. Clearly, L0 = 4. We also see that 2, 3 /∈ L1 since bothplayers would join without player 4. However, C∗(∅) = 1, that is the borderfrom Lemma 3.5 is not strict.

An important class of control structures are those where individuals havethresholds at which they become active. Interestingly, for these games theborder is sharp as the following example shows. But it is worth mentioningthat in models where players have thresholds according to which the decide onwhether or not to become active rule out free-riders.

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Example 3.6. Suppose that ui (S) = θi for all S ⊆ N \ i, that is T willadopt the innovation if and only if ui (S ∪ T ) ≥ θi for all i ∈ T . Then foreach i ∈ N \ L∗ there is T ⊇ L∗ such that uj (N \ T ) ≤ uj (N \ L∗) for allj ∈ T \ L∗. Since ui (N \ T ) = θi = ui (∅), it holds that ui (N \ L∗) ≥ ui (∅) forall i ∈ N \ L∗, so that N \ L∗ ∈ Cu (∅). This means that the border in Lemma3.5 is in fact sharp. Note that the same conclusion would hold if ui(S) ≤ ui(T )for all T ⊆ S ⊆ N \ i, that is if inactive individuals would prefer fewer peoplebeing active (i.e. if there were no free riders).

4 Diffusion

In this section I shall consider probability distributions over the set of states.The previously developed information cascade allowed for transitions from onestate to another, but it can be used to define a transition from one probabilitydistribution to another. For any (S,S) ∈ Υ, denote by XSS (t) the probabilitythat at time t ≥ 0

1. all members of S have switched, and nobody else has,

2. each individual i has the information that all members of Si have switched,and nobody else has.

Then X(t) =(XSS (t)

)(S,S)∈Υ

is a probability distribution over Υ for each t ≥ 0.

Suppose that state (T, T ) is the true state. Then for a control structure C andobservation probabilities β the probability of a transition from (T, T ) to (S,S)is given by

µS,ST,T =∑P∈Ω

β (P) ηPC (T, T ;S,S) . (8)

The change in the probability XSS is due to two causes: on the one hand side(S,S) might be the true state, in this case there is a (conditional) probability

of (1 − µS,SS,S that this state will be left; on the other side (S,S) might not bethe true state, then for each other state (T, T ) the (conditional) probability of

a transition from (T, T ) to (S,S) is given by µS,ST,T . Hence, the functions XSS (.)satisfy the differential equations

XSS (t) = −(

1− µS,SS,S)XSS (t) +

∑(T,T )6=(S,S)

µS,ST,TXTT (t) (9)

or in matrix form

X(t) = (M − I)X(t) (10)

with

M =

µN,(N,...,N)N,(N,...,N) . . . µ

N,(N,...,N)∅,(∅,...,∅)

.... . .

...

µ∅,(∅,...,∅)N,(N,...,N) . . . µ

∅,(∅,...,∅)∅,(∅,...,∅)

. (11)

12

So, X is a Markov process with transition matrix M − I. It can easily be seenthat M is a column stochastic matrix (Lemma B.2 in the appendix). Hence, ifX(0) is a probability distribution over Υ, then X(t) is a probability distributionover Υ for all t ≥ 0.

The next theorem gives a solution of the Differential Equation (10) in casethat the starting condition is such that only consistent states without misinfor-mation have a positive probability. Recall that for such states only transitionsto larger states (with respect to the order in Equation 2) are possible. Clearly,the theorem would hold true if one replaced Υ∗ by Υ.

Theorem 4.1. Let X satisfy the Differential Equation (10) and let XSS (0) = 0whenever (S,S) /∈ Υ∗. Then XSS (t) = 0 for each (S,S) ∈ Υ \Υ∗ and

XSS (t)=e−(1−µS,SS,S)t

XSS (0) +∑

(T,T )≺(S,S)

µS,ST,T

∫ t

0

e(1−µS,SS,S)uXTT (u)du

for all (S,S) ∈ Υ∗ and all t ≥ 0.

Example 4.2. Let N = 1, 2, 3, let β be defined as in Example 2.4, and letC(S) = ∅, 12, 13, 23, 123 for all S ⊆ N . Let X be the process defined by C,β, and the Starting Condition

XSS (0) =

1, if S = ∅ and Si = ∅ for all i ∈ N,0, otherwise.

(12)

The only states that can be reached with positive probability in this process are

1) (N, (N,N,N)) 5) (12, (12, 12, ∅))2) (N, (N, 12, 13)) 6) (13, (13, ∅, 13))

3) (N, (12, 23, N)) 7) (23, (∅, 23, 23))

4) (N, (12, 23, N)) 8) (∅, (∅, ∅, ∅)) .

For these eight states (in that order) the matrix M is given by

1 25

25

25

15

15

15

15

35

15

15

35

15

15

35

15

15

25

15

25

15

25

15

15

.

13

Using the formula in Theorem 4.1 one finds the following functions

X8 (1) = e−45 t

X7(t) = X6(t) = X5(t) = e−35 t − e 4

5 t

X4(t) = X3(t) = X2(t) = e−25 t − 2e−

35 t + e−

45 t

X1(t) = 1− 3e−25 t + 3e−

35 t − e− 4

5 t.

The only steady state of the process is the distribution that puts probability 1on state 1, and limt→X1(t) = 1.

4.1 Monotonicity

Let x, x be probability distributions over Υ. Distribution x first order stochas-tically dominates x if ∑

(S′,S)(S,S)

xS′

S′ ≥∑

(S′,S)(S,S)

xS′

S′

for all (S,S) ∈ Υ. This shall be denoted by x D x. A process X is calledmonotonic if X(t)DX(t′) for all t > t′ ≥ 0.

Given that for states is Υ∗ only transition to larger states are possible, itis not a great surprise that over time the probability of larger states becomeslarger, and the probability of small states vanishes.

Theorem 4.3. Let X satisfy the Differential Equation (10) and let XSS (0) = 0for all (S,S) ∈ Υ \Υ∗. Then X is monotonic.

Recall that Υ ⊆ Υ∗. Hence, the case of self-sufficient states is alreadycontained in Theorem 4.3.

4.2 Steady States and Long Run Outcome

For general Markov processes it can be difficult to find all starting conditionfor which the process converges. The next theorem shows that the process inEquation (10) will converge for every consistent starting condition and, if allobservations are possible, will converge even for all starting conditions.

