Imagine a citizen with antiregime feelings who is considering becoming involved in a revolutionary movement. He or she only wants to mobilize if he or she believes the movement is sufficiently likely to succeed. Success depends on many people mobilizing. So his or her assessment of the likelihood of success depends on his or her beliefs about how many of his or her fellow citizens will mobilize.

Because he or she dislikes the government, he or she suspects that many of his or her fellow citizens dislike the government as well, although he or she faces uncertainty about his or her fellow citizens' views. The more extreme his or her own antigovernment feelings, the more confident he or she is that his or her fellow citizens also oppose the regime. Hence, more extreme citizens are doubly more willing to join the revolutionary movement. They care more about replacing the regime. And they are more confident that others are ready to join, so they believe the movement is more likely to succeed.

A revolutionary vanguard wants to persuade citizens to mobilize. To do so, it must convince our citizen (and others like him or her) that the probability of success is sufficiently high. To do this, it must convince him or her that his or her fellow citizens are in fact quite antigovernment.

The tool that the vanguard has at its disposal is insurgent violence, such as guerilla or terrorist attacks. These attacks may be persuasive to our citizen because she believes that the vanguard cannot produce a high level of violence without the support of the surrounding population. Thus, high levels of vanguard violence suggest, to our citizen, a high level of antigovernment sentiment in the population as a whole.

Suppose that our citizen observes a series of unexpectedly successful vanguard attacks. These attacks convince him or her that his or her fellow citizens are quite hostile to the regime and, thus, likely to mobilize. As such, although his or her views of the regime have not changed, he or she becomes more willing to participate because he or she thinks the odds of the movement succeeding are higher. This is precisely the goal of the vanguard. (Of course, if vanguard violence had been lower than expected, the result would have been the opposite.) Thus, the vanguard has incentives to invest in violence to try to persuade citizens to participate.

I model such an informational role for vanguard violence and explore the incentives and strategic dynamics it creates. I do so within a coordination model of regime change with a stage in which a revolutionary vanguard (e.g., insurgents, terrorists, guerillas) engages in publicly observable political violence before population members decide whether to mobilize.

The mechanism I consider differs from standard accounts in several ways. First, although my model has multiple equilibria, I explicitly assume that vanguard violence cannot create focal points. Hence, to the extent that vanguard violence influences mobilization, it does so via the information it communicates, not by changing citizens' fundamental conjectures about one another.Footnote ¹ Second, the vanguard has no private information about factors related to the likely success of the revolution, such as antigovernment sentiment or regime capacity. As such, this is not a model of the vanguard signaling private information.Footnote ² Instead, the idea here is that vanguard violence is inherently informative because the existence of antigovernment sentiment is crucial in its production. The vanguard are, to paraphrase Mao, fish swimming in the sea of the people—depending on the population for material support, safe havens, information, and recruits. [See Berman, Shapiro, and Felter (Reference Berman, Shapiro and Felter2009) and Kalyvas (Reference Kalyvas1999) for discussions of the relationship between insurgents and populations.]

In addition to explicating a novel mechanism by which vanguard violence may foment revolution, the results also contribute more broadly to ongoing debates regarding the origins of revolution and political violence. Structural theories of revolution argue that revolutions are caused by constellations of structural factors—regime capacity, international pressure, grievances, the economy, and so on—that make a society ripe for revolution (Skocpol Reference Skocpol1979). Yet, as Geddes (Reference Geddes1990), DeNardo (Reference DeNardo1985), and others point out, the structural conditions often identified as root causes of revolution occur far more often than do revolutions themselves. My model, although not itself a structural account of regime change, casts some doubt on the logic of this empirical critique. The model has multiple equilibria—one in which structural factors influence expected mobilization and one in which they do not. Hence, the model is consistent with the possibility that structural factors are causes of revolution (or effect the probability of revolution) in some societies but not in others.

This fact suggests a general problem both for empirical assessments of root causes and for policy making. Much of the variation in the data may be due to whatever cultural or historical factors determine equilibrium selection, rather than those structural factors that we often believe are of first-order importance for explaining political violence and instability. Structural factors may matter (for a given equilibrium selection) but be difficult to detect empirically because we cannot observe which equilibrium a society is playing. From the perspective of policy making, this implies that, even though the data are not well explained by structural variation, it may be that, within a given society, changing key structural factors would reduce political violence or the likelihood of violent regime change. The model provides some suggestions for ways forward in empirical work, in light of the challenges posed by multiple equilibria.

