RELATED FORMAL LITERATURE

Our paper studies the main features of mass purges. By contrast, the literature has emphasized elite purges: the shaping of an autocrat’s inner circle and the resulting contests for power (e.g., Acemoglu et al. Reference Acemoglu, Wolitzky, Egorov and Sonin2008; Bueno de Mesquita and Smith Reference Bueno de Mesquita and Smith2015; Svolik Reference Svolik2009). Mass purges can complement power struggles (e.g., screening agents favorable to her opponents), but need not (e.g., cleansing the single-party of opportunists). A critical aspect of our analysis is that mass purges are an instrument of accountability used to motivate and select subordinates.

As such, our paper is in close connection with a large literature on agency problems in economics (see Laffont and Martimort Reference Laffont and Martimort2009 for a textbook treatment) and political science (see Ashworth Reference Ashworth2012 for a review). Most of this literature consider the relationship between one principal and one (possibly representative) agent. Some papers, however, analyze settings in which a principal faces multiple agents, as in our framework. Beginning with Holmström (Reference Holmström1982), several scholars have examined the problem of moral hazard in teams. Others have looked at agents’ effort on an individual project when promotion is the only tool available to the principal (e.g., Green and Stokey Reference Green and Stokey1983; Lazear and Rosen Reference Lazear and Rosen1981). Yet, another strand of the literature studies the ex-ante principal’s choice of one agent among many in democracy (e.g., Dewan and Hortala-Vallve Reference Dewan and Hortala-Vallve2011; Dewan and Myatt Reference Dewan and Myatt2010; Gailmard and Patty Reference Gailmard and Patty2012) or in autocracy (e.g., Egorov and Sonin Reference Egorov and Sonin2011; Zakharov Reference Zakharov2016). Few of these works focus on the relationship between performance and selection when the principal faces many agents as is the case in our paper (one exception is Jiang, Montagnes, and Wolton Reference Jiang, Montagnes and Wolton2017). Further, our paper innovates on the literature by looking at the principal’s choice of replacing a mass of subordinates and, as a result, studies different strategic interactions than previous works.

Our paper also joins a small literature interested in the impact of violence. In Esteban, Morelli, and Rohner (Reference Esteban, Morelli and Rohner2015), a leader can kill a mass of opponents to increase her chances of survival, but, by doing so, lowers her rents in the future. In bargaining settings, Bloch and Rao (Reference Bloch and Rao2002) and Dal Bó, Dal Bó, and Di Tella (Reference Dal Bó, Dal Bó and Di Tella2006) highlight how violence, or the threat thereof, strengthens a player’s position and thus improves his payoff. Acemoglu and Wolitzky (Reference Acemoglu and Wolitzky2011) model slavery as a principal-agent relationship in which the slave-owner can reduce the value of her slave’s outside option and, thus, his wage using coercion. Landa and Tyson (Reference Landa and Tyson2017) show how coercive leadership is necessary to coordinate subordinates who are uncertain of their leader’s preferences. All of these works focus on various moral hazard problems, ignoring adverse selection, and so cannot examine the effect of violence on screening and selection as our paper does.

EVIDENCE ON MASS PURGES

Before proceeding, we summarize historical evidence on mass purges, highlighting some of their key characteristics. Due to the volume of secondary sources available, we primarily focus on the USSR and China, and discuss purges in other countries at the end of the section.

The Communist Party of the Soviet Union (CPSU) experienced mass purges (“chistka”—a sweeping, a cleansing) in 1919, 1921–23, 1924, 1928, 1929, 1931, 1933–34, 1935, 1936, 1949, 1951–53, and 1971 (Brzezinski Reference Brzezinski1956; Getty Reference Getty1987; Rigby Reference Rigby1968; Schapiro Reference Schapiro1977). In China, mass purges (“quingchu”—to weed out) were part of rectification campaigns, which happened under Mao Zedong in 1947–48, 1950, 1951–54, 1953, 1957, 1957–58, 1959–60, 1960–61, 1962–63, and 1964–65. Due to their specificities, these lists do not include the Great Terror in the USSR (1936–38) and the Chinese Cultural Revolution (1966–76). We return to these two events in the discussion section when we discuss the implications of our results.

The large number of purges in both countries is not accidental. Purges were a system of government (Brzezinski Reference Brzezinski1956) and leaders in the USSR and China sought to regulate their periodicity (Getty Reference Getty1987; Teiwes Reference Teiwes1993). The goals of these purges were twofold. First, they provided the necessary momentum to accomplish the grand designs of the totalitarian regimes (Brzezinski Reference Brzezinski1956) and sustain a high level of activity (Teiwes Reference Teiwes1993). Second, they were meant to “cleanse the system in anticipation” (Brzezinski Reference Brzezinski1956, 19). By removing a proportion of party members, mass purges allowed for the influx of new members (Brzezinski Reference Brzezinski1956; Teiwes Reference Teiwes1993) drawn from the pool of candidates to the party (Rigby Reference Rigby1968).

Mass purges did not target specific individuals. All rank-and-file party members, representing millions of individuals (Teiwes Reference Teiwes1993), were affected. The most common accusation against purged members was one of opportunism, having joined the party for the social and economic benefits—reserved positions, special shops, etc.—associated with membership (Brzezinski Reference Brzezinski1956; Getty Reference Getty1987; Teiwes Reference Teiwes1993). Getty (Reference Getty1987, 38, emphasis in original) asserts that “[i]n the majority of purges, political crimes or deviations pertained to a minority of those expelled” from the CPSU. Accusations of opportunism, however, may simply have been a way to easily get rid of recalcitrant members and to hide substantial opposition within the party. Indeed, Rigby (Reference Rigby1968) argues that the purges of the CPSU in the 1930s paved the way for the Great Terror and the show trials of 1936–38. Whatever the justification for the purges, congruence with the autocrat seems to have been difficult to observe. Communist leaders had to use indirect signals such as work performance. Any failure in a member’s professional activity “automatically [became] a case of political accountability” (Brzezinski Reference Brzezinski1956, 86).

