Strategic Reporting: A Formal Model of Biases in Conflict Data

During violent conflict, governments may acknowledge their use of illegitimate violence (e.g., noncombatant casualties) even though such violence can depress civilian support. Why would they do so? We model the strategic incentives affecting government disclosures of illegitimate violence in the face of potential NGO investigations, where disclosures, investigations, and support are endogenous. We highlight implications for the analysis of conflict data generated from government and NGO reports and for the emergence of government transparency. Underreporting bias in government disclosures positively correlates with underreporting bias in NGO reports. Furthermore, governments exhibit greater underreporting bias relative to NGOs when NGOs face higher investigative costs. We also illustrate why it is difficult to estimate negative effects of illegitimate violence on support using government data: with large true effects, governments have incentives to conceal such violence, leading to strategic attenuation bias. Finally, there is a U-shaped relationship between NGO investigative costs and government payoffs.

D uring violent conflict, governments may acknowledge their use of illegitimate violence (e.g., noncombatant casualties) even though such violence can depress civilian support. Why would they do so? We model the strategic incentives affecting government disclosures of illegitimate violence in the face of potential NGO investigations, where disclosures, investigations, and support are endogenous. We highlight implications for the analysis of conflict data generated from government and NGO reports and for the emergence of government transparency. Underreporting bias in government disclosures positively correlates with underreporting bias in NGO reports. Furthermore, governments exhibit greater underreporting bias relative to NGOs when NGOs face higher investigative costs. We also illustrate why it is difficult to estimate negative effects of illegitimate violence on support using government data: with large true effects, governments have incentives to conceal such violence, leading to strategic attenuation bias. Finally, there is a U-shaped relationship between NGO investigative costs and government payoffs. I n the chaos of violent conflict, it can be difficult to determine whether combatants abuse civilians and which side perpetrated the abuse. State security forces can present themselves in plain clothes; rebels hide their identities. Air strikes present particularly difficult attribution problems. When one warring party is deemed responsible for illegitimate violence, such as indiscriminate violence, mass rape, or the destruction of critical infrastructure, it can lose the support of civilians who care about the armed actor's battlefield behavior (Benmelech, Berrebi, and Klor 2015;Condra and Shapiro 2012;Lyall, Blair, and Imai 2013). This support is critical for leaders: support among the selectorate ensures political survival (Bueno de Mesquita et al. 1999;Prorok 2016;Weeks 2012), and support among civilians in the conflict zone entails battlefield resources such as recruits, information, or supplies (Kalyvas 2006;Shaver and Shapiro 2021). The result is that, for warring governments in particular, there are strong incentives to conceal unpopular collateral damage.
Despite this, governments can, and sometimes do, preemptively disclose illegitimate violence. The Obama administration, for example, acknowledged many civilian causalities resulting from its use of drone strikes in a targeted-killings program outside active military theater. Likewise, in its conflict with Naxalite rebels, the Indian government started publishing a list of violent encounters in the South Asian Terror Portal, where it originally recorded whether government or rebel forces perpetrated the violence and whether the violence led to noncombatant causalities. 1 Critical to both these examples is the presence of watchdog NGOs. The Bureau of Investigative Journalism began systematically documenting drone-strike civilian casualties in 2011, four years before the Obama administrations published its own list. Likewise, in India, a collection of watchdog NGOs, including the Asian Centre for Human Rights and Forum for Fact Finding, Documentation, and Advocacy, began investigating the veracity of the government's list in 2007. In both cases, these reports differed substantially from the governments' accounts.
Given the incentives to conceal illegitimate violence, how and why does government transparency about this type of violence arise? What are the implications for conflict research that uses government and NGO reports as data?
Scholars often rely on government-or NGO-provided data to study the microfoundations of violence. Yet data from different sources can paint significantly different pictures about the extent of illegitimate violence in a conflict. Given the choice between government-and NGO-provided data, it is not clear which should be more accurate a priori. Furthermore, we do not know how relevant background variables (e.g., government popularity or NGO investigative costs) map onto data quality measures like underreporting bias or affect the ability of researchers to estimate parameters of interest (e.g., the degree to which illegitimate violence suppresses support). It is difficult to assess these considerations empirically because we cannot compare the observed data to some true account of the conflict. Consequently, we adopt a formal approach in this paper. Specifically, we model how governments and NGOs strategically report illegitimate violence. In the model, a government discloses the legitimacy of violence, where illegitimate violence captures acts that potential supporters would find distasteful (e.g., noncombatant casualties). A watchdog NGO allocates costly effort to investigate the veracity of the government's disclosure, receiving an additional benefit when it exposes a coverup. Both the government's disclosure and the NGO's investigation affect third-party support for the government, which the government seeks to maximize. The third party represents an actor or group on whose support the government relies, either for wartime information or for support among the selectorate (e.g., civilians in conflict zones or voters in democracies). Optimal disclosures, investigations, and support are described by three equilibria: a truthful equilibrium in which the government correctly discloses the state of violence, a never-admit-fault equilibrium in which the government never discloses illegitimate violence, and a partially truthful equilibrium in which the government mixes between disclosing and concealing illegitimate violence. To motivate our theoretical framework, we draw on two disparate examples: the Naxalite insurgency in India and the US drone-strike program.
