The case without technology is our benchmark scenario. Our first result shows that when information about other individuals is not available (i.e., when decisions are taken without knowing what other individuals have done) the efficient outcome in which individuals revolt may fail to materialize, even if it is known that sufficient willing individuals exist.
This is in line with the usual result of multiplicity of equilibria in coordination problems. Since \(u_{w,r,R}>u_{w,s}>u_{w,r,F}\), for the willing individuals it is optimal to revolt if the other willing individuals are revolting, while it is optimal to stay at home if nobody else is revolting. If a willing individual believes that the other willing individuals will participate in the revolution, then she best responds to this belief by participating as well. However, if they hold the opposite belief, then staying at home is the best response. In fact, there are two symmetric equilibria: one in which all willing individuals participate in the revolution; and another one in which no willing individual goes to the streets.
The previous result does not depend on whether type is a public information or not. However, when a communication technology is available this distinction becomes relevant as shown next.
It is instructive to see how the existence of information affects the outcome of revolts in a perfect information setup in which the willingness to revolt (that is, the type of individuals) is transmitted by the communication technology. In this setting, an individual who observes an action will know whether a willing or an unwilling individual took it. This can be the case, for instance, when the people willing to overthrow the dictator belong to the same social group (e.g. religious association, ethnic groups or social classes), so that individuals know the type of the people who have decided previously. For example, in Egypt the youth in general was unsatisfied with the regime, and so were also the Copts.
We model this situation by introducing the type of the predecessors in the information available to each individual. More specifically, the available information is
\(\varphi _{i}=\{ \tau _{i},\rho _{i},i-\rho _{i}-1,W_{i}\} \) in the case of mass media, where
\(W_{i}\) denotes the amount of willing individuals up to (but excluding) individual
i, that have already decided.
15 As for social media, the assumption about publicly observed types implies that the information of individual
i becomes
\(\varphi _{i}=\{ \tau _{i},\{ a_{j},\forall j<i\} ,\{ \tau _{j},\forall j<i\} \} \). In this setting, both the type and decision of each preceding individual are observed.
Individual i’s strategy is conditioned on the available information. It is defined as \(\mathbf {\sigma }_{i}:\varphi _{i}\rightarrow \{ r,s\} .\) Let \(\mathbf {\Sigma }=\{r,s\}^{n}\) be the game’s strategy space, and let \( \mathbf {\sigma }\in \mathbf {\Sigma }\) be a strategy profile, that is, \( \mathbf {\sigma }=(\mathbf {\sigma }_{1},\ldots ,\mathbf {\sigma }_{n}).\) Let \( h_{i} \) be the history of decisions before individual i, \(h_{i}=\{ a_{1},\ldots ,a_{i-1}\} \).
We find that given a type vector the unique subgame perfect equilibrium with the two communication technologies is the one in which the revolution succeeds, and every willing individual chooses to revolt. In this case, both technologies generate the same behavior in equilibrium.
The rationale for this result is that a willing individual chooses to join the revolution if she observes that already \(t-1\) individuals have revolted. Given this fact, a willing individual who observes \(t-2\) people participating in the revolution decides to revolt if she knows that after her there is at least one more willing individual. Since predecessors’ types are publicly observable, she can infer if there is a willing individual behind her. Iterating this reasoning, a willing individual decides to revolt when up to her sufficient willing individuals have chosen to do so and she anticipates that enough willing citizens behind her will follow suit. The conditions ensuring that this requirement is met at any position imply that all willing citizens choose to participate in the revolution.
As commented above, our assumption on predecessors’ type being public information is plausible in environments where the people willing to overthrow the dictator can be associated to particular groups. Under these circumstances, it is likely that when individuals acquire information they know both the actions and the types of those who have already decided. In this case, the existence of any of the communication technologies guarantees that the revolution triumphs in our simplified environment. Arguably, the uncertainty about how many individuals would participate in a revolution is the main barrier and it makes most of the individuals who are discontented to stay at home. This uncertainty comes—at least partly—from the uncertainty about types that we consider next.
We study now the case where type is private information. Given the untrust and fear generated by dictators in repressive regimes, however, types cannot be observed, what makes the setup more plausible. As argued by several authors (e.g. Ginkel and Smith
1999; Kuran
1991,
1995), decision making in any revolution is clouded by a considerable amount of uncertainty. This uncertainty blurs the information about the public discontent due to several reasons, e.g. the lack of free press, falsified preference revelation to official public opinion polls or the presence of informants penetrating all layers of the society, among others.
