Esteban, Morelli, and Rohner 2015, “Strategic Mass Killings”
Esteban, Morelli, and Rohner (2015) model the dynamic problem of mass killings, where a group in power deliberately decimates the population of a rival group. Even if we imagine that the ruling group somehow does not have a moral revulsion to such killings, it is puzzling that they occur. From a sheer economic point of view, it would be less costly for the ruling group to keep the opposition alive and tax its surplus.
The analysis in this paper locates the motivation for mass killing in a kind of commitment problem (Powell 2006). If the current ruling group were guaranteed to stay in power, there would be no motivation to eliminate the opposition rather than tax the opposition. But in fact, the opposition poses an ongoing threat to seize power from the ruling group. Because the opposition cannot commit not to try to overthrow the empowered group, the ruling group may decide the one-time cost of eliminating the opposition is lower than the cost of deterring a revolutionary threat forever.
The broad design of the model captures important features of this dynamic strategic interaction. I am surprised a model like this wasn’t published earlier. The main comparative static — that the likelihood of mass killing increases with the resource rents available to the group in power — is well explained, and I suspect would emerge from a much broader class of models with similar dynamics. Other bits of the analysis seem highly specific to arbitrary choices of solution method and functional forms.
Formal model
Two players, groups \(i\) and \(j\), interact over infinite discrete periods with a common discount factor \(\delta \in [0, 1)\). Each period is characterized by three state variables:
- Which group is in power, \(h^t \in \{i, j\}\). (Then \(k^t\) is used to denote the group not in power.)
- Population size of group \(i\), \(N_i^t \geq 0\).
- Population size of group \(j\), \(N_j^t \geq 0\).
In each period, the society produces output \[S^t = \beta (N_i^t + N_j^t) + R,\] where \(\beta > 0\) represents labor productivity and \(R \geq 0\) represents the per-period output of a natural resource stock.
The sequence of play in each period is as follows.
The group in power offers a share \(x_h^t \in [0, 1]\) of the period’s surplus to the other group. There may be a “fairness” constraint on such offers, namely that \(x_h^t \geq \underline{\lambda} \frac{N_k^t}{N_h^t + N_k^t}\) for some \(\underline{\lambda} \in (0, 1]\).
Both groups simultaneously decide whether to have peace or conflict.
If both choose peace, then the offer is implemented and the incumbent remains in power.
If either chooses conflict, then this period’s surplus is reduced by \(d > 0\). Each group’s probability of winning the conflict is proportional to its size. The winner consumes the entire surplus this period.
Whichever group is in power may eliminate members of the other group. There is a cumulative upper bound \(\overline{M}\) on such killings. (It’s not entirely clear from the model description whether this is summed across players, though I would think the natural assumption would be no?)
The basic tradeoff regarding mass killings is clear enough. Reducing the other group’s size decreases the credibility of their conflict threat and, when the fairness constraint on offers is present, effectively loosens that constraint. However, doing so also permanently reduces the size of the surplus that is produced each period. Clearly this part of the tradeoff does not bite hard when resource rents \(R\) are large relative to labor productivity \(\beta\).
Equilibrium analysis
From the setup I expected an analysis of Markov perfect equilibria, but that is not what the authors do. They focus on efficiency bounds, looking for the social welfare maximizing equilibrium in the set of subgame perfect equilibria. That seems fine enough if the purpose is to ask “To what extent are mass killings and the associated inefficiencies avoidable?” It’s a bit odder when the goal is to connect to large-N empirics — as in the back half of this paper — as it’s a bit hard to imagine a scenario where mass killings are a live option and yet distinct groups have coordinated their expectations around a mutually beneficial equilibrium.
Unconstrained case
First the authors consider the situation in which \(\overline{M}\) is arbitrarily high and \(\underline{\lambda} = 0\), so that there are no constraints on offers or killings.
The aim is to support an efficient equilibrium through threats of reversion to a very bad equilibrium, so the first step is to characterize what a very bad equilibrium looks like. Typically in models with simultaneous choices of whether to fight, there’s an “always fight” equilibrium (e.g., Ramsay 2011; Schram 2021), as it’s trivially a best response for me to fight if you’re certain to choose to fight. But this is not true in the present model with the possibility of unlimited killings. Suppose the strategy is for both players to fight forever with no mass killings, and consider the decision node beginning where \(i\) has just won the conflict in a particular period. Its present-period consumption is fixed at this point, so we can just consider future payoffs. If the winner proceeds as proposed, its continuation utility is \[\beta N_i + \frac{N_i}{N_i + N_j} (R - d)\] (see below for a derivation). Suppose instead the winner were to eliminate the opponent entirely. This would reduce the per-period surplus to \(\beta N_i + R\), but would eliminate the cost of conflict as well as the risk of losing in future periods. So the continuation utility would be \(\beta N_i + R\), making the deviation strictly profitable.
