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American Political Science Review Vol. 104, No. 3 August 2010

doi:10.1017/S0003055410000274

Regime Change and Revolutionary EntrepreneursETHAN BUENO DE MESQUITA University of Chicago

Istudy how a revolutionary vanguard might use violence to mobilize a mass public. The mechanismis informational—–the vanguard uses violence to manipulate population member’s beliefs about thelevel of antigovernment sentiment in society. The model has multiple equilibria, one equilibrium in

which there may be revolution and another in which there is certain not to be. In the former, structuralfactors influence expected mobilization, whereas in the latter they do not. Hence, the model is consistentwith structural factors influencing the likelihood of revolution in some societies but not others, offeringa partial defense of structural accounts from common critiques. The model also challenges standardarguments about the role of revolutionary vanguards. The model is consistent with vanguard violencefacilitating mobilization and even sparking spontaneous uprisings. However, it also predicts selectioneffects—–an active vanguard emerges only in societies that are already coordinated on a participatoryequilibrium. Hence, a correlation between vanguard activity and mass mobilization may not constituteevidence for the causal efficacy of vanguards—–be it through creating focal points, providing selectiveincentives, or communicating information.

Imagine a citizen with antiregime feelings who isconsidering becoming involved in a revolutionarymovement. He or she only wants to mobilize if he

or she believes the movement is sufficiently likely tosucceed. Success depends on many people mobilizing.So his or her assessment of the likelihood of successdepends on his or her beliefs about how many of his orher fellow citizens will mobilize.

Because he or she dislikes the government, he orshe suspects that many of his or her fellow citizensdislike the government as well, although he or she facesuncertainty about his or her fellow citizens’ views. Themore extreme his or her own antigovernment feelings,the more confident he or she is that his or her fellowcitizens also oppose the regime. Hence, more extremecitizens are doubly more willing to join the revolution-ary movement. They care more about replacing theregime. And they are more confident that others areready to join, so they believe the movement is morelikely to succeed.

A revolutionary vanguard wants to persuade citizensto mobilize. To do so, it must convince our citizen (andothers like him or her) that the probability of successis sufficiently high. To do this, it must convince himor her that his or her fellow citizens are in fact quiteantigovernment.

The tool that the vanguard has at its disposal is in-surgent violence, such as guerilla or terrorist attacks.These attacks may be persuasive to our citizen becauseshe believes that the vanguard cannot produce a highlevel of violence without the support of the surround-ing population. Thus, high levels of vanguard violence

Ethan Bueno de Mesquita is Associate Professor, Harris School ofPublic Policy Studies, University of Chicago, 1155 East 60th Street,Suite 108, Chicago, IL 60637 ([email protected]).

I received valuable comments from Scott Ashworth, SandeepBaliga, Eli Berman, Amanda Friedenberg, Catherine Hafer, RichardHolden, Ron Krebs, Dimitri Landa, Andrew Litttle, Jim Morrow,Roger Myerson, Erica Owen, Gerard Padro i Miguel, Canice Pren-dergast, Alberto Simpser, Matthew Stephenson, Lars Stole, andseminar participants at Amsterdam, Berkeley, Chicago, ECARES,Minnesota, New York University, Northwestern, and Rotterdam.

suggest, to our citizen, a high level of antigovernmentsentiment in the population as a whole.

Suppose that our citizen observes a series of un-expectedly successful vanguard attacks. These attacksconvince him or her that his or her fellow citizens arequite hostile to the regime and, thus, likely to mobilize.As such, although his or her views of the regime havenot changed, he or she becomes more willing to partic-ipate because he or she thinks the odds of the move-ment succeeding are higher. This is precisely the goalof the vanguard. (Of course, if vanguard violencehad been lower than expected, the result would havebeen the opposite.) Thus, the vanguard has incentivesto invest in violence to try to persuade citizens toparticipate.

I model such an informational role for vanguard vio-lence and explore the incentives and strategic dynamicsit creates. I do so within a coordination model of regimechange with a stage in which a revolutionary vanguard(e.g., insurgents, terrorists, guerillas) engages in pub-licly observable political violence before populationmembers decide whether to mobilize.

The mechanism I consider differs from standard ac-counts in several ways. First, although my model hasmultiple equilibria, I explicitly assume that vanguardviolence cannot create focal points. Hence, to the ex-tent that vanguard violence influences mobilization,it does so via the information it communicates, notby changing citizens’ fundamental conjectures aboutone another.1 Second, the vanguard has no private in-formation about factors related to the likely successof the revolution, such as antigovernment sentimentor regime capacity. As such, this is not a model ofthe vanguard signaling private information.2 Instead,the idea here is that vanguard violence is inherentlyinformative because the existence of antigovernment

1 As discussed here, the informational mechanism is also differentthan existing information models, such as Lohmann (1994) or Chwe(2000).2 For signaling models of vanguard activity, see Baliga and Sjostrom(2009) and Ginkel and Smith (1999).

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American Political Science Review Vol. 104, No. 3

sentiment is crucial in its production. The vanguardare, to paraphrase Mao, fish swimming in the sea ofthe people—–depending on the population for materialsupport, safe havens, information, and recruits. [SeeBerman, Shapiro, and Felter (2009) and Kalyvas (1999)for discussions of the relationship between insurgentsand populations.]

In addition to explicating a novel mechanism bywhich vanguard violence may foment revolution, theresults also contribute more broadly to ongoing de-bates regarding the origins of revolution and politi-cal violence. Structural theories of revolution arguethat revolutions are caused by constellations of struc-tural factors—–regime capacity, international pressure,grievances, the economy, and so on—–that make asociety ripe for revolution (Skocpol 1979). Yet, asGeddes (1990), DeNardo (1985), and others point out,the structural conditions often identified as root causesof revolution occur far more often than do revolutionsthemselves. My model, although not itself a structuralaccount of regime change, casts some doubt on thelogic of this empirical critique. The model has multipleequilibria—–one in which structural factors influenceexpected mobilization and one in which they do not.Hence, the model is consistent with the possibility thatstructural factors are causes of revolution (or effect theprobability of revolution) in some societies but not inothers.

This fact suggests a general problem both for em-pirical assessments of root causes and for policy mak-ing. Much of the variation in the data may be due towhatever cultural or historical factors determine equi-librium selection, rather than those structural factorsthat we often believe are of first-order importance forexplaining political violence and instability. Structuralfactors may matter (for a given equilibrium selection)but be difficult to detect empirically because we cannotobserve which equilibrium a society is playing. Fromthe perspective of policy making, this implies that,even though the data are not well explained by struc-tural variation, it may be that, within a given society,changing key structural factors would reduce politicalviolence or the likelihood of violent regime change.The model provides some suggestions for ways forwardin empirical work, in light of the challenges posed bymultiple equilibria.

The model also highlights a difficulty in assessing theefficacy of vanguards. The literature points to manymechanisms—–providing selective incentives, buildingeffective institutions, creating focal points, spreadinginformation, and so on—–by which vanguards may playa role in fomenting revolution.3 Proponents of sucharguments point to a variety of examples of vanguardsengaging in violence that appears to have inspired alarger insurrection. For instance, the FLN’s (NationalLiberation Front’s) terrorist campaign helped sparkthe Algerian War of Independence (Kalyvas 1999).Violence by Argentine guerilla groups such as the Mon-

3 See DeNardo (1985), Hardin (1996), Kurrild-Klitgaard (1997),Migdal (1974), Popkin (1988), Selbin (1993), and Tilly (1975), amongothers, for a variety of arguments about the role of vanguards.

toneros and the ERP (People’s Revolutionary Army)in the late 1960s and early 1970s led to much larger-scale insurgency by the mid-1970s (Gillespie 1995).Terrorist tactics and other forms of violent agitationby Russian revolutionaries helped set the stage for the“spontaneous” uprisings of 1905 and 1917 (DeNardo1985; Hardin 1996).

My model predicts that vanguard violence may facil-itate mobilization and even spark spontaneous, large-scale uprisings. However, it also suggests that empiricalarguments for the efficacy of vanguards may be lessconvincing than previously thought. In equilibrium,there are selection effects. The vanguard engages inhigher levels of violence in those societies that arecoordinated on a participatory equilibrium. Such so-cieties would be relatively more likely to experienceregime change, even in the absence of a vanguard.Hence, a correlation between vanguard activity andmass mobilization may not constitute evidence for thecausal efficacy of vanguards—–be it through creatingfocal points, providing selective incentives, or commu-nicating information.

The article proceeds as follows. The first section re-lates my informational mechanism to existing literatureon vanguards and revolution. The next several sectionslay out the game and solve for the equilibria. I thendiscuss implications of the model. Finally, I considertwo extensions—–one in which the regime can strate-gically invest in countering the vanguard and anotherin which vanguard violence directly damages regimecapacity—–and conclude.

RELATIONSHIP TO THECONCEPTUAL LITERATURE

I build on the familiar idea that credible revolutionarythreats are at least as much a problem of coordinationas of collective action (Schelling 1960). The theoret-ical literature on the role of vanguards in coordina-tion models of revolution has two key strands, whichI call pure coordination and informational models ofrevolution.

Pure Coordination Models

On the pure coordination view, the key to the rev-olutionary threat is equilibrium selection. Consider acomplete information model of regime change in whichthe regime falls if enough people mobilize. People onlywant to participate if the regime will fall. In this envi-ronment, there is an equilibrium with no participationand an equilibrium with full participation.

The role of a revolutionary vanguard, in such amodel, is to shift a society’s focal equilibrium. If van-guard activity can somehow change people’s funda-mental conjectures about each other’s intentions, thenit can move society from a nonrevolutionary equilib-rium to a revolutionary equilibrium. Hence, Hardin(1996) describes the protests that led to the fall ofthe Romanian dictator Ceausescu in 1989 as “tippingevents” that coordinated the mass of people. Kuran

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Regime Change and Revolutionary Entrepreneurs August 2010

(1989) develops a dynamic model of such tipping be-havior, integrating the idea that the costs of participa-tion are decreasing in total participation.

An important challenge to such focal point modelscomes from the recent literature on “global games”(Carlsson and van Damme 1993; Morris and Shin 1998,2000, 2003). This research shows that introducing smallamounts of uncertainty into coordination games canoften generate a unique equilibrium prediction. Forinstance, Morris and Shin (2004) consider the gameof regime change described previously, but introduceuncertainty regarding the number of people needed tooverthrow the regime. They show that if players receivecorrelated, private signals about the level of mobiliza-tion required for regime change that are sufficientlyinformative, then the game has a unique equilibrium.In it, a player participates if and only his or her privateinformation is sufficiently encouraging about regimeweakness (i.e., if and only if his or her signal lies be-yond some cutoff rule). In such a model, there is nopossibility of a vanguard creating a focal point becausethere is a unique equilibrium.

The fact that the equilibria of such incomplete infor-mation coordination models involve cutoff rules sug-gests a different role for vanguards. Vanguards wouldlike to manipulate the equilibrium cutoff rule itself.Suppose vanguard violence can lead the population touse a less stringent cutoff rule (i.e., convince people toparticipate if and only their signal is above some lowerthreshold than before). Then, at least on the margin, thevanguard can foment revolution by increasing partici-pation. It does so not by convincing people to shift froma nonparticipatory to a participatory equilibrium, butby encouraging more participation within a participa-tory equilibrium. My model suggests an informationalmechanism by which a vanguard might do preciselythis.4

It turns out that, despite the presence of incom-plete information, the game I analyze has multipleequilibria—–each one involving the population using adifferent cutoff strategy. However, my solution conceptrules out the possibility of the vanguard creating focalpoints. This allows me to focus on vanguards using in-formative violence to manipulate the cutoff rule usedby the population.

