Gibilisco 2021, “Decentralization, Repression, and Gambling for Unity”
Trying a new thing where I put a bit more structure on my personal notes about papers I teach, to make those notes suitable for public consumption.
Gibilisco (2021) studies a central tradeoff for the use of repression against secessionist movements — or, really, against marginalized political groups generally. Repression may help the government maintain control now, but it threatens the government’s future power by generating grievances in the repressed group. As grievances increase, so does the populace’s ability to organize for collective action against oppressive rule. Indefinite repression makes an autocrat’s job harder and harder, as the threat of collective resistance becomes more and more credible. The paper builds a formal model around this tradeoff, analyzing the dynamics of governance and grievance.
The most striking result here is that the government may sometimes “gamble for unity”, refraining from repression despite knowing the populace will mobilize against central control. The key insight is that just as grievances may build up with repression, they may also dissipate in the absence of repression. By tolerating some danger to its control in the short run, the government can wait for grievances to get low enough that the dissenters no longer have a credible threat to mobilize opposition. This could end badly for the ruler, with the government losing control in a costly revolution. But if the government is lucky enough to weather the temporary storm, then it is sure to maintain control without even having to use repressive measures.
This paper is an impressive example of how to model strategic interaction in politics without committing to a narrow homo economicus model of human motivation (e.g., Ashworth and Bueno de Mesquita 2014). Repression makes people mad, and that anger persists over time. It’s coherent to acknowledge that emotions play a real role in public responses to repression and that strategic calculations play a role too. If nothing else, I love this paper for knocking down the notion of opportunistic “greed” and irrational “grievance” as mutually exclusive explanations for civil conflict.
Formal model
Two players, Center and Periphery, interact over infinite discrete periods. Each period \(t\) is characterized by a state variable \(g^t\), a nonnegative integer denoting the extent of grievances. The other state variable here is whether P has successfully gained independence. Since there are no choices to be made in that case, I’ll call a period active if P hasn’t gained independence and inactive if it has.
In an active period, the Center first chooses whether to grant independence to the Periphery, repress the Periphery, or do nothing (“hands off”). The sequence of actions from there depends on the Center’s initial choice:
If the Center granted independence, then the game moves to the independence state, and all subsequent periods are inactive.
If the Center repressed, then it retains control this period and there are no further choices to make. Grievances increase next period: \(g^{t + 1} = g^t + 1\).
If the Center did nothing, then the Periphery chooses whether to mobilize for secession. If it does, then it wins — moving the game to the independence state — with probability \(F(g^t)\), a nondecreasing function of the current grievance level.
If the Center mobilizes and loses, or if it doesn’t mobilize, then the game remains active next period. Grievances decrease next period, unless there were none already: \(g^{t + 1} = \max \{g^t - 1, 0\}\).
Gibilisco assumes \(F(0) = 0\), so there is no chance of mobilization succeeding in the absence of grievances. To make my life easier in these notes, I’ll furthermore assume that \(F(\cdot)\) is strictly increasing on \(\{0, \ldots, G\}\) for some \(G > 0\), and that \(F(g) = F(G) := p\) for all \(g \geq G\).
At the end of any period where C still controls the territory, C receives \(\pi_C^C > 0\) and P receives \(\pi_P^C\). At the end of any period where P has gained independence, C receives \(0\) and P receives \(\pi_P^P > \pi_P^C\). (For reasons I don’t understand at this point, Gibilisco declines to normalize P’s payoff from its less-preferred outcome in the way that C’s has been normalized.) Repression costs \(\kappa_C > 0\) for C, and mobilization costs \(\kappa_P > 0\) for P.
If mobilization occurs and succeeds, C suffers an additional cost \(\psi > 0\) — so that if C were certain that mobilization would occur and succeed, it would rather just grant independence in advance. This is modeled as a one-time cost rather than a perpetual loss. I was initially concerned that this meant the cost would become irrelevant as the discount factor \(\delta \to 1\), but Equation 2 below rules this out by placing bounds that are functions of the discount factor.
Gibilisco imposes two assumptions to focus on interesting cases of the model.
