The Setup

Consider a society populated by a unit mass of citizens, indexed by $ i $ . They are not satisfied with the status quo policy $ {y}_0 $ . In general, citizens prefer a policy, denoted by $ y $ , that is aligned with the current situation in society, which we refer to as the “state” and denote by $ \theta $ . However, citizens have heterogeneous preferences regarding the appropriate policy for society. The preference of citizen $ i $ is parameterized by $ {x}_i $ , which is uniformly distributed on $ \left[0,1\right] $ . The payoff to the citizen with preference $ {x}_i $ when the policy is $ y $ and the state is $ \theta $ is

$$ u\left(y,\theta, {x}_i\right)=\overline{u}-|{x}_i+\theta -y|, $$

where $ \overline{u} $ is a constant. The most preferred policy of citizen $ i $ is $ {x}_i+\theta $ , which we refer to as her ideal policy. Because preferences are different, the interests of citizens are not perfectly aligned. We say that citizen $ i $ is more radical if her preference $ {x}_i $ is higher.Footnote ⁹

We assume that the political situation or the state $ \theta $ is uncertain and unknown to citizens. Specifically, there are two states, $ {\theta}_H $ and $ {\theta}_L $ . The prior probabilities of the two states are $ {\pi}_H $ and $ {\pi}_L $ , respectively. The status quo policy and the two states are such that $ {y}_0<{\theta}_L<{\theta}_H $ . As a result, regardless of the state of the world, there exists a mismatch between the status quo policy and the state, which leads to dissatisfaction and drives demand for reform. The difference is that in the high state $ {\theta}_H $ , the discrepancy between the current state and the status quo policy is larger and the discontent is stronger.

In addition to the group of citizens, there exists an opposition leader, with preference $ {x}^m $ , who formulates and proposes an alternative policy $ {y}_1 $ to incite a revolt, intending to replace the status quo $ {y}_0 $ . Such a policy can be interpreted as the reform agenda, political demands, or a blueprint for the new society. We say that a reform agenda is more radical if $ {y}_1 $ is higher. The key assumption is that the leader observes the state $ \theta $ , but she cannot produce objective verifiable evidence that credibly transmits her private information to citizens. The leader chooses a reform agenda $ {y}_1={y}_1^L $ if the state is low and $ {y}_1={y}_1^H $ if the state is high. Once agenda $ {y}_1 $ is announced, citizens observe it and make an inference about the state.

Our assumption about information asymmetry across the leader and citizens is reasonable. Individual citizens have little influence on the outcome; hence, the incentive for citizens to obtain accurate information about the state in society is small.Footnote ¹⁰ The leader, on the other hand, has a much greater stake in reform. She has to organize the masses, formulate a policy, and convince citizens that her alternative is superior to the status quo. Because her choice of agenda has a material effect on the outcome of the mass movement, the leader tends to spend more time and effort learning about the environment. In this static model, we abstract from the information acquisition decision and assume that the leader is endowed with private information about the state. The assumption that the leader of an organization possesses private information unavailable to followers is common in the literature on leadership (e.g., Hermalin Reference Hermalin1998).

Once citizens observe the opposition leader’s reform agenda $ {y}_1 $ and make inferences about the state $ \theta $ , they need to decide whether to participate in a mass movement against the existing regime ( $ {a}_i=1 $ ) or not ( $ {a}_i=0 $ ). We label a citizen who chooses $ {a}_i=1 $ a follower of the movement and a citizen who chooses $ {a}_i=0 $ a bystander. All citizens agree that some reform is desirable, since every citizen’s ideal policy is larger than $ {y}_0 $ , but they disagree on which reform is best. Indeed, if a reform agenda is farther from a citizen’s ideal policy than the status quo is, this citizen will prefer to maintain the status quo.

The success probability of the movement depends on the total mass of citizens who choose to follow the opposition. Let $ A $ represent the mass of citizens who choose $ {a}_i=1 $ , and let $ G(A) $ be the probability of success. If the movement succeeds, the reform agenda $ {y}_1 $ is implemented; otherwise, the status quo $ {y}_0 $ prevails.Footnote ¹¹

The utility from regime change and the cost incurred differ across three types of citizens: bystanders, followers, and leaders. Under the assumption that each citizen is atomistic, her participation decision has no influence on the total size of the attack in this model. The payoff of a bystander of the movement, for a given size of attack $ A $ and a given reform agenda $ {y}_1 $ , is

$$ {U}_b\left({x}_i\right)=u\left({y}_0,\theta, {x}_i\right)+G(A)\left(u\left({y}_1,\theta, {x}_i\right)-u\Big({y}_0,\theta, {x}_i\Big)\right). $$

The utility difference to citizen $ i $ under the alternative policy $ {y}_1 $ and under the status quo $ {y}_0 $ is the reward from success (i.e., $ u\left({y}_1,\theta, {x}_i\right)-u\left({y}_0,\theta, {x}_i\right) $ ). For any $ {y}_1>{y}_0 $ , the reward from success increases in $ \theta $ and in $ {x}_i $ . Bystanders may gain or lose in the new society, depending on whether the reward from success is positive or negative. If the reform agenda $ {y}_1 $ is far to the right of the ideal policy of bystanders, they may be worse off when the movement succeeds.