Intriguingly, despite ruling out focal point effects, mymodel yields the same correlation between vanguardactivity and the population playing an equilibrium fa-vorable to mobilization as would a model with focalpoints. Here the correlation is derived as an equilib-rium phenomenon within the game.

4 My argument is related to work that considers other types of in-formation manipulation in coordination games of incomplete infor-mation. Angeletos, Hellwig, and Pavan (2007) and Edmond (2007,2008) both explore ways in which regimes may attempt to manipulateinformation in global games of regime change. In models focused ongovernance rather than regime change, Dewan and Myatt (2007,2008) study conditions under which leaders may alter the behaviorof followers when there is uncertainty about characteristics of theleader.

Informational Models of Revolution

Lohmann (1994) presents a model in which peopleparticipate in costly collective action in order to com-municate their desire for policy change. The citizensin Lohmann’s model are differentiated both by howextreme their preferences are and by differences ininformation about the actual impact of a policy shift.Players with moderate preferences condition their be-havior on their information and dynamically adjusttheir beliefs and behavior based on the level of partic-ipation in prior periods. Hence, early participation canlead to a cascade of future participation. Importantly,the most extreme players have a dominant strategy toparticipate. Because their only communication tool isparticipation itself, and their participation decision isnot sensitive to their information, extremists cannotconvey any information to others. As such, to the ex-tent that early participators cause future participation,it moderates who does so.

The key idea of my model is that extremists havetools for transmitting information other than just theparticipation decision itself (e.g., insurgent attacks). (Itis also worth noting that, in my model, even the mostextreme antigovernment types do not have a dominantstrategy to participate.) Hence, extremists in my modelcommunicate information that may lead moderates toparticipate by using tactics that are outside the scopeof Lohmann’s analysis. (Similarly, the information cas-cades that are Lohmann’s focus are outside the scopeof my analysis.)

Chwe (1999, 2000) studies how close a communica-tion network needs to get to creating common knowl-edge of payoffs in order to make coordinated outcomesfeasible. In Chwe’s model, increasing the informative-ness of a network is always beneficial for coordination.In particular, even adding information that informssome players that others are quite reluctant to partici-pate makes coordination more feasible. As such, whenChwe examines the role of insurgents in facilitatingcoordinated outcomes, he is concerned with how effi-ciently they spread information, even if that informa-tion is in some sense “bad news.” Thus, the role of an in-formative vanguard in Chwe’s model is fundamentallydifferent than in mine. In my model, the vanguard isattempting to manipulate players’ beliefs to make themmore willing to mobilize. The information generatedby the vanguard can increase or decrease mobilization,depending on the nature of the information.

Other Roles for Vanguards

The accounts described previously, and the model pre-sented here, focus on relatively limited roles for revo-lutionary vanguards. In my case, the focus is exclusivelyon communicating information about antigovernmentsentiment. The goal, in so doing, is to understand thetype of incentives that such an informational role cre-ates for a revolutionary vanguard. I do not intend tosuggest that this is the only, or even the most important,thing that vanguards do in the revolutionary process.Indeed, my model abstracts away from at least two

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vanguard functions that have been central themes inthe literature on revolutions.

A major focus in the rational choice tradition is onthe idea that vanguards provide selective incentives toovercome collective action problems (Lichbach 1995;Olson 1965; Popkin 1979; Tullock 1971, 1974). Suchselective incentives can be either positive or negative.Positive selective incentives include giving supportersof a revolutionary movement special access to socialservices, protection, and so on. Negative selective in-centives might include intimidation of people who failto support the revolutionary movement.

Another literature emphasizes a different type ofcommunication from vanguards to the population. Inparticular, vanguards, in attempting to mobilize a pop-ulation, must find ways to signal the type of regimethat they will put in place once they take power(Finkel, Muller, and Opp 1989; Migdal 1974; Tilly1975; Wickham-Crowley 1992). This may deter van-guards from engaging in too much intimidation, butit may also lead them to construct quasigovernmen-tal institutions—–a strategy followed by many Maoistgroups—–to demonstrate a capacity to govern.

Clearly, these (and other) considerations create im-portant additional incentives and constraints for revo-lutionary vanguards. As such, the analysis here shouldbe understood as only one step toward a more completemodel of the role of vanguards in revolution.

THE MODEL

There are a revolutionary vanguard (labeled E for “ex-tremists”) and a continuum of population members ofmass 1. At the beginning of the game, each member ofthe population, i , learns his or her type �i = � + �i ,which I interpret as his or her level of antigovern-ment sentiment. The common component � is drawnby nature from a normal distribution with mean m andvariance �2

� . The idiosyncratic components �i are inde-pendent draws by nature from a normal distributionwith mean 0 and variance �2

� . Members of the popula-tion observe only �i , not � or �i . After each populationmember observes his or her type, the vanguard, whichhas no private information, chooses a level of effortto expend on a campaign of violence (e.g., terroristor guerrilla attacks). Individuals observe the level ofviolence and then decide whether to join an attemptedrevolution against the government. The game ends withthe government either being overthrown or remainingin place.

I refer to the stage of the game in which the revolu-tionary vanguard engages in violence as the vanguardstage and the stage in which population members de-cide whether to participate as the revolution stage.

The “number” of people who join the revolution isN. The regime is replaced if and only if N is greater thanor equal to a threshold T ∈ (0, 1), which is commonlyknown.

A member of the population, i , takes an actionai ∈ {0, 1}, where ai = 1 is the decision to participate. Aperson’s type, �i , determines how much he or she values

regime change. He or she derives a portion, � ∈ (0, 1],of that value only if the revolution succeeds and he orshe participated. The other portion, 1 − � , is realizedby all players, if the revolution succeeds, whether thatplayer personally participated in the revolution. Par-ticipating imposes a cost k > 0 on the individual. Thepayoff to a failed revolution is 0. The following matrixgives the von Neuman-Morgenstern expected utilityfunction for a population member i .

Player i

N < T N ≥ T

ai = 0 0 (1 − γ )θi

ai = 1 −k θi− k

Payoffs for a representative population member i .

Denote by t ∈ [t,∞) (with t ≥ 0) the level of effortexerted by the revolutionary vanguard. The total levelof vanguard violence is v = t + � + �, where � is ran-dom noise drawn by nature from a normal distributionwith mean 0 and variance �2

� . Vanguard violence is in-creasing in both vanguard effort and the average levelof antigovernment sentiment in society. As discussedat the beginning of this article, the idea is that whenthe public has a higher level of antigovernment sen-timent, it is easier for the revolutionary vanguard toproduce violence, because it will have the support ofthe population.

The vanguard benefits from regime change and findseffort costly. The vanguard’s payoffs are given by thefollowing von Neuman-Morgenstern expected utilityfunction:

UE(t, N) ={

1 − cE(t) if N ≥ T−cE(t) if N < T,

where cE represents the costs of vanguard effort. I as-sume cE(t) = 0 and cE is strictly increasing, convex,and satisfies c′

E(t) = 0 and limt→∞ c′E(t) = ∞.

A Comment on the Primitives

Several assumptions merit further comment. First,there is some portion (�) of the payoffs from regimechange that can only be accessed by those who par-ticipate in the revolution. (This relaxes the standardcollective action problem.) Substantively, this could bebecause those who actively participate in revolutiongain privileged status after regime change occurs orbecause there are expressive benefits to having partic-ipated in a victorious uprising.

Second, there is heterogeneity in the level of antigov-ernment sentiment, but population members’ views arepositively correlated. The idea is that particularly bad(resp. good) governments are likely, on average, to gen-erate more (resp. less) antigovernment sentiment.

Third, it is worth pointing out that, although the rev-olution stage of this model is similar to a global game

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of regime change (Angeletos, Hellwig, and Pavan 2006,2007; Edmond 2007; Morris and Shin 1998, 2000), it isnot a global game. In particular, the model here doesnot satisfy the two-sided “limit dominance” propertyof global games (Morris and Shin 1998)—–there is no �isuch that participation is a dominant strategy.5

Fourth, vanguard violence is an increasing functionof both vanguard effort and antigovernment sentimentin society. As mentioned previously, a supportive popu-lation is important for the operation of a revolutionaryvanguard for a variety of reasons. The vanguard is likelyto rely on the surrounding population for intelligence,safe houses, recruits, and resources. Moreover, it isdifficult for vanguards to function in an environmentwhere the surrounding population is hostile to theirefforts and likely to turn them in to the authorities.The additive functional form of v is a tractable reducedform.6

It is also worth noting that the outcome v can beinterpreted more broadly. Any action that is publiclyobservable and increasing in both effort and antigov-ernment sentiment could play a similar role. In repres-sive regimes, violence may be one of the few strategiesavailable that satisfies these conditions.

Finally, the vanguard chooses its effort without a pri-vate signal of the level of antigovernment sentiment.(The informational structure is related to Holmstrom’s(1999) model of “career concerns.”) The idea is to focuson violence as an inherently informative act (becauseits production requires support) and the incentives thatcreates. If the vanguard had private information the is-sue would be muddied because the vanguard’s strategyitself might be informative within a separating equi-librium. Situations in which vanguards have privateinformation are certainly of interest. [See Baliga andSjostrom (2009) for a model with cheap talk and Ginkeland Smith (1999) for a model with costly signaling.]However, because the situation in which the vanguarddoes not have a large informational advantage over thepopulation is also descriptive of many cases, it is alsoworth studying the pure informational value of vio-lence generation in the absence of private information.

Equilibrium Concept

A pure strategy for the vanguard is a choice of effortdirected at violence, t . A pure strategy for a member of

5 For another application with one-sided limit dominance but uncor-related signals, see Baliga and Sjostrom (2004).6 Additivity preserves normality of the posteriors, which is importantfor tractability. One could also preserve normality with a multiplica-tive functional form (v = �t + �), which allows for the perhaps morenatural assumption that effort and public support are complementsrather than substitutes. I do not do so for two reasons. First, it in-creases complexity (the variance of the posteriors becomes a functionof beliefs about effort choice) without adding insight. Second, andmore important, introducing a complementarity into the violenceproduction function might be a new source of equilibrium multiplic-ity. Because the existence of multiple equilibria is a major theme ofthis article, it is important that it be clear that this is not driven bycomplementarities in the vanguard stage. Indeed, it is a fact about therevolution stage that is preserved even in the absence of a vanguardstage.

the population is a mapping s(�i , v) : R × R → {0, 1};from personal antigovernment sentiment and van-guard violence into a decision of whether to participate.

The solution concept is pure strategy perfectBayesian equilibrium (PBE). Here this solution con-cept simply requires that beliefs be consistent with thestrategy profile and Bayes’ rule, and that strategies besequentially optimal given beliefs and the strategies ofthe other players.