Our main contribution is to use the model and its equilibrium characterization to derive implications for conflict research that uses data coded from government disclosures or NGO reports. First, we explore underreporting bias-that is, the difference between the baseline frequency of illegitimate violence and the frequency that a source reports illegitimate violence in equilibrium. Both government and NGO reports have underreporting bias, but the causes differ. Whereas the government has incentives to conceal illegitimate violence, the NGOs may fail to produce tangible results when they invest limited investigative effort. We characterize when government disclosures will have less underreporting bias than NGO reports in equilibrium and vice versa. We show that as NGOs face higher investigative costs, both the NGO and government data will underreport illegitimate violence, but the bias in government data will be more extreme. When investigative costs are large, NGOs invest less effort and are therefore unlikely to expose cover-ups. In exactly this situation, governments have large incentives to conceal illegitimate violence. An implication is that the underreporting bias in both data sources is positively correlated across cases; researchers cannot simply trade a bad information source for a good one.
Second, we study when conflict researchers can correctly identify the third-party's distaste for illegitimate violence. The analysis is motivated by empirical work that estimates the degree to which noncombatant causalities depress popular support for counterinsurgent forces (Lyall, Blair, and Imai 2013;Lyall, Shiraito, and Imai 2015;Shaver and Shapiro 2021). We show that when the government is truthful in equilibrium, studying variation in support after different types of government disclosures correctly estimates the third-party's distaste of illegitimate violence. In contrast, when the government conceals illegitimate violence with positive probability in equilibrium, an identical design underestimates the true effect of illegitimate violence on thirdparty support. This attenuation bias arises because uninformed third-party observers, anticipating government cover-ups in equilibrium, temper support after seeing a report of legitimate violence relative to the truthful baseline. The magnitude of the bias increases as the true distaste for illegitimate violence increases, and the bias exists even when government reports and third-party support are observed without measurement error.
Third, we explore when governments have incentives to manipulate the environment in which civil society operates. For instance, governments might weaken or strengthen transparency institutions such as freedom of information (FOI) laws or press protections more broadly (Colaresi 2012;Egorov, Guriev, and Sonin 2009;Grigorescu 2003;Lorentzen 2014), which affects the investigative costs of NGOs. We find that decreasing NGO investigative costs increases the likelihood that a cover-up is exposed and, as a secondorder effect, makes the government less likely to conceal illegitimate violence. This latter effect creates positive belief spillovers that enhance third-party support via equilibrium beliefs. Therefore, governments benefit from transparency institutions when the second-order effect dominates the first. In particular, moderately strong transparency institutions can leave the government the least well-off because they do not induce the government to truthfully disclose but still help NGOs expose cover-ups.

RELATED LITERATURE
Research on underreporting bias in conflict data focuses on nonstrategic sources: inconsistent media coverage (Hendrix and Salehyan 2015;Weidmann 2015), aggregation bias from combining sources (Cook and Weidmann 2019), or geographic biases associated with cellphone coverage (Weidmann 2016). In contrast, we explore the interdependent strategic forces that determine underreporting bias in government-and NGO-provided data, considering the motivations of each actor when reporting unpopular violence. Thus, our paper relates to Drakos and Gofas (2006), who study how terrorists anticipate media coverage, creating underreporting bias. Whereas their work is largely empirical and focuses on the decision of terrorist groups to attack, this paper is largely theoretical and focuses on the decision of governments to acknowledge illegitimate violence. Our analysis demonstrates how incentives to report and investigate illegitimate violence affect underreporting bias even when the underlying frequency of illegitimate violence is exogenous.
To do this, we construct a formal model similar to those in the literature on auditing and risk disclosure (Avenhaus, Von Stengel, and Zamir 2002;Dobler 2008). Auditing games appear in the study of arms control (Arena and Wolford 2012;Baliga and Sjöström 2008), covert affairs (Spaniel and Poznansky 2018), and cyberwarfare with attribution problems (Baliga, Bueno de Mesquita, and Wolitzky 2020). Besides our focus on the production and analysis of conflict data, the model below departs from this literature in two noticeable ways. First, whereas previous theoretical work has one uninformed audience, in our model the government (our inspectee) sends a report to two different uninformed audiences with different preferences over outcomes, the NGO (our inspector) and a third-party observer. Ex post, both types of governments benefit from third-party support, but only those perpetrating legitimate violence benefit from NGO investigations. Nonetheless, we show that governments benefit ex ante from stronger NGOs if the resulting investigations are thorough enough to commit governments to the truth.
Second, in our model the government's actions are reports or disclosures. As such, they only affect government payoffs indirectly through endogenous thirdparty support instead of having a direct effect or altering the structure of strategic interaction. Thus, lying costs are fully endogenous in our model and arise via the observer's distaste for cover-ups and equilibrium beliefs. In contrast, previous work treats lying costs as a black box by appealing to long-term reputational costs (e.g., Crescenzi et al. 2011;Smith 2021). This distinction is important because we show that when lying costs are endogenous, it becomes more difficult to observe them in data generated from the model's equilibrium. The difficulty arises because uninformed third-parties internalize the possibility of coverups when the government conceals illegitimate violence on the equilibrium path-even truthful governments are affected by lying costs.
More broadly, our analysis contributes to the literature examining how media shapes regime accountability-see Graber (2003) and Baum and Potter (2008) for reviews. Briefly, a robust civil society provides information to citizens that allows them to hold leaders accountable for unpopular decisions. Even autocrats may adopt partial press freedoms to incentivize local or bureaucratic officials (Egorov, Guriev, and Sonin 2009;Lorentzen 2014) or balance coup threats (Boleslavsky, Shadmehr, and Sonin 2021;Hollyer, Rosendorff, and Vreeland 2019). In the domain of national security, Colaresi (2012) and Bell and Martinez Machain (2018) argue that increasing transparency institutions is a win-win for democratic governments: strong transparency institutions simultaneously satisfy temporary secrecy demands and long-term accountability demands through retrospective oversight by the media. 2 Implicit in these accounts is an assumption that "the media serve primarily as a linkage mechanism rather than as an independent, strategic actor in the policymaking process" (Baum and Potter 2008, 50). In contrast, we treat NGOs as strategic actors who allocate investigative effort according to budgetary pressures and expectations about government behavior. Doing so helps to elucidates the relationship between transparency institutions and actual information disclosed by governments and also accounts for why governments adopt these institutions.