Mass purges occurred in other communist regimes across the world (e.g., Albania in 1948–52, Bulgaria in 1948–50, East Germany in 1950–53 and 1956–60, Hungary in 1950, 51, and 55, North Korea repeatedly before 1957; see Janda Reference Janda1980) and even in communist parties in democracies (e.g., the exiled Spanish communist party in France and the French communist party; see Wolton Reference Wolton2017). But mass purges were not just a communist phenomenon. In Nazi Germany, the Sturmabteilung (SA) was purged in 1935–36 following the “Röhm putsch,” the Schutzstaffel (SS) was purged in 1933–35 (Hohne Reference Höhne1981), and the party itself (NSDAP) was purged in 1938 (Orlow Reference Orlow1969). Lack of commitment to the Nazi cause seems to have been the main driver of the purges. Even in the SA in the years following the Night of Long Knives, “nearly all of those expelled or punished in the SA’s own internal purge were found guilty of moral failings” (Campbell Reference Campbell1993, 660). In Fascist Italy, the party experienced a mass purge in 1931, with opportunism again the primary cause of removal (Morgan Reference Morgan2012). In the 1980s, the Syrian Ba'ath party also removed from its ranks members and (principally) supporters who became “part of the regime, with the aim of profiting from material and other advantages which it supposedly offered” (Van Dam Reference Dam2011, 128). By contrast, Saddam Hussain seems to have used the purge of 1979 in Iraq to tighten his hold on power (Coughlin Reference Coughlin2005). In all of these historical cases, directly observable traits did not seem to play a major role in the autocrat’s purging decision (though new members were targeted disproportionately). Instead, the authoritarian leader arguably had to rely on indirect indicators, as in the framework we now describe.

SETUP

This section and the five that follow present and analyze our theoretical framework as well as various extensions. In the discussion section, we summarize our main findings and use historical and recent events to illustrate them. A non-technical reader may wish to proceed directly to this section before returning to the formal part of the paper.

We study a two-period (t ∈ {1, 2}) model with an autocrat (A) and a [0, 1] continuum of agents, indexed by the superscript i. Each agent is characterized by a type τ ∈ {c, nc}, where τ = c denotes a congruent type and τ = nc non-congruent. An agent’s type is his private information. However, it is common knowledge that there is a proportion λ of congruent types within current subordinates.

Each period, agent i exerts effort $e_t^i \in \left[ {0,1} \right]$ on an individual and independent project at cost ${{{{\left( {e_t^i} \right)}^2}} \over 2}$. The project can succeed (denoted ω _t = S) or fail (ω _t = F). The probability agent i’s project is successful equals $e_t^i$. The autocrat observes the first-period outcome of the agent’s project, but not his effort. We denote α _F and α _S the proportion of agents who fail and succeed, respectively.

After observing all first-period project outcomes, the autocrat decides to purge a proportion κ _F of agents who failed and κ _S of those who succeeded, yielding a purge breadth of κ = α _F × κ _F + α _S × κ _S. A purge is costly. This cost is captured by the function C(κ) with C(0) = 0 and (for ease of exposition) marginal cost C′(κ) = C ₀ + C ₁ × κ, C ₀ ≥ 0, C ₁ > 0. When a subordinate is purged, he is replaced by a new agent drawn from the replacement pool. The proportion of congruent types among the replacement pool is r with r = λ (we relax this assumption when we discuss another form of purges and in our extensions).

A purged agent faces (i) the loss of privileges associated with being a regime insider and (ii) a direct loss L which corresponds to the “intensity of violence” of the purge. The autocrat determines the intensity of violence at the beginning of the game at a cost ζ(L) with ζ(0) = 0 and marginal cost ζ′(L) = ζ ₀ + ζ ₁L, ζ ₀ ≥ 0 and ζ ₁ > 0.

In period 1, an agent enjoys a benefit R ≥ 0, which captures all special privileges accorded to regime insiders in autocracies. In addition, if he is not purged from the party, k ⁱ = 0, an agent obtains a payoff v(ω, τ), which depends on the outcome of his project and his type. We assume that v(F, τ) = 0, v(S, c) > 0, and v(S, nc) < v(S, c). To avoid dealing with corner solutions that only complicate the analysis, we further impose v(S, nc) ≥ 0. When purged, k ⁱ = 1, an agent suffers the loss L ≥ 0. Agent i’s first-period payoff assumes the following form:

(1)$$u_1^i\left( {e_1^i,\tau } \right) = R + \left( {1 - {k^i}} \right) \times v\left( {{\omega _1},\tau } \right) + {k^i} \times \left( { - L} \right) - {{{{\left( {e_1^i} \right)}^2}} \over 2}.$$

In period 2, there is no subsequent purge and a (surviving or new) subordinate i’s payoff can be expressed as the sum of the benefit R and the net gain from a successful project:

(2)$$u_2^i\left( {e_2^i,\tau } \right) = R + v\left( {{\omega _2},\tau } \right) - {{{{\left( {e_2^i} \right)}^2}} \over 2}.$$

To simplify the exposition, we assume throughout that agents do not discount the future.