Followers who attack the regime need to bear a cost of participating, $ {c}_f>0 $ . Followers join the movement because they benefit from the new policy—that is, the reward from success is positive. Further, we assume that they attach a higher weight, $ {k}_f>0 $ , to the reward from success than do bystanders. The payoff to a follower of the mass movement is

$$ {U}_f\left({x}_i\right)={U}_b\left({x}_i\right)+{k}_fG(A)\left(u\left({y}_1,\theta, {x}_i\right)-u\Big({y}_0,\theta, {x}_i\Big)\right)-{c}_f. $$

The additional weight $ {k}_f $ represents the extra psychological reward received by the revolutionaries and captures the notion of the “pleasure in agency,” which is conceptualized in Wood (Reference Wood2003) and formalized subsequently in Morris and Shadmehr (Reference Morris and Shadmehr2020). It refers to “the value they [revolutionaries] put on being part of the making of history” (Wood Reference Wood2003, 38). Morris and Shadmehr (Reference Morris and Shadmehr2020) stress that such a subjective value arises from the authorship of the changes in society, even though each participant cannot wield influence on the likelihood of success. Our formulation of the payoff structure corresponds to this notion.Footnote ¹² We denote the follower’s premium as $ FP\left({x}_i\right) := {U}_f\left({x}_i\right)-{U}_b\left({x}_i\right) $ . Given $ \theta $ , $ A $ , and $ {y}_1 $ , citizen $ i $ chooses $ {a}_i=1 $ if and only if $ FP\left({x}_i\right)\ge 0 $ .

The leader is involved in organizing the opposition movement and formulating a policy alternative $ {y}_1 $ and incurs a cost of $ {c}_l>0 $ for these leadership activities. The leadership role is also more rewarding: heavier involvement in the movement entails that the pleasure in agency has a stronger intensity, represented by an extra weight $ {k}_l>0 $ attached to the reward from success. The payoff to a leader is

(1) $$ {U}_l\left({x}_i\right)=\max \left\{{U}_b\left({x}_i\right),{U}_f\left({x}_i\right)\right\}+{k}_lG(A)\left(u\left({y}_1,\theta, {x}_i\right)-u\Big({y}_0,\theta, {x}_i\Big)\right)-{c}_l. $$

We denote the leader’s premium as $ LP\left({x}_i\right) := {U}_l\left({x}_i\right)-\max \left\{{U}_b\left({x}_i\right),{U}_f\left({x}_i\right)\right\} $ . The leader’s premium plays a role in determining whether a citizen chooses to be a leader or not, a decision that we analyze in the next section. In this section, the leader chooses a reform agenda $ {y}_1 $ to maximize $ {U}_l\left({x}^m\right) $ , while her preference $ {x}^m $ is known to citizens.

Throughout this paper, we maintain the assumption that the probability of success $ G(A) $ is strictly increasing and weakly log-concave in $ A $ , with $ G(0)>{c}_f/{k}_f $ . Diminishing returns from having more attackers implies log-concavity of $ G $ . It is plausible, however, that successful revolts may require a critical mass of attackers, meaning that the success determination technology may exhibit increasing and then decreasing returns. Our assumption can accommodate this type of success determination technology because log-concavity is consistent with increasing returns as long as $ {G}^{\prime }(A)/G(A) $ is nonincreasing. The second part of our assumption (i.e., $ G(0)>{c}_f/{k}_f $ ) is a sufficient condition that guarantees the existence of nontrivial equilibria with a positive mass of attackers. This condition requires that $ G(0) $ be positive. A regime facing mass discontent may implode or collapse due to many forces (e.g., internal strife among elites, economic pressure, or foreign intervention) other than the actions of revolutionary leaders and followers. We can interpret $ G(0) $ as the probability that these “background factors” bring down the regime.Footnote ¹³

In this game with asymmetric information, nature picks the state first. Then the leader observes the state and proposes an agenda. Citizens observe the agenda, make inferences about the state, and make participation decisions. Finally, the revolt can be a success or failure, depending on the mass of attackers and some random factors. The timeline of the game is summarized in Figure 1.

Figure 1.

Timeline of the Game with Asymmetric Information

Equilibrium of the Protest Game

In the following, we solve the model backwards. We first deal with the last stage, where citizens correctly infer about the state from the agenda proposed by the opposition leader and decide to attack the regime or not, taking the agenda as given. We then analyze the agenda decision of the leader under full information and asymmetric information in the last two subsections.

Given the status quo policy $ {y}_0 $ and the reform agenda $ {y}_1 $ , citizens decide whether to attack the regime or not in a simple protest game when the state $ \theta $ is learned and becomes common knowledge. For any expected size of attack $ A $ , citizen $ i $ chooses $ {a}_i=1 $ , if and only if the follower’s premium (defined in the previous subsection) $ FP\left({x}_i;A,{y}_1\right)\ge 0 $ . The function $ FP\left(\cdot; A,{y}_1\right) $ is weakly increasing; therefore, there exists a marginal attacker, denoted $ {x}^f $ and satisfying $ FP\left({x}^f;A,{y}_1\right)=0 $ , such that citizen $ i $ attacks if and only if $ {x}_i\ge {x}^f $ . Figure 2 illustrates the determination of the marginal attacker $ {x}^f $ given a pair of policies $ {y}_0 $ and $ {y}_1 $ .