When type is private information, communication technologies only transmit to individuals the actions of predecessors. For the mass media technology, the information of individual i becomes \(\varphi _{i}=\{ \tau _{i},\rho _{i},i-\rho _{i}-1\} \), so the amount of citizens of each type who have already decided is not known. In the case of social media, citizen i’s information is given by \(\varphi _{i}=\{ \tau _{i},\{ a_{j},\forall j<i\} \} \). Hence, citizen i cannot distinguish perfectly the type of her predecessors although she knows the exact sequence of decisions. Remember that observing that somebody revolts indicates unambiguously that she is of the willing type, since unwilling citizens always stay at home. However, since willing individuals may choose to stay at home, observing that someone has chosen not to participate in the revolt does not imply that she is unwilling.
Since types cannot be observed, we look for Bayesian Nash equilibrium. The following proposition highlights the importance of the communication technology. We find that under social media being truthful is the unique equilibrium profile; therefore, staying at home when an individual is unwilling and revolting when she is willing is the unique Bayesian Nash equilibrium. With mass media the revolution succeeds only when certain conditions are met.
The logic behind our result is that “identifications of types” is possible under social media, but not under mass media. More specifically, any individual that observes a history of decisions under social media knows that all willing (unwilling) predecessors decided to protest (revolt) in equilibrium. With mass media, there may be situations in which willing individuals find it optimal to stay at home, thus a citizen that observes a history of decisions will only revolt if she is sure that there are enough willing to revolt behind her. Hence, a willing individual that observes
\(t-2\) revolts will always revolt in the case of social media (even if only one individual is left to decide). This is because the individual that observes
\(t-2\) protests can infer the types of the predecessors and knows that there is (at least) one willing individual to decide. In the case of mass media, an individual that observes
\(t-2\) revolts does not know how many willing individuals decided to revolt, thus she can only be sure that the revolution will triumph if she revolts and there are at least
\(n-W+1\) individuals left to decide. While these considerations are embedded in the second part of Proposition
3, the following example is aimed to clarify the mechanism why social media promote revolutions more than mass media. We consider the simple society of Fig.
1 in which there are
\(n=4\) individuals, and three of them are willing to overthrow the dictator (
\(W=3\)). We assume that the revolution will be successful in this society if and only if at least three individuals decide to revolt (
\(t=3\)). We then focus on the worst possible scenario and construct an equilibrium in which everyone stays at home under mass media. We show that there is a unique equilibrium where all willing individuals revolt in the case of social media.
Example 1
Consider the case of
\(n=4,W=3,t=3\).
16 When a communication technology exists, the optimal decision of a willing individual in the last position is obvious. If she observes two people revolting (
\(\rho _{4}=2\)), then she revolts and the revolution triumphs. Otherwise she stays at home. The same is true for a willing individual in position 3 if
\(\rho _{3}=2\) (she best responds by revolting). As a consequence, a willing citizen in position 2 observing one revolting individual (
\(\rho _{2}=1\)) revolts as well, because she anticipates that if she decides to revolt, then the last willing individual (either in position 3 or 4) will follow suit. Thus, in any equilibrium a willing citizen revolts when
\(\rho _{4}=2;\rho _{3}=2\) or
\(\rho _{2}=1\), and stays at home when
\( \rho _{4}\in \{ 0,1\} \) or
\(\rho _{3}=0\). In these last cases, a willing individual knows that the revolution is doomed to fail, so she does not join. Thus, we are left with the following information sets for which a willing citizen’s optimal action is not clear:
\(\rho _{1};\rho _{2}=0\) and
\( \rho _{3}=1\). In words, we do not know yet what a willing citizen does when she is the first to decide; when she is in the second position and observes no protester and when third in the sequence of decision and observes one protester. We show an equilibrium for the case of mass media where nobody revolts on the equilibrium path for some payoffs. Then, we show that it cannot be the case for social media.