The model also highlights a difficulty in assessing the efficacy of vanguards. The literature points to many mechanisms—providing selective incentives, building effective institutions, creating focal points, spreading information, and so on—by which vanguards may play a role in fomenting revolution.Footnote ³ Proponents of such arguments point to a variety of examples of vanguards engaging in violence that appears to have inspired a larger insurrection. For instance, the FLN's (National Liberation Front's) terrorist campaign helped spark the Algerian War of Independence (Kalyvas Reference Kalyvas1999). Violence by Argentine guerilla groups such as the Montoneros 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 Reference Gillespie and Crenshaw1995). Terrorist tactics and other forms of violent agitation by Russian revolutionaries helped set the stage for the “spontaneous” uprisings of 1905 and 1917 (DeNardo Reference DeNardo1985; Hardin Reference Hardin1996).

My model predicts that vanguard violence may facilitate mobilization and even spark spontaneous, large-scale uprisings. However, it also suggests that empirical arguments for the efficacy of vanguards may be less convincing than previously thought. In equilibrium, there are selection effects. The vanguard engages in higher levels of violence in those societies that are coordinated on a participatory equilibrium. Such societies would be relatively more likely to experience regime change, even in the absence of a vanguard. Hence, a correlation between vanguard activity and mass mobilization may not constitute evidence for the causal efficacy of vanguards—be it through creating focal points, providing selective incentives, or communicating information.

The article proceeds as follows. The first section relates my informational mechanism to existing literature on vanguards and revolution. The next several sections lay out the game and solve for the equilibria. I then discuss implications of the model. Finally, I consider two extensions—one in which the regime can strategically invest in countering the vanguard and another in which vanguard violence directly damages regime capacity—and conclude.

RELATIONSHIP TO THE CONCEPTUAL LITERATURE

I build on the familiar idea that credible revolutionary threats are at least as much a problem of coordination as of collective action (Schelling Reference Schelling1960). The theoretical literature on the role of vanguards in coordination models of revolution has two key strands, which I call pure coordination and informational models of revolution.

THE MODEL

There are a revolutionary vanguard (labeled E for “extremists”) and a continuum of population members of mass 1. At the beginning of the game, each member of the population, i, learns his or her type θ_i = θ + ε_i, which I interpret as his or her level of antigovernment sentiment. The common component θ is drawn by nature from a normal distribution with mean m and variance σ²_θ. The idiosyncratic components ε_i are independent draws by nature from a normal distribution with mean 0 and variance σ²_ε. Members of the population observe only θ_i, not θ or ε_i. After each population member observes his or her type, the vanguard, which has no private information, chooses a level of effort to expend on a campaign of violence (e.g., terrorist or guerrilla attacks). Individuals observe the level of violence and then decide whether to join an attempted revolution against the government. The game ends with the government either being overthrown or remaining in place.

I refer to the stage of the game in which the revolutionary vanguard engages in violence as the vanguard stage and the stage in which population members decide whether to participate as the revolution stage.

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

A member of the population, i, takes an action a_i ∈ {0, 1}, where a_i = 1 is the decision to participate. A person's type, θ_i, determines how much he or she values regime change. He or she derives a portion, γ ∈ (0, 1], of that value only if the revolution succeeds and he or she participated. The other portion, 1 − γ, is realized by all players, if the revolution succeeds, whether that player personally participated in the revolution. Participating imposes a cost k > 0 on the individual. The payoff to a failed revolution is 0. The following matrix gives the von Neuman-Morgenstern expected utility function for a population member i.

Payoffs for a representative population member i

Denote by t ∈ [t, ∞) (with t ≥ 0) the level of effort exerted by the revolutionary vanguard. The total level of vanguard violence is v = t + θ + η, where η is random noise drawn by nature from a normal distribution with mean 0 and variance σ²_η. Vanguard violence is increasing in both vanguard effort and the average level of antigovernment sentiment in society. As discussed at the beginning of this article, the idea is that when the public has a higher level of antigovernment sentiment, it is easier for the revolutionary vanguard to produce violence, because it will have the support of the population.

The vanguard benefits from regime change and finds effort costly. The vanguard's payoffs are given by the following von Neuman-Morgenstern expected utility function:

where c_E represents the costs of vanguard effort. I assume c_E(t) = 0 and c_E is strictly increasing, convex, and satisfies c′_E(t) = 0 and lim_{t → ∞} c′_E(t) = ∞.

Beliefs

Applying Bayes' rule for the case of normal priors and normal signals, a population member of type θ_i, after observing his or her type but not the level of violence v, has posterior beliefs about θ that are distributed normally with mean

and variance

with (DeGroot Reference DeGroot1970). Substantively, the more antigovernment an individual is, the more antigovernment he or she believes society is likely to be (i.e., m_i is increasing in θ_i.)

