Protests, cascades, and revolutions
In a previous post, we looked at tipping in a business context. We showed, for example, how products with strong network effects can dominate the market in equilibrium. But tipping is not confined to strategic situations in business. They’re common in cultural and political settings too. In this post, we illustrate the role that critical mass can play in protests and social revolutions.
Let’s imagine there is some social issue that affects a small village. Maybe there’s an autocratic mayor that the town finds undesirable. Let’s say also that there’s little hope for change unless people partake in some mass demonstration. What does it take to make this happen?
Intuitively, we know that social revolutions are more effective when people participate in large numbers. In this game, we’ll say that the payoff to each individual from protesting is equal to 10 x P, where P is the number of people protesting. And the payoff from non-participation is zero. *
Individual payoff = 10 x P
* A zero payoff from non-participation implies that even if the protest is successful, the non-participator will not reap the benefits. One interpretation of this game is that protestors fight only for their own rights and wants. Variants of this game introduce the option for free riding, which can produce equally interesting results.
Radicals and plebeian
While protests can generate results, they invoke a personal cost. Protestors might face the risk of arrest or reputation damage. Others face an opportunity cost. After all, that’s time they could spend at work or with their family in this otherwise quaint village. Let’s say there are two distinct groups of people in this game: 100 radicals and 1,000 plebs. The personal cost of protesting is 500 to radicals, and 5000 to plebs. That is to say, plebs, for whatever reason, prefer not to protest with greater intensity.
Critical mass
Remember that people will only protest when the payoff from participation exceeds the personal cost of doing so. Because the payoff depends on the total number of participants, we can derive the implied critical mass that induces participation from each social group.
The table below summarises the players, group size, personal cost and implied critical masses of the game. It tells that the critical mass to induce radical participation (50 protestors) is much lower than the critical mass to induce plebeian participation (500 protestors).
Individual cost from protesting
| Number of citizens | Personal cost | Implied critical mass | |
| Radicals | 100 | 500 | 50 |
| Plebs | 1,000 | 5,000 | 500 |
Finding Nash equilibria
So, what actually happens in this game? As you can guess, it depends a lot on the population’s beliefs and expectations. For example, if the total population believes that P < 50, we get a Nash equilibrium in which nobody protests. In this scenario, the payoff from protesting remains below the personal cost of protesting, both for the radicals and the plebs.
However, if the population believes that 50 ≤ P < 500, we have a Nash equilibrium in which every radical participates, but none of the plebs do. Here, the payoff exceeds the cost of protesting for the radicals. But the net payoff remains negative for the plebs. Finally, if the population believes that P ≥ 500, then every citizen joins the protest in Nash equilibrium. Beyond the critical mass of 500, every town member finds it in his or her best interest to participate.
Coordination failure and social strategies
This game exhibits multiple Nash equilibria because payoffs rise with participation, while personal costs differ between social groups. The equilibrium that arises depends mightily on the collective’s beliefs. In this way, expectations can create a self-fulfilling prophecy. If people have little faith in one another, the coordination necessary for successful social movements breaks down.
It follows that to sustain participation, radicals must find ways to generate critical mass, and common knowledge about critical mass. Social movements tend to begin in tight-knit community groups for this very reason, where it is less ‘costly’ to establish early momentum.
By contrast, an authoritarian regime that seeks to preserve the status quo may seek to crush beliefs and the formulation of common knowledge, and to raise the personal cost of participation. Dictators will control the military, manipulate the press, and contort the judicial systems for these very reasons.
Free riding
The simplified game we presented ignores the temptation to free ride. That is, if the benefit of protesting accrues to everybody, there is additional incentive not to participate given the potential benefit and personal cost. More advanced games account for this nuance (try specifying and solving the game for yourself).
Cascades
Some variants of the game allow for the possibility of cascades. That is, once critical mass is reached at the lowest level, it can generate sufficient participation such that critical mass is reached at the next level, generating even greater participation, and so on.
We can find instances of behavioural cascades in financial markets. For example, as Robert Shiller illustrates in Irrational Exuberance, newfound excitement for an asset class (say cryptocurrencies) may increase investor demand and purchases. This raises the asset’s price, increasing investor enthusiasm and expectations further. This may induce new investors to join the buying frenzy, fuelling the positive loop once again.
Automatons
People aren’t always calculative automatons. They make decisions on many dimensions, many of them emotional. People don’t always make choices on payoffs to themselves, but on the payoffs to their families, neighbours and future generations. The simple cold world of game theory may not always capture the essence of human decision making. You have to specify their payoffs and choices carefully.