The autocrat gets a positive payoff—normalized to 1—when a subordinate’s project is successful, and she receives 0 otherwise. The autocrat, thus, seeks to maximize the proportion of successful projects, which is equal to agents’ average effort in each period. In the first period, the autocrat also bears the costs of investing in the intensity of violence and of purging. Letting ${\bar{e}_t}$ denote the average effort in period t ∈ {1, 2}, we can thus express the autocrat’s first-period and second-period payoffs, respectively, as follows:

(3)$$u_1^A\left( {\kappa ,L} \right) = {\bar{e}_1} - C\left( \kappa \right) - \zeta \left(L \right);$$

(4)$$u_2^A = {\bar{e}_2}.$$

The autocrat discounts the future with factor β ∈ (0, 1), which captures, among other things, the risk (perceived or real) of losing power between the two periods.

To summarize, the timing of the game is:

Period 1:

1. 1. The autocrat chooses the intensity of violence L ≥ 0;

2. 2. Agent i chooses his effort $e_1^i$ after privately observing his type τ ⁱ ∈ {c, nc};

3. 3. Each project outcome (ω ₁ ∈ {S, F}) is determined by Nature and observed by the autocrat. The autocrat chooses to purge a proportion κ _F of agents who failed and κ _S of agents who succeeded;

4. 4. Purged agents are replaced, and the first-period payoffs are realized.

Period 2:

1. 1. Surviving or new agent i chooses effort $e_2^i$;

2. 2. Each project outcome (ω ₂ ∈ {S, F}) is determined;

3. 3. The game ends and the second-period payoffs are realized.

The equilibrium concept is Perfect Bayesian Equilibrium (PBE), which requires that each agent correctly anticipates the autocrat’s purging decision and other agents’ efforts when choosing his own effort, and, in turn, the autocrat correctly anticipates the level of effort by each type when determining her investment in violence and her purging strategy. For simplicity, we assume that agents are anonymous, all agents who fail face an identical probability of being purged, and all agents who succeed face the same risk of being removed. To avoid uninteresting PBE in which all agents exert no effort, we further impose that when the autocrat observes success as an out-of-equilibrium event, she treats the deviation as a mistake and does not distinguish between the agent who exerted effort and other subordinates who followed their prescribed strategy.^{Footnote 1} If, after these restrictions, multiple PBE arise, we select the one which maximizes the purge incidences, that is, κ _F and κ _S. Our main insights and comparative statics are robust to change in this last criterion. In what follows, the term “equilibrium” refers to the PBE which satisfies all of our refinements.

Before proceeding to the analysis, a few remarks on the setup are in order. First, we remain agnostic about what constitutes non-congruent types. It could correspond to opportunistic agents attracted to the regime for the associated benefits of being an insider (Getty Reference Getty1987), subordinates who lacked “a wholehearted commitment to the Party’s cause” (Teiwes Reference Teiwes1993, 114–5) or to the leader, or agents not satisfied with the regime’s line (Gregory Reference Gregory2009). In our theory, it only matters that the autocrat prefers some types of subordinates over others.

We also suppose that the autocrat is relatively well informed. As it is common in the agency literature, the autocrat does not observe her agents’ effort. She, however, learns the project outcome of all agents (in Online Appendix F.1, we show that our results hold when the autocrat only learns about a subset of outcomes). Historically, communist leaders in China or the USSR were sending special commissions or work teams from higher up to evaluate local performance (Rigby Reference Rigby1968; Teiwes Reference Teiwes1993). Hence, project outcomes can be understood as fulfilling quotas, dealing with problem cases, or ensuring the provision of local services.

Finally, both carrying out the mass purge and investing in the intensity of violence are costly. Mass purges entail a loss in terms of human capital and organizational knowledge as well as the cost of potentially deporting agents or delay in finding suitable replacements for the purged subordinates. When it comes to the intensity of violence, fines—which can correspond to a low L from being purged—requires less infrastructure and personnel than killing subordinates, deporting their spouse, and sending their children to orphanages, a very high L quite common in Stalin’s USSR (Brzezinski Reference Brzezinski1956; Conquest Reference Conquest2008).

Note that the autocrat commits to an intensity of violence at the beginning of the game. This assumption is not innocuous. Absent commitment, at the time of the purge, the unique PBE features no violence if violence is costly (violence has no effect on selection and effort then) or any intensity can be part of a PBE if violence is costless. With commitment, we offer one possible rationale for the variation in the intensity of violence of purges across countries and over time. From the onset, let us stress that all results from the baseline model described here remain unchanged if the autocrat can commit to both the intensity of violence (L) and the purge incidences (κ _F and κ _S) at the beginning of the game (see Online Appendix F.2).

Early commitment to an intensity of violence can be thought of as earmarking funds for the purge (like in the case of the Great Terror, Wolton Reference Wolton2015). This assumption can also be interpreted as investment in the infrastructure of violence such as the Solovski prison camp (opened in the USSR in 1923) or the Qincheng Prison (opened in China in 1958). The cost function C(κ) then corresponds to the cost of filling these coercive structures with purged subordinates (in this case, C ₀ and C ₁ may be a function of L; our model can easily accommodate this possibility and all of our results would hold as long as the two parameters do not increase too fast with the intensity of violence).