I further restrict the set of equilibria in two ways.First, I restrict attention to those pure strategy PBE ofthe full game in which, in the revolution stage, playersuse cutoff strategies of the form, “choose ai = 1 if andonly if �i ≥ �(v, t∗),” where v is vanguard violence andt∗ is the population’s common belief about the level ofeffort by the vanguard. (I allow for the possibility ofinfinite cutoff rules.)

Second, for some values of v and t∗, the revolu-tion stage has multiple equilibria in cutoff strategies:one with an infinite cutoff (i.e., no participation) andtwo with finite cutoffs. I focus on equilibria of the fullgame in which players play the same selection—–i.e.,the lower cutoff, the higher cutoff, or the infinitecutoff—–whenever there are multiple equilibria. Thisimposes continuity of the cutoff rule in v (except, atmost, at one jump).

The idea behind this second requirement is twofold.First, equilibrium selection is likely to be a fact aboutthe culture and history of a society (Chwe 1998, 2001). Ido not allow small changes in revolutionary violence toalter the fundamental conjectures players have aboutone another’s behavior—–that is, to change society’s fo-cal equilibrium. Second, as already emphasized, themodel is concerned with how revolutionary violencecan affect mobilization by communicating informationabout antigovernment sentiment. To study this phe-nomenon, I want to explicitly rule out the possibility ofthe vanguard affecting mobilization by creating focalpoints.

I refer to a pure strategy PBE that satisfies these twocriteria as a cutoff equilibrium.

Beliefs

Applying Bayes’ rule for the case of normal priors andnormal signals, a population member of type �i , afterobserving his or her type but not the level of violencev, has posterior beliefs about � that are distributednormally with mean

mi = ��i + (1 − �)m

and variance

�21 = ��2

� ,

with � = �2�

�2� + �2

�(DeGroot 1970). Substantively, the

more antigovernment an individual is, the moreantigovernment he or she believes society is likely tobe (i.e., mi is increasing in �i .)

Suppose it is common knowledge that populationmembers believe the level of effort by the vanguard

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FIGURE 1. For a fixed cut-off rule (�(v − t∗)), the shaded portion [which represents participation,N(�, �(v − t∗))] is increasing in antigovernment sentiment (�). Thus, because � > � ′, participation ishigher in (a) than in (b)

θ

θ′ θ(v − t∗)

θ(v − t∗)

N(θ, θ(v − t∗)) = 1 − Φ(

θ(v−t∗)−θσε

)

N(θ′, θ(v − t∗)) = 1 − Φ(

θ(v−t∗)−θ′σε

)

(a)

(b)

Density of θis

Antigovernment Sentiment (θi)

Antigovernment Sentiment (θi)Participate

Participate

is t∗. Then population members believe that v − t∗ isa mean � normally distributed random variable withvariance �2

� . After observing a level of violence, a per-son of type �i has posterior beliefs about � that arenormally distributed with mean

mi,v−t∗ = � (v − t∗) + (1 − � )mi (1)

and variance

�22 = � �2

�

with � = �21

�21 + �2

�. Here we see that violence commu-

nicates information. The more violence the vanguardgenerates relative to expectations, the more antigov-ernment sentiment a population member believesthere is in society (i.e., mi,v−t∗ is increasing in v − t∗).

THE REVOLUTION STAGE

Denote by Pr(N ≥ T | �i , v − t∗, s−i ) an individual’s as-sessment of the probability of regime change, given hisor her type (�i ), vanguard violence (v), beliefs aboutvanguard effort (t∗), and a strategy profile for all otherplayers (s−i ). Comparing the expected payoff from par-ticipating and not participating, a population memberof type �i participates if and only if

Pr(N ≥ T | �i , v − t∗, s−i )��i ≥ k. (2)

That is, player i participates if his or her incrementalbenefit from participating (the left-hand side) is at leastas large as his or her incremental cost from participat-ing (the right-hand side).

The first fact to note is that there is always an equilib-rium with zero participation. If an individual believesthat no one else will participate, regardless of his or hertype, he or she believes the regime will not fall. Hence,he or she too will not participate, as formalized here.

LEMMA 1. There is always an equilibrium of the gamecharacterizing the revolution stage in which no playerparticipates (i.e., with �(v − t∗) = ∞ for all v − t∗).

All proofs are in the Appendix.There may also be cutoff equilibria with positive par-

ticipation. To solve for such an equilibrium, I follow thefollowing four steps:

1. Conjecture a mapping, �(·) that gives a cut-off rule,�(v − t∗), for each level of vanguard violence andbeliefs about vanguard effort (v − t∗).

2. Compute a player i ’s subjective belief about theprobability of regime change, Pr(N ≥ T | �i , v −t∗, �(v − t∗)), given v − t∗ and the belief that allother players, j , participate if and only if �j ≥�(v − t∗).

3. Find which players will participate given the subjec-tive belief from step 2. That is, for which players isPr(N ≥ T | �i , v − t∗, �(v − t∗))��i ≥ k?

4. To be part of an equilibrium, the following must betrue of the mapping �(·). For each value of v − t∗,the answer to the question in step 3 is that players oftype �i ≥ �(v − t∗) will participate and no one elsewill.

Begin with step 1 by conjecturing a mapping fromlevels of unexpected violence (v − t∗) into cutoff rules,�(·) : R → [−∞,∞]. Fix a v − t∗. A player j using thecutoff rule �(v − t∗) participates if �j = � + � j ≥ �(v −t∗). Put differently, a player j participates if he or sheis antigovernment enough, which can be reexpressedas � j ≥ �(v − t∗) − �.

Now proceed to step 2—–computing a player i ’s sub-jective belief about the probability of regime change,given v − t∗ and the belief that all other players usethe cutoff rule �(v − t∗). As we have just seen, fromplayer i ’s perspective, if all other players use the cutoffrule �(v − t∗), then total participation is the mass ofplayers j with � j ≥ �(v − t∗) − �. Refer to either panelof Figure 1. Here, we see that, for a given �, this mass

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is equal to

N(�, �(v − t∗)) = 1 − �

(�(v − t∗) − �

��

),

where � is the cumulative distribution function of thestandard normal.

The revolution succeeds if enough people parti-cipate—–in particular, if N(�, �(v − t∗)) ≥ T. Referagain to Figure 1. Comparing the two panels, we seethat, for a fixed �(v − t∗), participation is strictly in-creasing in �—–the more antigovernment sentiment insociety, the more participation. This implies that, fora given �(v − t∗), there is a minimal level of antigov-ernment sentiment necessary for regime change to beachieved. This is precisely the amount that makes themass of people who mobilize equal to the government’sthreshold for withstanding revolution, T. Call this min-imal level of antigovernment sentiment necessary forachieving regime change �∗(�(v − t∗)). It is implicitlydefined by

N(�∗, �(v − t∗)) = 1 − �

(�(v − t∗) − �∗

��

)= T,

which can be rewritten

�∗(�(v − t∗)) = �(v − t∗) − �−1(1 − T)�� . (3)

So, suppose player i observed v − t∗ and believesall other players use the cutoff rule �(v − t∗). His orher subjective belief about the probability of regimechange is simply how likely he or she believes it is that� is greater than �∗(�(v − t∗)). Recall that he or she be-lieves that � is distributed normally with mean mi,v−t∗ ,which is increasing in �i and v − t∗. That is, the moreextreme is player i , the larger he or she believes � is. Themore (unexpected) violence the vanguard generates,the larger he or she believes � is. Formally, player i ’ssubjective belief about the likelihood of regime changeis given by

Pr(N ≥ T | �i , v − t∗, �(v − t∗))

= Pr(� ≥ �∗(�(v − t∗)) | �i , v − t∗)

= 1 − �

(�∗(�(v − t∗)) − mi,v−t∗

�2

).

Now turn to step 3—–determining which players willparticipate, given v − t∗ and the belief that every-one else uses the cut-off rule �(v − t∗). Substitutingthe Pr(N ≥ T | �i , v − t∗, �(v − t∗)) just calculated intoequation (2), a player i who believes that everyone elseis using the cutoff rule �(v − t∗) will participate if[

1 − �

(�∗(�(v − t∗)) − mi,v−t∗

�2

)]��i ≥ k. (4)

The interpretation is the same as equation (2). Player iparticipates if and only if his or her incremental benefit

from participating given v − t∗ and a belief that every-one else uses the cut-off rule �(v − t∗) (the left-handside) is greater than her incremental cost from partic-ipating (the right-hand side). This incremental benefitwill be important. As such, I introduce the followingnotation:

IB(�i , �(v − t∗), v − t∗)

≡[

1 − �

(�∗(�(v − t∗)) − mi,v−t∗

�2

)]��i .

LEMMA 2. Fix v − t∗ and a finite �. For �i > 0,IB(�i , �, v − t∗) is increasing in its first argument.

Lemma 2 says that, among players who are at allantigovernment (i.e., �i > 0), if all players use a cut-off strategy, the incremental benefit of participating ishigher for more antigovernment members of the po-pulation. This is true for two reasons. First, more anti-government types have higher payoffs from regimechange. Second, more antigovernment types believethe probability of regime change is higher, becausethey believe there is more antigovernment sentimentin society. Lemma 2 implies that, if player i believesall other players use a cutoff rule, then player i willuse a cutoff rule. That is, he or she will only participateif he or she is sufficiently antigovernment. (Note thata player with �i ≤ 0 has a dominant strategy not toparticipate.)

Finally, step 4 says that it is not enough for playeri to want to use any cutoff rule, given that he or shebelieves everyone else uses the cutoff rule �(v − t∗).To make an equilibrium, player i must want to use thatsame cutoff rule, �(v − t∗). For this to be the case, thefollowing must be true. If �(v − t∗) is a finite cutoff rule,then a player whose type is right at the cutoff (i.e., �i =�(v − t∗)) is exactly indifferent between participatingand not. If this holds, then population members whoare more (resp. less) antigovernment than �(v − t∗) willhave a strict preference to (resp. not to) participate.Formally, then, equilibrium requires

IB(�(v − t∗), �(v − t∗), v − t∗) = k. (5)

The left-hand side of this condition is the incre-mental benefit from participation to a player of type�i = �(v − t∗), when �(v − t∗) is used as a cutoff ruleby all other players. Because this quantity is critical tocharacterizing the equilibrium, I notate it as follows:

IB(�(v − t∗), v − t∗) ≡ IB(�(v − t∗), �(v − t∗), v − t∗).

Finding a mapping, �(·), that is consistent witha cutoff equilibrium is now straightforward. First,whenever it yields a finite cutoff rule, it must satisfyIB(�(v − t∗), v − t∗) = k. Second, if for some v − t∗,max� IB(�, v − t∗) < k, then there is no finite cutoffrule consistent with equilibrium. In such a case, thepopulation members can do only one thing in equilib-rium: not participate. Third, except at values of v − t∗,where max� IB(�, v − t∗) = k, the mapping must be

452


FIGURE 2. The solid curve represents IB(�, v − t∗)—–the incremental benefit to a player of type �,should � be adopted as the cutoff, given a v − t∗. The dashed line represents the costs ofparticipation. Intersections, where IB(�, v − t∗) = k, are finite cutoff rules that are consistent withequilibrium for that value of v − t∗

IB ,v' t

θH

Cutoff Rule

k

Utility

IB ,v'' t

θ v'' t θH

Cutoff Rule

k

Utility

IB ,v''' t

θL v''' t θM v''' t θH

Cutoff Rule

k

Utility

continuous. These facts are summarized in the follow-ing result.