Figure 2.

The Determination of the Marginal Attacker $ {x}^f $ .

Note: The utility difference between the solid line and the dashed line represents the reward from success for citizens with preference $ {x}_i $ . The follower’s premium is nonnegative if and only if $ {x}_i\ge {x}^f $ .

Under this decision rule and the uniform distribution assumption, the total mass of attackers is $ A=1-{x}^f $ . The equilibrium of the protest game can be characterized by the indifference condition for the marginal attacker, which can be written as

(2) $$ FP\left({x}^f;1-{x}^f,{y}_1\right)={k}_fG\left(1-{x}^f\right)\left(u\left({y}_1,\theta, {x}^f\right)-u\Big({y}_0,\theta, {x}^f\Big)\right)-{c}_f=0. $$

To emphasize its dependence on the reform agenda and the state, we use $ {x}^f\left({y}_1;\theta \right)\in \left(0,1\right) $ to denote the equilibrium marginal attacker that satisfies condition (2). If $ FP\left(0;1,{y}_1\right)\ge 0 $ , then $ {x}^f\left({y}_1;\theta \right)=0 $ , and all citizens attack. If $ FP\left(1;0,{y}_1\right)<0 $ , then $ {x}^f\left({y}_1;\theta \right)=1 $ , and no one attacks.

In case (a) of Lemma 1, the reward from success is small if the reform agenda $ {y}_1 $ is close to the status quo $ {y}_0 $ . Given a positive cost of participation, no one chooses to participate in the movement. In case (b), the reform agenda $ {y}_1 $ is sufficiently far from the status quo, which guarantees at least a fraction of the relatively more radical citizens would find it worthwhile to attack—that is, a nontrivial equilibrium exists. In this case, there may be multiple equilibria, one with $ {x}^f\le {y}_1-\theta $ and another with $ {x}^f>{y}_1-\theta $ .Footnote ¹⁴ Throughout the paper, we focus on the equilibrium with the largest equilibrium attack size (i.e., the one with $ {x}^f\le {y}_1-\theta $ ). This is the only interior equilibrium in which the cutoff strategy (i.e., attack if and only if $ {x}_i \ge {x}^f $ ) can be reasonably justified, and it is stable with meaningful comparative statics.

In a nontrivial equilibrium of the protest game, a more radical reform agenda $ {y}_1 $ suppresses participation. First, because $ {x}^f+\theta \le {y}_1 $ , the reform agenda is already to the right of the ideal policy of the marginal attacker. Raising $ {y}_1 $ further would make the agenda even less appealing to this citizen. Thus, holding $ A $ fixed, the marginal attacker shifts to the right. Second, as fewer citizens participate (i.e., $ A $ decreases), the follower premium falls for every citizen, which shifts the marginal attacker to the right even further. Consequentially, the marginal attacker becomes more radical and the probability of success falls correspondingly.

Equally important, a larger mismatch between the state and status quo policy causes more citizens to prefer the proposed policy $ {y}_1 $ to the status quo $ {y}_0 $ because the reward from success is increasing in the state $ \theta $ for any $ {y}_1>{y}_0 $ . Lemma 1 shows that the size of the attack is larger when $ \theta $ is higher, which implies that the probability of success, $ G\left(1-{x}^f\left({y}_1;\theta \right)\right) $ , increases in $ \theta $ . Such a comparative static is particularly useful when the agenda decision is analyzed.

Strategic Reform Agenda in the Full-Information Benchmark

Now we turn to the agenda-making stage. The crux of our theory is that the agenda proposed by the leader has the dual role of informing citizens about the state and laying down a policy alternative that matters for their welfare after the regime change. In this section, we study a benchmark case in which we disentangle the informational role from the material role of the agenda by assuming that the state is known even before the leader announces the agenda. The results obtained will be utilized and contrasted with those obtained in the subsequent section, where the effect of asymmetric information is studied.

When the state $ \theta $ is common knowledge, the leader simply chooses $ {y}_1 $ to maximize her own payoff $ {U}_l\left({x}^m\right) $ , taking into account its influence on the protest game, characterized in Lemma 1. The leader’s maximization problem can be written as

(3) $$ \underset{y_1}{\max} u\left({y}_0,\theta, {x}^m\right)+\kappa G\left(1-{x}^f\left({y}_1;\theta \right)\right)\left[u\left({y}_1,\theta, {x}^m\right)-u\Big({y}_0,\theta, {x}^m\Big)\right]-{c}_l, $$

where $ \kappa =1+{k}_f+{k}_l $ if she chooses to attack in the protest game and $ \kappa =1+{k}_l $ if she chooses not to attack.