We’ll work backward to get to the assumption. Imagine that P would decline to mobilize every period, even with very high grievances. Then P’s payoff would be \(\pi_P^C\) in perpetuity. With sufficiently high grievances, a one-period deviation to mobilization would yield an expected utility of \[p \pi_P^P + (1 - p) \pi_P^C - (1 - \delta) \kappa_P.\] So to rule out the boring cases where P’s threat to mobilize is never credible, Gibilisco assumes the deviation would be profitable when grievances are high enough: \[ \begin{multlined} p \pi_P^P + (1 - p) \pi_P^C - (1 - \delta) \kappa_P > \pi_P^C \\ \quad\Leftrightarrow\quad \pi_P^P - \pi_P^C > \frac{(1 - \delta) \kappa_P}{p}. \end{multlined} \tag{1}\]
We’ll work backward again. Suppose grievances are arbitrarily high, C goes hands-off every period, and P mobilizes every period. C’s long-run expected utility \(V_C\) must solve \[V_C = p [(1 - \delta) (-\psi) + \delta (0)] + (1 - p) [(1 - \delta) \pi_C^C + \delta V_C],\] and therefore \[V_C = \frac{(1 - \delta) [(1 - p) \pi_C^C - p \psi]}{1 - (1 - p) \delta}.\] Immediately granting independence is strictly preferable if \(V_C < 0\), or equivalently \[\psi > \frac{(1 - p) \pi_C^C}{p}.\] Perpetual repression is strictly preferable if \(V_C < \pi_C^C - \kappa_C\), or equivalently \[\psi > \frac{[1 - (1 - p) \delta] \kappa_C - p \pi_C^C}{p (1 - \delta)}.\] The assumption is that at least one of these two alternatives is preferable: \[\psi > \min \left\{\frac{\pi_C^C (1 - p)}{p}, \frac{[1 - (1 - p) \delta] \kappa_C - p \pi_C^C}{p (1 - \delta)} \right\} \tag{2}\]
Equilibrium analysis
Gibilisco solves for Markov perfect equilibria, in which play in each active period depends solely on the level of grievances \(g\) at the outset of the period.
I do wonder how philosophically Markovian the equilibrium here is. I want to interpret the relationship between grievances and the probability of mobilization success in terms of a coordination problem — past repression essentially creates a focal point, making it easier to turn people out for a revolt. But then the implicit underlying story there is one where past actions with no direct payoff relevance are affecting equilibrium selection in the present, violating the Markov spirit. So I think a “strong” Markovian would have to go with something like the final microfoundation Gibilisco gives on p. 1355, where repression has direct effects discouraging participation in the formal economy.
Low grievances
The first main result is that there’s no repression, mobilization, or independence along the path of play when grievances are low enough. The result is that grievances gradually dissipate until eventually hitting bottom, and the state stays together without the need for repression.
First let’s establish that this is an MPE when \(g = 0\). Gibilisco assumes that mobilization has no chance of leading to independence in this case, i.e., \(F(g) = 0\). Under the strategy profile proposed here, C’s long-run expected utility is \(V_C(0) = \pi_C^C\) and P’s is \(V_P(0) = \pi_P^C\). Conditional on C doing nothing, it is best for P not to mobilize, as doing so would be costly with no chance of success. Formally, a one-shot deviation would yield a payoff of \[F(0) \cdot \pi_P^P + [1 - F(0)] \cdot [(1 - \delta) \pi_P^C + \delta V_P(0)] - (1 - \delta) \kappa_P = \pi_P^C - (1 - \delta) \kappa_P.\] Finally, because C receives its first-best payoff of \(\pi_C^C\) under this strategy profile, it cannot strictly benefit from any deviation.
Now think about whether this type of equilibrium is sustainable at \(g = 1\). Again because C is getting its first-best, we only need to worry about a possible deviation to mobilization by P. It is no longer trivial to rule out a profitable deviation by P:
The chance of mobilization succeeding is now \(F(1) > 0\), making it worthwhile if \(F(1)\) is high enough and \(\kappa_P\) is low enough.
There is an imperative to mobilize now, because otherwise grievances will dip to \(g = 0\) next period and then we will be stuck in the equilibrium where P has no credible threat and C maintains control forever. (I’m taking for granted that the equilibria characterized here are unique.)
A one-shot deviation to mobilization is weakly unprofitable for P iff \[ \begin{aligned} \MoveEqLeft{} F(1) \cdot \pi_P^P + [1 - F(1)] \cdot \pi_P^C - (1 - \delta) \kappa_P \leq \pi_P^C \\ &\quad\Leftrightarrow\quad F(1) [\pi_P^P - \pi_P^C] \leq (1 - \delta) \kappa_P \\ &\quad\Leftrightarrow\quad F(1) \leq \frac{(1 - \delta) \kappa_P}{\pi_P^P - \pi_P^C}. \end{aligned} \tag{3}\]
If Equation 3 fails, then equilibrium behavior at \(g = 1\) will have to differ. So suppose for the moment that Equation 3 holds, and think about the sustainability of this same strategy at the next highest level of grievances, \(g = 2\). Then C is again getting its first-best, and we just need to check a no-deviation condition for P not to mobilize. Given that not mobilizing will send the game down to \(g = 1\), at which Equation 3 guarantees that the hands-off/no-mobilization path will persist forever, the no-deviation condition turns out to be nearly identical: \[F(2) \leq \frac{(1 - \delta) \kappa_P}{\pi_P^P - \pi_P^C}.\]
Proceeding inductively this way, we can characterize a cutpoint \(g^-\) that determines the sustainability of the hands-off/no-mobilization equilibrium: \[g^- := \max \left\{g \mid F(g) \leq \frac{(1 - \delta) \kappa_P}{\pi_P^P - \pi_P^C}\right\}. \tag{4}\] Assumption 1 implies that \(F(G) := p > \frac{(1 - \delta) \kappa_P}{\pi_P^P - \pi_P^C}\), and thus \(g^- < G\).