Assume the existence of mass media and the following payoffs:
\(u_{w,r,R}=1\),
\(u_{w,s}=0\) and
\(u_{w,r,F}=-10\), that satisfy
\(u_{w,r,R}>u_{w,s}>u_{w,r,F}\). If nobody chooses to revolt in the previous information sets (
\(\rho _{1};\rho _{2}=0\) and
\(\rho _{3}=1\)) and acts optimally at the other information sets (i.e., revolts when
\(\rho _{4}=2;\rho _{3}=2\) or
\(\rho _{2}=1\), and stays at home when
\(\rho _{4}\in \{ 0,1\} \) or
\(\rho _{3}=0\)) then, we end up in an equilibrium without revolution. To show that nobody has a profitable unilateral deviation, take the first individual. Her deviation consists in revolting instead of staying at home. This is profitable if the second individual is willing, because the first individual induces the second one to revolt as well by the arguments we have seen before. In this case, the revolution triumphs and the highest utility is obtained. When the second individual is unwilling (which has conditional probability
\(\frac{1}{3}\)), then the proposed strategies imply that subsequent willing individuals will stay at home and the revolution fails. Therefore, the unilateral deviation is not profitable if and only if
$$\begin{aligned} u_{w,s}>\frac{2}{3}u_{w,r,R}+\frac{1}{3}u_{w,r,F}, \end{aligned}$$
which holds for the proposed payoffs. In the same vein, it is easy to check that if a willing individual in position 2 observes that nobody has revolted yet (
\(\rho _{2}=0\)), then she does not have a profitable unilateral deviation given the prescribed strategy. Consider now a willing individual in position 3 who observes that just one citizen has revolted (
\(\rho _{3}=1\)). According to the prescribed strategy, it is an off-equilibrium decision. Thus, consistent beliefs may include that a willing individual followed the strategy and one willing individual deviated and revolted. Those beliefs imply that she will be followed by the unwilling citizen. Along these lines, it is the inability of a willing individual in position 3 to distinguish whether the first or the second individual stayed at home (and her pessimism) that makes it optimal for the willing individual in position 1 to stay at home. Note also that the equilibrium in which each willing individual participates exists if we simply consider a strategy profile that establishes that willing individuals should revolt when
\(\rho _{1};\rho _{2}=0\) and
\(\rho _{3}=1\).
Restrictions on off-the-equilibrium beliefs could eliminate the equilibrium in which citizens do not revolt. For instance, assume that the first individuals to decide were willing ones with certainty, so that the type vector is (w, w, w, x). In this case it is profitable to deviate unilaterally from staying at home when a willing individual is the first to decide as given the best responses indicated above the revolution will triumph. In fact, our result suggests that repressive regimes may attempt actively to increase uncertainty about the public discontent (even in the form to hinder citizens to have detailed information about participation in protests) so that multiple equilibria, and hence potentially no protests remains an equilibrium.
Next, we show that under social media there is a unique equilibrium in which all willing individuals revolt and succeed in overthrowing the dictator. For a willing individual in the last position the previous arguments apply. Thus, upon observing that two other citizens have revolted (the order does not matter) she revolts as well, otherwise she stays at home. When in position 3, a willing individual joins the protest when observing two protesters. As a consequence, a willing individual in the second position, who observes that the first citizen decided to protest, will revolt as well, anticipating that the last willing individual (either in position 3 or 4) will join the protests as well.
As a next step, let us consider what happens if a willing citizen observes that the first citizen revolted, whereas the second one stayed at home. We denote this by (r, s). By previous reasoning this individual can be sure that the second individual was the unwilling one (a willing individual in the second position would have joined the uprising upon observing that the first citizen revolted), so she knows that the last citizen is willing and anticipating her reaction to observing a history with two individuals revolting she decides to protest as well.
Given the previous argument, a willing individual in the first position chooses optimally to protest, since any history starting with a revolting citizen leads to a successful revolution (either if she is followed by a willing individual who protests herself or when followed by an unwilling one who stays at home, but then the next individual will join the protest and in any case the last willing individual will revolt as well). As a consequence, when observing that the first individual has stayed at home, willing citizens know that she must have been the unwilling one (i.e., individuals get to know that the type vector is (x, w, w, w) and by backward induction they play the unique equilibrium in which all of them revolt). That is, since the willing individual in position 4 (3) revolts when observing two (one) previous revolts, the willing citizen in position 2 will revolt and thus the revolution triumphs, and the dictator is overthrown. Importantly, these arguments apply for any payoffs such that \(u_{w,r,R}>u_{w,s}>u_{w,r,F}\) . It is also worth noting that with social media the outcome is unique because individuals are able to distinguish the histories (r, s) and (s, r) , while with mass media individuals may believe that (with some positive probability) the one who stayed at home is a willing individual.
Proposition
3 establishes also a relationship between the threshold
t and the number of willing individuals in the society
W such that it is possible to construct an equilibrium where the revolution does not triumph in the presence of mass media. In the following example, we show the quantitative difference in the effectiveness of social media versus mass media in fostering revolutions. In particular, Example
2 shows that all willing citizens revolting can be achieved as an equilibrium with mass media only if a relatively low proportion of the society is required to overthrow the dictator.