Throughout, we use the following notation. V ₂(τ) denotes agent i’s expected payoff in period 2 as a function of his type. Simple algebra yields ${V_2}\left( \tau \right) = R + {{{{\left( {v\left( {S,\tau } \right)} \right)}^2}} \over 2}$. For the autocrat, W ₂(τ) denotes her second-period expected payoff induced by an agent of type τ ∈ {c, nc}, with W ₂(τ) = v(S, τ). To limit the number of cases, we impose βr(W ₂(c) − W ₂(nc)) < C ₀ + C ₁ so the autocrat never purges all agents. Finally, to simplify the analysis, we assume that the highest feasible intensity of violence, which we denote $\overline{L}$, satisfies $\overline{L}: = 1 - v\left( {S,c} \right) - {V_2}\left(c \right)$ so that all subordinates exert effort less than one.

EFFORT AND INCENTIVE TO PURGE

In this section, we determine the agents’ equilibrium effort and the purge incidences κ _F, κ _S for a given intensity of violence L. Our first result, quite intuitively, states that in any equilibrium, the autocrat only purges from the failure pool (i.e., the set of agents who fail in their project).

Lemma 1. In equilibrium, κ _F ∈ [0, 1] and κ _S = 0.

Proof. All proofs are collected in the Online Supplemental Appendix.

Due to our equilibrium refinements, there is no equilibrium in which agents exert zero effort.^{Footnote 2} When they exert effort, agents endogenously sort into failure and success pools, which constitute, with the proportion (λ) of congruent types among current subordinates and potential replacements, the only information available to the autocrat at the time of her purging decision. Given that congruent types receive a greater intrinsic benefit from succeeding, they always exert more effort than their non-congruent counterparts and are more likely to belong to the success pool (i.e., the single-crossing condition holds). Consequently, success becomes a positive signal of congruence, and failure a negative one. The autocrat never purges successful agents then since she (correctly) believes the success pool is strictly better than the replacement pool.

When deciding his level of effort, a subordinate takes into account his present and future payoff from succeeding (since he survives the purge) and the risk of facing a loss L and of forgoing future payoffs when he fails. This risk is proportional to the (anticipated) proportion of failures being purged, κ _F. Greater κ _F, therefore, translates into higher effort.

Lemma 2. A type τ ∈ {c, nc} agent i chooses effort:

(5)$$e_1^i\left( \tau \right) = v\left( {S,\tau } \right) + {\kappa _F}\left( {{V_2}\left( \tau \right) + L} \right).$$

Having characterized the agents’ effort choices, we now turn to the autocrat’s incentive to purge. After observing all project outcomes, the autocrat forms a posterior μ _F that a subordinate who fails (ω ₁ = F) is congruent, with μ _F < λ as discussed above. The autocrat then chooses the purge incidence, κ _F, to equate the marginal cost of purging an additional agent—C ₀ + C ₁α _F × κ _F—with its associated marginal benefit. The marginal benefit is a function of two elements: (i) the increased probability that the replacement is congruent, λ − μ _F, and (ii) the expected payoff gain from having a congruent agent rather than a non-congruent subordinate in period 2, β(W ₂(c) − W ₂(nc)). The autocrat’s expected marginal benefit from purging an additional failure is:

(6)$${{\cal W}^F} = \left[ {r - {\mu ^F}} \right]\left( {{W_2}\left(c \right) - {W_2}\left( {nc} \right)} \right).$$

In our baseline model, there exists a clear demarcation between purged and non-purged agents. Success is always a guarantee of safety and failures risk being replaced. The purge is thus “discriminate.” Project outcomes, however, are only a noisy signal of types. A congruent agent exerts effort less than 1, while a non-congruent subordinate exerts positive effort [see equation (5)]. Hence, in any purge, some congruent agents are purged and some non-congruent subordinates survive.

As purges are costly, not all failures may be removed (κ _F ∈ (0, 1)). This case exhibits an interesting property. To describe it, suppose that κ _F increases exogenously. Both congruent and non-congruent types then raise their effort in response to a higher threat of being purged conditional on failure. Congruent agents, however, have more to lose from being purged and so increase their effort relatively more [see equation (5)]. Further, congruent types are also less likely to fail to start with. The combination of these two effects implies that more congruent agents (in relative and absolute terms) exit the failure pool. The autocrat’s evaluation of subordinates who fail then deteriorates (μ _F decreases) and her marginal benefit of purging increases [see equation (6)].

The analysis above highlights a feedback mechanism whereby an anticipated increase in the purge incidence yields higher effort by all agents and, in turn, a greater incentive to purge for the autocrat. We label this reinforcing effect the “pool makeup effect.” As the pool makeup effect plays a crucial role below, we introduce the following definition as a point of reference.

Definition 1. The pool makeup effect corresponds to the negative effect of the purge incidence κ _F ∈ [0, 1) (as anticipated by the agents) on the autocrat’s posterior upon failure: ${{\partial {\mu _F}} \over {\partial {\kappa _F}}} \lt \,0$.

Having examined the agents’ efforts and the autocrat’s incentive to purge, we turn in the next section to the consequences of (for now) exogenous changes in the intensity of violence.

THE CONSEQUENCES OF VIOLENCE

Greater intensity of violence increases subordinates’ efforts [see equation (5)]. This direct effect of L on agents’ action, in turn, induces a series of indirect effects that needs to be carefully unpacked.