LEMMA 3. A strategy profile in which all popula-tion members use the same strategy s : R × R → {0, 1},which is not the strategy “never participate,” is consistentwith a cutoff equilibrium if and only if:

s(�i , v − t∗)

={

1 if max� IB(�, v − t∗) ≥ k and �i ≥ �(v − t∗)0 else,

with �(v − t∗) satisfying

1. IB(�(v − t∗), v − t∗) = k for all v − t∗ such thatmax� IB(�, v − t∗) ≥ k

2. Continuity in v − t∗ for all v − t∗ such thatmax� IB(�, v − t∗) > k

Given all this, what does the mapping �(·) look like?That is, what are the equilibrium cut-off rules used bythe population? The answer depends on the shape ofthe function IB.

As formalized in Lemma 4 and illustrated in eachpanel of Figure 2, IB(·, v − t∗) is single peaked andgoes to zero as the cut-off rule goes to zero or infin-ity. This implies that there could be multiple mappingsconsistent with cut-off equilibrium because IB couldcross k more than once.

The intuition for why IB is nonmonotonic is as fol-lows. For a fixed v − t∗, increasing � (i.e., making thecutoff rule more stringent) has three competing effectson the incremental benefit to a player of type �i = �.First, when the cutoff rule is more stringent, a playerwhose type equals the cutoff rule has a higher privatebelief about the level of antigovernment sentiment (i.e.,mi,v−t∗ is increasing in �i ). Hence, he or she believes thatfor any given cut-off rule, more people will participate.This implies that he or she believes the probability ofsuccessful regime change is higher, increasing his orher incremental benefit from participation. Call this thebeliefs effect of increased stringency. Second, when thecutoff rule is more stringent, a player whose type equalsthe cutoff rule simply has higher personal payoffs from

regime change, which increases his or her incrementalbenefit from participation. Call this the payoff effectof increased stringency. Third, when the cutoff rule ismore stringent, given a belief about �, the amount ofparticipation a player anticipates is lower (Figure 1).This implies that the probability of successful regimechange is lower, decreasing the incremental benefitfrom participation. Call this the participation effect ofincreased stringency. The beliefs effect and the payoffeffect tend to make the function IB increasing in �.The participation effect tends to make the function IBdecreasing in �. Together, these competing effects leadto nonmonotonicity, as formalized here.

LEMMA 4. For all parameter values and all v − t∗,IB(·, v − t∗) has the following properties:

1. Any � satisfying IB(�, v − t∗) = k is strictly positive.2. IB(·, v − t∗) is single peaked in positive values of its

first argument.3. lim�→∞ IB(�, v − t∗) = 0.

4. IB(0, v − t∗) = 0.

As illustrated in Figure 2, the shape of IB impliesthat, for a given value of v − t∗, there are generically ei-ther zero (the first panel) or two (the third panel) finitecutoff rules consistent with equilibrium. For values ofv − t∗ where two finite cutoff rules are consistent withequilibrium, I label the lower �L(v − t∗) and the higher�M(v − t∗) (for low and middle). The second panel ofFigure 2 shows the knife-end case where there is onefinite cut-off rule consistent with equilibrium. Recallthat there is also always a cutoff equilibrium with noparticipation [i.e., �H(v − t∗) = ∞].

I can now characterize equilibrium behavior in therevolution stage.

Proposition 1. There are three strategies for the popu-lation members in the revolution stage that are consistentwith a cutoff equilibrium of the full game:

s∞(�i , v − t∗) = 0 for all �i and v − t∗

453


sM(�i , v − t∗)

={

1 if max� IB(�, v − t∗) ≥ k and �i ≥ �M(v − t∗)0 else.

sL(�i , v − t∗)

={

1 if max� IB(�, v − t∗) ≥ k and �i ≥ �L(v − t∗)0 else.

The game is only interesting if it is possible for thereto be positive participation. Hence, I assume that theparameters of the game are such that, for some real-ization of v − t∗, the strategies sM and sL actually yielda finite cutoff rule.

Assumption 1. maxv−t∗ max� IB(�, v − t∗) ≥ k.

The result in proposition 1 is reminiscent of standardresults for games with strategic complements. There area highest, and lowest equilibrium (here s∞ and sL) thatbehave intuitively, and there is a middle equilibriumthat does not. The middle equilibrium seems unstablein a way that is analogous to the instability of the mixedstrategy equilibrium in a standard complete informa-tion coordination game (Echenique and Edlin 2004).To see this, imagine a simple learning dynamic, suchas players playing best responses to the distribution ofplay in a previous round. Suppose v − t∗ is such thatthere is a finite cutoff strategy in equilibrium. If playis slightly perturbed away from �M(v − t∗), then thelearning dynamics do not return play to �M(v − t∗). Ifa few too many players participate, then players withtypes slightly less than �M(v − t∗) want to participate,making more players want to participate, until every-one with a type greater than �L(v − t∗) is participating.Similarly, if a few too few players participate, thenplayers with types slightly higher than �M(v − t∗) donot want to participate, making more players not wantto participate, until no one is participating. Given this,I restrict attention to the strategies s∞ or sL.

Assumption 2. Population members do not play thestrategy sM.

VANGUARD VIOLENCE AND REVOLUTION

Before studying how the vanguard behaves, it will beuseful to understand how the population responds tochanges in vanguard violence. If the population playsthe strategy s∞ so that no one ever participates, thenvanguard violence has no effect on population mem-bers’ behavior. But if population members play sL,violence can affect their behavior. How does it do this?

Producing violence requires public support. Hence,the more vanguard violence there is relative to expec-tation (i.e., higher v − t∗), the more antigovernmentsentiment each population member believes there is insociety (i.e., mi,v−t∗ is increasing in v − t∗). As the twopanels of Figure 1 make clear, for a fixed cut-off rule,the higher � is, the more people will participate. As

a result, the higher population members believe � is,the more likely they believe it is that participation willbe sufficient to achieve regime change. Hence, higherlevels of v − t∗ make population members believe thatrevolution is more likely to succeed. This increases theincremental benefit of participation.

Substantively, the idea would be something like thefollowing. Imagine an IRA sympathizer who is on thefence about whether to participate in some mass action.On the one hand, he or she sympathizes with the IRA’santiregime stance. On the other hand, he or she is notsure how well attended the mass action will be (and,consequently, how likely it is to achieve its goals) andis concerned about government reprisal. Then he orshe observes a series of unexpectedly successful IRAattacks that he or she does not believe could have beenachieved without support from the surrounding popu-lation. He or she concludes that his or her neighborssupport the cause and, thus, that the mass action islikely to be well attended. This change in his or herbeliefs about his or her fellow citizens’ views does notchange his or her preferences over regime change. Butit changes his or her beliefs about the likely efficacious-ness of the mass action. This leads him or her (andothers like him or her) to participate. Put differently,the cut-point shifts down as a result of unexpected van-guard violence.

More formally, unexpected vanguard violence hastwo effects on behavior at the revolution stage. First,the greater v − t∗, the more likely it is that IB(·, v − t∗)crosses k (so that positive participation is consistentwith equilibrium), as formalized in the followinglemma.

LEMMA 5. If the population plays sL, then for any� + t − t∗, there exists a unique, finite �(� + t − t∗) suchthat there is positive participation in the revolution stageif and only if � ≥ �(� + t − t∗). Moreover �(� + t − t∗)is decreasing in � + t − t∗.

Second, as argued previously, given that a finite cut-off rule exists, vanguard violence changes the cutoffitself. In particular, unexpected violence by the van-guard convinces population members that there is ahigher level of antigovernment sentiment, resulting ina lower cutoff rule and more participation. This factis illustrated in Figure 3 (where �L(·) decreases withan increase in v − t∗) and formalized in the followinglemma.

LEMMA 6. Fix � ≥ �(� + t − t∗). �L(v − t∗) is strictlydecreasing in v − t∗.

Two further points are worth mentioning. First,(unexpected) vanguard violence affects participationwithout creating focal points. When v − t∗ increases,�L(v − t∗) decreases, leading to more participation onthe margin. Increased vanguard violence does not con-vince players to switch from playing �H to �L, whichwould be a focal point effect. Second, as discussedin greater detail later, if the population plays sL, itis possible for society to jump discontinuously from

454


FIGURE 3. The solid and dashed curvesrepresent IB(·, v − t∗) for lower and higherrealizations of v − t∗, respectively (i.e.,v ′′ > v ′). The horizontal line represents thecost of participation. An equilibrium finitecutoff rule for a given v − t∗ is a point wherethe line and the relevant curve intersect. Thelower cutoff rule, �L (·), is decreasing in v − t∗

IB , v ' t

IB , v '' t

Cutoff Rule

k

Utility

no mobilization to high mobilization. This happens atthe one point where a finite cutoff rule goes from notexisting to existing, as formalized in lemma 5.

THE VANGUARD STAGE

The vanguard is only willing to invest in costly violenceinsofar as doing so increases the probability of regimechange. If the population is playing the strategy s∞ suchthat there will be no participation in the revolutionstage no matter what, then this is not possible, so thevanguard will not engage in violence.

LEMMA 7. If the population members use the strat-egy s∞, then the revolutionary vanguard exerts minimaleffort in the vanguard stage (t = t).

Suppose, instead, the population uses the strategysL so that positive participation is possible. Two con-ditions must be met to achieve regime change. First,a finite cutoff rule consistent with equilibrium mustexist. As shown in lemma 5, such a rule only existsif � ≥ �(� + t − t∗). Second, participation must exceedthe threshold for regime change, T. As shown in equa-tion (3), this is true if and only if � is greater than�∗

L(� + t − t∗) implicitly defined by

�∗L = �L(�∗

L + � + t − t∗) − �−1(1 − T)�� . (6)

The regime will fall as long as � is larger thanmax{�(� + t − t∗), �∗

L(� + t − t∗)}. Which constraintbinds depends on the realization of � + t − t∗ as for-malized in the following result.

LEMMA 8. Suppose the population uses the strategysL.

1. For any realization of � + t − t∗, there is a unique, fi-nite �L(� + t − t∗) = max{�∗

L(� + t − t∗), �(� + t −t∗)} such that there will be regime change if and onlyif � ≥ �L(� + t − t∗).

2. �L(� + t − t∗) is decreasing.

3. There exists a unique �(t − t∗) such that �∗L(� + t −

t∗) > �(� + t − t∗) if and only if � > �(t − t∗).

Figure 4 illustrates the three possible outcomes whenthe population plays sL:

1. Successful revolution: Successful revolution occursif (�, �) lies northeast of the curve defined by �(� +t − t∗) (the dashed line) and �∗

L(� + t − t∗) (the solidcurve).

2. Failed revolution: A failed revolution (i.e., an up-rising but not regime change) occurs if (�, �) liesbetween �(� + t − t∗) and �∗

L(� + t − t∗).3. No mobilization: No mobilization occurs if (�, �) lies

to the southwest of �(� + t − t∗).

The probability of each outcome is found by inte-grating the area corresponding to that outcome withrespect to the distributions of � and �.