THEORETICAL APPROACH AND ILLUSTRATIVE CASES
Three premises constitute the foundation of our formal model. First, warring parties rely on the support of third-party noncombatants. In civil war, localized support can produce resources or information about the tactical strategies of the opponent (Condra and Shapiro 2012;Kalyvas 2006;Shaver and Shapiro 2021). In international conflict, support from locals in theater provides similar benefits, but leaders in both autocracies and democracies need the support of their selectorate at home to ensure political survival (Bueno de Mesquita et al. 1999;Prorok 2016;Weeks 2012). Over the course of a conflict, a government may, intentionally or otherwise, perpetrate violence that they expect their potential supporters to find abhorrent, should they learn of it. Because violent conflict is messy, potential supporters do not have complete information about the nature of violence. Consequently, governments may attempt to conceal illegitimate violence to ensure continued support in the immediate term.
Second, although governments would like to conceal their use of illegitimate violence, they also wish to avoid getting caught in a cover-up, which could further undermine support. Societies with a minimally free press present the potential that watchdog NGOs or newspapers may also investigate the conflict and, in doing so, they may uncover evidence that contradicts the government's official narrative about its wartime behavior. Watchdog, transparency, or human rights organizations view their primary purpose as gathering and disseminating information, and they derive monetary benefit via donors from doing so. For example, Human Rights Watch and Global Witness rely on donations from the public to continue fact finding and publishing reports on human rights violations both in and outside the context of violent conflict. These benefits might be even larger when their reports contradict information provided by the government. Therefore, when the government has concealed illegitimate violence, it cannot ensure that the violence will not be exposed by this kind of watchdog NGO. Such exposure may cause the government to lose supporters who care about the government's honesty.
Third, NGOs and governments provide different types of information to observers. Governments have more information on the state of violence, given their participation in the event and the chain of command that regularly allows for the transfer of information through the ranks. However, even if the government would like to release verifiable information about the nature of a conflict event, constraints on military intelligence may prevent them from doing so. Nongovernmental organizations do not know the nature of violence at the outset 2 Colaresi (2012) uses "institutions of oversight" to encompass FOI laws and press protections, whereas we use transparency institutions. They also include national-security legislative oversight powers, which are outside the scope of our analysis. Strategic Reporting: A Formal Model of Biases in Conflict Data but need to exert costly effort to investigate, providing verifiable information. 3 This creates an incentive for governments to conceal illegitimate violence that affects their support in the short term, so long as the potential cost of lying is not perceived to be too great in the long run. Of course, NGO investigations may not always be successful. Their likelihood of success depends on the amount of devoted resources, which is strategically allocated according to expectations about what an investigation could find. Although NGOs may be more or less driven by this expectation, they are unlikely to fabricate illegitimate violence because their survival and funding is contingent on their reputation as credible actors.
This theoretical framework captures primary empirical features of cases as disparate as the Naxalite conflict in India and civilian deaths from US drone strikes abroad. These cases motivate our main assumptions and illustrate how the model works on the ground. We do not use the cases to test the model or in a processtracing exercise. Instead, we rely on them to identify prominent contextual features that we believe a model of strategic reporting should include. Our model is, in that sense, motivated by the cases. The model further provides a precise and internally consistent account of how reporting incentives of governments and NGOs interact, producing a set of intuitive as well as counterintuitive implications for conflict scholars.

Naxalite Conflict, India
The Naxalite conflict is a Maoist insurgency in India that originated in 1967. Initially, Naxalites were a small, ideological group that split from the main Marxist party in India. In 2004, the conflict escalated from a low-level skirmish with tens of fatalities annually to a full-scale insurgency with thousands of casualties per year. The states most affected were parts of Jharkand, Bihar, Andra Pradesh, Orissa, and most of Chhattissgarh. The escalation was in part due to the emergence of village-level militias, called the Salwa Judum, fighting to support the government. The warring parties-the Naxalites, the Salwa Judum, and the Indian military forces-allegedly engaged in firefights, rape, targeted killings, and destruction of villages. For civilians, it is difficult to differentiate between the Salwa Judum and the Indian military. Moreover, Naxalites are not regularly dressed in recognizable uniforms and belong to no easily identifiable tribe or caste. Consequently, conflict events that occur in the rural regions of Chhattisgarh, for instance, are difficult to attribute although much of the territory outside the urban centers in Chhattisgarh is recognized as Naxal-held territory during the time of interest.
The Indian government considers the Naxalites one of the most dangerous threats to internal security.
Consequently, it began maintaining a list of conflict events and associated fatalities in an online platform called the South Asia Terror Portal (SATP), which is widely used by academics and policy makers. The platform initially provided the location (state) of the event, the date, and some contextual details that might have included the event's perpetrator and civilian casualties. These events often include smaller skirmishes after rebels attack police outposts or when government forces patrol villages. The resulting government narrative suggests Naxalites are the primary instigators in months and areas with the largest number of casualties and fatalities.