Suppose it is common knowledge that population members believe the level of effort by the vanguard is t*. Then population members believe that v − t* is a mean θ normally distributed random variable with variance σ²_η. After observing a level of violence, a person of type θ_i has posterior beliefs about θ that are normally distributed with mean

(1)

and variance

with . Here we see that violence communicates information. The more violence the vanguard generates relative to expectations, the more antigovernment sentiment a population member believes there is in society (i.e., m_{i, v − t}* is increasing in v − t*).

THE REVOLUTION STAGE

Denote by Pr(N ≥ T | θ_i, v − t*, s_−i) an individual's assessment of the probability of regime change, given his or her type (θ_i), vanguard violence (v), beliefs about vanguard effort (t*), and a strategy profile for all other players (s_−i). Comparing the expected payoff from participating and not participating, a population member of type θ_i participates if and only if

(2)

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

The first fact to note is that there is always an equilibrium with zero participation. If an individual believes that no one else will participate, regardless of his or her type, 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 game characterizing the revolution stage in which no player participates (i.e., with (v − t*) = ∞ for all v − t*).

All proofs are in the Appendix.

There may also be cutoff equilibria with positive participation. To solve for such an equilibrium, I follow the following four steps:

1. 1. Conjecture a mapping, (⋅) that gives a cut-off rule, (v − t*), for each level of vanguard violence and beliefs about vanguard effort (v − t*).

2. 2. Compute a player i's subjective belief about the probability of regime change, Pr(N ≥ T | θ_i, v − t*, (v − t*)), given v − t* and the belief that all other players, j, participate if and only if θ_j ≥ (v − t*).

3. 3. Find which players will participate given the subjective belief from step 2. That is, for which players is Pr(N ≥ T | θ_i, v − t*, (v − t*))γ θ_i ≥ k?

4. 4. To be part of an equilibrium, the following must be true of the mapping (⋅). For each value of v − t*, the answer to the question in step 3 is that players of type θ_i ≥ (v − t*) will participate and no one else will.

Begin with step 1 by conjecturing a mapping from levels of unexpected violence (v − t*) into cutoff rules, (⋅): ℝ → [−∞, ∞]. Fix a v − t*. A player j using the cutoff rule (v − t*) participates if θ_j = θ + ε_j ≥ (v − t*). Put differently, a player j participates if he or she is antigovernment enough, which can be reexpressed as ε_j ≥ (v − t*) − θ.

Now proceed to step 2—computing a player i's subjective belief about the probability of regime change, given v − t* and the belief that all other players use the cutoff rule (v − t*). As we have just seen, from player i's perspective, if all other players use the cutoff rule (v − t*), then total participation is the mass of players j with ε_j ≥ (v − t*) − θ. Refer to either panel of Figure 1. Here, we see that, for a given θ, this mass is equal to

where Φ is the cumulative distribution function of the standard normal.

FIGURE 1.

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

The revolution succeeds if enough people participate—in particular, if N(θ, (v − t*)) ≥ T. Refer again to Figure 1. Comparing the two panels, we see that, for a fixed (v − t*), participation is strictly increasing in θ—the more antigovernment sentiment in society, the more participation. This implies that, for a given (v − t*), there is a minimal level of antigovernment sentiment necessary for regime change to be achieved. This is precisely the amount that makes the mass of people who mobilize equal to the government's threshold for withstanding revolution, T. Call this minimal level of antigovernment sentiment necessary for achieving regime change θ*((v − t*)). It is implicitly defined by

which can be rewritten

(3)

So, suppose player i observed v − t* and believes all other players use the cutoff rule (v − t*). His or her subjective belief about the probability of regime change is simply how likely he or she believes it is that θ is greater than θ*((v − t*)). Recall that he or she believes that θ is distributed normally with mean m_{i, v − t*}, which is increasing in θ_i and v − t*. That is, the more extreme is player i, the larger he or she believes θ is. The more (unexpected) violence the vanguard generates, the larger he or she believes θ is. Formally, player i's subjective belief about the likelihood of regime change is given by

Now turn to step 3—determining which players will participate, given v − t* and the belief that everyone else uses the cut-off rule (v − t*). Substituting the Pr(N ≥ T | θ_i, v − t*, (v − t*)) just calculated into equation (2), a player i who believes that everyone else is using the cutoff rule (v − t*) will participate if

(4)

The interpretation is the same as equation (2). Player i participates if and only if his or her incremental benefit from participating given v − t* and a belief that everyone else uses the cut-off rule (v − t*) (the left-hand side) is greater than her incremental cost from participating (the right-hand side). This incremental benefit will be important. As such, I introduce the following notation:

Lemma 2. Fix v − t* and a finite . For θ_i > 0, IB(θ_i, , v − t*) is increasing in its first argument.

