On the tendency of revolutions to devour their own children

Like Saturn, the Revolution devours its own children.

Jacques Mallet du Pan (1793, p. 80).

Abstract

Genuine revolutions often use violence not only against their enemies but also against their friends. This paper argues that using violence against the proponents of a revolution functions as a way to boost collective action. A weakest-link punishment against the revolutionaries contributing the least to the revolutionary cause can be a way to solve Tullock’s paradox of revolutions. Further implications are developed. First, we show that those benefiting the most from the revolution and facing the lowest cost of punishing will self-select into the group using punishment to boost contributions to the revolutionary cause. Second, we explain that weakest-link punishment can be over-provided when its provision is decentralized, in which case centralizing punishment may be an efficient response. We use the French Revolution as a case study to illustrate our theory. While the proximate cause of the Terror during the French Revolution was to avoid free-riding, its underlying cause may not have been the “blank-slate” mindset of the revolutionaries but the reactionary and inflexible nature of the Ancien Régime’s institutions.

Mathematical appendix

This appendix reproduces the results in Sect. 3.3 in greater detail. Starting with Eq. 10, we have:

$$\begin{aligned} \upsilon ^{kd}(e_\ell ) = \frac{\gamma ^k}{N}\bigg [e_\ell + \sum _{\ell \ne i}e^h_i(P^h_i) +\sum _{\ell \ne j}e^l_j(P^l_j)\bigg ]- \frac{e_\ell ^2}{2} - c^d p_\ell - \frac{P^k_\ell }{N^k_{min}} \end{aligned}$$

(14)

Using 9 according to which $e^k_\ell =\frac{\gamma ^k}{N}+2 {P^k_\ell }^{0.5}$, individuals in the group putting a high value on the Revolution ($\gamma ^h$) and having a low cost of punishment ($c^l$) have the following utility:

$$\begin{aligned} \upsilon ^{hl}(e_\ell ) = \frac{\gamma ^h}{N}\bigg [e_\ell + (N^h-1)\big [\frac{\gamma ^h}{N}+2 {P^h_\ell }^{0.5}\big ] +N^l\big [\frac{\gamma ^l}{N}+2 {P^l_\ell }^{0.5}\big ]\bigg ]- \frac{e_\ell ^2}{2} - c^l p_\ell - \frac{P^h_\ell }{N^h_{min}} \end{aligned}$$

(15)

Keeping in mind that $P^h_{\ell } = \sum _{s\ne \ell }p^h_{i}$ and $p_\ell = p^h_\ell + p^l_\ell$ in the equation above, the solution to the unconstrained maximization problem with respect to $p_\ell ^h$ is:

$$\begin{aligned} \frac{\partial \upsilon ^{hl}}{\partial p^h_\ell }=0=\frac{\gamma ^h}{N}(N^h-1){P^h_\ell }^{-0.5}-c^l \end{aligned}$$

$$\begin{aligned} p^h_\ell =\bigg (\frac{(N^h-1)}{N}\frac{\gamma ^h}{c^l}\bigg )^2 -\sum _{s\ne \ell } p^h_s \end{aligned}$$

(16)

Equation 16 holds only when $(\frac{(N^h-1)}{N}\frac{\gamma ^h}{c^l})^2 > \sum _{s\ne \ell } p^h_s$, otherwise we have a corner solution and $p^h_\ell =0$. Similarly:

$$\begin{aligned} \frac{\partial \upsilon ^{hl}}{\partial p^l_\ell }=0=\frac{\gamma ^h}{N}N^l{P^l_\ell }^{-0.5}-c^l \end{aligned}$$

$$\begin{aligned} p^l_\ell =\bigg (\frac{N^l}{N}\frac{\gamma ^h}{c^l}\bigg )^2 -\sum _{s\ne \ell } p^l_s \end{aligned}$$

(17)

Equation 16 holds only when $(\frac{N^l}{N}\frac{\gamma ^h}{c^l})^2 > \sum _{s\ne \ell } p^l_s$, otherwise we have a corner solution and $p^l_\ell =0$.

The same mathematical steps apply to the three other groups. The levels of punishment for members of the group with a high value given to the Revolution ($\gamma ^h$) and having a high cost of punishment ($c^h$) are:

$$\begin{aligned} p^h_\ell =\bigg (\frac{(N^h-1)}{N}\frac{\gamma ^h}{c^h}\bigg )^2 -\sum _{s\ne \ell } p^h_s \end{aligned}$$

(18)

$$\begin{aligned} p^l_\ell =\bigg (\frac{N^l}{N}\frac{\gamma ^h}{c^h}\bigg )^2 -\sum _{s\ne \ell } p^l_s \end{aligned}$$

(19)

The levels of punishment for those giving a low value to the Revolution ($\gamma ^h$) are:

$$\begin{aligned} p^h_\ell =\bigg (\frac{N^h}{N}\frac{\gamma ^l}{c^d}\bigg )^2 -\sum _{s\ne \ell } p^h_s \end{aligned}$$

(20)

$$\begin{aligned} p^l_\ell =\bigg (\frac{(N^l-1)}{N}\frac{\gamma ^l}{c^d}\bigg )^2 -\sum _{s\ne \ell } p^l_s \end{aligned}$$

(21)