In 2007, multiple NGOs started investigating the veracity of the SATP between 2005 and 2007 and gathering their own data about the conflict with a focus on Chhattisgarh, the state hardest hit by the insurgency. Through fact-finding missions in suspected areas of violence, the NGOs compiled lists of conflicts, detailing their locations and the names of individual casualties. The NGO data differ significantly from the SATP data, although the latter has greater temporal coverage than the former. Furthermore, in the years since the NGOs' reporting, the SATP has become less detailed, providing only aggregate numbers of deaths on an annual basis. 4 Recall that investigating NGOs rely on external funding resources to operate, some of which is contingent on the truthfulness or revelatory nature of their findings. Because of the large number of NGOs in India, there is significant competition for these resources. Thus NGOs cannot afford to fabricate data outright or they risk damaging their reputation and losing their funding.

The US Drone Strikes and Targeted-Killings Program
After the September 11 attacks, the Bush administration began a secretive targeted-killing program in regions without ongoing hostilities, including Pakistan, Somalia, and Yemen (at the time). Relying primarily on drone strikes to carry out assassinations of alleged terrorists, President Barack Obama continued this program throughout his administration. Information about civilian casualties was originally limited. Unlike the lists of civilian casualties in active military theater, the administration produced no such list for the targeted-killing program. It was not until Obama's second term that the program's existence was even acknowledged. Although the use of assassinations outside of active war zones appears to have recently gained greater acceptance among governments, the noncombatant casualties associated with these tactics remain a point of moral outrage. According to the Pew Research Center (2015), 60% of US citizens support drone strikes targeting extremists, but 80% are concerned about the attacks endangering the lives of innocent civilians.
The US government acknowledges that it has verifiable information about the number of fatalities associated with these targeted strikes: "government post-strike reviews involve the collection and analysis of multiple sources of intelligence before, during and after a strike, including video observations, human sources and assets, signals intelligence, geospatial intelligence, accounts for local officials on the ground, and open source reporting" (Director of National Intelligence 2016, 2). It cannot release such information due to military and intelligence constraints. It was only in 2015 that the Obama administration passed an executive order requiring an annual report of the number of noncombatants killed during the program. Two reports were issued; one covers 2009-2015 and another 2016 (Director of National Intelligence 2015; 2016). 5 The reports are vague, providing a single numerical estimate of aggregate noncomabatant fatalities across strikes in all three countries, similar to the current SATP list. Unlike the Naxalite case, the Obama administration refused to acknowledge the existence of the drone program and in doing so withheld information about noncombatant casualties until these reports were issued. Although security hawks may view this as a lie of omission, security doves may view it more nefariously.
Several years before the Obama administration passed the executive order, the Bureau of Investigative Journalism (BIJ), the Long Wars Project, and the New America Foundation began gathering data on the annual number of civilian casualties associated with targeted drone strikes in Pakistan, Somalia, and Yemen. Drawing on media reports and contacts on the ground, these NGOs have each developed a list of strikes, the total number of fatalities, and the number of noncombatant fatalities in each of these countries from 2004 onward. Although these sources differ from each other in their accounting, they differ further still from the two government reports: every independent investigation of drone strikes has found more noncombatant deaths than admitted by the administration (Shane 2015). The NGOs and a variety of media outlets believe that it was this investigative reporting that led Obama to issue the executive order requiring reporting on the targeted-killings program. Although the costs of not revealing this program in the face of NGO reports may be difficult to observe, Obama continues to face criticism from the left, even postpresidency, for failing to justify noncombatant casualties resulting from the drone strikes (e.g., Friedersdorf 2016; Williams 2017).

FORMAL FRAMEWORK
The model consists of three actors: a government, a watchdog NGO, and a third-party observer who is a potential supporter of the government, labeled G, N, and O, respectively. The government is involved in an ongoing violent conflict. At the beginning of the interaction, the current state or type of violence is v ∈ 0, 1 f g: The state v = 1 denotes an occurrence of illegitimate violence-for example, violations of human rights, violence against noncombatants, or destruction of critical infrastructure. In contrast, the state v = 0 means no illegitimate violence occurred. Initially, the state of violence is known only to the government, and the probability that violence is illegitimate is Pr v = 1 ð Þ= q. The parameter q captures at least three sources of illegitimate violence including how often the government chooses to violate laws or norms of war, agency problems between leaders and on-the-ground troops, and mere bad luck. These last two sources are outside the government's purview, so q > 0.
After observing state v, the government chooses whether to acknowledge that illegitimate violence occurred (denoted m = 1Þ or not (denoted m = 0Þ: The message m = 1 corresponds to the government disclosing its use of illegitimate violence to reflect the deaths of civilians, for instance. We interpret m = 0 as the business-as-usual message where the government does not update its list of government-perpetrated noncombatant killings or its list of drone strikes with civilian casualties. The government's disclosure decision may not occur immediately after the state of violence is revealed, but this disclosure phase occurs before NGOs can investigate. Both the third-party and the NGO observe the government's message m. Subsequently, the observer chooses an initial level of support for the government s 1 ∈ R. The NGO then chooses a level of effort e ∈ 0, 1 ½ for investigating the state of violence. With probability e, enough information is uncovered to publish a report revealing the type of violence (r = 1). With probability 1−e, the investigation fails to uncover enough information to publish (r = 0). If a report is published (r = 1), then the type of violence v is revealed to the observer. If the report is not released (r = 0), then the type of violence remains unknown. 6 In other words, NGOs find and disseminate evidence that verifies either type of violence v ∈ 0, 1 f g. 7 The likelihood that they find such evidence depends on their chosen effort e. The observer then chooses a second level of support s 2 ∈ R.