Theorem 4.4. Let X be defined by the Differential Equation (10). If XSS (0) = 0for all S ∈ Υ \ Υ∗ then X(t) converges. If, additionally, β > 0 then X(t)converges for all starting conditions.

In models of innovation diffusion without coordination individual players decidewhether or not to be active. In a steady state, with probability 1 no playercan unilaterally improve by deviating, i.e. steady states are Nash equilibria inthat sense. Here, the stability of steady states is much stronger. Let P be anobservation. A clique is a coalition T with T ⊆ Pi for all i ∈ T . A state (S,S) iscalled P-Nash stable if for any clique T ⊆ N and any T ′ ⊆ T there is i ∈ T such

14

that ui ((Si ∪ T ) \ T ′) ≤ ui (Si). That is for any clique T and any deviationits members might choose (some may become active, some others may becomeinactive) there is at least one individual i ∈ T who does not profit from thedeviation. With the insights on self-sufficient states from the previous Section3, it is not difficult to show that in a steady state only such states can have apositive probability that a P-Nash stable for all observations P that occur withpositive probability.

Theorem 4.5. Let C = Cu for some monotonic utility functions ui, let X bedefined by the Differential Equation (10), and let XSS (0) = 0 whenever S ∈ Υ\Υ.Then X converges towards a steady state x so that xSS = 0 for all (S,S) thatare not P-Nash stable for all P ∈ Ω with β (P) > 0.

Many models of informational cascades assume that there is a coalition of playerswho have adopted the innovation at t = 0. For the process defined by Equation(10) more general starting conditions can be used. Since players can coordinate,a process can even start instantaneously provided that C (∅) 6= ∅. (Note that thestate (∅, (∅, . . . , ∅)) is both consistent and self-sufficient) Also, it is not necessaryto specify an initial state, but rather an initial distribution over states. However,if a process starts from a single consistent state with probability 1, it will alsoconverge to a single consistent state with probability 1.

Corollary 4.6. Let X be defined by the Differential Equation (10), and letXSS (0) = 1 for some S ∈ Υ∗. The there is (S∗,S∗) with limt→∞XS

∗

S∗ (t) = 1

The final outcome in Corollary 4.6 depends both on β and on C. However,if β (P) > 0 for all P ∈ Ω then S∗ = S∗i = C∗(S) for all i ∈ N .

4.3 Comparative Statics and Robustness

For later applications it will be useful to analyze how the process X can beaffected by changing C or β. Intuitively, one expects that increasing the controlstructure C, i.e. increasing C(S) for all S ⊆ N , should imply that the processevolves faster. And the same should happen if observations of many players atthe same time become more likely.

Theorem 4.7. Let C,D be control structures with C(S) ⊆ D(S) for all S ⊆ Nand let β, γ be probability distributions over Ω such that β first order stochasti-cally dominates γ. Let X be the process generated by C and β, and let Y be gen-erated by D and γ. Let X(0) = Y (0) and let XSS (0) = 0 for all (S,S) ∈ Υ \Υ∗.Then X(t) first order stochastically dominates Y (t) for all t ≥ 0.

The cascade models of Granovetter (1978) or Schelling (1978) imply that thefinal outcome of a cascade heavily depends on the dynamics early in the process.Chwe (1999) on the other hand argues that this is true only if individuals cannotcommunicate. Since observation and influence are separated here, their effectson the long run outcome shall be considered separately. The next result isalmost immediate given the insights from the previous subsection.

15

Theorem 4.8. Let C,D be two control structures and let β (P) > 0 for allP ∈ Ω. Let X and Y the processes that correspond to C and D, respectively,

with X(0) = Y (0) and X(∅,...,∅)∅ (0) = 1. Then limt→∞X(t) = limt→∞ Y (t) if

and only if C∗(∅) = D∗(∅).

Let (T, T ) , (S,S) ∈ Υ∗. An adoption chain from (T, T ) to (S,S) of length n is asequence (T0, T0) ≺ . . . ≺ (Tn, Tn) such that (T0, T0) = (T, T ), (Tn, Tn) = (S,S),

and µTk,TkTk−1,Tk−1

> 0 for all k = 1, . . . , n. The distance d (T, T ;S,S) is the length

of the maximal adoption chain (recall that there are no cycles in Υ∗) from (T, T )to (S,S), or ∞ if such an adoption chain does not exist. For a subset Υ0 ⊆ Υlet d (Υ0;S,S) be the length of the longest adoption chain from any element ofΥ0 to (S,S), or ∞ if no such adoption chain exists.The following theorem gives an upper bound for the multiplicative error if theobservation probabilities are (multiplicatively) perturbed. A distribution γ iscalled an ε-approximation of a distribution β if

(1− ε)β(P) ≤ γ(P) ≤ (1 + ε)β(P)

for all P ∈ Ω. Since the probabilities of certain events may be very small, therelative error might be more relevant than the absolute error. Note that if γ isan ε-approximation of β and β (P) = 0 for some P ∈ Ω then γ (P) = 0 as well.

Theorem 4.9. Let C be a control structure, let β be a probability distributionover Ω, let ε > 0, and let γ be an ε-approximation of β. Let X and Y be theprocesses corresponding to β and γ respectively, let X(0) = Y (0), and denoteby Υ0 ⊆ Υ∗ the collection of minimal states (with respect to ) (S,S) withXSS (0) > 0. Then for all (S,S) ∈ Υ and all t ≥ 0 it holds that XSS (t) > 0 if andonly if Y SS (t) > 0. This is the case if and only if d (Υ0;S,S) <∞. In this caseit holds that

(1− ε)d(Υ0;S,S)e−εtXSS (t) ≤ Y SS (t) ≤ (1 + ε)

d(Υ0;S,S)eεtXSS (t).

for all t ≥ 0.

Although in the long run the boundaries in Theorem 4.9 are not very strong,they show that the distribution x continuously depends on β for β > 0. Inparticular, in the short run small perturbations in β will not entirely destroysolution x.