Let us first consider the indirect effect of violence on effort through the purge incidence, assuming not all failures are purged (κ _F ∈ (0, 1)). To do so, fix the purge breadth κ = α _F × κ _F (the proportion of agents who fail times the proportion of failures purged). As the intensity of violence increases, all subordinates exert more effort, and the pool of failures shrinks (α _F ↓). As the purge breadth is held constant by design, the purge incidence must necessarily increase (κ _F ↑), which, in turn, induces greater effort. Thus, the indirect effect through purge incidence is positive. Higher purge incidence then generates a pool makeup effect (Definition 1). The autocrat has a greater incentive to purge more failures, further increasing the purge incidence, which again induces even greater effort.

As efforts increase at a relatively high rate with the intensity of violence due to the direct and indirect effects, the failure pool is quickly exhausted (fewer subordinates fail and more failures are purged). Consequently, there exists a threshold of violence—denoted L ^full ≥ 0—above which all failures are purged. Using the superscript “*” to denote equilibrium values, we thus obtain:

Proposition 1. There exists L ^full > 0, unique whenever ${L^{full}} \lt \,\overline{L}$, such that:

1. (i) Some agents who fail survive ( $\kappa _F^*\left(L \right) \in \left( {0,1} \right)$) if and only if L < L ^full;

2. (ii) All agents who fail are purged ( $\kappa _F^*\left(L \right) = 1$) if and only if $L \in \left[ {{L^{full}},\overline{L}} \right]$ (possibly an empty interval).

To simplify the exposition, we assume in what follows that ${L^{full}} \lt \,\overline{L}$ so all failures may be purged if violence is high enough (we provide a precise condition for this inequality to hold in the Online Appendix).

As noted above, as the intensity of violence increases, the autocrat’s assessment of the failure pool deteriorates (μ ^F decreases). When some failures survive, the equilibrium purge breadth is determined by equating the marginal cost and the marginal benefit of purging. Greater marginal benefit then yields greater purge breadth. When the autocrat purges all failures, the breadth equals the size of the failure pool, which shrinks with the intensity of violence. The relationship between the breadth κ*(L) and the intensity of violence L is, thus, non-monotonic.

Proposition 2. The equilibrium purge breadth, κ*(L), satisfies:

1. (i) For L < L ^full, κ*(L) is strictly increasing with L;

2. (ii) For $L \in \left[ {{L^{full}},\overline{L}} \right]$, κ * (L) is strictly decreasing with L.

We now turn to the impact of violence on effort. While greater violence always generates better first-period performance, the responsiveness of effort to L depends on how many failures are purged and, thus, the original intensity of violence (see Proposition 1). When some failures survive, as violence increases, direct and indirect effects complement each other, inducing a relatively large response in subordinates’ efforts. In turn, when all failures are purged, there is no indirect effect. Effort then increases linearly with violence [equation (5) with κ _F = 1].

Proposition 3. The total derivative of average effort with respect to violence ${{{\rm{d}}\bar{e}\left( L \right)} \over {{\rm{d}}L}}$ is always strictly positive. Further, there exist L ^eff ≥ 0, unique if L ^eff < L ^full, such that the derivative satisfies:

1. (i) ${{{\rm{d}}\bar{e}\left(L \right)} \over {{\rm{d}}L}} > 1$ for all L ∈ (L ^eff, L ^full);

2. (ii) ${{{\rm{d}}\bar{e}\left(L \right)} \over {{\rm{d}}L}} = 1$ for all $L \in \left[ {{L^{full}},\overline{L}} \right]$.

We finally consider the relationship between violence and selection. As a first observation, note that a new subordinate is always more likely to be congruent than a purged agent. Thus, the purge always raises the proportion of congruent types among second-period subordinates. Here, we study the impact of increasing the intensity of violence on screening and selection.

Consider first the consequence of greater violence on screening (i.e., the proportion of congruent types among surviving subordinates). When some failures survive ($\kappa _F^*\left(L \right) \in \left( {0,1} \right)$, equivalently L < L ^full), a greater intensity of violence is associated with fewer agents failing and more failures being purged. Among surviving subordinates, a greater proportion of agents thus belongs to the success pool, which is of higher quality than the failure pool, resulting in better screening. When surviving agents all belong to the success pool ($\kappa _F^*\left(L \right) = 1$, or equivalently L ≥ L ^full), screening worsens because the success pool becomes more tainted as more non-congruent types enter the pool relative to the stocks of both types.

Let us now study the relationship between violence and selection (i.e., the proportion of congruent types among second-period agents, both old and new). When some failures survive (L < L ^full), selection strictly improves with the intensity of violence. Higher intensity yields better screening of surviving agents and more failures are replaced by subordinates of higher quality. When all failures are purged (i.e., L ≥ L ^full), new subordinates are still better than purged agents, but there are fewer of them and screening deteriorates. In general, the overall consequences of violence on selection would be undetermined. Under our assumptions, the positive and negative effects fully cancel out and violence has no effect on selection.

Proposition 4.

1. (i) The proportion of congruent types among subordinates surviving the purge strictly increases with the intensity of violence if and only if L < L ^full and strictly decreases otherwise.

2. (ii) The proportion of congruent types among second-period agents strictly increases with the intensity of violence if and only if L < L ^full and is constant otherwise.

Greater violence always engenders higher effort in the first period (Proposition 3). The relationship between violence and selection is more nuanced, and violence may impede screening (Proposition 4). With these comparative statics, we can now turn to the autocrat’s optimal choice of violence.