Figure 4 illustrates two key points. First, increasingantigovernment sentiment (the y-axis) or unexpectedvanguard violence (the x-axis) moves the outcome inthe direction of revolution. Second, increasing van-guard effort (t) relative to expectation (t∗) shifts thecurves to the southwest, increasing mobilization andthe probability of successful revolution (see lemma 8,item 2.) This is precisely because (unexpected) vio-lence communicates information to the population,and thereby changes population members’ behaviorin the manner described in the previous section. This iswhat gives the vanguard incentives to invest in violence.

Given this, the vanguard chooses a level of effort, t ,to solve the following:

maxt

∫ �(t−t∗)

−∞

∫ ∞

�(�+t−t∗)

1��

�

(� − m

��

)1��

�

(�

��

)d� d�

+∫ ∞

�(t−t∗)

∫ ∞

�∗(�+t−t∗)

1��

�

(� − m

��

)

× 1��

�

(�

��

)d� d� − cE(t), (7)

which implies the following.

LEMMA 9. If a cutoff equilibrium in which the pop-ulation plays the strategy sL exists, then the vanguard’saction, t∗, is characterized by:∫ �(0)

−∞�

(�(�) − m

��

)�

(�

��

)d�

−∫ ∞

�(0)�

(�∗(�) − m

��

)∂�∗(�)

∂t�

(�

��

)d�

= ��c′E(t∗).

There is a unique such t∗.The lemma states that, if there is a cutoff equilibrium,

then the vanguard’s effort is given by the first-order con-dition in lemma 9. Such a cutoff equilibrium exists ifthe cost function is sufficiently convex. The next lemmashows that there are such cost functions.

455


FIGURE 4. If the population plays sL , then there is successful regime change when the realizationof (�, �) lies to the northeast of the curve defined by � (the dashed line) and �∗ (the solid curve).Increasing t shifts this curve to the southwest, thereby increasing the probability of successfulregime change

η

θ

η

θ

θ∗L

Successful Revolution

FailedRevolution

NoM

obilization

–

LEMMA 10. There exists a nonempty set of cost func-tions, C, such that a cutoff equilibrium in which thepopulation plays the strategy sL exists.

Assumption 3. The cost function cE is in the set C.

Lemma 9 shows the vanguard’s fundamental trade-off. Increased violence increases the population’s be-liefs about the level of antigovernment violence. Thishas two effects. First, as shown in Lemma 5, there isa greater probability of positive participation. Second,as shown in Lemma 6, given positive participation, thelevel of mobilization is increasing in violence. Botheffects increase the likelihood of regime change, whichis a marginal benefit for the vanguard. However, thereare costs to resources expended on violence.

Given this analysis, the following result describes thecutoff equilibria of the game.

Proposition 2. Given assumptions 1–3, there are ex-actly two cutoff equilibria:

1. The vanguard chooses minimal effort, t , and the pop-ulation plays s∞.

2. The vanguard chooses a level of effort, t∗ > t , andthe population plays sL.

COMPARATIVE STATICS

In the equilibrium where the population plays thestrategy sL, the level of mobilization and the proba-bility of successful regime change are decreasing in the

government’s capacity to withstand an uprising (T) andto impose costs on those who organize against it (k).They are increasing in the extent to which the benefitsassociated with regime change only go to participants(�). Intuitively, when the probability of or payoff tosuccess increases (i.e., T decreases or � increases) orthe cost of participation decreases (i.e., k decreases), itbecomes more attractive to participate.

Proposition 3. In the equilibrium in which the pop-ulation uses the strategy sL, the number of people whomobilize and the probability of successful regime changeare decreasing in T and k and increasing in � .

These comparative statics highlight the fact thatstructural factors can affect the level of mobilization inequilibrium. They are also useful for determining howthe model relates to standard intuitions about historicalcases. For instance, in the early 20th century, Russiaexperienced two major attempts at revolution: a faileduprising in 1905 and the successful revolution of 1917.The fact that the 1905 uprising occurred suggests thatthe success of the 1917 revolution was not caused bythe population shifting to a new focal equilibrium. Inboth cases, the people were willing to mobilize (i.e.,they were playing a strategy akin to sL).

Rather, the model is consistent with a familiar struc-turalist account (Skocpol 1979). The 1905 failure canbe attributed to the uprising being insufficient to over-come the repressive capacity of the state. That is, Tand k were too large, relative to the level of antigov-ernment sentiment (�) and the success of extremists in

456


producing vanguard violence (v − t∗). By 1917, WorldWar I had diminished the government’s capacity torepress (k) and withstand (T) rebellion. Thus, with-out changing focal equilibria, the revolutionary van-guard and the Russian population were now able towage a successful revolution because structural condi-tions had become more favorable to regime change.These structural changes simultaneously increased thepopulation’s willingness to mobilize and the likeli-hood that such mobilization would lead to the govern-ment’s downfall. In terms of Figure 4, the two curvesshifted down, moving the situation from one (in 1905)that was between � and �∗, to one (in 1917) that wasabove �∗.

VANGUARDS AND REGIME CHANGE

The model has implications both for empirical researchand for conceptual debates over the causes of revolu-tion and political violence.

Structure versus Culture:The Problem of Root Causes

Structural theories of revolution argue that revo-lutions are caused by constellations of structuralfactors—–regime capacity, international pressure, grie-vances, the economy, and so on—–that make a societyripe for revolution (Skocpol 1979). Yet, as DeNardo(1985), Geddes (1990), and others point out, the struc-tural conditions often identified as root causes ofrevolution occur far more often than do revolutionsthemselves.

My model, although not itself a structural accountof regime change, casts some doubt on the logic ofthis empirical critique. The discussion of comparativestatics highlighted several parameters of the model thatcan be interpreted as representing structural featuresof a society. Proposition 3 shows that, in the equilibriumwhere the population plays the strategy sL, structuralfactors influence mobilization and the likelihood of asuccessful revolution. Hence, these structural factorscan be viewed as causes of revolution in a society play-ing that equilibrium. But they are not structural causesof revolution in a society playing the other equilib-rium. Moreover, if two structurally identical societiesplay different equilibria, then they have very differentlikelihoods of revolution occurring.

More concretely, return to the structuralist claim thatWorld War I was a cause of the Russian Revolution. Asdiscussed previously, my model is consistent with thisargument. Geddes (1990) critiques this claim by notingthat in other settings, international pressure does notseem to cause revolution. But my model is also con-sistent with this observation. Both claims can be true,simultaneously, if the population is playing the strategysL in those societies where international pressure in-creases the risk of revolution, but is playing the strategys∞ in those societies where international pressure doesnot increase the risk of revolution. (Of course, whether

this is indeed the reason for the differential impact offoreign pressure on revolution across countries is anopen question.)

This argument suggests a quite general problem bothfor the empirical literature on the root causes of polit-ical violence and for policy making. In a world char-acterized by multiple equilibria, much of the variationin the data may be due to whatever cultural or his-torical factors determine equilibrium selection, ratherthan those structural factors that we often believe areof first-order importance. Thus, structural factors maymatter (for a given equilibrium selection) but be dif-ficult to detect empirically because we cannot observewhich equilibrium a society is playing. Moreover, fromthe perspective of policy making, this implies that, eventhough the data are not well explained by structuralvariation, it may be that, within a given society (playingits particular equilibrium), changing key structural fac-tors would reduce political violence or the likelihoodof violent regime change.

Given this argument, how might we proceed to studythe root causes of violence and regime change empiri-cally? The model offers two modest suggestions.

First, if one believes that countries are unlikely toswitch equilibria, then one could study the effects ofboth vanguards and of structural variation within acountry. [For instance, Dube and Vargas (2009) studythe effects of economic variation on violent mobiliza-tion in Colombia, exploiting within-country geographicvariation in economic shocks.] On this logic, Geddes(1990) specific finding that some countries in her sam-ple did have uprisings at some point but that inter-national pressure did not seem to be a correlate is amore compelling piece of evidence than the generalcritique.

Second, the behavior of actors within the model canserve as indicators of equilibrium selection for the em-pirical researcher. For instance, my model predicts thata vanguard will only be active in the event that thepopulation is playing the strategy sL. Although this pre-diction may be a bit stark, given the stylized nature ofthe model, it does suggest a strategy for addressing theempirical challenges associated with theoretical mod-els yielding multiple equilibria, by using the predictionsof the theoretical models themselves.

In particular, suppose there are two players in amodel, A and B. B has two strategies consistent withequilibrium, each with different comparative statics.An empirical researcher (or policymaker) needs toknow which strategy B is following in order to knowthe empirical predictions (or policy implications) ofthe model. (This is the case in my model, where thepopulation has two equilibrium strategies.) Unfortu-nately, B’s strategy choice may not be observable. Sup-pose the theoretical model predicts that player A be-haves differently depending on B’s selection (as doesthe vanguard in my model, depending on the popula-tion’s strategy), and A’s behavior is observable. Thenan empirical researcher can determine the equilibriumselection, and proceed to study the fit of B’s behav-ior to the theoretical model, by using A’s observedbehavior to inform the researcher about B’s strategy.

457


Put differently, the theoretical model allows the re-searcher to exploit the “expertise” of player A (whichcomes from being an actual player in the game) to learnwhat the theoretical model predicts about the behaviorof player B.

Vanguards, Selection Effects,and Focal Points

As discussed at the beginning of this article, manyobservers and theorists of revolution argue that theemergence of vanguards is a critical factor that dif-ferentiates “structurally ripe” societies that do or donot experience mass political violence. [See Goldstone(2001) for a summary of this and many other issues re-lated to revolution.] The model, however, suggests thatthe fact that the level of vanguard activity is a predictorof a society having a high risk of revolution should notnecessarily be interpreted as evidence that vanguardshelp cause revolutions. In particular, the model predictsthat in equilibrium there will be selection effects—–evencontrolling for all relevant structural factors, vanguardswill be active in societies that would have been morelikely to have successful regime change, even without avanguard. This is true because violence is only useful tothe vanguard if the population’s mobilization decisionis responsive to the level of vanguard violence. And thisis only the case if the population uses the strategy sL

(see lemma 7). Hence, the fact that the presence of anactive revolutionary vanguard appears to empiricallydistinguish societies that do and do not experience vio-lent regime change (all else equal) may not constituteevidence for the causal efficacy of vanguards in anysimple way.

This point is particularly striking when one considersits implications for focal point arguments (Hardin 1996;Schelling 1960). Such arguments suggest that there willbe a correlation between vanguard violence and revo-lution because vanguard activity somehow coordinatespeople on believing others will play a participatoryequilibrium, thereby causing the revolution.

In the context of my model, a focal point argumentwould go as follows. Consider two levels of violence, v′

and v′′, such that �L(v − t∗) exists for both. For one levelof violence, the population uses the cutoff rule �H(v′ −t∗) = ∞ and for the other (perhaps higher) level ofvanguard violence, the population uses the cutoff rule�L(v′′ − t∗). That is, by changing the level of violence,the vanguard is able to persuade population membersto change their fundamental conjectures about eachother’s behavior.

Recall that I explicitly rule out such focal pointsthrough my selection criteria.7 Instead, vanguard vi-olence plays a more modest role. If the populationplays sL, then increased vanguard violence can in-crementally increase mobilization by shifting the cut

7 In particular, from proposition 1, the population has only twostrategies consistent with cutoff equilibrium: sL and s H. In sL, thepopulation uses �L(v − t∗) whenever v − t∗ is such that a finite cutoffrule exists. In s H, the population always uses �H(v − t∗) = ∞.

point that the population uses from �L(v′ − t∗) to�L(v′′ − t∗) (Figure 3). But it cannot convince the pop-ulation to shift from no mobilization to mobilizationby convincing population members that their fellowcitizens have qualitatively changed the strategies theyfollow.