Consider how this setup captures the cases. In the Naxalite conflict, government troops regularly encounter rebels when patrolling villages. During these encounters, noncombatants may be targeted or killed by government forces. The third-party observer is the local, noncombatant population that determines the degree to which to support government forces-for example, by providing tactical information that could 5 Trump ended the practice by executive order. 6 We assume that the observer sees the NGO report if it is published, but this assumption can be relaxed, e.g., the report is read with a fixed probability. In this version, the equilibrium characterization would not substantively change, but the government would be less likely to disclose illegitimate violence in equilibrium. 7 Because we focus on verifiable information, there is no possibility that the NGO lies in the model, reflecting the NGO's incentive to maintain legitimacy to secure funding. In Appendix H we consider a version of the model in which only illegitimate violence is verifiable. be used to defeat the rebels. If noncombatants are killed in an encounter, either accidentally or because they were incorrectly labeled as Naxalites, then the locals may consider the violence to be illegitimate. Initially, only the Indian government knows whether its forces engaged in illegitimate violence. Because war is messy and information is incomplete, locals outside of those immediately affected by the violence do not know its legitimacy. The government decides whether to report noncombatant causalities (i.e., disclose the legitimacy of violence) when updating its list of rebel encounters. The local population then decides whether to lend support to the government-for example, by providing more or less useful tips to the government about the insurgents' tactical operations. At this point, NGOs may decide to investigate whether the government is being truthful in its description of the conflict. After their reports are published, potential supporters may change their level of support based on the findings.
In the US's targeted-killing program, drone strikes may or may not entail noncombatant causalities or the destruction of critical civilian infrastructure. In contrast to the Naxalite case, the observer represents a constituency that decides the degree to which to support the Obama administration at the voting booth or in the court of public opinion. If the administration's goal is to create broad support, then the observer could be a representative US citizen. In contrast, if the goal is to motivate the progressive base, then it could be a representative member of the Democratic Party. The model accommodates either interpretation, although the preferences of the specific observer-which we describe below-would change across interpretations. Strikes that only destroy munitions stockpiles or terrorist cells are likely to be considered legitimate violence, but those that kill noncombatants or destroy hospitals, for example, are less likely to be viewed as legitimate. When these events occur, US citizens have limited information about them. The government, in contrast, has explicitly stated that it knows the civilian cost of each of its targeted attacks (Director of National Intelligence 2016). If citizens gain information about the degree of noncombatant casualties associated with drone strikes through NGO reports, such as those published by the BIJ, then public opinion of the administration or Obama's legacy within the Democratic Party may change.
For payoffs, the government wants to maximize support from the observer-support offered before the NGO publishes and support offered after any potential dishonesty is revealed. Its payoff is Above, the function g : R ! R maps support into some benefit. The function g is strictly increasing and continuously differentiable with a nonvanishing derivative (g 0 s ð Þ > 0 for all s). These benefits naturally depend on the nature of the conflict and the interpretation of the third-party. In the Naxalite conflict, support comes in the form of tactical information reported to the government as tips from the locals that can be used to defeat the insurgency (Kalyvas 2006;Lyall, Shiraito, and Imai 2015;Shaver and Shapiro 2021). In the drone-strike case, support refers to Obama's poll numbers that can be used as political capital required for reelection or his progressive legacy upon leaving office. The parameter δ > 0 captures the relative importance of timing. If δ < 1, then the government prioritizes immediate support. If δ > 1, then the government prioritizes final support. The observer has an ideal level of supportŝ that depends on the state of violence, the government's message, and a baseline popularity level: Above, β ∈ R is the baseline support for the government. The parameter γ > 0 denotes the observer's distaste for the government using illegitimate violence, and κ > 0 is the observer's distaste for the government after it hides illegitimate violence. 8 With ideal support levelŝ, the observer's payoffs are which is the quadratic loss between the chosen support and ideal support. 9 Finally, the watchdog NGO wants to discover enough information to publish a report subject to some cost of investigating. Its payoff is Above, we normalize the NGO's benefit of revealing the state of violence to one but divide this benefit into two components. The first term λ ∈ 0, 1 ð captures the proportion of benefit from releasing reports regardless of whether there is a cover-up. The second term 1−λ captures the proportion of benefit that arises from catching the government in a cover-up. If λ is close to zero, then a substantial proportion of NGO publication benefits depend on exposing government cover-ups. If λ is close to 1, then NGO benefits depend on releasing reports regardless of their salaciousness.
The term ρ 2 e 2 is the cost of exerting effort, and the parameter ρ captures NGO efficiency. For a fixed probability of success, less efficient NGOs (larger ρ ) pay higher investigative costs than do more efficient ones. Efficiency likely varies across NGOs, depending on funding and transparency institutions. Better funded NGOs will be more efficient, as they are not likely to face binding budget constraints and thus large opportunity costs, so ρ should be smaller. Likewise, NGOs operating in countries with transparency institutions such as FOI laws and press protections will face lower investigative costs because it is easier to gather information (Colaresi 2012).
Our two cases help to motivate these payoffs. Many watchdog NGOs rely on charitable donations for funding. The size and frequency of these donations relate to their ability to publish visible reports and to their provision of information that differs from the prevailing state narrative. Some NGOs benefit from a surprise dividend-that is, additional resources following their revelation of government cover-ups. Not all NGOs are similarly reliant on this surprise dividend and thus may have different preferences, representing cases where λ is closer to one. We expect λ to be small when there is competition among NGOs for attention and donations, as in India, which has one of the largest numbers of NGOs per capita. At least one of the NGOs investigating the Naxalite conflict was particularly resource scarce and relied heavily on a surprise dividend, attempting to be the first to reveal dramatic information contradicting the government's narrative. 10 Notice that the government potentially trades off initial and final support, where δ captures the relative importance of final support to initial support. In the Naxalite conflict, local support in the Chhattissgarh region might be instrumental for the government's military success. Thus, we might suspect that δ is close to or smaller than one as the government might want to end the conflict as fast as possible. In the US case, δ might be correlated with the time until the next election. If the presidential election is far off, δ would be greater than 1, but if an election is immediate, δ is close zero. In addition, δ could capture the degree to which Obama prioritizes his postpresidential legacy. If this is sufficiently valued, δ would be greater than one.