We restrict our attention to the interesting case where the leader’s ideal policy is large enough (i.e., $ {x}^m+\theta >{y}_{\mathrm{min}} $ ).Footnote ¹⁵ The leader would never find it optimal to choose an agenda that is more radical than her ideal policy (i.e., $ {y}_1>{x}^m+\theta $ ) because she can always choose a less radical policy that is closer to her ideal policy and that gives rise to a higher probability of success. She would not choose a very moderate agenda, either (i.e., $ {y}_1<{y}_{\mathrm{min}} $ ), because such an agenda would induce no attack in the protest game. For $ {y}_1\in \left[{y}_{\mathrm{min}},{x}^m+\theta \right] $ , the objective function (3) is quasiconcave; therefore, the optimal agenda $ {y}_1^{\ast } $ is unique and is characterized by the first-order condition. We also show that it is not optimal to choose $ {y}_1={y}_{\mathrm{min}} $ . The optimal agenda $ {y}_1^{\ast } $ satisfies

(4) $$ 1-\frac{G^{\prime}\left(1-{x}^f\left({y}_1^{\ast };\theta \right)\right)}{G\left(1-{x}^f\left({y}_1^{\ast };\theta \right)\right)}\frac{\partial {x}^f\left({y}_1^{\ast };\theta \right)}{\partial {y}_1}\left({y}_1^{\ast }-{y}_0\right)\ge 0, $$

with $ {y}_1^{\ast }={x}^m+\theta $ if (4) holds as a strict inequality.

The leader faces a trade-off between the reward from success and the chances of success. The marginal utility from having a policy closer to the leader’s ideal policy is $ 1 $ . The marginal cost is that the chance of obtaining the reward from success is lowered as $ \partial {x}^f/\partial {y}_1 $ is positive by Lemma 1. The first-order condition (4) optimally balances this trade-off. Because this optimal agenda $ {y}_1^{\ast } $ reflects the leader’s concern about the equilibrium outcome in the subsequent protest game, we sometimes refer to it as a strategic agenda, and we write $ {y}_1^{\ast}\left({x}^m;\theta \right) $ to emphasize its dependence on the leader’s preference and on the state.

Proposition 1. For any leader with $ {x}^m\ge {y}_{\mathrm{min}}\left(\theta \right)-\theta $ , there exists a unique cutoff $ \hat{x}\left(\theta \right) $ such that the strategic agenda is

(5) $$ {y}_1^{\ast}\left({x}^m;\theta \right)=\left\{\begin{array}{cc}{x}^m+\theta & if\;{x}^m<\hat{x}\left(\theta \right);\\ {}\hat{x}\left(\theta \right)+\theta & if\;{x}^m\ge \hat{x}\left(\theta \right).\end{array}\right. $$

Further, a higher state makes the strategic agenda more radical, but increases the probability of success in the protest game.

When the leader’s preference $ {x}^m $ is relatively moderate (i.e., $ {x}^m<\hat{x}\left(\theta \right) $ ), she can afford to choose her own ideal policy (i.e., $ {x}^m+\theta $ ) without suppressing citizens’ support very much—that is, the optimal solution $ {y}_1^{\ast}\left({x}^m;\theta \right) $ is a corner solution. When the leader’s preference is relatively radical (i.e., $ {x}^m\ge \hat{x}\left(\theta \right) $ ), choosing her ideal policy would discourage too many moderate citizens from joining the mass movement. Therefore, the leader must compromise by choosing the agenda $ {y}_1^{\ast }=\hat{x}\left(\theta \right)+\theta $ , which is less radical than her ideal policy. In general, the strategic agenda $ {y}_1^{\ast}\left({x}^m;\theta \right) $ is weakly increasing in $ {x}^m $ .Footnote ¹⁶

The second part of the proposition says that the opposition leader chooses a more radical agenda in response to a higher state, but would never raise the equilibrium agenda $ {y}_1^{\ast } $ to the point where it hurts the chances of success. In other words, $ G\left(1-{x}^f\left({y}_1^{\ast}\left({x}^m;\theta \right);\theta \right)\right) $ always increases in $ \theta $ . In connection to the two states specified in our model, we let $ {y}_1^{L\ast} := {y}_1^{\ast}\left({x}^m;{\theta}_L\right) $ and $ {y}_1^{H\ast} := {y}_1^{\ast}\left({x}^m;{\theta}_H\right) $ represent the strategic agenda chosen by the leader with preference $ {x}^m $ in the low and high states, respectively, under full information. Further, let $ {G}^{L\ast} := G\left(1-{x}^f\left({y}_1^{L\ast };{\theta}_L\right)\right) $ and $ {G}^{H\ast} := G\left(1-{x}^f\left({y}_1^{H\ast };{\theta}_H\right)\right) $ represent the corresponding probabilities of success. Proposition 1 states that $ {y}_1^{H\ast }>{y}_1^{L\ast } $ and $ {G}^{H\ast }>{G}^{L\ast } $ .

The implication of this result is important for our subsequent analysis. Once citizens believe that the environment is more favorable for a revolt (i.e., the state is high), the leader can choose a more radical agenda, but the chances of success are also greater. That provides the leader an incentive to mislead citizens when the leader knows the state but citizens do not know. In the next section, such an incentive will be the focus of our analysis. We will also show that, under some circumstances, asymmetric information may cause the leader to “overreact”: she chooses a policy so radical that it reduces the probability of success, despite the state being more favorable to the mass movement.