The intuition behind this result is the following. If an individual can be sure that the revolution succeeds, then she joins the protests. Whenever she may believe that with positive probability the revolution fails, it is possible to find a punishment that is sufficiently large to deter individuals from participating in the protests. A willing individual at position 7, 8, 9 or 10 cannot be sure that among the subsequent citizens there is a willing one, since possibly all of them are unwilling. Hence, if she revolts, in the worst case the number of participants increases only by 1. Thus, a willing individual at these positions only revolts if she observes at least \(t-1\) previous revolts. That is, she revolts if only one more revolting individual is needed to bring the uprising to triumph. At position 6, a willing individual knows that there is for sure one more willing individual behind her and she can convince her to revolt with certainty if she will observe \( t-1\) revolts. Thus, the willing citizen at position 6 revolts if she observes \(t-2\) revolts. A willing individual at position 5 knows that there are at least two willing citizens behind her, but she cannot make sure that both of them will revolt if she decides to revolt. This is the case because possibly the first of the willing individuals is at position 9 and then by previous arguments this citizen cannot be sure that there is another willing individual behind her. Hence, a willing citizen at position 5 knows that by revolting she can prompt for sure one more willing individual to participate, so she revolts if she observes \(t-2\) revolts. The same reasoning applies to willing individuals at position 4 and 3. A willing individual at position 2 knows that there are at least 5 willing individuals behind her. In the worst case, the next one is at position 6 and by previous reasoning even she knows that there is one more willing citizen behind her. So, at position 2 a willing individual knows that she can induce two more willing citizens. Hence, if she observes that \(t-3\) individuals have already revolted, then she joins the protest. The same is true for a willing individual at position 1. Note that a willing individual at position 1 cannot observe anybody revolting, so the threshold that enables a successful revolution is 3 or less citizens. Probably, the unwilling individuals are at the first 3 positions. However, rational actors understanding the game infer, that if with a threshold of 3 none of the first 2 citizens revolted, then it must be due to the fact that those individuals were unwilling. But it reveals at the same time, that there are enough willing individuals behind, so a willing individual at position 3 (or 4) will revolt. Note that with a threshold of 3 (or less) all willing individuals will revolt by the previous arguments.
With higher thresholds, the revolution may fail. In particular, suppose that payoffs are
\(u_{w,r,R}=1\),
\(u_{w,s}=0\),
\(u_{w,r,F}=-10^{100}\) and
\(t=4\). Hence, the dictator would punish very strongly (say, execute) the participants of a failed revolution. Consider the following strategy for willing individuals
where
\(\rho _{i}\) is the number of participants that have chosen
r before individual
i. A willing individual revolts at the information sets specified above. We have to prove that the strategy is optimal in the rest of information sets. Because a willing individual at position 1 observes nobody revolting, the proposed strategy profile states that she should stay at home. What if she deviates? Conditional on the first individual being willing, the probability that the second citizen is willing too is 6 out of 9. Thus, with probability
\(\frac{2}{3}\) the deviation is successful and if the subsequent citizens follow the above strategy, then the revolution triumphs. However, with probability
\(\frac{1}{3}\) the next citizen is unwilling and she will stay at home. Then, if the subsequent willing individuals act according to the proposed strategy, the revolution fails. Given the payoffs it is easy to calculate that the expected utility of a willing individual in position 1 is negative, so the deviation is not profitable. The same argument can be applied for the rest of information sets to show that no deviation is profitable.
In our example, mass media communication guarantees that the revolt succeeds only if \(t\le 3\) individuals’ participation is necessary to overthrow the dictator. To get an idea how our results affect mobilization, we scale up the numbers. In a society of 100 individuals in which 51 people are required to participate in the protests to change the regime, mass media facilitate a successful revolution (with all willing individuals revolting) for any payoffs only if it is known that all individuals (\(W =100\)) are in favor of the revolution. If the required threshold were \(t=49\), mass media would guarantee that the revolution succeeds only if it was commonly known that at least \(W =99\) individuals are of the willing type. With social media, both \(W=51\) and \(W=49\) ensure that the revolt triumphs, respectively. Thus, mass media lead to a successful revolution only if there is a huge amount of people willing to participate in the protests, or if the dictator is very weak (the threshold is very low). In any other case, the dictator could implement a sufficiently high punishment (\(u_{w,r,F}\)) so that revolts may not occur in equilibrium.