Lemma 2 says that, among players who are at all antigovernment (i.e., θ_i > 0), if all players use a cutoff strategy, the incremental benefit of participating is higher for more antigovernment members of the population. This is true for two reasons. First, more antigovernment types have higher payoffs from regime change. Second, more antigovernment types believe the probability of regime change is higher, because they believe there is more antigovernment sentiment in society. Lemma 2 implies that, if player i believes all other players use a cutoff rule, then player i will use a cutoff rule. That is, he or she will only participate if he or she is sufficiently antigovernment. (Note that a player with θ_i ≤ 0 has a dominant strategy not to participate.)

Finally, step 4 says that it is not enough for player i to want to use any cutoff rule, given that he or she believes everyone else uses the cutoff rule (v − t*). To make an equilibrium, player i must want to use that same cutoff rule, (v − t*). For this to be the case, the following must be true. If (v − t*) is a finite cutoff rule, then a player whose type is right at the cutoff (i.e., θ_i = (v − t*)) is exactly indifferent between participating and not. If this holds, then population members who are more (resp. less) antigovernment than (v − t*) will have a strict preference to (resp. not to) participate. Formally, then, equilibrium requires

(5)

The left-hand side of this condition is the incremental benefit from participation to a player of type θ_i = (v − t*), when (v − t*) is used as a cutoff rule by all other players. Because this quantity is critical to characterizing the equilibrium, I notate it as follows:

Finding a mapping, (⋅), that is consistent with a cutoff equilibrium is now straightforward. First, whenever it yields a finite cutoff rule, it must satisfy ((v − t*), v − t*) = k. Second, if for some v − t*, (, v − t*) < k, then there is no finite cutoff rule consistent with equilibrium. In such a case, the population members can do only one thing in equilibrium: not participate. Third, except at values of v − t*, where (, v − t*) = k, the mapping must be continuous. These facts are summarized in the following result.

Lemma 3. A strategy profile in which all population members use the same strategy s: ℝ × ℝ → {0, 1}, which is not the strategy “never participate,” is consistent with a cutoff equilibrium if and only if:

with (v − t*) satisfying

1. 1. ((v − t*), v − t*) = k for all v − t* such that (, v − t*) ≥ k

2. 2. Continuity in v − t* for all v − t* such that (, v − t*) > k

Given all this, what does the mapping (⋅) look like? That is, what are the equilibrium cut-off rules used by the population? The answer depends on the shape of the function .

As formalized in Lemma 4 and illustrated in each panel of Figure 2, (⋅, v − t*) is single peaked and goes to zero as the cut-off rule goes to zero or infinity. This implies that there could be multiple mappings consistent with cut-off equilibrium because could cross k more than once.

FIGURE 2.

The solid curve represents (, 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 of participation. Intersections, where (, v − t*) = k, are finite cutoff rules that are consistent with equilibrium for that value of v − t*

The intuition for why is nonmonotonic is as follows. For a fixed v − t*, increasing (i.e., making the cutoff rule more stringent) has three competing effects on the incremental benefit to a player of type θ_i=. First, when the cutoff rule is more stringent, a player whose type equals the cutoff rule has a higher private belief about the level of antigovernment sentiment (i.e., m_{i, v − t*} is increasing in θ_i). Hence, he or she believes that for any given cut-off rule, more people will participate. This implies that he or she believes the probability of successful regime change is higher, increasing his or her incremental benefit from participation. Call this the beliefs effect of increased stringency. Second, when the cutoff rule is more stringent, a player whose type equals the cutoff rule simply has higher personal payoffs from regime change, which increases his or her incremental benefit from participation. Call this the payoff effect of increased stringency. Third, when the cutoff rule is more stringent, given a belief about θ, the amount of participation a player anticipates is lower (Figure 1). This implies that the probability of successful regime change is lower, decreasing the incremental benefit from participation. Call this the participation effect of increased stringency. The beliefs effect and the payoff effect tend to make the function increasing in . The participation effect tends to make the function decreasing in . Together, these competing effects lead to nonmonotonicity, as formalized here.

Lemma 4. For all parameter values and all v − t*, (⋅, v − t*) has the following properties:

1. 1. Any satisfying (, v − t*) = k is strictly positive.

2. 2. (⋅, v − t*) is single peaked in positive values of its first argument.

3. 3. (, v − t*) = 0.

4. 4. (0, v − t*) = 0.

As illustrated in Figure 2, the shape of implies that, for a given value of v − t*, there are generically either zero (the first panel) or two (the third panel) finite cutoff rules consistent with equilibrium. For values of v − t* where two finite cutoff rules are consistent with equilibrium, I label the lower _L(v − t*) and the higher _M(v − t*) (for low and middle). The second panel of Figure 2 shows the knife-end case where there is one finite cut-off rule consistent with equilibrium. Recall that there is also always a cutoff equilibrium with no participation [i.e., _H(v − t*) = ∞].