Radical Reform Agenda in Separating Equilibrium

In this section, we analyze the case of asymmetric information, in which only the leader knows the state but citizens do not. In this setting, the leader lacks an effective means of convincing citizens that it is her information rather than her own interest that drives her choice of agenda. She has to rely on the reform agenda to credibly signal her private information. Such a signaling mechanism is costly, because the agenda has to be distorted to command credibility.

In the following formal analysis, we restrict our attention to leaders with a preference higher than a threshold (specifically, $ {x}^m\ge {y}_{\mathrm{min}}\left({\theta}_L\right)-{\theta}_L $ ) to ensure that in each state, there is a positive mass of attackers in the equilibrium of the protest game.

The leader in the low state may have an incentive to mislead citizens into believing that the state is high, and the more radical her preference is, the greater the incentive to mislead. First, for the same reform agenda, the mass of attackers is larger if citizens believe that the state is high (Lemma 1) when it is actually low. Second, when the leader misleads citizens with $ {y}_1^{H\ast } $ , the reward from success could be either larger or smaller, depending on whether $ {y}_1^{H\ast } $ or $ {y}_1^{L\ast } $ is closer to her ideal policy. Specifically, by choosing $ {y}_1^{H\ast } $ instead of $ {y}_1^{L\ast } $ in the low state, the reward from success would change from $ {y}_1^{L\ast }-{y}_0 $ to $ 2{x}^m+2{\theta}_L-{y}_1^{H\ast }-{y}_0 $ . For a leader with low $ {x}^m $ , the policy $ {y}_1^{H\ast } $ is too far from her ideal policy in the low state, and the net change is negative. For a leader with high $ {x}^m $ , the policy $ {y}_1^{H\ast } $ can be closer to her ideal policy in the low state, and the net change is positive. In general, the net change is larger when the leader is more radical. Therefore, in the low state $ {\theta}_L $ , a leader with a moderate preference (small $ {x}^m $ ) faces a trade-off between a higher chance of success and a lower reward from success when she lies. However, both are higher for a leader with a radical preference (large $ {x}^m $ ). Lemma 2 shows that there exists a unique $ {x}^{\dagger } $ such that the concern over a lower reward dominates the benefit of a higher chance of success if and only if $ {x}^m\le {x}^{\dagger } $ . We say that the leader is moderate if $ {x}^m\le {x}^{\dagger } $ and radical, otherwise.

Suppose the leader’s preference is moderate ( $ {x}^m\le {x}^{\dagger } $ ); she thus has no incentive to mislead, and citizens know this. In equilibrium, the leader would choose the strategic agenda $ {y}_1^{L\ast } $ in the low state and the strategic agenda $ {y}_1^{H\ast } $ in the high state, and citizens correctly infer about the state. In other words, the leader indeed behaves “optimally”—that is, as she would behave under full information.

Suppose the leader’s preference is radical ( $ {x}^m>{x}^{\dagger } $ ); she thus has an incentive to mislead by proposing $ {y}_1^{H\ast } $ in the low state. Of course, citizens would not be fooled into believing that the state is high simply because they observe the leader choosing the reform agenda $ {y}_1^{H\ast } $ . In this case, the full-information outcome cannot be supported as an equilibrium outcome under asymmetric information. In the following proposition, we characterize the least-cost separating equilibrium that satisfies the D1 criterion (Banks and Sobel Reference Banks and Sobel1987; Cho and Kreps Reference Cho and Kreps1987). We show that the agenda chosen must be even more radical than $ {y}_1^{H\ast } $ in the high state, so the leader would not have an incentive to mimic the high state by choosing this very radical policy when the state is low. In other words, the radical leader cannot behave “optimally” in the high state; she has to resort to “irrational radicalism” to separate herself from leaders who may otherwise bluff in order to boost support.

To formalize, let $ ({\hat{y}}_1^L,{\hat{y}}_1^H) $ represent the agenda choices in the separating equilibrium in the respective states. Let $ {\hat{G}}^L := G(1-{x}^f({\hat{y}}_1^L;{\theta}_L)) $ and $ {\hat{G}}^H := G(1-{x}^f({\hat{y}}_1^H;{\theta}_H)) $ represent the corresponding probabilities of success.

Part (i) says that, in a separating equilibrium, the leader would not propose an agenda different from $ {y}_1^{L\ast } $ in the low state. Any agenda other than optimal strategic policy would give her a lower payoff as long as the low state is correctly inferred by citizens.

Part (ii) establishes a contrasting result in the high state. For any leader with $ {x}^m>{x}^{\dagger } $ , proposing an agenda equal to $ {y}_1^{H\ast } $ in the high state will not be an equilibrium (Lemma 2). To prevent her from overstating the favorability of the state, the equilibrium policy $ {\hat{y}}_1^H $ must exceed $ {y}_1^{H\ast } $ .Footnote ¹⁷ Such an equilibrium agenda is so high that it is to the right of the leader’s ideal policy $ {x}^m+{\theta}_L $ in the low state, and it significantly reduces the probability of success, which makes proposing $ {\hat{y}}_1^H $ unattractive in the low state. In other words, in the high state, to convince citizens of the state, a radical leader must pursue a policy that is too radical and too costly at the expense of losing too many followers.