Strategies and beliefs are straightforward. For the government, a strategy is a function σ G : 0, 1 f g ! 0, 1 ½ , where σ G v ð Þ is the probability that the government admits that it used illegitimate violence after violence state v. For the observer, a strategy is a function σ O : Þ is the support O gives the government after message m when the state of violence is unknown (v = ∅), revealed to be legitimate (r = 1 and v = 0), or revealed to be illegitimate (r = 1 and v = 1 ). Finally, a strategy for the NGO is a function σ N : 0, 1 f g ! 0, 1 ½ , where σ N m ð Þ is the amount of effort N chooses after message m. In addition, μ m is the belief that conflict involved illegitimate violence after message m-that is, μ m = Pr v = 1jm ð Þ . We focus on perfect Bayesian equilibria where beliefs satisfy a version of the D1 criterion, referred to as equilibrium hereafter. Specifically, an equilibrium is an assessment σ, μ ð Þwhere (a) σ = σ G , σ O , σ N ð Þ is a sequentially rational strategy profile given beliefs μ = μ 0 , μ 1 ð Þ and (b) beliefs μ are consistent with the strategies and updated via Bayes rule whenever possible. In addition, for any message m not sent with positive probability in equilibrium, the belief μ m satisfies a version of D1 modified to account for endogenous verification of the sender's type. 11 In the analysis, the refinement removes an equilibrium in which the government always admits to illegitimate violence regardless of v. Given the rarity of governments admitting fault in military combat, this is a substantively appealing criteria. Before proceeding, it is important to remember that when illegitimate violence occurs, the business-as-usual message represents the government concealing the true state of violence. This concealment may take two forms, however. The government may omit the presence of illegitimate violence or may explicitly claim violence was legitimate. If the interpretation is the former, then we expect the lying costs κ to be comparatively smaller in magnitude than if the interpretation is the latter. This approach is justified for three reasons. First, our substantive implications focus on the frequency with which the government admits illegitimate violence, so the exact nature of the concealment is not a first-order concern. This is similar to empirical work that uses counts or indicators of illegitimate violence. Second, the type of concealment is difficult to classify in our cases. For example, in the Naxalite case, when the government does not report a clash with the rebels that resulted in noncombatant fatalities, this could be interpreted as concealment via omission. Another interpretation would be that, because the government reports no clashes, there could not have been illegitimate violence, which represents concealment via a lie. Third, in a version of the model with three messages-representing acknowledge illegitimate violence, omit discussing violence, and say explicitly no illegitimate violence occurred-the government has peculiar incentives after legitimate violence in nonseperating equilibria. Specifically, it might want to send unexpected messages suggesting that it concealed illegitimate violence, in which case the NGO would have greater incentives to investigate (as it expects a coverup), which increases the probability that legitimate violence will be exposed and then increases final support for the government. This incentive seems disconnected from our cases where we do not observe governments, after claiming legitimate violence, trying to convince NGOs that there was indeed illegitimate violence to encourage investigations.

ANALYSIS
NGO's effort. To see how NGOs investigate, note that after the government sends message m = 0, the probability of a cover-up is μ 0 . Then the NGO selects an effort level after message m such that 10 Author personal correspondence and experience. 11 We assume that in any subgame after the NGO releases a report (r = 1) the observer has correct beliefs (i.e., knows the state) even if the subgame is off the equilibrium path. See the proof of Lemma 1 for details.

Strategic Reporting: A Formal Model of Biases in Conflict Data
and its equilibrium effort therefore takes the form If λ < 1, then a proportion of NGO publication benefits depends on exposing government cover-ups. In this case, Equation 1 says that as the NGO expects the government to more frequently conceal illegitimate violence (i.e., μ 0 increases), it allocates more investigative effort because it is more likely to expose a coverup, which entails 1−λ of additional benefit. Likewise, because countries with press protections and FOI laws will have smaller investigative costs ρ, Equation 1 says that NGOs in these countries will exert more effort than those in countries without such transparency institutions, all else equal.
Observer's support. When the observer chooses support (either initial s 1 or final s 2 Þ it may not know whether the government concealed illegitimate violence. Also, note that the observer does not have private information and its actions do not influence the NGO's equilibrium incentives in Equation 1. Thus, when the observer does not know the value of vbut sees message m, its equilibrium support satisfies for t = 1,2. In contrast, when the observer knows v-for example, when the NGO successfully investigated the government-then the observer can choose its level of support to matchŝ. Overall, this discussion implies that in equilibrium, third-party support takes the form: Equation 2 illustrates why it is difficult to use variation in observed support over time to identify the distaste of cover-ups and illegitimate violence. Suppose violence is illegitimate and the government conceals it by sending message m = 0: In equilibrium, initial support is s 1 = β− γ þ κ ð Þμ 0 . If the NGO does not release a report, then final support is also uninformed, s 2 = s 1 , but if the NGO releases a report, then s 2 = β−γ−κ. After an investigation reveals a cover-up, the change in support is therefore s 2 −s 1 = γ þ κ ð Þ μ 0 −1 ð Þ, which is muted by equilibrium beliefs μ 0 . When the observer anticipates coverups and illegitimate violence, μ 0 is large, so a smaller shift in support occurs than suggested by γ and κ. Government's message. The government sends message m to maximize its expected benefits given the type of violence v and assessment σ, μ ð Þ. The first result says that the government is truthful in equilibrium after legitimate violence (v = 0). Lemma 1. If violence was legitimate, then the government sends the business-as-usual message-that is, σ G 0 ð Þ = 0 in every equilibrium σ, μ ð Þ . After the business-as-usual message, equilibrium beliefs are which is strictly decreasing in σ G 1 ð Þ and weakly increasing in the prior q.