This suggests a challenge to the focal point view,to the extent that that view hinges on the empiricalclaim that vanguard activity is correlated with societyplaying an equilibrium favorable to mobilization (sug-gesting that vanguard violence is what coordinates soci-ety on this equilibrium). Precisely the same correlationemerges in my model. The vanguard engages in vio-lence if and only if the population plays the equilibriumstrategy favorable to mobilization, sL. Yet, the van-guard is not creating a focal point. Rather, it only findsengaging in violence profitable if it believes society hasalready coordinated on the strategy sL. Hence, an em-pirical correlation between vanguard violence and so-ciety coordinating on a high mobilization equilibriumdoes not constitute evidence for vanguards creatingfocal points. Indeed, in my model, such a correlationexists because focal points “create” vanguards—–thevanguard only invests in violence if it anticipates thatsociety has coordinated on a participatory equilibrium.

The Efficacy of the Revolutionary Vanguard

The previous subsection points out that any correlationbetween the vanguard activity and the probability ofregime change could be a selection effect. This raisesthe question: is the vanguard able to increase mobiliza-tion and make regime change more likely?

When the population uses the strategy sL, at leastfrom an ex post perspective, the answer is “yes.” Ahigher level of (unexpected) vanguard violence in-creases mobilization and the likelihood of successfulrevolution in two ways: it increases the probability ofa finite cutoff rule consistent with equilibrium existing(see lemma 5) and, conditional on one existing, it in-creases mobilization by decreasing the cutoff rule (seelemma 6).

These two effects are illustrated in Figure 5. Figure 5shows what happens to the level of mobilization fora fixed � as � increases (thereby increasing the levelof unexpected violence) when the population plays sL.For low levels of unexpected violence, there is no finitecutoff rule consistent with equilibrium, and mobiliza-tion is zero. When the level of unexpected violence be-comes high enough, a finite cutoff rule consistent withequilibrium exists, leading to a discontinuous jump inparticipation. The level of mobilization then increasescontinuously as vanguard violence increases. Thus, themodel is consistent with cases, such as Russia or Al-geria, where successful vanguards seem to ignite orfurther inflame mass uprisings against a government.

The previous discussion focuses on the ex post ef-fects of higher levels of violence. To assess the ex-pected efficacy of vanguards, we must take an exante view. From this perspective, the vanguard is notefficacious in the following sense. Ex post, higher

458


FIGURE 5. Equilibrium mobilization as a function of unexpected vanguard violence (i.e., ofchanges in � for a fixed �). The simulations assume m = 1, � = 1, T = 0.3, � = 0.5, �2

ε = �2� = �2

� = 1

−0.8 0.0 0.8 1.6 2.4 3.2 3.8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

unexpected violence (η)

N

Mobilization given sL

η

than expected levels of vanguard violence increasemobilization and lower than expected levels ofvanguard violence decrease mobilization. Ex ante,these two possibilities are equally likely. (Put dif-ferently, on the equilibrium path, the expectedvalue of v − t∗ is �, the true level of antigovern-ment sentiment.) So, in expectation, the vanguardis equally likely to increase mobilization (by gener-ating greater-than-expected violence) or to decreasemobilization (by generating lower-than-expected vio-lence). The vanguard nonetheless exerts effort because,if it did not do so the population would likely observelower-than-expected levels of violence and concludethat the level of antigovernment sentiment is lowerthan it is in reality. However, if the vanguard couldcommit to low effort, the population would then up-date based on that commitment. As a result, the exante probability of a successful revolution would beunchanged.

Although, within the current model, the vanguardwould like to (but cannot) commit to devoting min-imal effort toward violence, it would be an overin-terpretation to conclude that this means vanguardsshould never emerge. The informational mechanismstudied here constitutes only one role for vanguardsin organizing violence. The vanguard may be more ef-ficacious, even from an ex ante perspective, at othertasks such as providing selective incentives (Lichbach1995; Popkin 1979; Tullock 1971, 1974), constructinga highly committed revolutionary movement (Berman2009; Berman and Laitin 2008; Migdal 1974; Tilly 1975),provoking the government (Bueno de Mesquita andDickson 2007; Siqueira and Sandler 2007), drawinginternational attention, or degrading government ca-pacity (a possibility explored later in the article). Onceone of these other mechanisms motivates a vanguardto form, the informational mechanism studied here willcome into play.

Vanguards and Spontaneous Revolution

Scholars of revolution are particularly interested in un-derstanding how vanguards can account for the seem-ingly spontaneous nature of many revolutions (Hardin1996; Kuran 1989, 1991; Lichbach 1995; Opp, Voss, andGern 1995). The model presented here is consistentwith vanguards sparking such spontaneous revolutions,through a mechanism that weds informational andspark-and-tinder models. To see this, consider Figure4. A population that plays the strategy sL is primed forrevolution. Nonetheless, if the realization of (�, �) liesto the southwest of �, then there will be no mobilizationat all. If the realization of these two random variablesjust crosses the dashed line in Figure 4, then there willsuddenly be mass mobilization. (This mobilization mayor may not successfully overturn the regime, dependingon the relationship to �∗.) Recall that an increase invanguard violence relative to expectations shifts thesecurves to the southwest. Thus, a small increase in thelevel of vanguard violence can, without changing thefocal equilibrium, spontaneously move a populationfrom no mobilization to mass mobilization and even tosuccessful regime change. This effect also can be seenin Figure 5, where, at one critical point, an increase inunexpected vanguard violence leads to a discontinuousjump in mobilization.

EXTENSIONS

In this section, I briefly consider extensions to showhow the model can be enriched without undoing theprevious key findings.

Efficacious Vanguard

The fact that the vanguard is not efficacious from an exante perspective might be troubling because it suggests

459


that vanguard organizations should not emerge. Here Iaddress this concern by considering an extension wherethe vanguard is ex ante efficacious and show that noneof my key results are altered.

Suppose vanguard violence degrades the govern-ment’s capacity to withstand an uprising, in addi-tion to communicating information. In particular,suppose T is a decreasing, differentiable function ofv. An argument identical to that surrounding equa-tion (3) shows that, for a given mapping and a fixedv and t∗, the revolution will succeed if � is greater than�∗(�(v − t∗, T(v)), T(v)) = �(v − t∗, T(v)) − �−1(1 −T(v))�� . Given this, an equilibrium cutoff rule mustsatisfy:

[1−�

(�∗(�(v − t∗, T(v)), T(v)) − m�(v−t∗,T(v)),v−t∗

�2

)]

× ��(v − t∗, T(v)) = k.

The left-hand side of this equality behaves essentia-lly identically to the function IB from the main analysis.As such, the qualitative structure of equilibrium in therevolution stage is unaffected.

How do the vanguard’s incentives change? Becausevanguard violence now directly degrades governmentcapacity, increased vanguard violence (even if not un-expected) makes participation more attractive becauseit makes revolutionary success more likely. So, violenceis more valuable to the vanguard.

Slightly more formally, when the population playsthe strategy analogous to sL, increases in vanguard vi-olence now increase the probability of regime changein two ways that they did not before. First, an increasein vanguard violence heightens the probability of therebeing positive participation (i.e., lowers �) by decreas-ing T (see lemma 15 in the Appendix). Second, condi-tional on � ≥ �, an increase in vanguard violence nowdecreases the cutoff rule the population uses, �L, fortwo reasons: (1) the informational effect discussed inthe main analysis and formalized in lemma 6 and (2)because an increase in vanguard violence decreases T,which decreases �L. (See lemma 14 in the Appendix.)

The preceding argument has two implications. First,if vanguard violence can directly diminish governmentcapacity, the vanguard has an additional marginal ben-efit from violence. Second, and more important, van-guard violence degrades government capacity indepen-dent of the population’s expectations. That is, T is afunction of v, not of v − t∗. As such, the vanguard ben-efits from this aspect of violence (when the populationplays sL) even from an ex ante perspective. Hence, theability to degrade government capacity could serve asan ex ante rationale for forming a vanguard organiza-tion. Once it is created, the informational incentivesmodeled in the main analysis come into play and allthe key results—–the importance of structural factors inonly one of the equilibrium, selection effects, the pos-sibility of spontaneous revolutions—–continue to hold.

Counterterrorism

The regime was left unmodeled in the main analysis.However, governments are obviously a key strategicplayer here. As such, I consider an extension in whichthe regime can invest resources in trying to preventvanguard violence.

Suppose the game is played just as in the main anal-ysis, except that at the same time that the vanguardchooses t , the regime invests in counterinsurgency mea-sures r ∈ [r ,∞) [at cost cR(r)], with r ≥ 0. The totallevel of vanguard violence is now v = � + � + t − r .

Population behavior in the revolution stage will bejust as in the original game, except that they will nowcondition their beliefs on v − t∗ + r∗. That is, popu-lation members will still learn about antigovernmentsentiment from vanguard violence, but they will nowhave to filter out both vanguard and regime effort toobtain an unbiased signal.

The vanguard’s objective is

maxt

∫ �(t−t∗−r+r∗)

−∞

∫ ∞

�(�+t−t∗−r+r∗)

1��

�

(� − m

��

)

× 1��

�

(�

��

)d� d�

+∫ ∞

�(t−t∗−r+r∗)

∫ ∞

�∗(�+t−t∗−r+r∗)

1��

�

(� − m

��

)

× 1��

�

(�

��

)d� d� − cE(t).

Assuming the regime wants to minimize the probabilityof regime change, its objective is

maxr

1 −∫ �(t−t∗−r+r∗)

−∞

∫ ∞

�(�+t−t∗−r+r∗)

1��

�

(� − m

��

)

× 1��

�

(�

��

)d� d�

−∫ ∞

�(t−t∗−r+r∗)

∫ ∞

�∗(�+t−t∗−r+r∗)

1��

�

(� − m

��

)

× 1��

�

(�

��

)d� d� − cR(r).

In equilibrium, population members’ beliefs aboutthe vanguard’s choice of t and the regime’s choice ofr are correct. Hence, in a pure strategy equilibrium,vanguard behavior is described by the same first-ordercondition as in the main analysis. Regime counterter-rorism is described by an analogous first-order condi-tion. (An argument identical to that in lemma 10 showsthat we can find a sufficiently convex cost function forthe regime.) In any such equilibrium, behavior by thevanguard and the population is just as described in themain analysis, and the key results continue to hold.

This extension, of course, only considers one aspectof counterrevolutionary policy—–minimizing vanguardviolence. In reality, such policies are more compli-cated. For instance, governments must consider the

460


possibility that repressive measures will backfire, in-creasing, rather than decreasing, mobilization. Suchconcerns, although of clear interest, are outside thelimited scope of this extension. They are discussed indetail elsewhere. [For formal models, see, for instance,Bueno de Mesquita (2005), Bueno de Mesquita andDickson (2007), Lichbach (1987), Rosendorff and San-dler (2004), and Siqueira and Sandler (2007).]