INTENSITY OF VIOLENCE

Recall that effort is very responsive to violence at low intensity (i.e., L < L ^full) due to the direct effect (on payoff) and indirect effects (especially, the pool makeup effect) of violence. In addition, greater intensity also improves selection. As a result, the marginal benefit from increased violence is large and increasing in L for L < L ^full. The marginal cost can then intersect the marginal benefit more than once, as Figure 1 illustrates. If so, the autocrat chooses between two possible maxima: the lowest intersection L ¹ in Figure 1 and L ^full (the highest intersection L ² is a local minimum). The presence of multiple maxima implies that small changes in the underlying fundamentals can be associated with a large swing in equilibrium values, including the intensity of violence or the purge breadth.

FIGURE 1.

Marginal Benefit of Violence in a Partially Discriminate Purge

The solid blue line is the marginal benefit, the dashed black line the marginal cost. Parameter values: λ = 1/3, r = 2/3, R = 0, v(S, c) = 1/4, v(S, nc) = 0, β = 0.9, C ₀ = 0, C ₁ = 0.17, ζ ₀ = 0.5, and ζ ₁ = 4.8.

Remark 1. Small changes in the cost of purging (C ₀, C ₁) or in the cost of investing in violence (ζ ₀, ζ ₁) can be associated with large shifts in the observed intensity of violence and purge breadth.

At L = L ^full, the marginal benefit drops. There is no pool makeup effect, so effort increases linearly with violence (Proposition 3), and the selection effect is null (Proposition 4). The benefit is then concave in the range $\left[ {{L^{full}},\overline{L}} \right]$. Consequently, the intensity of violence is such that all failures are purged (L* ≥ L ^full) if and only if the cost parameters determining the investment in violence are not too high.

Remark 2. All failures are purged (L* ≥ L ^full) only if the cost parameters of investing in violence ζ ₀ and ζ ₁ are below some thresholds $\zeta _0^{disc} > 0$ and $\zeta _1^{disc}\left( {{\zeta _0}} \right) > 0$.

ANOTHER FORM OF PURGES

So far, we have assumed that the proportion of congruent types is the same among existing subordinates and their possible replacements. In this section, we relax this assumption and suppose that the pool of replacement is better than the pool of first-period agents: r > λ.

When the replacement pool is of sufficiently high quality (r sufficiently large relative to λ), the autocrat may decide to purge agents from the success pool (i.e., κ _S > 0). The purge then appears somewhat indiscriminate: Now, even success does not inoculate a subordinate against the purge. However, as before, success is a signal of congruence, and failure is a signal of non-congruence. Hence, the autocrat always replaces all failures before purging agents belonging to the success pool. Formally, in any equilibrium, the purge incidence conditional on success satisfies κ _S > 0 only if κ _F = 1. This form of purges is not exactly random, and we label it as “semi-indiscriminate.”

Turning to the agents’ first-period effort, anticipating a semi-indiscriminate purge, each subordinate now anticipates that success is no longer sufficient to survive the purge. A type-τ agent then exerts effort (τ ∈ {c, nc}):

(7)$$e_1^i\left( \tau \right) = \left( {1 - {\kappa _S}} \right)\left( {v\left( {S,\tau } \right) + {V_2}\left( \tau \right) + L} \right).$$

Quite intuitively, effort is decreasing with the purge incidence κ _S since a greater risk of being purged conditional on success reduces the value of effort.

The autocrat’s marginal benefit of purging a successful agent is, in turn:

(8)$${{\cal W}^S} = \left[ {r - {\mu ^S}} \right]\left( {{W_2}\left(c \right) - {W_2}\left( {nc} \right)} \right).$$

Using equations (7) and (8), we proceed as in the section on the consequences of violence and describe the main observable features of semi-indiscriminate purges. A first characteristic of semi-indiscriminate purges is that they tend to be violent. For all intensity of violence, the marginal benefit of purging a successful agent is strictly lower than the marginal benefit of purging a failure [using μ _S > λ > μ _F and comparing equations (6) and (8)]. When the intensity of violence is such that all failures are purged (i.e., at L = L ^full), the autocrat does not immediately start purging successful agents. Rather, there is a range of intensity of violence such that the purge remains discriminate.

As the intensity of violence increases, two combined forces make a semi-indiscriminate purge more attractive to the autocrat. First, greater violence yields the same increase in effort from congruent and non-congruent subordinates [see equation (5) with κ _F = 1]. The success pool then becomes more tainted as relatively more non-congruent agents enter the pool. Consequently, the autocrat’s posterior, μ _S, decreases with L, and the marginal benefit of purging a successful agent increases. Second, greater effort by all subordinates reduces the size of the failure pool and the purge breadth, and the marginal cost of purging an additional agent decreases. Thus, there exists a threshold L ^ind > L ^full such that the purge is semi-indiscriminate if and only if the intensity of violence is above L ^ind.

A second characteristic of semi-indiscriminate purges is that they are not necessarily large. Indeed, they occur only if the failure pool is sufficiently depleted. However, once a semi-indiscriminate purge has begun (L ≥ L ^ind), the breadth again is positively correlated with the intensity of violence. As violence increases, the success pool is more tainted (μ _S ↓), the marginal benefit of purging is greater, and, as a consequence, the purge breadth is wider.^{Footnote 3}

Proposition 5. There exists L ^ind > L ^full, unique whenever ${L^{ind}} \lt \overline{L}$, such that:

1. (i) The purge is semi-indiscriminate ( $\kappa _F^*\left(L \right) = 1$ and $\kappa _S^*\left(L \right) > 0$) if and only if L > L ^ind and discriminate ( $\kappa _S^*\left(L \right) = 0$) otherwise; and

2. (ii) The purge breadth κ*(L) is strictly increasing with L for all L ≥ L ^ind.