5 The Emergence of Revolutions

5.1 Critical Mass Games and Opportunists

Suppose now that the members of the society N have to decide whether ornot to engage in anti-government protests. Such a protest leads to a successfulrevolution if sufficiently many people take part in it. Let F denote the collectionof coalitions such that a revolution is successful if and only if a stage (S,S) is

16

reached with S ∈ F . In the following assume that F ′ ∈ F for each F ∈ F andeach F ′ ⊇ F . This set up is essentially a critical mass game where a change ofthe status quo is achieved if a critical mass is reached. It shall also hold thatN \ L∗ ∈ F , i.e. the success of a revolution is not impossible.These games are naturally accompanied by coordination problems: becomingactive might be a good option only if sufficiently many other people are ac-tive as well. The decision on whether or not to become active will depend onquestions such as: Who is already active? or: Who will follow me? (RecallMustafa Nayem’s Facebook post) It also depends on the perceived probabilityof a success. All these factors can contribute to and be represented by the utilityfunctions described in Example 2.1. As before let ui be the (monotonic) utilityfunction of individual i. Call i ∈ N an opportunist if

ui (S ∪ i) < ui (S) and ui (1;F ∪ i) ≥ ui (F )

for all S, F ⊆ N \i with S ∪i /∈ F and F ∪i ∈ F . That is an opportunistwants to be active only if the revolution is successful. Let O denote the set ofall opportunists.

5.2 Sticks and Carrots

A regime is able to influence people’s utility functions by taking certain actions,e.g. prosecuting activists. Suppose that A is the status quo and utility functionsare ui. Let B be a strategy that induces utility functions vi. Then B is calleda stick if

vi (S ∪ i) ≤ ui (S ∪ i) and vi (S) = ui (S) (13)

for all i ∈ N and all S ⊆ N \ i with S ∪ i /∈ F . B is called a carrot if

vi (S ∪ i) = ui (S ∪ i) and vi (S) ≥ ui (S) (14)

for all i ∈ N and all S ⊆ N \i with S∪i /∈ F . Hence, under a stick activistsare punished, whereas under a carrot non-activists are rewarded. I assume thata regime has no power over people’s utility if a revolution is successful. That is

ui (F ∪ i) = vi (F ∪ i) (15)

for all i ∈ N and all F ∈ F .In the following let C = Cu and D = Cv. For a probability distribution β

over Ω the matrices M =(µT,TS,S

)and L =

(λT,TS,S

)shall be defined according to

Equations (8) and (11), and X and Y shall denote the Markov processes definedby Equations (10) and (12). Denote by L∗A and L∗B the sets of skeptics under Aand B respectively. The first insight is that sticks cannot increase the numberof skeptic individuals.

Lemma 5.1. Let B be a stick. Then L∗A = L∗B.

17

The following theorem is an application of several previous results and doesnot need any further discussion. It simply says that a stick can make a massprotests less likely in the short run, but not necessarily in the long run.

Theorem 5.2. Let B be a stick. Then X(t) first order stochastically dominatesY (t) for all t ≥ 0. If, additionally, ui (S) = θi for all i ∈ N and all S ⊆ N \ iand β > 0 then the long run probability of a revolution is the same under A andB.

Suppose that at the status quo for each F ∈ F there is an opportunist i ∈ F ,that is N \ (L∗A ∪OA) /∈ F . For (S,S) ∈ Υ∗ with S /∈ F define ΩC (S,S) asthe collection of observations P for which the successor (F, T ) of (S,S) via Psatisfies F ∈ F ; let ΩD (S,S) be defined accordingly. Let P ∈ ΩC (S,S) andlet (F, T ) ∈ Υ∗ be the successor of (S,S) via P (according to C). Then thereis a coordination unit K ⊆ F with i ∈ K. Let R be defined as in Equation(3). By the definition of coordination units K ∈ C (Rj) for all j ∈ K, that isall j ∈ K think it is a good idea that K becomes active given their individualinformation. Since i ∈ K is an opportunist, this means that K ∪Rj ∈ F for allj ∈ K, since otherwise j would not believe that i would join K. Therefore

vk (K ∪Rj) = uk (K ∪Rj) ≥ uk (Rj \ k) = vk (Rj \ k)

for all j ∈ K and all k ∈ K ∪ Rj , where the first equality comes from thegovernment’s inability to affect players’ utilities after a successful revolution,and the last equality comes from the definition of a stick. (Note that the overallinequality would hold even if a stick would negatively affect those who are notprotesting). By definition of D, it now holds that K ∈ D (Rj) for all j ∈ K.But since K ⊆ Pj ∪ Rj for all j by definition, K is a coordination unit underD as well. Let now T be such that ηPD (S,S;T, T ) = 1. Since (S,S) ∈ Υ∗ itholds by Lemma 3.3 that Rj ∪K ⊆ S ∪K ⊆ T for all j ∈ K, and hence T ∈ F .Therefore ΩC (S,S) ⊆ ΩD (S,S). The converse of this inclusion is clearly truesince C(S) ⊆ D(S), hence ΩC (S,S) = ΩD (S,S). Therefore∑F∈F

∑T :(F,T )∈Υ∗

µF,TS,S =∑

P∈ΩC(S,S)

β (P) =∑

P∈ΩD(S,S)

β (P) =∑F∈F

∑T :(F,T )∈Υ∗

λF,TS,S ,

that is the probability of a transition from (S,S) to a state (F, T ) with F ∈ Fis not affected by a stick! On the other hand λS,SS,S ≥ µ

S,SS,S since each column in

L first order stochastically dominates the corresponding column in M . Hence,

1

1− λS,SS,S

∑F∈F

∑T :(F,T )∈Υ∗

λF,TS,S ≥1

1− µS,SS,S

∑F∈F

∑T :(F,T )∈Υ∗

µF,TS,S

that is the conditional probability of a transition into a successful revolutiongiven a transition is at least the same under a stick.

Theorem 5.3. Let B be a stick. If for each F ∈ F there is an opportunisti ∈ F then the conditional probability of a jump from a state (S,S) ∈ Υ∗ withS /∈ F to a state (F, T ) with F ∈ F given that there is a jump out of (S,S) ishigher under B than under A.

18

The comparative statics in Theorem 5.3 are not obvious and deserve somediscussion. In order to be precise, I shall divide the set of states (S,S) intothree groups: F-states are those with S ∈ F , intermediate states are those thathave a successor (T, T ) with T ∈ F , and beginning states are those without asuccessor in F .