CONCLUSION

I study how a vanguard may use violence to mobilizemembers of a mass public by convincing them thatantigovernment sentiment is high. The model is consis-tent with the idea that violence by vanguards can affectmobilization and sometimes even spark spontaneousuprisings. However, the model also suggests that themicrofoundations of revolution, in general, and therole of vanguards, in particular, are complicated andsubtle.

The model has an equilibrium where successfulregime change is possible and another where it is not.Structural factors affect the likelihood of revolution inthe former equilibrium but not the latter. These find-ings imply that it may be difficult to empirically identifyroot causes of political violence or instability. More-over, it suggests that the standard empirical critique ofstructural accounts—–that many more societies possessthe putative structural causes of revolution than actu-ally experience revolution—–may have weaker logicalfoundations than the current literature acknowledges.The model also suggests some ways forward in termsof empirical assessment.

The model also predicts the presence of selectioneffects. Revolutionary vanguards only emerge in so-cieties that are already prone to regime change. Thus,even if vanguard violence is ineffective, a society with amore active vanguard will be more likely to have a suc-cessful revolution (all else equal) than a society withoutone. These selection effects complicate attempts to em-pirically validate various theories about the role of van-guards in causing revolution. Moreover, they suggestthat one should be cautious when interpreting corre-lations between vanguard activity and mobilization asevidence that vanguards are a cause of revolution—–beit through creating focal points, providing selective in-centives, or communicating information.

APPENDIX

Notation

Throughout the Appendix, I make use of the following nota-tions not introduced in the text:� = (1−(1 − � )�)

�2

� = (1 − � )(1 − �)m+ ��−1(1 − T)+� (v − t∗)�2

� f (�, v − t∗) ≡ � − � x∗(v − t∗) ≡ arg max� IB(�, v − t∗)

Rremark 1. IB(�, v − t∗) = [1 − �( f (�, v − t∗))]��.

Proof. Follows immediately from substituting for m�,v−t∗

and �∗(�(v − t∗)) in IB(�, v − t∗). �

Additional Results

I also make use of three additional results.

LEMMA 11. For � > 0, IB(�, v − t∗) is increasing in � if andonly if 1 − �( f (�,v − t∗))

��( f (�,v − t∗))≥ .

Proof. Follows directly from differentiating and rear-ranging. �

LEMMA 12. IB(x∗(v − t∗), v − t∗) is increasing in v − t∗.

Proof. Let IB2 be the partial derivative of IB withrespect to its second argument. By the envelope the-

orem, dIBd(v−t∗) (x∗(v − t∗), v − t∗) = IB2(x∗(v − t∗), v − t∗) =

�( f (x∗(v − t∗), v − t∗)) � � x∗(v−t∗)�2

> 0. �

LEMMA 13. IB1(�L(v − t∗), v − t∗) > 0.

Proof. By lemma 4, IB is single peaked in its first argu-ment. The combination of single peakedness and the factthat IB(�, v − t∗) = k at all equilibria with finite �, impliesthat �L(v − t∗) < x∗(v − t∗) < �M(v − t∗). Because x∗(v − t∗)is the maximum and IB is single peaked, IB is increasing inits first argument to the left of x∗(v − t∗). �

Proofs of Numbered Results

Proof of Lemma 1. Suppose players play a strategy profilewith ai = 0 for all �i . The probability of victory is 0. If anindividual were to consider deviating to participation, thenthe probability of victory would still be zero, because all in-dividuals are measure 0. Thus, the payoff to the deviation is−k, whereas the payoff to not participating is 0 > −k. �Proof of Lemma 2. Differentiating, we have IB1(�i ,

�(v − t∗), v − t∗) = �( �∗(�(v − t∗))−mi,v − t∗�2

) 1�2

∂mi,v − t∗∂�i

��i + (1 −�( �∗(�(v − t∗))−mi,v − t∗

�2))�. Now the result follows directly from

the fact that mi,v − t∗ is increasing in �i . �Proof of Lemma 3. The argument in the text demon-strates that, given v − t∗, a necessary condition for a cut-off rule �(v − t∗) being part of an equilibrium is IB(�(v −t∗), v − t∗) = k. Thus, if max� IB(�, v − t∗) < k, then thestrategy must assign the choice ai = 0. Furthermore, thesecond equilibrium selection criterion requires that ifthe strategy uses a finite cut-off rule for any v − t∗ satisfyingmax� IB(�, v − t∗) ≥ k, then it must use a finite cutoff rulefor all such v − t∗. Hence, if s ever chooses a finite cutoffrule, then it must do so whenever IB(x∗(v − t∗), v − t∗) ≥ kas in the strategy in the lemma. The definition of equilibriumfurther requires that the cutoff rule be continuous in v − t∗,except at the v − t∗ satisfying max� IB(�, v − t∗) = k.

All that remains is to show sufficiency of IB(�(v − t∗), v −t∗) = k. For sufficiency, consider a profile in the revolutionstage where all players employ such a cutoff rule. Fix v − t∗

such that a finite cutoff rule consistent with equilibriumexists, and consider a player with type �i < �(v − t∗). Bylemma 2, IB(�i , �(v − t∗), v − t∗) < k, so there is no prof-itable deviation to participating. Now consider a player withtype �i > �(v − t∗). By lemma 2, IB(�i , �(v − t∗), v − t∗) > k,

461


so there is no profitable deviation to not participating. Byconstruction, a player of type �(v − t∗) is indifferent. �

Proof of Lemma 4.

1. Because k > 0, at any � that satisfies equation (5),IB(�, v − t∗) must be positive. Because 1 − �( f (�, v −t∗)) > 0 for all �, this means that for IB(�, v − t∗) =[(1 − f (�, v − t∗))]�� to be positive, � must be positive.

2. From lemma 11, IB(�, v − t∗) is increasing in � if andonly if 1−�( f (�,v−t∗))

��( f (�,v−t∗))is greater than the finite positive con-

stant . Furthermore, f (�, v − t∗) is clearly increasing in�. Now, notice that because the normal density satisfies themonotone hazard rate property, 1−�( f (�,v − t∗))

�( f (�,v − t∗))is decreasing

monotonically in �, which implies that 1−�( f (�,v − t∗))��( f (�,v − t∗))

is de-

creasing monotonically in �, for � > 0. Thus, to prove thatIB(�, v − t∗) = [1 − �( f (�, v − t∗))]�� is single peaked in� for � > 0, it is sufficient to show that there exists a � suf-ficiently small that 1−�( f (�,v − t∗))

��( f (�,v − t∗))> and a � sufficiently

large that 1−�( f (�,v − t∗))��( f (�,v − t∗))

< . If this is true, then the fact

that IB(�, v − t∗) is continuous and its slope is strictlydecreasing will imply single peakedness.I start by showing that lim�→0

1−�( f (�,v − t∗))��( f (�,v − t∗))

= ∞. To see

this, note that the limit of the numerator as � goes to 0 issome positive finite number and the limit of the denomina-tor is zero. Thus, for � > 0 sufficiently small, IB(�, v − t∗)is increasing.Next I show that there is a sufficiently large � thatIB(�, v − t∗) is decreasing. To see this, first note thatIB(1, v − t∗) is strictly positive. Next the following chainof inequalities shows that lim�→∞

1−�( f (�,v − t∗))��( f (x�,v − t∗))

= 0:

lim�→∞

1 − �( f (�, v − t∗))

��( f (�, v − t∗))

= lim�→∞

−�( f (�, v − t∗)) f1(�, v − t∗)

�( f (�, v − t∗)) + ��′( f (�, v − t∗)) f1(�, v − t∗)

= lim�→∞

× −�( f (�, v − t∗)) f1(�, v − t∗)

�( f (�, v − t∗)) − � f (�, v − t∗)�( f (�, v − t∗)) f1(�, v − t∗)

= lim�→∞

f1(�, v − t∗)

� f (�, v − t∗) f1(�, v − t∗) − 1

= 0,

where f1(�, v − t∗) = is the partial derivative of f withrespect to its first argument (�). The first equality is dueto l’Hopital’s rule, whereas the second equality uses thefact that �′(x) = −x�(x), the third equality is algebra, andthe fourth equality follows from the fact that f (�, v − t∗)is increasing in � and f1(�, v − t∗) = is constant in �.These equalities show that somewhere between � = 1 andthe limit, as � goes to infinity, IB(�, v − t∗) is decreasing.The fact that the derivative of IB(�, v − t∗) is strictly de-creasing for positive � implies that whenever IB(�, v − t∗)first slopes down, it slopes down forever after, establishingthat there is a single peak for positive �.

3.

lim�→∞

(1 − �( f (�, v − t∗)))�

= lim�→∞

(1 − �( f (�, v − t∗)))1�

= lim�→∞

�( f (�, v − t∗)) f1(�, v − t∗)�2

= lim�→∞

f1(�, v − t∗)�2

ef (�,v − t∗)2

2

√2�

= lim�→∞

f1(�, v − t∗)2�

f (�, v − t∗) f1(�, v − t∗)ef (�,v − t∗)2

2√

2�

= lim�→∞

2�

(� − )e(�−)2

2√

2�

= lim�→∞

2

2e(�−)2

2√

2� + (� − )22e(�−)2

2√

2�

= 0,

where, in order, the equalities follow from (1) simple re-arrangement, (2) l’Hospital’s rule, (3) the definition of thePDF of the standard normal, (4) l’Hospital’s rule, (5) thedefinition f (�, v − t∗) = � − , (6) l’Hospital’s rule, and(7) the numerator is a positive constant and the denomi-nator goes to infinity.

4. This point is immediate. �Proof of Proposition 1. Lemma 1 shows that s∞ is consis-tent with cutoff equilibrium.

From lemma 3, we have that a strategy, s, that includes afinite cutoff rule is consistent with cutoff equilibrium if andonly if the cutoff rule satisfies equation (5) and

s(�i , v − t∗)

={

1 if max� IB(�, v − t∗) ≥ k and �i ≥ �(v − t∗)0 else.

Because s M and sL take this form and, by definition, �L and �M

satisfy equation (5), both s M and sL are strategies consistentwith cutoff equilibrium.

To see that there are no other cut-off rules consistent withcutoff equilibrium, it suffices to show that for any v − t∗

such that IB(x∗(v − t∗), v − t∗) ≥ k, if IB(�, v − t∗) = k, then� ∈ {�L(v − t∗), �M(v − t∗)}. By lemma 4, item 1, it sufficesto consider positive cutoff rules. Now, recall from lemma 4that IB(0, v − t∗) = 0 and the limit of IB(�, v − t∗) as � goesto infinity is also 0. Because IB is continuous and singlepeaked in its first argument, with peak IB(x∗(v − t∗), v − t∗),it follows that, for � ∈ (0,∞), IB(�, v − t∗) takes all values in(0, IB(x∗(v − t∗), v − t∗)) exactly twice. Because IB(x∗(v −t∗), v − t∗) > k, this implies that IB(�, v − t∗) takes thevalue k exactly twice for � ∈ (0, ∞), at �L(v − t∗) and�M(v − t∗). �Proof of Lemma 5. From proposition 1, an equilibriumfinite cutoff rule exists if and only if IB(x∗(v − t∗), v − t∗) ≥k. From lemma 12, IB(x∗(v − t∗), v − t∗) is strictly increasingin v. Thus, if there is a �(� + t − t∗), it must satisfy

IB(x∗(� + � + t − t∗), � + � + t − t∗) = k. (8)

462


From the definition of IB, it is immediate thatlimv→−∞ IB(x∗(v − t∗), v − t∗) = 0. Because v = � + � + t ,and � and � have full support on the real line, v has fullsupport on the real line. Thus, for sufficiently small realiza-tions of � + � a finite cutoff rule consistent with equilibriumdoes not exist. Moreover, because IB(x∗(v − t∗), v − t∗) isstrictly increasing (by lemma 12) and continuous in v, andbecause by assumption 1. there exists some v such thatIB(x∗(v − t∗), v − t∗) ≥ k, then there is some v where theinequality holds with equality and for any larger v it continuesto hold strictly, which establishes that a �(� + t − t∗) exists.