Assume both players choose to fight in all periods, with no killings by the winner. Because the incumbent doesn’t have any advantage in case of conflict, each player’s continuation value is the same regardless of whether they start the period in power. Letting \(V_i\) denote the continuation payoff, we have \[ \begin{aligned} \MoveEqLeft{} V_i = \frac{N_i}{N_i + N_j} \left\{(1 - \delta) [\beta (N_i + N_j) + R - d] + \delta V_i\right\} + \frac{N_j}{N_i + N_j} \delta V_i \\ &\quad\Leftrightarrow\quad V_i = (1 - \delta) \beta N_i + \frac{(1 - \delta) N_i}{N_i + N_j} (R - d) + \delta V_i \\ % &\eqq (1 - \delta) V_i = (1 - \delta) \beta N_i + \frac{(1 - \delta) N_i}{N_i + N_j} (R - d) \\ &\quad\Leftrightarrow\quad V_i = \beta N_i + \frac{N_i}{N_i + N_j} (R - d). \end{aligned} \]
It strikes me as strange that the tradeoff always cuts in favor of mass killing rather than continued conflict. If \(d\) is small and \(N_i\) is high, it seems like it would be better to fight forever and continue to extract from the other group than to eliminate part of the surplus. I’m pretty sure the stark result here comes from the functional form assumption that productivity and contest success probability are both exactly proportional to group size. With a more general contest success function or production function, I think the result would be more conditional.
In any case, rather than a forever war, the reversion equilibrium here will be one in which war occurs in the first period, and any war winner fully exterminates the opponent. A player’s expected utility from entering the reversion equilibrium is \[V_i^o = \frac{N_i}{N_i + N_j} [(\beta N_i + R) + (1 - \delta) (\beta N_j - d)].\] As mutual fighting is trivially an equilibrium, the only thing to check here is that full killing is optimal. Consider when \(i\) wins the conflict and makes a one-stage deviation to killing \(M \in [0, N_j)\) of the losing side. Present period consumption is sunk, so we just need to compare continuation values. The continuation value from the proposed full killing strategy is \(\beta N_i + R\). The continuation value from the deviation is \[ \begin{aligned} V_i' &= \frac{N_i}{N_i + N_j - M} \left\{(1 - \delta) [\beta (N_i + N_j - M) + R - d] + \delta [\beta N_i + R]\right\} \\ &= \frac{N_i}{N_i + N_j - M} \left[(\beta N_i + R) + (1 - \delta) (\beta N_j - \beta M - d)\right] \\ &< \frac{N_i}{N_i + N_j - M} \left[(\beta N_i + R) + (\beta N_j - \beta M)\right] \\ &= \beta N_i + \frac{N_i}{N_i + N_j - M} R \\ &< \beta N_i + R. \end{aligned} \] Thus, a deviation to less than full killing is unprofitable. Once again, the congruity between the production function and the contest success function is critical to obtain the result.
The question now is whether a peaceful equilibrium, with no conflict or killings, can be sustained through a threat of reversion to a conflictual equilibrium of this form. Suppose \(i\) is in power in the initial period. Consider a strategy profile in which:
- Along the path of play:
- \(i\) offers \(x_i\) in each period
- Both players choose peace following an offer of \(x_i\)
- There are no mass killings
- If any deviation has occurred:
- Both players choose conflict following any offer
- The winner at the end of the period fully eliminates the other group
We need to check for no profitable deviation at each stage of the interaction here.
No deviation to mass killing. Consider \(i\)’s decision after both players have chosen peace. Present period consumption is sunk, so we just need to compare continuation values. The proposed strategy profile gives \(i\) a continuation payoff of \((1 - x_i) [\beta(N_i + N_j) + R]\). A deviation to full mass killing (the arguments above ruling out partial mass killing as optimal carry over here) would yield a continuation payoff of \(\beta N_i + R\). Hence an equilibrium of the form proposed here requires \[x_i \leq \frac{\beta N_j}{\beta N_i + \beta N_j + R}.\]
No deviation to conflict by i. It seems like this should be covered by the previous case, but I guess not if the discount factor is super low. It also does not seem to be considered in the text of the paper; the authors only consider deviations to mass killing and to “exploitation” where the governing group makes a lowball offer at the start of the period. Since this condition is almost certainly redundant as \(\delta \to 1\), I’ll skip it.