Unstable regimes Theorem 5.3 does not say that a revolution is more likelyto occur under a repressive regime. The contrary is true, as seen in Theorem5.2. However, if a protest has already reached an intermediate state, a morerepressive regime makes transition into another intermediate state less likely,while the probability of a transition into an F-state is not affected. Hence,instead of a smooth gradual increase in the number of protest participants,we would expect to see rare but large jumps. This is one of the key findingsof Kuran (1989), namely that revolutions under repressive regimes are quickand unanticipated. Both the Ukrainian revolution 2013/2014 and the Tunisianrevolution in 2010/2011 were events that occurred in PTS3 countries and veryquickly turned into revolutions.

Very severe sticks There are PTS-5-countries where in the last 20 years nota single protest occurred (for instance North Korea). Although it seems con-tradictory to Theorem 5.3, it can be explained using my model: The transitionprobability from an intermediate state to an F-state cannot be affected by astick. However, a stick can severely reduce the probability to leave a beginningstate. In particular, if a severe stick is applied before a small protest (in a be-ginning state) turns into a large protest (in an intermediate state), the dangerof a revolution can be banned. Hence, regimes that prohibit any form of eventhe smallest protest can survive using a stick.

Communication devices Some regimes try to secure their power by blockingcertain social media in their countries. These media are the devices that increaseβ since it is easy to communicate with large groups. Hence, disconnectingpeople from social media makes large jumps less likely. This in turn meansthat decreasing β could make a transition from an intermediate state to anF-state impossible. Hence, the set of intermediate set becomes smaller, whilethe set of beginning states increases, and the set of F states in unaffected.However, since beginner states are the ones affected by sticks, we see that aless severe stick is very powerful in countries where mass communication is verydifficult. The People’s Republic of China (PTS 4) would be an example of astick in combination with limited communication, but also recent initiatives ofthe president of Turkey (PTS 3-4) would lead the country into that direction.

Concessions In the model of Ginkel and Smith (1999) a regime that makesconcessions to protesters sends a signal of weakness (since a strong regime hasnot reason to accommodate them). This in turn increases the protesters’ beliefthat a revolution can be successful and might have the undesired effect to even

19

increase the number of protesters since no even some free-riders might join. Mymodel has a different view on concessions: After a certain threshold has beenexceeded (i.e. after an intermediate state has been reached), a stick would nothave any effect on the probability of a transition into a revolution. Concessions(hence a carrot) on the other hand can only increase the number of skeptics andthereby reduce the number of F-states that are reachable from an intermediatestate. Hence, we can expect that concessions are made when it is too late fora stick, but that they can be successful only if they are going far enough. Thisis similar the model of (Acemoglu and Robinson, 2000, 2001) where a non-democratic regime of the rich can extend the franchise in order to prevent thepoor from (always successfully) revolting. In practice, Yanukovych’s concessionsto the Ukrainian demonstrators in 2014 would only have restored the status quobefore the protests and was therefore rejected. The mass protests in France 1995(PTS 1-2) were called off after the government agreed not to implement theirretirement reform plans.

Free Riders As we have seen in Example 3.6 the state (∅, (∅, . . . , ∅)) is anintermediate state if there are no free-riders. Hence, in this case there are nobeginning states at all. This means that a stick cannot reduce the probabil-ity that a protest turns into a revolution for any protest. Consequently, theeffectiveness of a stick relies upon the presence of free riders.

Example 5.4. Let N = 1, 2, 3, let F = N, and suppose that all utilityfunctions satisfy

ui (i, j) ≥ ui (0, ∅) = θi > ui (i)

for all all i, j ∈ N , i 6= j, i.e. there are no loyals. Then each i, j have the jointlybest response to join the revolution if they meet each other. By the monotonicityof ui for all i the same is true if all players meet at once. Hence these utilityfunctions induce the control structure C with C(S) = ∅, 12, 13, 23, 123 for allS ⊆ N , that has already been investigated in Example 4.2.Suppose now that a stick is imposed such that

vi (N) ≥ θi > vi (i, j)

for all i, j ∈ N . That is, still no player is loyal to the regime, but they wouldjoin a riot only if all other players do so. Given Starting Condition (12) theonly states that can be reached with positive probability are (N, (N,N,N))and (∅, (∅, ∅, ∅)). The corresponding transition matrix is given by 1 1

5

0 45

and hence, Y2(t) = e−

15 t and Y1(t) = 1 − e−

15 t. As expected X first order

stochastically dominates Y . Also, the condition probability of a jump to Ngiven a jump is 1 for all states (even for those states that are not included inthis matrix).

20

If B is a carrot, it still holds that D(S) ⊆ C(S) for all S ⊆ N , and hence X(t)first order stochastically dominates Y (t) for all t ≥ 0. And clearly, L∗A ⊆ L∗B .8

However, even in case of threshold utility functions L∗A and L∗B need not tocoincide as the following example shows.

Example 5.5. Suppose that in Example 5.4 instead of a stick a carrot is usedsuch that v1 = u1, v2 (j) = θ2 > v2 (2, j) for j = 1, 3, and v3 (1, 2) = θ3 >v3 (N). Then 3 has become loyal, 2 has become skeptic and the probability ofa riot is 0. Note that the change in player 2’s utility function could have beenreached with a stick as well, but for the change in 3’s utility function a carrotis necessary.

6 Concluding Remarks

The instruments derived in this article can be used to model coordinated behav-ior in very general network games. They can explain for instance the dynamicsthat have been observed during mass protests and revolutions around the globeduring the last 40 years. However, their application is not restricted to thisvery extreme form of coordination problems. A general conclusion that can bedrawn from Theorem 4.7 (and many articles before) is that (perceived) risk willslow down the diffusion of social innovations. But this effect can be overcomeif communication channels and coordination devices are improved.There are innovations that have great impacts on a society if they are adoptedby a critical mass, but that are perceived as very dangerous by some individuals.The latest example in this category are vaccinations – both in the US and in Eu-rope where activists group reject the vaccination of their children. The insightsfrom the previous Section suggest that it might be easier to convince wholegroups of close people to vaccine their children than to convince individuals.There are also cases where policy makers aim for a stop of the widespread of aninnovation, for instance if home grown terrorism is interpreted as an innovation(in the sense of this article). A conclusion from Section 5 is that instrumentswhich improve the status quo are better suited to extinct terrorism than harderpunishment of those involved: the critical mass a terror cell needs for an attackis very low, hence a stick that avoids that an intermediate state be reached mustbe very harsh. In particular, stricter counter-terror laws, as they are discussedin Europe after the attacks in Paris and Brussels, cannot increase the numberof those who are loyal to a state or skeptical towards terrorism.This paper provides a new method to analyze coordinated behavior within socialnetworks where individuals can decide between being active and inactive. Butobviously in many conflicts there are more than two parties and coordinatedaction can take place in any direction. This is a natural avenue of furtherresearch that should be followed over the next years.