Figure 2 illustrates the non-monotonic relationship between the intensity of violence and the purge breadth combining our earlier analysis (Proposition 2) with the result described in Proposition 5(ii). The purge breadth is increasing with L when the intensity of violence is relatively low (L < L ^full) or relatively high (L ≥ L ^ind). In turn, the correlation is negative for intermediary intensity of violence when the purge is discriminate and all agents are purged (L ∈ [L ^full, L ^ind]).

FIGURE 2.

Equilibrium Purge Breadth and Intensity of Violence

Parameter values: λ = 1/3, r = 2/3, R = 0, v(S, c) = 1/4, v(S, nc) = 0, β = 0.9, C ₀ = 0, and C ₁ = 0.17.

We now turn to effort and selection. In a semi-indiscriminate purge, the direct effect of violence on payoffs implies that greater intensity always increases effort. This positive impact, however, is dampened by the increased purge incidence κ _S. Indeed, as the purge breadth increases, so does the risk that success is not enough to inoculate an agent against the purge. As a result, a subordinate’s effort is generally less responsive to violence in a semi-indiscriminate purge than in a discriminate purge.

In a semi-indiscriminate purge, selection worsens as the intensity of violence increases. Greater violence yields a more tainted success pool from which survivors are drawn and, thus, leads to less efficient screening. Further, the purge breadth tends to increase slowly with violence (due to the low marginal benefit of purging). Combining these two effects, the overall congruence of second-period subordinates decreases with L as long as the quality of the replacement pool is not too high (r ≤ 2λ, a sufficient condition).

The higher quality of the replacement pool relative to existing subordinates also implies that greater violence decreases the proportion of congruent second-period subordinates when the purge is discriminate and all failures are purged (L ∈ [L ^full, L ^ind]). Then, the purge breadth decreases with violence so that existing agents, who are worse on average than their potential replacements, constitute a greater proportion of the second-period subordinates. Overall, when the quality of the replacement pool is high (r > λ), an increase in violence improves selection only in a discriminate purge as more agents who fail are purged.

Proposition 6. The total derivative of average effort with respect to violence ${{{\rm{d}}\bar{e}\left(L \right)} \over {{\rm{d}}L}}$ satisfies $0 \,\lt {{{\rm{d}}\bar{e}\left(L \right)} \over {{\rm{d}}L}} \lt 1$ for all $L \in \left[ {{L^{ind}},\overline{L}} \right]$.

If r ∈ (λ, 2λ], the proportion of congruent types among second-period agents decreases with L for all L ≥ L ^full.

Figures 3 and 4 illustrate the consequences of violence on effort and selection, combining the results of this section and those of our baseline model (Section “The consequences of violence”). Figure 3 shows that effort increases at a high rate in a discriminate purge when some failures survive (for L close enough to L ^full), at a relatively low rate in a semi-indiscriminate purge (L > L ^ind), and at an intermediate speed when all failures are purged. Figure 4 highlights that screening (dashed line) and selection (solid line) improve with the intensity of violence only if violence is low and some failures survive.

FIGURE 3.

Violence and Effort

Parameter values: λ = 1/3, r = 2/3, R = 0, v(S, c) = 1/4, v(S, nc) = 0, β = 0.9, C ₀ = 0, and C ₁ = 0.17.

FIGURE 4.

Violence and Selection

The solid line corresponds to the proportion of congruent types among second-period subordinates, the dashed line to the proportion of congruent types among surviving agents. Parameter values: λ = 1/3, r = 2/3, R = 0, v(S, c) = 1/4, v(S, nc) = 0, β = 0.9, C ₀ = 0, and C ₁ = 0.17.

Overall, when the replacement pool is of high quality relative to existing subordinates, our analysis suggests that, at a high intensity of violence, the autocrat faces a trade-off between effort and selection when choosing L. Further, violence has a relatively low effect on first-period performance. So can semi-indiscriminate purges actually occur? The next remark states three conditions for a purge to be semi-indiscriminate.

Remark 3. A purge is semi-indiscriminate (L* > L ^ind) if the following conditions are satisfied:

1. 1. The proportion of congruent types in the replacement pool r is strictly higher than some r ≥ λ;

2. 2. The cost parameters C ₀ and C ₁ are, respectively, strictly below some $\overline {{C_0}} \left(r \right) > 0$ and $\overline {{C_1}} \left( {r,{C_0}} \right) > 0$; and

3. 3. The cost parameters ζ ₀ and ζ ₁ are, respectively, strictly below some $\overline {{\zeta _0}} \left(r \right) > 0$ and $\overline {{\zeta _1}} \left( {r,{\zeta _0}} \right) > 0$.

The three conditions are relatively intuitive. First, the replacement pool must be of sufficiently good quality and in particular, better on average than existing subordinates (condition 1). Second, the cost of purging (C ₀, C ₁) must be sufficiently low to compensate for the relatively low marginal benefit of purging a successful agent (condition 2). Finally, the cost of investing in violence must also be relatively small so that the autocrat is willing to choose a high L despite the low marginal benefit of doing so (condition 3).

EXTENSIONS

In this section, we briefly discuss three extensions of our baseline model, with the full formal analysis available in Online Appendix E.

Carrots and Sticks

So far, we have assumed that the autocrat can only choose the intensity of violence, the sticks. In practice, she can also vary the material benefits available to insiders, the carrots. In some situations, such as democratic regimes, positive incentives may be the only tools available to the principal. This subsection studies the role of carrots in addition to sticks.