I can now characterize equilibrium behavior in the revolution stage.

Proposition 1. There are three strategies for the population members in the revolution stage that are consistent with a cutoff equilibrium of the full game:

The game is only interesting if it is possible for there to be positive participation. Hence, I assume that the parameters of the game are such that, for some realization of v − t*, the strategies s^M and s^L actually yield a finite cutoff rule.

Assumption 1. max_{v − t*} (, v − t*) ≥ k.

The result in proposition 1 is reminiscent of standard results for games with strategic complements. There are a highest, and lowest equilibrium (here s ^∞ and s^L) that behave intuitively, and there is a middle equilibrium that does not. The middle equilibrium seems unstable in a way that is analogous to the instability of the mixed strategy equilibrium in a standard complete information coordination game (Echenique and Edlin Reference Echenique and Edlin2004). To see this, imagine a simple learning dynamic, such as players playing best responses to the distribution of play in a previous round. Suppose v − t* is such that there is a finite cutoff strategy in equilibrium. If play is slightly perturbed away from _M(v − t*), then the learning dynamics do not return play to _M(v − t*). If a few too many players participate, then players with types slightly less than _M(v − t*) want to participate, making more players want to participate, until everyone with a type greater than _L(v − t*) is participating. Similarly, if a few too few players participate, then players with types slightly higher than _M(v − t*) do not want to participate, making more players not want to participate, until no one is participating. Given this, I restrict attention to the strategies s ^∞ or s^L.

Assumption 2. Population members do not play the strategy s^M.

VANGUARD VIOLENCE AND REVOLUTION

Before studying how the vanguard behaves, it will be useful to understand how the population responds to changes in vanguard violence. If the population plays the strategy s ^∞ so that no one ever participates, then vanguard violence has no effect on population members' behavior. But if population members play s^L, violence can affect their behavior. How does it do this?

Producing violence requires public support. Hence, the more vanguard violence there is relative to expectation (i.e., higher v − t*), the more antigovernment sentiment each population member believes there is in society (i.e., m_{i, v − t*} is increasing in v − t*). As the two panels of Figure 1 make clear, for a fixed cut-off rule, the higher θ is, the more people will participate. As a result, the higher population members believe θ is, the more likely they believe it is that participation will be sufficient to achieve regime change. Hence, higher levels of v − t* make population members believe that revolution is more likely to succeed. This increases the incremental benefit of participation.

Substantively, the idea would be something like the following. Imagine an IRA sympathizer who is on the fence about whether to participate in some mass action. On the one hand, he or she sympathizes with the IRA's antiregime stance. On the other hand, he or she is not sure how well attended the mass action will be (and, consequently, how likely it is to achieve its goals) and is concerned about government reprisal. Then he or she observes a series of unexpectedly successful IRA attacks that he or she does not believe could have been achieved without support from the surrounding population. He or she concludes that his or her neighbors support the cause and, thus, that the mass action is likely to be well attended. This change in his or her beliefs about his or her fellow citizens' views does not change his or her preferences over regime change. But it changes his or her beliefs about the likely efficaciousness of the mass action. This leads him or her (and others like him or her) to participate. Put differently, the cut-point shifts down as a result of unexpected vanguard violence.

More formally, unexpected vanguard violence has two effects on behavior at the revolution stage. First, the greater v − t*, the more likely it is that (⋅, v − t*) crosses k (so that positive participation is consistent with equilibrium), as formalized in the following lemma.

Lemma 5. If the population plays s^L, then for any η + t − t*, there exists a unique, finite (η + t − t*) such that there is positive participation in the revolution stage if and only if θ ≥ (η + t − t*). Moreover (η + t − t*) is decreasing in η + t − t*.

Second, as argued previously, given that a finite cutoff rule exists, vanguard violence changes the cutoff itself. In particular, unexpected violence by the vanguard convinces population members that there is a higher level of antigovernment sentiment, resulting in a lower cutoff rule and more participation. This fact is illustrated in Figure 3 (where _L(⋅) decreases with an increase in v − t*) and formalized in the following lemma.

FIGURE 3.

The solid and dashed curves represent (⋅, v − t*) for lower and higher realizations of v − t*, respectively (i.e., v″ > v′). The horizontal line represents the cost of participation. An equilibrium finite cutoff rule for a given v − t* is a point where the line and the relevant curve intersect. The lower cutoff rule, _L(⋅), is decreasing in v − t*

Lemma 6. Fix θ ≥ (η + t − t*). _L(v − t*) is strictly decreasing in v − t*.