8The verification of these facts is simple and left to the reader.

21

A Data Selection

Figure 1 depicts the numbers from Table 1. The time frame from 1976 to 2014is chosen, since data from all three data sets are available only in this period.

Table 1: Revolutions and Mass Protests 1976 - 2014

PTS∗ 1 2 3 4 5

Revolutions 0 2 12 11 4

Mass Protests 625 742 1339 1195 390

For each country and each year where Banks and Wilson (2015) contains dataon the number of anti-government protests, the corresponding PTS value fromGibney et al. (2015) is used for the classification in Table 1.9 Data points wherethe PTS value is not available are left out.Marshall and Marshall (2015) contains 39 events during this period that arelisted as Resignation of Executive due to Poor Performance or Loss of Authority.In seven cases the political change was due to military intervention or rebelmovements rather than mass protests10, in two cases the political leader wasimpeached11 and in one case the government stepped back after a lost war12.The remaining 29 events are contained in Table 1. The Political Terror Scalevalue is calculated as PTS∗ = m

12PTS0 + 12−m12 PTS−1 rounded to the closest

integer. Thereby, PTS0 is the PTS value in the year of the event and PTS−1

is the value in the previous year.13

B Mathematical Appendix

B.1 On Control Structures

Lemma B.1. Let C be a control structure and let D1 and D2 be extensionsof C. Then D1 ∩ D2 is an extension as well. If, additionally D1 and D2 areinvariant then D1 ∩D2 is invariant as well.

Proof of Lemma B.1. The first part is clear. Let D1 and D2 be invariant,let S ⊆ N , let T ∈ (D1 ∩D2) (S), and let R ∈ (D1 ∩D2) (T ). Then T ∈ Di(S)

9Banks and Wilson (2015) distinguishes between Germany and East Germany, while Gib-ney et al. (2015) distinguishes between Germany (from 1990), East Germany, and WestGermany (until 1989). PTS data from West Germany in the latter data set are used forprotests in Germany in the former.

10Bolivia 1982, East Timor 2006, Georgia 1992, Guinea-Bissau 1999, Honduras 2009,Lesotho 1990, Liberia 2003

11Lithuania 2004, Madagascar 199612Argentina 198213Except for the Iranian Revolution 1979 where PTS−1 is not available. Here, PTS∗ =

PTS0.

22

and R ∈ Di(T ) for i = 1, 2, and therefore R ∈ Di(S) by the invariance of Di.Hence R ∈ (D1 ∩D2) (S), and D1 ∩D2 is invariant.

Proof of Lemma 2.2. Clearly, Ck is monotonic and additive and Ck(S) ⊆Ck+1(S) for all k ≥ 1 and all S ⊆ N . As the player set is finite, there mustbe r ≥ 0 with Cr = Ck for all k ≥ r, and Cr is an extension of C. LetS ⊆ N , T ∈ Cr(S) and R ∈ Cr(T ). Then R ∈ Cr+1(S) = Cr(S). Hence, Cr

is invariant. Let D be an invariant extension of C and let S, T ⊆ N be suchthat T ∈ Cr(S). Then there are a natural number m ≤ r and T 1, . . . , Tm ⊆ Nsuch that T 1 ∈ C(S), Tm = T , and T k ∈ C

(T k−1

)for all k = 2, . . . ,m. As

D is an extension of C, it must hold that T 1 ∈ D(S) and T k ∈ D(T k−1

)for

all k = 2, . . . ,m. Hence T ∈ D∗(S) and, by the invariance of D, it follows thatT ∈ D (S). Therefore Cr(S) ⊆ D(S) for all invariant extensions of C.

Proof of Lemma 3.2. Let Ri be defined as in Equation (3). Since C = Cu

it holds that i ∈ Ti ⊆ Ri and i ∈ C (Ti) ⊆ C (Ri) for all i ∈ T . Hence,i ∈ K (R,P) and T ⊆

⋃K∈K(R,P)K = S. If i ∈ S then Ri ⊆ Ri ∪ κi = Si; if

i /∈ S then Si = Ri since i /∈ T and hence i /∈ Ri. In both cases Ti ⊆ Si.Assume there is a nonempty S′ ⊆ S such that ui (Si \ S′) > ui (Si) for alli ∈ S′. Let T ′ = S′ ∩ T and suppose that T ′ 6= ∅. By definition there is i ∈ T ′with ui (Ti) ≥ ui (Ti \ T ′). Since Ti ⊆ Si by the first part of the Lemma and(Ti \ T ) ∩ S′ = ∅ by definition, the monotonicity of u implies

ui (Si \ S′) > ui (Si)≥ ui (Ti ∪ (Si \ (Ti ∪ S′)))≥ ui ((Ti \ T ′) ∪ (Si \ (Ti ∪ S′)))− ui (Ti \ T ′) + ui (Ti)≥ ui ((Ti \ T ′) ∪ (Si \ (Ti ∪ S′)))= ui (Si \ S′) ,

which is impossible. So, let S′ ⊆ S\T and let i ∈ S′. Then there is K ∈ K (R,P)with i ∈ K and ui (Ti ∪K) ≥ ui (Ti). Since K, Ti ⊆ Si and Ti∩S′ = ∅, it follows

ui (Si \ S′) > ui (Si)≥ ui ((Ti ∪K) ∪ (Si \ S′))≥ ui (Ti ∪ (Si \ S′))− ui (Ti) + ui (Ti ∪K)

≥ ui (Ti ∪ (Si \ S′))= ui (Si \ S′) ,

which is impossible. Hence, (S,S) is self-sufficient.