The reform agenda in the high state under the least-cost separating equilibrium that satisfies the D1 refinement is the $ {\hat{y}}_1^H $ (greater than $ {y}_1^{H\ast } $ ) such that the leader’s incentive constraint in the low state is just binding:

(6) $$ {\hat{G}}^L(u({\hat{y}}_1^L,{\theta}_L,{x}^m)-u({y}_0,{\theta}_L,{x}^m\Big))-{\hat{G}}^H(u({\hat{y}}_1^H,{\theta}_L,{x}^m)-u({y}_0,{\theta}_L,{x}^m\Big))=0. $$

When the leader is indifferent between $ {\hat{y}}_1^L $ and $ {\hat{y}}_1^H $ in the low state, she strictly prefers $ {\hat{y}}_1^H $ to $ {\hat{y}}_1^L $ in the high state.Footnote ¹⁸ Figure 3 illustrates the equilibrium outcome.

Figure 3.

Strategic Agenda and Radical Agenda.

Note: The strategic agenda pair is $ \left({y}_1^{L\ast },{y}_1^{H\ast}\right) $ under full information. When $ {x}^m>{x}^{\dagger } $ , the leader chooses $ {\hat{y}}_1^L={y}_1^{L\ast } $ and $ {\hat{y}}_1^H>{y}_1^{H\ast } $ in a separating equilibrium, and $ {\hat{y}}_1^H $ strictly increases in $ {x}^m $ .

In the least-cost separating equilibrium, the chance of success in the high state must be lower than that in the full-information setting (i.e., $ {\hat{G}}^H<{G}^{H\ast } $ ). That is because the reform agenda in the high state is more radical, and the mass of attackers decreases with a more radical agenda (Lemma 1). In the high state, the leader cannot moderate her position, even though she knows that her radical agenda discourages citizens: citizens would interpret an agenda less radical than $ {\hat{y}}_1^H $ as an opportunistic deviation by a leader who attempts to exaggerate the state by choosing a high agenda in the low state.

Part (iii) of Proposition 2 characterizes the comparative statics of the equilibrium $ {\hat{y}}_1^H $ with respect to $ {x}^m $ . As the leader becomes more radical, her incentive to exaggerate the state and mislead citizens increases. As a result, she needs to propose even more radical agendas in the high state in order to remain credible. Interestingly, when the leader is very radical, the more favorable state does not necessarily guarantee a better chance of success because the agenda chosen to signal the state is too radical.

A corollary to Proposition 2 is that there exists $ {x}^{\dagger \dagger }>{x}^{\dagger } $ such that, if the leader’s preference is more radical than $ {x}^{\dagger \dagger } $ , then the chances of success are smaller in the high state than in the low state. Because $ {\hat{y}}_1^H $ increases with $ {x}^m $ at a nonvanishing rate for a large enough $ {x}^m $ , and $ {\hat{y}}_1^L $ is constant with respect to $ {x}^m $ for $ {x}^m>\hat{x}\left({\theta}_L\right) $ , we must have $ {\hat{y}}_1^H-2{\theta}_H>{\hat{y}}_1^L-2{\theta}_L $ when $ {x}^m $ exceeds a threshold value $ {x}^{\dagger \dagger } $ . By the indifference condition (2) for the marginal attacker, this inequality implies that $ {x}^f\left({\hat{y}}_1^H;{\theta}_H\right)>{x}^f\left({\hat{y}}_1^L;{\theta}_L\right) $ , and hence $ {\hat{G}}^H<{\hat{G}}^L $ . Such an equilibrium outcome under asymmetric information is qualitatively different from that under full information, where we show that $ {G}^{H\ast }>{G}^{L\ast } $ . That is, the leader who does not suffer from the citizens’ suspicion always proposes a more radical agenda given a higher state, but she never “overreacts” by choosing a policy so radical that it hurts the chances of success.

Leader Selection

We enrich the model in the previous section by allowing the opposition leadership to comprise a group of citizens instead of a single leader and by making the choice to join the leadership group endogenously. Specifically, before observing the state, each citizen only knows its distribution and can make an additional decision to join the leadership of the mass movement ( $ {l}_i=1 $ ) or not ( $ {l}_i=0 $ ). In each state, the payoff to any citizen who chooses to be a leader is given by Equation 1.Footnote ¹⁹

If no citizens choose to join the leadership, then the status quo will prevail, and therefore there will be no revolt. If a group of citizens decide to join the leadership, their preferences within the leadership group must be diverse. We assume that it is the one with the median preference among the leadership group who will formulate and propose the reform agenda for the mass movement. In other words, the task of drafting the policy proposal is delegated to the median citizen in the group, who will be referred to as the median leader in the following. We continue to use $ {x}^m $ to denote the preference of the median leader.