Only after illegitimate violence (v = 1Þ does the government have incentives to lie. On the one hand, the government can send a truthful message (m = 1Þ, thereby avoiding a lie but decreasing support. On the other hand, the government can lie (m = 0) to increase initial support in hopes that the NGO does not reveal the lie, in which case it enjoys uninformed support in both the immediate and long term.
The expected benefits of lying are thus endogenous to equilibrium behavior. After illegitimate violence, if the government lies by sending the business-as-usual message m = 0, then its payoff is In Equation 3, the government potentially receives two different levels of support if it sends the business-asusual message after illegitimate violence. The observer's first level of support (made before the NGO report) will be uninformed, Þμ 0 ρ and will be uninformed with complimentary probability.
In Equation 3 both the NGO's effort and the observer's uninformed support depend on equilibrium beliefs μ 0 : By Lemma 1, μ 0 ∈ 0, q ½ is strictly decreasing in σ G 1 ð Þ-that is, the truthfulness of the government. If the government is expected to lie -σ G 1 ð Þ close to zero-then the NGO exerts more effort to investigate and the observer reduces uninformed support. These forces decrease the government's benefit from lying. If the government is expected to tell the truth-σ G 1 ð Þ close to one-then the NGO exerts less effort to investigate and the observer increases uninformed support. These forces increase the government's benefit from lying. The following result details how the government balances these trade-offs in equilibrium.
Proposition 1. The government's behavior is unique in equilibrium: 1. The government is truthful-σ G v ð Þ = v-in equilibrium if and only if

The government never admits fault
3. The government admits fault after illegitimate violence with probability strictly between zero and one -σ G 1 ð Þ ∈ 0, 1 ð Þ-if and only if both inequalities in Equations 4 and 5 are not satisfied. Figure 1 illustrates the inequalities in Proposition 1 as functions of the cost of lying and the importance of future support. The main implication for conflict scholars is that as the observer's distaste of lying increases or the government prioritizes long-term rather than immediate support, then the government becomes more truthful in equilibrium. In Appendix H, we illustrate how the equilibrium characterization changes when only illegitimate violence is verifiable rather than both types of violence being verifiable as in the baseline model. The substantive features of the equilibrium characterization do not change, but the government is weakly less truthful in equilibrium.

Underreporting bias
Scholars often rely on government data because of its temporal span and ease of access. Yet domestic and international NGOs may criticize this data as incomplete or biased in favor of the government. For example, Human Rights Watch provides alternative accounts of the government's use of violence in the Naxalite conflict (Human Rights Watch 2008). In the drone-strike case as well, there are multiple lists recording the extent of illegitimate violence in the conflict. How can researchers or policy makers know which ones to prioritize and when?
To answer these questions, define the probability that actor i = N,G reports illegitimate violence given strategy profile σ: Then i's underreporting bias is B i σ ð Þ = q−PIV i σ ð Þ. In words, underreporting bias is the difference between the baseline frequency of illegitimate violence, q, and i's frequency of reporting illegitimate violence PIV i . In equilibrium, both actors have a tendency to underreport. 12 The government's source of underreporting bias is its incentive to conceal illegitimate violence. The NGO's source of underreporting bias is that it needs to exert costly effort to uncover the truth. Which source does the most damage? To answer this question, we first introduce the following assumption.
Assumption 1. The benefits of support are sufficiently responsive: there exists s ∈ R such that g s Comparing Assumption 1 to Equation 4, the assumption says that we can find a distaste of cover-ups, κ, that Distaste of lying, κ Importance of future support, δ Note: Example generated assuming g(s) = s, γ = 1, λ = 0.5, ρ = 1, and q = 0.2.
12 Scholars often use conflict event lists to count the number of incidences of illegitimate violence in a given region and period without observing the total number of events. We view these counts as aggregating several draws of outcomes from the equilibrium (σ, μ) that is determined by parameters that are fixed throughout a given region and period. Furthermore, when scholars do not observe the underlying events, it is also difficult to aggregate the lists to improve biases (Cook and Weidmann 2019). is large enough to ensure that the government is truthful in equilibrium. The assumption holds if g is concave, for example. The next result describes two cutpoints on the distaste of lying that demarcate the three equilibria.