Formally, we return to the case when the replacement pool does not depend on the purge breadth. We further assume that the replacement pool may be of higher quality than existing subordinates (i.e., r ≥ λ). To facilitate comparison with the original setup (baseline model with r ≥ λ), we suppose that the autocrat can only supplement the second-period material benefit at marginal cost ξ′(R ₂) = ξ ₀ + ξ ₁R ₂. The original model then corresponds to the case when R ₂ is constrained to zero (equivalently, ξ ₁ → ∞). Throughout, to simplify the analysis, we further impose that ξ ₀ = ζ ₀ and that the highest feasible endogenous reward $\overline {{R_2}}$ and intensity of violence $\overline{L}$ jointly satisfy $\overline {{R_2}} + \overline{L} = 1 - v\left( {S,c} \right) - {V_2}\left( {S,c} \right)$.

In this extension, an agent’s effort as a function of the purge incidences κ _F and κ _S is:

(9)$$e_1^i\left( \tau \right) = \left\{ {\matrix{ {v\left( {S,\tau } \right) + {\kappa _F}\left( {{V_2}\left( \tau \right) + L + {R_2}} \right)} \hfill &#x0026; {{\rm{if}}\;{\kappa _S} = 0} \hfill \cr {\left( {1 - {\kappa _S}} \right)\left( {v\left( {S,\tau } \right) + {V_2}\left( \tau \right) + L + {R_2}} \right)} \hfill &#x0026; {{\rm{if}}\;{\kappa _S} > 0.} \hfill \cr} } \right.$$

Essentially, the additional reward R ₂ is a perfect substitute for the intensity of violence L. As a result, when the autocrat has the ability to use both carrots and sticks, she can provide the same amount of total inducement at a strictly lower cost. Quite logically, the autocrat uses this lower cost to increase the overall incentives provided to her agents. This generates a few differences in equilibrium outcomes summarized in the next proposition.

Proposition 7. Suppose the autocrat can also offer an additional reward R ₂ in period 2. Compared to the original model, in equilibrium:

1. (i) The purge incidence is always weakly higher;

2. (ii) The purge breadth can be higher or lower; and

3. (iii) The intensity of violence can be higher or lower.

Point (i) parallels the analysis of the impact of greater intensity of violence in the original model. As total inducement (punishment and reward combined) increases, the target pool becomes more tainted, yielding a strictly higher marginal benefit of purging and weakly higher purge incidence (strictly if some failures survive or if some successful agents are purged). Point (ii) follows from the non-monotonic relationship between violence and breadth. Depending on the form of the purge, the greater total incentive can be associated with larger purge breadth (e.g., if some failures survive with and without carrots) or lower purge breadth (e.g., if all failures and no successes are purged with and without carrots).

Point (iii) of Proposition 7 is slightly more surprising. Even though the autocrat can use carrots (R ₂) in addition to sticks (L), the intensity of violence may actually be strictly higher than in the original model when no rewards are available. This is a consequence of the indirect effects present in our many-to-one accountability framework. Recall that absent carrots, the benefit of investing in violence is convex when some failures survive (i.e., L < L ^full, as shown in Figure 1). Given that reward and punishment are perfect substitutes, this property carries over to the case where the autocrat can also use carrots. With the lower cost of providing incentives, the equilibrium total inducement is then much larger than the equilibrium intensity of violence in our original model. Indeed, the difference may be such that, even though violence only constitutes a portion of the total incentives, the intensity of violence can be strictly higher with rewards than without.

Mass Purges and the Autocrat’s Survival

Our results are also robust to a change in the autocrat’s objective. Rather than caring about the first- and second-period performances, the autocrat may seek to maximize her survival probability, which, we assume, is a function of the first-period performance (e.g., economic performance) and the proportion of second-period non-congruent agents. Formally, the autocrat seeks to maximize

$$P\left( {survives} \right) = \gamma {\bar{e}_1} + \left( {1 - \gamma } \right)\beta \left( {1 - {\cal P}} \right),$$

with ${\cal P}$ the proportion of non-congruent agents after the purge, γ the weight on first-period performance, and β(·) a strictly decreasing and weakly concave function (in Online Appendix E.3, we study a more general objective function allowing for some complementarity between performance and the proportion of non-congruent subordinates).

The rest of the model is similar to the original model. In particular, for ease of comparison, we suppose that a type τ ∈ {c, nc} subordinate obtains a payoff v(S, τ) from a successful project in the first period. He also gets V ₂(τ) from surviving the purge, with V ₂(c) > V ₂(nc) (e.g., congruent agents receive a higher payoff when the autocrat survives).

Agents then behave as in the original model and the observable features of purges remain the same. For relatively low intensity of violence, the purge is discriminate and some failures survive, whereas the purge is semi-indiscriminate for high intensity if r is sufficiently large. The purge breadth still varies non-monotonically with the intensity of violence.

As before, effort increases with the intensity of violence (at a high rate when some failures survive and at a low rate when the purge is semi-indiscriminate), and selection is non-monotonic with L (assuming r > λ) whenever β(·) is not “too concave” (we provide exact conditions in Propositions E.3 and E.4). This last restriction guarantees that any second-order effect due to the shape of the survival function is dominated by the first-order direct and indirect effects of violence that we have identified in the analysis of our original model.

This extension thus suggests that the strategic interactions between one principal and a mass of subordinates, rather than the particular goal of the autocrat (performance or survival), are the key force behind the equilibrium features of purges and the trade-offs we identify.

SUPPLEMENTARY MATERIAL

To view supplementary material for this article, please visit https://doi.org/10.1017/S0003055419000455.