Medium grievances
At grievance levels above \(g^-\), we can no longer expect P never to mobilize after C does nothing. Gibilisco now considers a “gambling for unity” strategy, wherein C continues to do nothing, even knowing that mobilization will take place for time, in order to let grievances dissipate.
Following Gibilisco’s notation, let \(\tilde{V}_C(g)\) denote C’s continuation payoff from gambling for unity starting at state \(g\). The arguments above imply \(\tilde{V}_C(g) = \pi_C^C\) for all \(g \leq g^-\). For any \(g > g^-\), we can define this value inductively via \[\tilde{V}_C(g) = F(g) [(1 - \delta) (-\psi)] + [1 - F(g)] [(1 - \delta) \pi_C^C + \delta \tilde{V}_C(g - 1)].\] In the technical appendix to the paper, Gibilisco shows that \(\tilde{V}_C(\cdot)\) is strictly decreasing above \(g^-\), and that if Assumptions 1 and 2 hold then there is a cutpoint \(g^+\) such that \(\tilde{V}_C(g) < \max \{0, \pi_C^C - \kappa_C\}\) for all \(g \geq g^+\). These conclusions seem reasonable enough, so I’m not going to go through their technical backing.
Now consider a state in between the thresholds, \(g \in (g^-, g^+)\). To confirm that gambling for unity is an equilibrium here, we have to check three things:
P will mobilize. If C does nothing, the expected utility to P from mobilizing is \[\begin{align} \MoveEqLeft{} F(g) \pi_P^P + [1 - F(g)] [(1 - \delta) \pi_P^C + \delta V_P(g - 1)] - (1 - \delta) \kappa_P \\ &\geq F(g) \pi_P^P + [1 - F(g)] \pi_P^C - (1 - \delta) \kappa_P \\ &> \pi_P^C, \end{align}\] where the second inequality follows because \(g > g^-\).
C does not prefer granting independence. Because granting independence yields a payoff of \(0\), this follows from \(g < g^+\).
C does not prefer repression. This is the trickiest one to prove, as it requires a comparison to a state with higher grievances where we haven’t necessarily figured out optimal behavior yet.
The first part of the proof (Lemma 7 in the appendix) is that if the equilibrium involves repressing with positive probability at \(g\), then it cannot involve granting independence with positive probability at \(g + 1\). Independence can never be granted in equilibrium if \(\pi_C^C > \kappa_C\) since then it’s better for C to just repress forever. Otherwise, if \(\pi_C^C < \kappa_C\), then it’s better to grant independence today than to repress today (for net negative present payoff) and then grant independence tomorrow.
Now consider a one-shot deviation to repression. Because \(g < g^+\), this cannot be profitable for C if it would result in indefinite repression. So there must be some finite number \(n\) of periods (perhaps just one) during which repression will be chosen. By the argument above, independence can never be chosen in state \(g + n\), so by process of elimination C must choose hands-off in state \(g + n\). Because \(g + n > g^-\), P will then mobilize for certain. But if C doesn’t want to repress when P is certain to mobilize at state \(g + n\), then it also shouldn’t want to repress when P is certain to mobilize at the lower-grievance state \(g\). Altogether, then, there cannot be a profitable one-shot deviation to repression.
High grievances
Now consider the other end of the spectrum, where grievances are very high, \(g \gg G\). (This is not how things are organized in the paper but I think it makes sense for solving.) In other words, not only is P’s ability to succeed after mobilizing at its highest level, but it will take a very long time for it to come down below that level.
At this extreme high grievance level, Assumptions 1 and 2 together ensure that C would not choose a hands-off policy. If perpetual repression is better than independence, \(\pi_C^C - \kappa_C > 0\), then that will be the outcome.