Two further points are worth mentioning. First, (unexpected) vanguard violence affects participation without creating focal points. When v − t* increases, _L(v − t*) decreases, leading to more participation on the margin. Increased vanguard violence does not convince players to switch from playing _H to _L, which would be a focal point effect. Second, as discussed in greater detail later, if the population plays s^L, it is possible for society to jump discontinuously from no mobilization to high mobilization. This happens at the one point where a finite cutoff rule goes from not existing to existing, as formalized in lemma 5.

THE VANGUARD STAGE

The vanguard is only willing to invest in costly violence insofar as doing so increases the probability of regime change. If the population is playing the strategy s ^∞ such that there will be no participation in the revolution stage no matter what, then this is not possible, so the vanguard will not engage in violence.

Lemma 7. If the population members use the strategy s^∞, then the revolutionary vanguard exerts minimal effort in the vanguard stage (t = t).

Suppose, instead, the population uses the strategy s^L so that positive participation is possible. Two conditions must be met to achieve regime change. First, a finite cutoff rule consistent with equilibrium must exist. As shown in lemma 5, such a rule only exists if θ ≥ (η + t − t*). Second, participation must exceed the threshold for regime change, T. As shown in equation (3), this is true if and only if θ is greater than θ*_L(η + t − t*) implicitly defined by

(6)

The regime will fall as long as θ is larger than max{(η + t − t*), θ*_L(η + t − t*)}. Which constraint binds depends on the realization of η + t − t* as formalized in the following result.

Lemma 8. Suppose the population uses the strategy s^L.

1. 1. For any realization of η + t − t*, there is a unique, finite θ_L(η + t − t*) = max{θ*_L(η + t − t*), (η + t − t*)} such that there will be regime change if and only if θ ≥ θ_L(η + t − t*).

2. 2. θ_L(η + t − t*) is decreasing.

3. 3. There exists a unique η(t − t*) such that θ*_L(η + t − t*) > (η + t − t*) if and only if η > η(t − t*).

Figure 4 illustrates the three possible outcomes when the population plays s^L:

1. 1. Successful revolution: Successful revolution occurs if (η, θ) lies northeast of the curve defined by (η + t − t*) (the dashed line) and θ*_L(η + t − t*) (the solid curve).

2. 2. Failed revolution: A failed revolution (i.e., an uprising but not regime change) occurs if (η, θ) lies between (η + t − t*) and θ*_L(η + t − t*).

3. 3. No mobilization: No mobilization occurs if (η, θ) lies to the southwest of (η + t − t*).

The probability of each outcome is found by integrating the area corresponding to that outcome with respect to the distributions of θ and η.

FIGURE 4.

If the population plays s^L, then there is successful regime change when the realization of (η, θ) lies to the northeast of the curve defined by (the dashed line) and θ* (the solid curve). Increasing t shifts this curve to the southwest, thereby increasing the probability of successful regime change

Figure 4 illustrates two key points. First, increasing antigovernment sentiment (the y-axis) or unexpected vanguard violence (the x-axis) moves the outcome in the direction of revolution. Second, increasing vanguard effort (t) relative to expectation (t*) shifts the curves to the southwest, increasing mobilization and the probability of successful revolution (see lemma 8, item 2.) This is precisely because (unexpected) violence communicates information to the population, and thereby changes population members' behavior in the manner described in the previous section. This is what gives the vanguard incentives to invest in violence.

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

(7)

which implies the following.

Lemma 9. If a cutoff equilibrium in which the population plays the strategy s^L exists, then the vanguard's action, t*, is characterized by:

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 condition in lemma 9. Such a cutoff equilibrium exists if the cost function is sufficiently convex. The next lemma shows that there are such cost functions.

Lemma 10. There exists a nonempty set of cost functions, , such that a cutoff equilibrium in which the population plays the strategy s^L exists.

Assumption 3. The cost function c_E is in the set .

Lemma 9 shows the vanguard's fundamental trade-off. Increased violence increases the population's beliefs about the level of antigovernment violence. This has two effects. First, as shown in Lemma 5, there is a greater probability of positive participation. Second, as shown in Lemma 6, given positive participation, the level of mobilization is increasing in violence. Both effects increase the likelihood of regime change, which is a marginal benefit for the vanguard. However, there are costs to resources expended on violence.

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

Proposition 2. Given assumptions 1–3, there are exactly two cutoff equilibria:

1. 1. The vanguard chooses minimal effort, t, and the population plays s^∞.

2. 2. The vanguard chooses a level of effort, t* > t, and the population plays s^L.