Citizens choose to join the leadership before they know the state. After the median leader emerges from the leadership group, she observes the state, makes a proposal, and announces it to all the other citizens. Upon hearing the proposal, each citizen makes inferences about the state and decides whether to participate in attacking the regime or not. That is, once the median leader acquires information about the state, the rest of the game is similar to the benchmark model. The timing of the game with endogenous leadership is summarized in Figure 4. The equilibrium of this static game is a fixed point, in which the preference of the median leader is endogenously determined and known to other citizens.

Figure 4.

Timeline of the Game with Asymmetric Information and Endogenous Leadership

The assumptions related to the emergence of leaders deserve discussion. First, the assumption that the median leader chooses the agenda captures the idea that the proposed agenda should reflect the preferences of the leadership group as a whole. This can be formally justified by a democratic decision-making process within the leadership group, in which the median leader’s preference prevails in any pairwise vote between reform agendas. This assumption appears to be rather specific but is a useful shortcut for representing two general conditions: (i) the policy proposed by the leadership group is more radical than the policy preferred by the least radical leader in the group and (ii) the proposed policy becomes more radical if a fraction of the least radical leaders are removed from the group. Conceptually, the precise policy formulation mechanism matters little for the subsequent analysis and results, as long as conditions (i) and (ii) hold, but our setting is more tractable.Footnote ²⁰

Second, in this model, there will be a continuum of leaders. Only the median leader chooses the agenda; what she proposes affects the payoffs of all leaders but not the other way around because each leader is atomistic and her decision to become a leader or not has no influence on who will be selected as the median leader. This assumption is made to simplify the characterization. Even in a setting with a finite number of leaders, the agenda-making mechanism can still satisfy conditions (i) and (ii), in which case our results and analysis continue to hold.Footnote ²¹

Finally, regarding the timing, we assume that citizens do not observe the state before they choose to join the leadership, which is reasonable but not crucial for our results. This is reasonable because political mass movements are often sudden outbreaks, and individuals who become leaders may acquire information about the political situation after they assume their role.Footnote ²² It is not crucial, because we can also allow a small fraction of randomly drawn citizens to observe the state before they decide whether to participate in the leadership group. In a separating equilibrium where the rest of citizens still infer the state from the reform agenda, the main results and properties of leadership formation will be similar.

Equilibrium Analysis

Because citizens do not know the state at the time that they choose to become leaders, the determination of endogenous leadership is based on the expected value of the leader’s premium. Specifically, if the preference of the median leader is conjectured to be equal to $ {x}^m $ , citizens expect that the proposed agenda is $ {\hat{y}}_1^L\left({x}^m\right) $ or $ {\hat{y}}_1^H\left({x}^m\right) $ , depending on the state. The expected leader’s premium for a citizen with preference $ {x}_i $ is

$$ \hat{L}P\left({x}_i;{x}^m\right) := \sum \limits_{j=L,H}{\pi}_j LP\left({x}_i;{\hat{y}}_1^j\left({x}^m\right),{\theta}_j\right), $$

where

$$ LP\left({x}_i;{y}_1,\theta \right)=\left\{\begin{array}{cc}{k}_lG\left(1-{x}^f\left({y}_1;\theta \right)\right)\left(2{x}_i+2\theta -{y}_1-{y}_0\right)-{c}_l& \mathrm{if}\;{x}_i+\theta <{y}_1,\\ {}{k}_lG\left(1-{x}^f\left({y}_1;\theta \right)\right)\left({y}_1-{y}_0\right)-{c}_l & \mathrm{if}{x}_i+\theta \ge {y}_1.\end{array}\right. $$

For a given $ {x}^m $ , citizen $ i $ chooses to join the leadership group if and only if the expected leader’s premium is nonnegative. Since $ LP\left({x}_i;{y}_1,\theta \right) $ is weakly increasing in $ {x}_i $ , $ \hat{L}P\left({x}_i;{x}^m\right) $ is also weakly increasing in $ {x}_i $ , which justifies a monotone strategy. That is, if there exists a marginal leader $ {x}^l $ such that $ \hat{L}P\left({x}^l;{x}^m\right)=0 $ , then citizen $ i $ chooses $ {l}_i=1 $ if and only if $ {x}_i\ge {x}^l $ . If $ \hat{L}P\left(0;{x}^m\right)>0 $ , then all citizens choose to become leaders, and if $ \hat{L}P\left(1;{x}^m\right)<0 $ , then no citizen chooses to be a leader. Of course, the preference of the marginal leader depends on $ {x}^m $ , and we write $ {x}^l\left({x}^m\right) $ to emphasize this dependence.

Because the distribution of preferences $ {x}_i $ is uniform, if the marginal leader has preference $ {x}^l\left({x}^m\right) $ , then the median preference of the leadership group is

$$ M\left({x}^m\right)=\frac{1}{2}\left({x}^l\left({x}^m\right)+1\right). $$

The equilibrium in this model is characterized by a 6-tuple, $ \left({x}_{\ast}^m,{x}_{\ast}^l,{x}_{L\ast}^f,{x}_{H\ast}^f,{\hat{y}}_{1\ast}^L,{\hat{y}}_{1\ast}^H\right) $ , that satisfies the following requirements:

1. 1. The median leader’s preference satisfies $ {x}_{\ast}^m=M\left({x}_{\ast}^m\right) $ .