Proof of Lemma 3.3. Let Ri be defined as in Equation (3). Since no playeris misinformed, Ti ⊆ Ri ⊆ T for all i ∈ N . By the consistency of (T, T ) foreach i ∈ T there is Ki ⊆ N with Ki ⊆ Tj ⊆ Rj and Ki ∈ C (Tj) ⊆ C (Rj) forall j ∈ K. Therefore Ki ∈ K (R,P) and hence,

T =⋃i∈T

Ki ⊆⋃

K∈K(R,P)

K = S.

23

If i ∈ S then Ri ⊆ Ri ∪ κi = Si; if i /∈ S then Si = Ri since i /∈ T and hencei /∈ Ri. Therefore (S,S) (T, T ). Since Ri ⊆ T ⊆ S and κi ⊆ T ⊆ S (whereκi is defined as in Equation (6)), no player is misinformed. As each i ∈ S ismember of at least one coordination unit, (S,S) is consistent.

Proof of Lemma 3.5. Let T ⊆ C (N \ L∗). Then ui ((T ∩ L∗) ∪ (N \ L∗)) =ui (T ∪ (N \ L∗)) ≥ ui (N \ L∗) for all i ∈ T . Hence, by the monotonicity of ui

ui (N) = ui ((T ∩ L∗) ∪ (N \ L∗) ∪ (L∗ \ T ))

≥ ui ((N \ L∗) ∪ (L∗ \ T ))− ui (N \ L∗) + ui ((T ∩ L∗) ∪ (N \ L∗))≥ ui ((N \ L∗) ∪ (L∗ \ T ))

= ui (N \ (T ∩ L∗))

for all i ∈ T ∩ L∗. Therefore T ∩ L0 = (T ∩ L∗) ∩ L0 = ∅ by the definition ofL0. Let k ≥ 1 and suppose that T ∩ Lk−1 = ∅. Then

ui(N \ Lk−1

)= ui

((T ∩ L∗) ∪ (N \ L∗) ∪

(L∗ \

(T ∪ Lk−1

)))≥ ui

((N \ L∗) ∪

(L∗ \

(T ∪ Lk−1

)))− ui (N \ L∗)

+ ui ((T ∩ L∗) ∪ (N \ L∗))≥ ui

((N \ L∗) ∪

(L∗ \

(T ∪ Lk−1

)))= ui

(N \

(Lk−1 ∪ (T ∩ L∗)

))for all i ∈ T ∩L∗. Therefore T ∩Lk = (T ∩ L∗)∩Lk = ∅ by the definition of Lk.Repeating this argument shows that T ⊆ N \ L∗, and therefore C∗ (N \ L∗) =C (N \ L∗) by the definition of C∗. By the monotonicity of C∗, T ⊆ N \L∗ forall S ⊆ N \ L∗ and all T ∈ C∗ (S).

B.2 On the Diffusion Process

Lemma B.2. The matrix M is a column stochastic matrix.

Proof of Lemma B.2. Since for any (T, T ) ∈ Υ and each P ∈ Ω there isexactly one (S,S) ∈ Υ with ηPC (T, T ;S,S) = 1, it follows that∑

(S,S)∈Υ

µS,ST,T =∑

(S,S)∈Υ

∑P∈Ω

β (P) ηPC (T, T ;S,S)

=∑P∈Ω

β (P)∑

(S,S)∈Υ

ηPC (T, T ;S,S)

=∑P∈Ω

β (P)

= 1

for all (T, T ) ∈ Υ.

24

Proof of Theorem 4.1. Lemma 3.3 implies that XTT (t) = 0 if (T, T ) ∈ Υ\Υ∗,and that µS,ST,T = 0 if (T, T ) ∈ Υ∗ and (S,S) (T, T ). Hence Equation (9) isequivalent to

XSS (t) = −(

1− µS,SS,S)XSS (t) +

∑(T,T )≺(S,S)

µS,ST,TXTT (t) (16)

for all (S,S) ∈ Υ∗. Differentiating the formula in Theorem 4.1 delivers (16),and the uniqueness of the solution is clear from standard calculus.

Proof of Theorem 4.3. If (S,S) ∈ Υ \ Υ∗ then XSS (t) = 0 for all t ≥ 0 byTheorem 4.1. Let (S,S) ∈ Υ∗. Then by Equation (16)∑

(S′,S′)(S,S)

XS′

S′ (t) = −∑

(S′,S′)(S,S)

XS′

S′ +∑

(S′,S′)(S,S)

∑(T,T )(S′,S′)

µS′,S′

T,T XTT (t)

= −∑

(S′,S′)(S,S)

XS′

S′ +∑

(T,T )∈Υ∗

XTT (t)∑

(S′,S′)(S,S),(T,T )

µS′,S′

T,T

= −∑

(S′,S′)(S,S)

XS′

S′ +∑

(T,T )(S,S)

XTT (t)∑

(S′,S′)(T,T )

µS′,S′

T,T

+∑

(T,T )(S,S)

XTT (t)∑

(S′,S′)(S,S),(T,T )

µS′,S′

T,T

=∑

(T,T )(S,S)

XTT (t)∑

(S′,S′)(S,S),(T,T )

µS′,S′

T,T

≥ 0.

Note that the last equality follows from∑

(S′,S′)(T,T ) µS′,S′T,T = 1 for all (T, T ) ∈

Υ∗.

Proof of Theorem 4.4. Since X(t) is monotonic, the relation D is a total

order on the bounded set X(t) : t ≥ 0, and x = supDt≥0X(t) exists. Let ε > 0

and let

Bε(x) =

x ∈ ∆ : xE x and max(T,T )∈Υ∗

∑(S,S)(T,T )

∣∣xSS − xSS ∣∣ = ε

where ∆ denotes the simplex of RΥ∗ . As no x ∈ Bε(x) is an upper bound forX(t), there is t′ with xEX(t)E x for some x ∈ Bε(x) and all t ≥ t′. Since

‖z‖ = max(T,T )∈Υ∗

∑(S,S)(T,T )

∣∣zTS ∣∣defines a norm on ∆ and since ‖X(t)− x‖ ≤ ε for all t ≥ t′ by definition, X(t)is converging with limt→∞X(t) = x.If β > 0 then limt→∞