It’s a bit more complicated if \(\pi_C^C - \kappa_C < 0\). We can’t necessarily sustain an equilibrium where C grants independence with certainty. To see why, suppose \(\delta \approx 1\), and consider an equilibrium in which C grants independence for sure for all \(g \geq \tilde{g}\). If C were to make a one-shot deviation to hands-off for some \(g > \tilde{g}\), then P cannot credibly threaten to mobilize, as they are patient and will win independence at no cost next period anyway. But then it would be profitable for C to make this one-shot deviation and enjoy \(\pi_C^C\) for one more period before granting independence. In this case, the equilibrium at very high grievances must involve mixing. Gibilisco shows that the mixture will be such that the game never transitions back to low/medium grievances, and that P will eventually become independent. The argument for this is pretty finicky and technical so I’m not going to dive deep on it.
The necessity of mixing in equilibrium here is related to the argument in Mark Fey and Brad Smith’s working paper “Committing to Fight”, which unfortunately still has no public draft.
Part of why Gibilisco gets mixing here is the implicit assumption that it may take arbitrarily long for grievances to dissipate. I suspect the solution under high grievances would be simpler if there were simply some maximal feasible level of grievances. It’s a nice technical accomplishment that Gibilisco is able to solve the model even without being able to work backward from a highest possible level of grievance — though I do (genuinely) wonder how realistic it is to assume grievances can grow without bound. This also gets back to the question of just how philosophically Markovian we’re being here, in terms of not conditioning behavior on payoff-irrelevant histories.
On page 1362, Gibilisco interprets “cycles of repression and popular mobilization” observed in real-world cases as potentially reflective of the mixed strategy equilibrium here. I appreciate the effort to ground these in a Markovian model rather than some tit-for-tat folk theorem stuff (especially because the observed path of play under tit-for-tat ought to be all cooperation!) but I’m always a bit leery to attribute “sometimes this thing happens and other times this other thing happens” to mixed-strategy equilibria. Are the decision-makers really indifferent between their options at each point in time? (Or close enough to indifference that we can appeal to some kind of Harsanyi purification logic?)
Comparative statics
The comparative statics analysis is based on a restriction of the model with payoffs \[\begin{alignat}{2} \pi_C^C &= \pi - d \qquad &\pi_C^P &= 0, \\ \pi_P^C &= d &\pi_P^P &= \pi \end{alignat}\] where \(d \in (0, \pi)\) represents the level of “decentralization” in the absence of total independence for the periphery. The parameter \(d\) is inversely related to the stakes of independence for both sides, as we have \(\pi_i^i - \pi_i^{-i} = \pi - d\) for both players in this formulation.
This payoff formulation is the reason that Gibilisco doesn’t normalize \(\pi_P^C = 0\). Technically, all the results here would be the same if we set \(\pi_P^P = \pi - d\) and \(\pi_P^C = 0\). However, it is a bit trickier to see the substantive interpretation of \(d\) as status quo decentralization if it only enters P’s payoff in case of independence. Thanks to Mike for pointing this out to me when I sent him a grumpy text about not normalizing the payoffs.
One result (Observation 1 in the paper) is that decentralization expands the conditions for the “low grievances” equilibrium, where the government chooses hands-off, the periphery does not mobilize, and eventually grievances dissipate. In the decentralization formulation, the relevant cutpoint (defined by Equation 4) turns into \[g^- = \max \left\{g \mid F(g) \leq \frac{(1 - \delta) \kappa_P}{\pi - d}\right\}.\] The claim follows from the fact that the right-hand side strictly increases with \(d\). Intuitively, the result comes down to the credibility of the threat to mobilize — holding fixed the chances of success, it’s harder to mobilize when the stakes are smaller. (Seeing as the function governing the chances of success, \(F(\cdot)\), is a reduced form of an endogenous political process, I do wonder whether it’s sensible to treat this functional form as if it were independent of the degree of status quo decentralization.)
The comparative statics for the other cutpoint are not so straightforward (Observation 2). There’s a bunch of different effects here:
Decentralization lowers the value of the “low grievances” outcome for the center, thereby lowering \(\tilde{V}_C(\cdot)\). This would tend to lower \(g^+\), expanding the “high grievances” set.
Decentralization reduces the credibility of the periphery’s threat to fight, pushing \(g^-\) upward (per above) and thus increasing \(\tilde{V}_C(\cdot)\). This would tend to increase \(g^+\), shrinking the “high grievances” set. Yet at extreme levels (\(d \approx \pi\)), it would result in Assumption 1 being violated, such that all grievances are “small” (\(g^- = +\infty\)).
Decentralization reduces the value of indefinite repression. Up to a point, this would tend to increase \(g^+\), shrinking the “high grievances” set. However, this effect shuts off once the status quo is sufficiently unattractive for the center that it would rather grant independence than repress indefinitely.
This work is licensed under CC BY-NC-SA 4.0