COMPARATIVE STATICS

In the equilibrium where the population plays the strategy s^L, the level of mobilization and the probability of successful regime change are decreasing in the government's capacity to withstand an uprising (T) and to impose costs on those who organize against it (k). They are increasing in the extent to which the benefits associated with regime change only go to participants (γ). Intuitively, when the probability of or payoff to success increases (i.e., T decreases or γ increases) or the cost of participation decreases (i.e., k decreases), it becomes more attractive to participate.

Proposition 3. In the equilibrium in which the population uses the strategy s^L, the number of people who mobilize and the probability of successful regime change are decreasing in T and k and increasing in γ.

These comparative statics highlight the fact that structural factors can affect the level of mobilization in equilibrium. They are also useful for determining how the model relates to standard intuitions about historical cases. For instance, in the early 20th century, Russia experienced two major attempts at revolution: a failed uprising in 1905 and the successful revolution of 1917. The fact that the 1905 uprising occurred suggests that the success of the 1917 revolution was not caused by the population shifting to a new focal equilibrium. In both cases, the people were willing to mobilize (i.e., they were playing a strategy akin to s^L).

Rather, the model is consistent with a familiar structuralist account (Skocpol Reference Skocpol1979). The 1905 failure can be attributed to the uprising being insufficient to overcome the repressive capacity of the state. That is, T and k were too large, relative to the level of antigovernment sentiment (θ) and the success of extremists in producing vanguard violence (v − t*). By 1917, World War I had diminished the government's capacity to repress (k) and withstand (T) rebellion. Thus, without changing focal equilibria, the revolutionary vanguard and the Russian population were now able to wage a successful revolution because structural conditions had become more favorable to regime change. These structural changes simultaneously increased the population's willingness to mobilize and the likelihood that such mobilization would lead to the government's downfall. In terms of Figure 4, the two curves shifted down, moving the situation from one (in 1905) that was between and θ*, to one (in 1917) that was above θ*.

VANGUARDS AND REGIME CHANGE

The model has implications both for empirical research and for conceptual debates over the causes of revolution and political violence.

The Efficacy of the Revolutionary Vanguard

The previous subsection points out that any correlation between the vanguard activity and the probability of regime change could be a selection effect. This raises the question: is the vanguard able to increase mobilization and make regime change more likely?

When the population uses the strategy s^L, at least from an ex post perspective, the answer is “yes.” A higher level of (unexpected) vanguard violence increases mobilization and the likelihood of successful revolution in two ways: it increases the probability of a finite cutoff rule consistent with equilibrium existing (see lemma 5) and, conditional on one existing, it increases mobilization by decreasing the cutoff rule (see lemma 6).

These two effects are illustrated in Figure 5. Figure 5 shows what happens to the level of mobilization for a fixed θ as η increases (thereby increasing the level of unexpected violence) when the population plays s^L. For low levels of unexpected violence, there is no finite cutoff rule consistent with equilibrium, and mobilization is zero. When the level of unexpected violence becomes high enough, a finite cutoff rule consistent with equilibrium exists, leading to a discontinuous jump in participation. The level of mobilization then increases continuously as vanguard violence increases. Thus, the model is consistent with cases, such as Russia or Algeria, where successful vanguards seem to ignite or further inflame mass uprisings against a government.

FIGURE 5.

Equilibrium mobilization as a function of unexpected vanguard violence (i.e., of changes in η for a fixed θ). The simulations assume m = 1, θ = 1, T = 0.3, γ = 0.5, σ²_ε = σ²_η = σ²_θ = 1

The previous discussion focuses on the ex post effects of higher levels of violence. To assess the expected efficacy of vanguards, we must take an ex ante view. From this perspective, the vanguard is not efficacious in the following sense. Ex post, higher than expected levels of vanguard violence increase mobilization and lower than expected levels of vanguard violence decrease mobilization. Ex ante, these two possibilities are equally likely. (Put differently, on the equilibrium path, the expected value of v − t* is θ, the true level of antigovernment sentiment.) So, in expectation, the vanguard is equally likely to increase mobilization (by generating greater-than-expected violence) or to decrease mobilization (by generating lower-than-expected violence). The vanguard nonetheless exerts effort because, if it did not do so the population would likely observe lower-than-expected levels of violence and conclude that the level of antigovernment sentiment is lower than it is in reality. However, if the vanguard could commit to low effort, the population would then update based on that commitment. As a result, the ex ante probability of a successful revolution would be unchanged.