No deviation to conflict by j. The less trivial condition here is that the offer needs to be high enough to keep the disempowered group from resorting to conflict. Under the proposed strategy profile, \(j\)’s payoff is \(x_i [\beta(N_i + N_j) + R]\). A deviation to conflict would yield an expected payoff of \[\frac{N_j}{N_i + N_j} [\beta N_j + R + (1 - \delta) (\beta N_i - d)].\] Consequently, peace requires \[x_i \geq \frac{N_j}{N_i + N_j} \cdot \frac{\beta N_j + R + (1 - \delta) (\beta N_i - d)}{\beta (N_i + N_j) + R}. \tag{1}\] Combining this with the condition for no deviation to mass killing by \(i\), peace is sustainable only if \[ \begin{aligned} \MoveEqLeft{} \beta N_j \geq \frac{N_j}{N_i + N_j} [\beta N_j + R + (1 - \delta) (\beta N_i - d)] \\ &\quad\Leftrightarrow\quad\beta (N_i + N_j) \geq \beta N_j + R + (1 - \delta) (\beta N_i - d) \\ &\quad\Leftrightarrow\quad R \leq \delta \beta N_i + (1 - \delta) d \end{aligned} \tag{2}\] This is not quite the condition listed in Equation 4 in the paper; I can’t tell if that’s because I normalize per-period payoffs by \((1 - \delta)\) and they don’t; or because I made an algebra mistake; or because they made an algebra mistake.
I used one of my 10 monthly Deep Research queries to ask ChatGPT to get to the bottom of the difference in my Equation 2 and the authors’ Equation 4. Here’s the chat log. ChatGPT seems to think they derived their constraint just by seeing when the disempowered group’s reservation value exceeds 1, without considering that the ruling group’s reservation constraint also must be met. But that doesn’t check out either — the right-hand side of my Equation 1 is always strictly less than 1.
No deviation to a lowball offer by i. This would result in conflict, which we have already ruled out as unprofitable. (The paper instead seems to consider that the consequence would be acceptance this period followed by conflict in subsequent periods, but the upshot is the same.)
Comparative statics The key comparative statics come from the condition on \(R\), Equation 2.
Natural resource rents make mass killings more likely. This is a kind of standard “conflict resource curse” result (Hodler 2006). If the bulk of the surplus isn’t a function of endogenous labor, then the main disincentive to eliminate the other side disappears.
Labor productivity makes mass killings less likely. Basically the opposite of the logic of the previous comparative static.
Costliness of conflict makes mass killings less likely. Here the logic is more like bargaining models of conflict with commitment problems (Fearon 1995; Powell 2006). Peace here is sustained in part by fear of reversion to an equilibrium where conflict occurs. The greater the costs of that conflict, the wider the bargaining range is.
A larger group in power makes mass killings less likely. If the group in power is large, then the disempowered group’s threat to engage in conflict is weak. They are very likely to lose and be killed. Consequently, it costs less to buy them off, making it easier to have an equilibrium where the group in power stays in power.
Constrained case
Constraints on offers. Section V.A of the paper considers the case where \(\underline{\lambda} > 0\), so the group in power is constrained away from making certain lowball offers. The exact thresholds change in this case, but the qualitative results don’t really change.
Constraints on killings. On the other hand, a number of things may change if \(\overline{M}\) is small enough to be binding.
First off, the nature of the reversion equilibrium might change. It’s just generally a bit trickier to define the reversion equilibrium in this context, as it is no longer feasible to eliminate the other player and then remain at peace forever. So now the reversion equilibrium will necessarily entail both groups engaging in conflict forever after doing all the killing they’re going to do — the question is whether that killing is worth it in the first place. The answer turns out to be “yes” if \(R\) is large enough (surplus dominated by fixed resources) and “no” if \(R\) is small (surplus dominated by labor that goes away if killings are conducted).
When a peaceful equilibrium is not sustainable, the shape of the efficient non-peaceful equilibrium may also change. There is potential path dependence in the extent of mass killings, where the total occurrence may depend on who wins the conflict in the first period.
All that said, the important comparative statics on the parameters besides \(\underline{\lambda}\) and \(\overline{M}\) themselves seem to remain roughly the same in this case.
The comparative statics on \(\overline{M}\) itself are nonmonotone. The sustainability of an equilibrium with zero on-path killings is highest when \(\overline{M}\) is moderate. I found the explanation for this comparative static in the paper hard to follow, until I noticed the connection between this and Luo (2022). In his model, power aggrandizement by the government functions somewhat similarly to non-eliminative mass killings here, insofar as it permanently reduces the other player’s ability to carry out an overthrow threat. His main result is a similarly nonmonotone comparative static, where the likelihood that the government pushes all the way to the “power frontier” — akin to carrying out \(\overline{M}\) mass killings here — is highest when that frontier is either very low or very high.
This work is licensed under CC BY-NC-SA 4.0