∑(S,S)∈Υ\Υ∗ X

SS (t) = 0 and limt→∞

∑(S,S)∈Υ∗ X

SS (t) =

1. Hence, by the previous part of the proof, X(t) converges.

25

Proof of Theorem 4.5. Since Υ ⊆ Υ∗, X(t) converges towards a steady statex by Theorem 4.4. Let (S,S) ∈ Υ be such that xSS > 0. By Lemma 3.2(S,S) ∈ Υ. Let P ∈ Ω with β (P) > 0. Let R be defined according to Equation(3). Assume that there is a clique P that could deviate. Without loss ofgenerality suppose that all members of P \S become active and all members ofP ∩ S become inactive: otherwise consider the clique P ′ ⊆ P of all individualsthat change their strategy. Let P1 = P \ S, let P2 = P ∩ S, and supposethat P1 6= ∅. By the monotonicity of u and the definition of P it holds thatui (Si ∪ P1) ≥ ui ((Si \ P2) ∪ P1) > ui (Si) for all i ∈ P1. Therefore P1 ∈C (Si) ⊆ C (Ri) for all i ∈ P1. As P1 ∈ Pi for all i ∈ P1, P1 is a coordination

unit. Hence, there is (T, T ) ∈ Υ with S ∪ P1 ⊆ T and µT,TS,S ≥ β (P) > 0. SinceX(t) is monotonic and (T, T ) (S,S), x cannot be a steady state. So, letP1 = ∅, i.e. P ⊆ S. By the self-sufficiency of (S,S) there is i ∈ P with

ui (S \ P ) > ui (S) ≥ ui (S \ P )

which is impossible.

Proof of Corollary 4.6. Suppose that there is T 6= S∗ and T such thatlimt→∞XTT (t) > 0 and assume (without loss of generality) that there is i ∈S∗ \ T . Then there is a chain of observations P1, . . . ,Pm such that startingfrom (S,S) individual i becomes active. Since X is monotonic, it holds that(T, T ) (S,S). Hence, if the same observations P1, . . . ,Pm are made after(T, T ) is reached, i becomes active as well. Hence S∗ ⊆ T , and for the samereasons T ⊆ S∗, i.e. S∗ = T . But in this case it is clear that T = S∗.

Lemma B.3. [This Lemma is taken from Massey (1987).] Let X and Y be twomonotonic Markov processes, let the corresponding generating matrices M andL be such that column k of M first order stochastically dominates column k ofL for each k, and let X(0) first order stochastically dominate Y (0). Then X(t)first order stochastically dominates Y (t) for all t ≥ 0.

Proof of Theorem 4.7. By Theorem 4.3 both X and Y are monotonic. LetM and L be the transition matrices corresponding to X and Y respectively,

and let M∗ =(µS,ST,T

)(S,S)∈Υ∗

(T,T )∈Υ∗and L∗ =

(λS,ST,T

)(S,S)∈Υ∗

(T,T )∈Υ∗be the restrictions

of M and L on Υ∗ × Υ∗. By Lemma B.3 it is sufficient to show that eachcolumn of matrix M∗ dominates the corresponding column of matrix L∗. So,let (T, T ) ∈ Υ∗, let P ∈ Ω, and let (S,S) ∈ Υ∗ be the successor of (T, T ) via Pin D. Since each coordination unit in D is also a coordination unit in C, there

26

is (S′,S ′) (S,S) such that ηPC (T, T ;S′,S ′) = 1. Hence∑(S′,S′)(S,S)

µS′,S′

T,T =∑

(S′,S′)(S,S)

∑P∈Ω

β (P) ηPC (T, T ;S′,S ′)

=∑P∈Ω

β (P)∑

(S′,S′)(S,S)

ηPC (T, T ;S′,S ′)

≥∑P∈Ω

β (P)∑

(S′,S′)(S,S)

ηPD (T, T ;S′,S ′)

≥∑P∈Ω

γ (P)∑

(S′,S′)(S,S)

ηPD (T, T ;S′,S ′)

=∑

(S′,S′)(S,S)

λS′,S′

T,T

for all (T, T ) , (S,S) ∈ Υ∗, where the second inequality holds because∑(S′,S′)(S,S)

ηP′

D (T, T ;S′,S ′) ≥∑

(S′,S′)(S,S)

ηPD (T, T ;S′,S ′) .

whenever P ′ P.

Proof of Theorem 4.8. It is clear that X(t) converges towards x with

xSS =

1, if S = maxC∗(∅) and Si = S for all i ∈ N,0, otherwise.

Hence, both processes converge towards the same steady state if and only ifC∗(∅) = D∗(∅).

Proof of Theorem 4.9. By Theorem 4.1 it holds that XSS (t) = Y SS (t) = 0 forall (S,S) ∈ Υ \Υ∗. Further

1− λS,SS,S =∑

(S′,S′)(S,S)

λS′,S′

S,S ≤ (1 + ε)∑

(S′,S′)(S,S)

µS′,S′

S,S = (1 + ε)(

1− µS,SS,S)

for all (S,S) ∈ Υ∗, and for similar reasons 1−λS,SS,S ≥ (1−ε)(

1− µS,SS,S)

. Suppose

that d (Υ;S,S) = 0. Then (S,S) ∈ Υ0. Since there are no transitions betweenstates in Υ0 it holds that

Y SS (t) = e−(1−λS,SS,S)tXSS (0) ≥ e−(1−µS,S

S,S)(1+ε)tXSS (0)

≥ e−εte−(1−µS,SS,S)tY SS (0) = e−εtXSS (t)

and, similarly, Y SS (t) ≤ eεtXSS (t).