Although, within the current model, the vanguard would like to (but cannot) commit to devoting minimal effort toward violence, it would be an overinterpretation to conclude that this means vanguards should never emerge. The informational mechanism studied here constitutes only one role for vanguards in organizing violence. The vanguard may be more efficacious, even from an ex ante perspective, at other tasks such as providing selective incentives (Lichbach Reference Lichbach1995; Popkin Reference Popkin1979; Tullock Reference Tullock1971, Reference Tullock1974), constructing a highly committed revolutionary movement (Berman Reference Berman2009; Berman and Laitin Reference Berman and Laitin2008; Migdal Reference Migdal1974; Tilly Reference Tilly, Greenstein and Polsby1975), provoking the government (Bueno de Mesquita and Dickson Reference Bueno de Mesquita and Dickson2007; Siqueira and Sandler Reference Siqueira and Sandler2007), drawing international attention, or degrading government capacity (a possibility explored later in the article). Once one of these other mechanisms motivates a vanguard to form, the informational mechanism studied here will come into play.

EXTENSIONS

In this section, I briefly consider extensions to show how the model can be enriched without undoing the previous key findings.

Efficacious Vanguard

The fact that the vanguard is not efficacious from an ex ante perspective might be troubling because it suggests that vanguard organizations should not emerge. Here I address this concern by considering an extension where the vanguard is ex ante efficacious and show that none of my key results are altered.

Suppose vanguard violence degrades the government's capacity to withstand an uprising, in addition to communicating information. In particular, suppose T is a decreasing, differentiable function of v. An argument identical to that surrounding equation (3) shows that, for a given mapping and a fixed v and t*, the revolution will succeed if θ is greater than θ*((v − t*, T(v)), T(v)) = (v − t*, T(v)) − Φ⁻¹(1 − T(v))σ_ε. Given this, an equilibrium cutoff rule must satisfy:

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

How do the vanguard's incentives change? Because vanguard violence now directly degrades government capacity, increased vanguard violence (even if not unexpected) makes participation more attractive because it makes revolutionary success more likely. So, violence is more valuable to the vanguard.

Slightly more formally, when the population plays the strategy analogous to s^L, increases in vanguard violence now increase the probability of regime change in two ways that they did not before. First, an increase in vanguard violence heightens the probability of there being positive participation (i.e., lowers ) by decreasing T (see lemma 15 in the Appendix). Second, conditional on θ ≥ , an increase in vanguard violence now decreases the cutoff rule the population uses, _L, for two reasons: (1) the informational effect discussed in the 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 government capacity, the vanguard has an additional marginal benefit from violence. Second, and more important, vanguard violence degrades government capacity independent of the population's expectations. That is, T is a function of v, not of v − t*. As such, the vanguard benefits from this aspect of violence (when the population plays s^L) even from an ex ante perspective. Hence, the ability to degrade government capacity could serve as an ex ante rationale for forming a vanguard organization. Once it is created, the informational incentives modeled in the main analysis come into play and all the key results—the importance of structural factors in only one of the equilibrium, selection effects, the possibility of spontaneous revolutions—continue to hold.

Counterterrorism

The regime was left unmodeled in the main analysis. However, governments are obviously a key strategic player here. As such, I consider an extension in which the regime can invest resources in trying to prevent vanguard violence.

Suppose the game is played just as in the main analysis, except that at the same time that the vanguard chooses t, the regime invests in counterinsurgency measures r ∈ [r, ∞) [at cost c_R(r)], with r ≥ 0. The total level of vanguard violence is now v = θ + η + t − r.

Population behavior in the revolution stage will be just as in the original game, except that they will now condition their beliefs on v − t* + r*. That is, population members will still learn about antigovernment sentiment from vanguard violence, but they will now have to filter out both vanguard and regime effort to obtain an unbiased signal.

The vanguard's objective is

Assuming the regime wants to minimize the probability of regime change, its objective is

In equilibrium, population members' beliefs about the vanguard's choice of t and the regime's choice of r are correct. Hence, in a pure strategy equilibrium, vanguard behavior is described by the same first-order condition as in the main analysis. Regime counterterrorism is described by an analogous first-order condition. (An argument identical to that in lemma 10 shows that we can find a sufficiently convex cost function for the regime.) In any such equilibrium, behavior by the vanguard and the population is just as described in the main analysis, and the key results continue to hold.

This extension, of course, only considers one aspect of counterrevolutionary policy—minimizing vanguard violence. In reality, such policies are more complicated. For instance, governments must consider the possibility that repressive measures will backfire, increasing, rather than decreasing, mobilization. Such concerns, although of clear interest, are outside the limited scope of this extension. They are discussed in detail elsewhere. [For formal models, see, for instance, Bueno de Mesquita (Reference Bueno de Mesquita2005), Bueno de Mesquita and Dickson (Reference Bueno de Mesquita and Dickson2007), Lichbach (Reference Lichbach1987), Rosendorff and Sandler (Reference Rosendorff and Sandler2004), and Siqueira and Sandler (Reference Siqueira and Sandler2007).]