To see that �(� + t − t∗) is decreasing, notice that �, �, andt are substitutes in v and do not enter IB anywhere else. �Proof of Lemma 6. Implicitly differentiating equation (5),we have that

∂ �L(v − t∗)∂(v − t∗)

=−�(�L(v − t∗) − ) �

�2��L(v − t∗)2

IB1(�L(v − t∗), v − t∗).

The numerator is clearly negative. By lemma 13, the denom-inator is positive. �Proof of Lemma 7. If the population plays the equilibriumwith no participation, then the payoff to any level of violenceis simply −cE(t) and the optimal choice is t∗ = t . �Proof of Lemma 8.

1. This point follows from the argument in the text.2. �(� + t − t∗) = �(� + t − t∗) if and only if x∗(�(� + t −

t∗) + � + t − t∗) − �−1(1 − T)�� ≥ �(� + t − t∗). Fromlemma 5, �(� + t − t∗) is decreasing. This implies that theright-hand side of the previous inequality is decreasing in�. Now consider the left-hand side.

∂x∗(�(� + t − t∗) + � + t − t∗)∂�

= ∂x∗(�(� + t − t∗) + � + t − t∗)∂v

(∂ �

∂�+ 1

).

Differentiating equation (8), we have

∂ �

∂�= − IB1

∂x∗∂v

+ IB2

IB1∂x∗∂v

+ IB2= −1.

Substituting back in yields ∂x∗(�(�+ t − t∗) + �+ t − t∗)∂�

= 0, sothe left-hand side of the inequality is constant. Thus, theinequality holds for � sufficiently large. Label the minimal� as �(t − t∗), given by x∗(�(� + t − t∗) + � + t − t∗) −�−1(1 − T)�� = �(� + t − t∗).

3. If � = �, then it is decreasing ( ∂ �∂�

= −1.) Suppose instead

that � = �∗. Implicitly differentiating equation (6) yields:

∂�∗L

∂�=

∂ �L∂�

1 − ∂ �L∂�

.

It is immediate from equation (5) that ∂ �L∂�

= ∂ �L∂�

. Anargument identical to that in the proof of lemma 13shows that both derivatives are negative. Taken to-gether, this implies that the numerator in the previousdisplayed equation is negative and the denominator ispositive. �

Proof of Lemma 9. There cannot be a corner solution and tbecause cE(t) = c′(t) = 0. If there is an interior pure strategybest response, then it must satisfy the first-order conditions.Differentiating the objective function yields the followingfirst-order condition:

1��

[−

∫ �(t∗−t∗)

−∞�

(�((� + t∗ − t∗) − m

��

)

× ∂ �(� + t∗ − t∗)∂t

�

(�

��

)d�

+(∫ ∞

�(�(t∗−t∗)+t∗−t∗)�

(� − m

��

)�

(�(t∗ − t∗)

��

)d�

)

× ∂�(t∗ − t∗)∂t

−∫ ∞

�(t∗−t∗)�

(�∗(� + t∗ − t∗) − m

��

)

× ∂�∗(� + t∗ − t∗)∂t

�

(�

��

)d�

−(∫ ∞

�∗(�(t∗−t∗)+t∗−t∗)�

(� − m

��

)�

(�(t∗ − t∗)

��

)d�

)

× ∂�(t∗ − t∗)∂t

]− c′

E(t∗) = 0.

I use the following facts:

1. Implicitly differentiating equation (8) shows that

∂ �

∂t= IB1

∂x∗∂t + IB2

IB1∂x∗∂t + IB2

= −1.

2. �( �(t∗−t∗)��

) does not depend on the � over which we areintegrating in the first and fourth terms of the first-ordercondition.

3. In equilibrium, beliefs about t are correct.

Given these facts, we can rewrite the first-order conditionas follows:

1��

[∫ �(0)

−∞�

(�(�) − m

��

)�

(�

��

)d�

−∫ ∞

�(0)�

(�∗(�) − m

��

)∂�∗(�)

∂t�

(�

��

)d�

+ �

(�(0)��

)∂�(0)

∂t

(�

(�∗(�(0))

��

)

− �

(�(�(0))

��

)) ]= c′

E(t∗).

By the definition of �, �∗(�(0)) = �(�(0)), so the third termon the left-hand side is equal to 0. Thus, the first-order con-dition reduces to

1��

[∫ �(0)

−∞�

(�(�) − m

��

)�

(�

��

)d�

−∫ ∞

�(0)�

(�∗(�) − m

��

)∂�∗(�)

∂t�

(�

��

)d�

]= c′

E(t∗),

(9)

which is equivalent to the statement in the lemma.

463


To see that there can only be one such cutoff equilibrium,notice that the left-hand side of the equilibrium condition inequation (9) is finite and constant in t , while the right-handside is strictly increasing. Thus, there is only one t∗ satisfyingthis condition. �

Proof of Lemma 10. Differentiating the vanguard’s objec-tive function twice, the global second-order condition can bewritten:

−∫ �(t−t∗)

−∞�′

(�(� + t − t∗) − m

��

)1��

�

(�

��

)d�

+ �

(�(�(t − t∗) + t − t∗) − m

��

)�

(�(t − t∗)

��

)∂�(t − t∗)

∂t

×(

∂�(t−t∗)∂t + 1 + ��

��

)+

(�′

(�(t − t∗)

��

)1��

(∂�(t − t∗)

∂t

)2

+ �

(�(t − t∗)

��

)∂2�(t − t∗)

∂t

)(�

(�∗(�(t − t∗)+ t − t∗)−m

��

)

− �

(�(�(t − t∗) + t − t∗) − m

��

))

+ �

(�(t − t∗)

��

)∂�(t − t∗)

∂t�

(�∗(�(t − t∗)+ t − t∗)−m

��

)1��

× ∂�∗(�(t − t∗) + t − t∗)∂�

(∂�(t − t∗)

∂t+ 1

)

−∫ ∞

�(t−t∗)

[�′

(�∗(�t − t∗) − m

��

)1��

∂�∗(� + t − t∗)∂t

+ �

(�∗(� + t − t∗) − m

��

)∂2�∗(� + t − t∗)

∂t2

]�

(�

��

)d�

+ �

(�∗(�(t − t∗)) − m

��

)�

(�(t − t∗)

��

)

× ∂�∗(�(t − t∗))∂�

∂�(t − t∗)∂t

≤ ��c′′E(t),

for all t ∈ [t,∞). It is straightforward that for any finite t ,the left-hand side is finite. Thus, it is feasible to choose a costfunctions such that ��c′′

E(t) is greater than the left-hand sidefor all t . For any such cost function, the global second-orderconditions are satisfied. �

Proof of Proposition 2. Follows from proposition 1,lemma 7, and lemma 9. �

Proof of Proposition 3. I make use of the following lem-mata.

LEMMA 14. �L(� + �) is increasing in T, increasing in k, anddecreasing in � .

Proof. Implicitly differentiating equation (5), we have

∂ �L

∂T=

�(�L − )��L��2

(�−1)′(1 − T)

IB1(�L, v − t∗)> 0,

where the inequality follows from the facts that the

numerator is positive and the denominator is positive bylemma 13.

∂ �L

∂k= 1

IB1(�L, v − t∗)> 0,

where the inequality follows from the fact that the denomi-nator is positive by lemma 13.

∂ �L

∂�= −(1 − �(�L − ))�L

IB1(�L, v − t∗)< 0,

where the inequality follows from the fact that the numeratoris negative and the denominator is positive by lemma 13. �

LEMMA 15. �(�) is increasing in T, increasing in k, and de-creasing in �

Proof. Implicitly differentiating equation (8) and using thefact that because x∗(� + �) is a maximizer of IB with respectto x, we have IB1(x∗(� + �), v − t∗) = 0, yields

∂ �

∂T=

�(x∗(� + �) − ) ��2

(�−1)′(1 − T)� x∗(� + �)

�(x∗(� + �) − ) ��2

� x∗(� + �)> 0,

where the inequality follows from the fact that x∗ > 0 and(�−1)′ is positive because � is strictly increasing.

∂ �

∂k= 1

�(x∗(� + �) − ) ��2

� x∗(� + �)> 0.

∂ �

∂�= − (1 − �(x∗(� + �) − ))x∗(� + �)

�(x∗(� + �) − ) ��2

� x∗(� + �)< 0.

�

The number of people who mobilize is

N =∫ ∞

−∞

∫ ∞

�(�)

(1 − �

(�L(� + �) − �

��

))

× 1��

�

(� − m

��

)1��

�

(�

��

)d �d �.

Differentiating we have

∂ N∂T

=∫ ∞

−∞

∫ ∞

�(�)− �

(�L(� + �) − �

��

)∂ �L(� + �)

∂T

× 1��

1��

�

(� − m

��

)1��

�

(�

��

)d �d �

−∫ ∞

�(�)

(1 − �

(�L(�(�) + �) − �(�)

��

))

× 1��

�

(�(�) − m

��

)∂ �(�)∂T

1��

�

(�

��

)d � < 0,

where the inequality follows from the fact that ∂ �L(� + �)∂T > 0

and ∂ �(�)∂T > 0 from lemmata 14 and 15, respectively. Thus, as T

increases, mobilization decreases and the number of people

464


needed for victory increases, so the probability of victory alsodecreases.

∂ N∂k

=∫ ∞

−∞

∫ ∞

�(�)− �

(�L(� + �) − �

��

)∂ �L(� + �)

∂k

× 1��

1��

�

(� − m

��

)1��

�

(�

��

)d �d �

−∫ ∞

�(�)

(1 − �

(�L(�(�) + �) − �(�)

��

))

× 1��

�

(�(�) − m

��

)∂ �(�)

∂k1��

�

(�

��

)d � < 0,

where the inequality follows from the fact that ∂ �L(�+�)∂k > 0

and ∂ �(�)∂k > 0 from lemmata 14 and 15, respectively. Thus, as

k increases, mobilization decreases and T stays constant, sothe probability of victory also decreases.

∂ N∂�

=∫ ∞

−∞

∫ ∞

�(�)− �

(�L(� + �) − �

��

)∂ �L(� + �)

∂�

× 1��

1��

�

(� − m

��

)1��

�

(�

��

)d �d �

−∫ ∞

�(�)

(1 − �

(�L(�(�) + �) − �(�)

��

))

× 1��

�

(�(�) − m

��

)∂ �(�)∂�

1��

�

(�

��

)d � > 0,

where the inequality follows from the fact that ∂ �L(�+�)∂�

< 0

and ∂ �(�)∂�

< 0 from lemmata 14 and 15, respectively. Thus, as� increases, mobilization increases and T stays constant, sothe probability of victory also increases. �

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