2. 2. The marginal leader’s preference satisfies $ \hat{L}P\left({x}_{\ast}^l;{x}_{\ast}^m\right)=0 $ .

3. 3. The reform agenda in the low state solves the maximization problem (3) and is given by $ {\hat{y}}_{1\ast}^L={y}_1^{\ast}\left({x}_{\ast}^m;{\theta}_L\right) $ in Equation 5.

4. 4. If $ {x}_{\ast}^m\le {x}^{\dagger } $ , the reform agenda is given by $ {\hat{y}}_{1\ast}^H={y}_1^{\ast}\left({x}_{\ast}^m;{\theta}_H\right) $ in Equation 5; otherwise, it is given by the solution to the binding incentive constraint (6) for $ {x}^m={x}_{\ast}^m $ and $ {\hat{y}}_1^L={\hat{y}}_{1\ast}^L $ .

5. 5. The marginal attacker in state $ j\in \left\{H,L\right\} $ satisfies $ FP\left({x}_{j\ast}^f;1-{x}_{j\ast}^f,{\hat{y}}_{1\ast}^j,{\theta}_j\right)=0 $ .

In this equilibrium, citizen $ i $ chooses $ {l}_i=1 $ if and only if $ {x}_i\ge {x}_{\ast}^l $ , and she chooses $ {a}_i=1 $ in state $ j\in \left\{H,L\right\} $ if and only if $ {x}_i\ge {x}_{j\ast}^f $ . The probability of success in state $ j $ is $ G\left(1-{x}_{j\ast}^f\right) $ .

The key to our model is that the leadership premium $ \hat{L}P\left({x}_i;{x}^m\right) $ is weakly increasing in $ {x}_i $ , and any citizen with preference $ {x}_i\ge {x}^l\left({x}^m\right) $ will choose to be a leader. That means that the median leader $ {x}^m $ in general is more radical than the marginal leader $ {x}^l $ (i.e., condition (i) holds), and the median leader $ {x}^m $ is more radical if the marginal leader $ {x}^l $ is more radical (i.e., condition (ii) holds).

The two conditions combined give rise to a mechanism for the determination of the leadership, in which the mapping $ M\left({x}^m\right) $ is increasing in $ {x}^m $ . To understand the positive slope, consider the following comparative statics on an increase in the preference of the median leader (i.e., a larger $ {x}^m $ ). The median leader would propose more radical agendas—that is, $ {\hat{y}}_1^L\left({x}^m\right) $ and $ {\hat{y}}_1^H\left({x}^m\right) $ are both higher (Propositions 1 and 2), and this, in turn, hurts the marginal leader who has less radical preferences than the median one. As a result, a more radical citizen would become the marginal leader—that is, $ {x}^l\left({x}^m\right) $ shifts to the right as $ {x}^m $ increases. The distribution of preferences of the leadership group therefore becomes more radical, which, consequentially, gives rise to a more radical median leader. In other words, the leadership group will be more radical if the conjectured median leader becomes more radical. The solid line in Figure 5 illustrates the mapping $ M\left({x}^m\right) $ .

Figure 5.

Fixed Point.

Note: The solid black curve represents $ M\left({x}^m\right) $ , and $ {x}_{\ast}^m $ is the fixed point. When the cost of joining the leadership is higher, $ M\left({x}^m\right) $ shifts up and the equilibrium $ {x}_{\ast}^m $ increases.

To ensure the existence of a nontrivial equilibrium in the protest game, we require that $ {c}_f/{k}_f $ be smaller than a threshold.Footnote ²³ To ensure that the leadership is not empty in equilibrium, we require that $ {c}_l/{k}_l $ be smaller than a threshold so that a citizen with preference $ {x}_i\in \left[1/2,1\right] $ will choose to be a leader if she expects herself to be the median leader who proposes the reform agendas.Footnote ²⁴ Under these conditions, Proposition 3 establishes that $ M\left({x}^m\right) $ is increasing and that a fixed point of the mapping exists.

The fixed point of $ M\left(\cdot \right) $ is unique and stable if its slope is less than one. Intuitively, $ {M}^{\prime}\left({x}^m\right) $ will be smaller if fewer leaders on the left end of the leadership drop out when the median leader becomes more radical. This is true when the proposed policy $ {\hat{y}}_1^H\left({x}^m\right) $ does not increase too much when the leader’s preference $ {x}^m $ becomes more radical. Proposition 3 shows that the condition $ {\theta}_H-{\theta}_L\le 1/3 $ is sufficient to ensure that $ \partial {\hat{y}}_1^H\left({x}^m\right)/\partial {x}^m $ is so small that the slope of $ M\left(\cdot \right) $ is less than one. Therefore, the fixed point is also a globally stable one: convergence to the fixed point from any likely perturbations is guaranteed.

We have characterized the determination of the endogenous leadership group. In the following section, we investigate the conditions under which citizens with radical preferences take on the leadership roles and how this mechanism of leadership selection and the mechanism of signaling reinforce each other.