27

Let now d = d (Υ0;S,S) ≥ 1 and let the claim be true for all (T, T ) withd (Υ0;T, T ) < d. By definition d (Υ0;T, T ) < d implies that (T, T ) ≺ (S,S) ord (T, T ;S,S) =∞. Hence, with Theorem 4.1,

Y SS (t) = e−(1−λS,SS,S)t

Y SS (0) +∑

(T,T )≺(S,S)

λS,ST,T

∫ t

0

e(1−λS,SS,S)uY TT (u)du

= e−(1−λS,S

S,S)tY SS (0) +∑

(T,T )≺(S,S)

λS,ST,T

∫ t

0

e(1−λS,SS,S)(u−t)Y TT (u)du

≥ e−(1+ε)(1−µS,SS,S)tXSS (0) +

∑(T,T )≺(S,S)

(1− ε)µS,ST,T∫ t

0

e(1−µS,SS,S)(u−t)eε(u−t) (1− ε)d(Υ0;T,T )

e−εuXTT (u)du

≥ e−εt(e−(1−µS,S

S,S)tXSS (0)

+∑

(T,T )≺(S,S)

(1− ε)d(Υ0;T,T )+1µS,ST,T

∫ t

0

e(1−µS,SS,S)(u−t)XTT (u)du

≥ e−εt (1− ε)d(Υ0;T,T )+1e−(1−µS,S

S,S)tXSS (0) +∑

(T,T )≺(S,S)

µS,ST,T

∫ t

0

e(1−µS,SS,S)(u−t)XTT (u)du

≥ (1− ε)d(Υ0;S,S)

e−εtXSS (t)

and for similar reasons Y SS (t) ≤ (1 + ε)d(Υ0;S,S)

eεtXSS (t).

B.3 On Revolutions

Proof of Lemma 5.1. Clearly L∗A ⊆ L∗B . Let i ∈ N \ L∗A. Then for eachT ⊆ N with i ∈ T there is j such that

vj (N \ L∗A) = uj (N \ L∗A) ≥ uj (N \ (L∗A ∪ T )) = vj (N \ (L∗A ∪ T ))

where the first equality follows from N \ L∗A ∈ F , the inequality follows fromthe definition of L∗A and the last equality follows from B’s being a stick. Hence,i ∈ N \ L∗B , that is L∗A = L∗B .

Proof of Theorem 5.2. The control structures C and D satisfy D(S) ⊆ C(S)for all S ⊆ N . Hence, X(t) first order stochastically dominates Y (t) for allt ≥ 0 by Theorem 4.7. Under the additional conditions C∗ (∅) = N \ L∗A andD∗ (∅) = N \L∗B as seen in Example 3.6. Therefore C∗ (∅) = D∗ (∅) by Lemma5.1.

28

References

Acemoglu, D., Dahleh, M., Lobel, I., Ozdaglar, A., 2011. Bayesian learning insocial networks. Review of Economic Studies 78, 1201–1236.

Acemoglu, D., Robinson, J., 2000. Why did the west extend the franchise?democracy, inequality, and growth in historical perspective. The QuaterlyJournal of Economics 115, 1167–1199.

Acemoglu, D., Robinson, J., 2001. A theory of political transitions. The Amer-ican Economic Review 91, 938–963.

Aliprantis, C., Camera, G., Puzzello, D., 2007. A random matching theory.Games and Economic Behavior 59, 1–16.

Aumann, R., 1967. A Survey of Cooperative Games Without Side Payments,in: Shubik, M. (Ed.), Essays in Mathematical Economics in Honor of OskarMorgenstern. Princeton University Press, pp. 3–27.

Bala, V., Goyal, S., 1998. Learning from neighbours. Review of EconomicStudies 65, 595–621.

Banks, A.S., Wilson, K.A., 2015. Cross-national time-seriesdata archive. Databanks International. Jerusalem, Israel, seehttp://www.databanksinternational.com.

Chwe, M., 1999. Structure and strategy in collective action. American Journalof Sociology 105, 128–156.

Edmond, C., 2013. Information manipulation, coordination and regime change.Review of Economic Studies 80, 1422–1458.

Galeotti, A., Goyal, S., , Jackson, M.O., Vega-Redondo, F., Yariv, L., 2010.Network games. The Review of Economic Studies 77, 218–244.

Gibney, M., Cornett, L., Wood, R., Haschke, P., Arnon, D., 2015. The politicalterror scale 1976-2015. Date Retrieved, from the Political Terror Scale website:http://www.politicalterrorscale.org.

Ginkel, J., Smith, A., 1999. So you say you want a revolution: A game theoreticexplanation of revolution in repressive regimes. Journal of Conflict Resolution43, 291–316.

Grabisch, M., Rusinowska, A., 2011. Influence functions, followers and commandgames. Games and Economic Behavior 72, 123–138.

Granovetter, M., 1973. The strength of weak ties. American Journal of Sociology78, 1360–1380.

Granovetter, M., 1978. Threshold models in collective behavior. AmericanJournal of Sociology 83, 1420–1443.

29

Hu, X., Shapley, L., 2003. On authority distributions in organizations: Control.Games and Economic Behavior 45, 153–170.

Jackson, M., Rogers, B., 2007. Relating network structures to diffusion proper-ties through stochastic dominance. BE Journal of Theoretical Economics 7,1–13.

Karos, D., Peters, H., 2015. Indirect control and power in mutual control struc-tures. Games and Economic Behavior 92, 150–165.

Kuran, T., 1989. Sparks and prairie fires: A theory of unanticipated politicalrevolution. Public Choice 61, 41–74.

Lim, Y., Ozdaglar, A., Teytelboym, A., 2015. A simple model of cascades innetworks. Working Paper.

Lohmann, S., 1994a. The dynamics of informational cascades: The mondaydemonstrations in leipzig, east germany, 1989-1991. World Politics 47, 42–101.

Lohmann, S., 1994b. Information aggregation through costly political action.American Economic Review 84, 518–530.

Lopez-Pintado, D., 2008. Diffusion in complex social networks. Games andEconomic Behavior 62, 573–590.

Marshall, M.G., Marshall, D.R., 2015. Coup d’etat events 1946 - 2014. Centerfor Systemic Peace.

Massey, W., 1987. Stochastic orderings for markov processes on partially orderedspaces. Mathematics of Operations Research 12, 350–367.

McAdam, D., Paulson, R., 1993. Specifying the relationship between social tiesand activism. American Journal of Sociology 99, 640–667.

Morris, S., 2000. Contagion. Review of Economic Studies 67, 57–78.

Olson, M., 1971. The Logic of Collective Action. Harvard University Press.

Schelling, T., 1971. Dynamics of segregation. Journal of Mathematical Sociology1, 143–186.

Schelling, T., 1978. Micromotives and Macrobehavior. W.W. Norton and Com-pany.

Young, P., 2009. Innovation diffusion in heterogenous polpulations: Contagion,social influence, and social learning. American Economic Review 99, 1899–1924.

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