In the left panel of Figure 2, we graph the underreporting bias for each actor as a function of κ. If the distaste of lying is large enough (κ ≥ κ ), then the government is always truthful. This corresponds to the government having a bias of zero and the NGO having a bias of q 1− λ ρ > 0. In contrast, if the government's cost of lying is small (κ ≤ κ), then the government never admits fault. In this case, its bias is q and the NGO's bias is q 1− λþ 1−λ ð Þq ρ < q. In the intermediate range κ∈ κ, κ ð Þ, the government admits fault after illegitimate violence with probability strictly between zero and one. This probability is strictly increasing in κ, so the government's bias decreases to zero as κ gets larger. As the government becomes more truthful, however, the NGO is less likely to catch the government in a cover-up, so it invests less effort, thereby increasing its bias. As the distaste of cover-ups κ moves from κ to κ, the government's bias becomes smaller than the NGO's bias at the point κ * . Implication 1. Under Assumption 1, there exists cutpoint κ * > 0 such that the NGO's underreporting bias is smaller than the government's if and only if κ < κ * . Furthermore, ∂κ * ∂ρ > 0 if g is concave and ρ 1−q Notice that the cutpoint κ * can increase as the NGO's cost of effort, ρ, increases, which is illustrated in Figure 2's right panel. In words, the NGO's underreporting bias is more likely to be smaller than the government's (in the set inclusion sense) as the NGO becomes less effective at investigating conflict events. 13 Implication 1 states a sufficient condition for this relationship, which is more likely to hold when q is small and δ is sufficiently large. This captures situations where the government's use of illegitimate violence is not rampant and the government cares about its long-term prospects.
One could imagine the opposite result: greater investigative costs disincentivize NGO effort, thereby making NGO data less reliable relative to government data. This story misses the strategic interplay between the NGO and the government, however. When the costs of investigating increase, two effects emerge in equilibrium. In the direct effect, the NGO invests less effort, leading to greater underreporting bias in NGO data. In the indirect effect, the government anticipates the direct effect and also becomes less truthful, leading to greater underreporting bias in government data. Thus, both data sources become more biased after an Note: Left panel graphs the actors' equilibrium level of underreporting bias B i as a function of κ. Right panel graphs κ * as a function of ρand λ. Graphs generated assuming g s ð Þ = s, γ = 1, and q = 0:2. In the left panel, we fix ρ = 1:5 and λ = 0:5, implying that κ * ≈1:83.
increase in investigative costs, but the indirect effect dominates under the sufficient condition in Implication 1. In other words, as investigative costs increase, both government and NGO reports will exhibit more underreporting bias, but the effect will be larger for the government. For similar reasons, κ * can be decreasing in λ-that is, the proportion of NGO publication benefits that depends on releasing information regardless of cover-ups, as illustrated in Figure 2's right panel.
Overall, the analysis suggests two important considerations for conflict researchers. First, it establishes conditions under which NGO data should exhibit less underreporting bias relative to government data: (a) when NGO investigations are relatively inefficient (large ρ) and (b) when NGOs rely heavily on surprise dividends (small λ). As mentioned above, the first condition likely holds when NGOs do not have longterm, robust funding or when NGOs operate in countries without transparency institutions. The second likely holds with substantial competition among NGOs for influence and attention when, for example, there are many NGOs per capita.
Second, underreporting bias in NGO and government data should be positively correlated across cases. When NGOs are substantially underreporting illegitimate violence, there are few incentives for governments to truthfully disclose illegitimate violence, as the likelihood of being exposed in a cover-up is small. Inversely, when NGOs correctly report illegitimate violence, then the government has stronger incentives to tell the truth to avoid cover-ups. Thus, the model suggests that combining government and NGO data will have limited benefits when addressing underreporting bias. When one source consistently misses violence incidents, it is likely the other source will as well.

Illegitimate Violence and Support
Recent work in the counterinsurgency literature estimates the effect of noncombatant casualties on local support for the side responsible. Broadly, state-caused collateral damage can depress support for the government, but the effect is attenuated when examining insurgent-caused damage and support for insurgent groups (Condra and Shapiro 2012;Lyall, Blair, and Imai 2013;Shaver and Shapiro 2021). To measure violence against noncombatants, researchers use selfreported exposure in surveys (e.g., Lyall, Blair, and Imai 2013) or NGO-reported conflict events (e.g., Condra and Shapiro 2012;Shaver and Shapiro 2021). In addition, Lyall, Shiraito, and Imai (2015) measure exposure to violence using data from the International Security Assistant Force, a coalition of NATO governments charged with securing Afghanistan against the Taliban insurgency (842). With these data, they find "no consistent association between indirect exposure to violence and individual attitudes" (844-5). 14 Inspired by these studies and our cases, we compare the observed effect of illegitimate violence on equilibrium support to the true effect when governments strategically report illegitimate violence. We begin by assuming that researchers observe government messages m and final support s 2 from several draws from one equilibrium. 15 For example, they observe whether the government reports causing collateral damages or not and the resulting level of civilian support. With such data, researchers can compare expected support after the government reports illegitimate violence (noncombatant causalities in this context) to expected support after the government reports no illegitimate violence, all else equal. 16 Definition 1 states this comparison formally.
Definition 1. Given a strategy profile σ , the observed effect of illegitimate violence using government data is The observed effect underestimates the distaste of illegitimate violence when Γ σ ð Þ < γ and correctly estimates the distaste when Γ σ ð Þ = γ. Table 1 computes the observed effect of illegitimate violence in the truthful and partially truthful equilibria. In the never-admit-fault equilibrium, the government never sends message m = 1, so Γ is undefined. The rows enumerate all possible combinations of messages and violence states. Given m, v ð Þ, the column E s 2 jm, v, σ ½ refers to the expected level of support following message m and violence state v. The NA values correspond to pairs m, v ð Þ that never appear on the equilibrium path. In the truthful equilibrium, government disclosures completely reveal its type so observed support includes no distaste of lying. In the partially truthful equilibrium, if the government admits illegitimate violence, then it is truthful and the observer correctly anticipates illegitimate violence. In contrast, if the government sends the business-as-usual message, then it is potentially lying. In this case, unobserved support is biased downward when violence was legitimate but biased upward when violence was illegitimate. Expected support after each message then follows from the law of total expectation. Thus, the observed effect of illegitimate violence on support correctly estimates γ only when the government is truthful. In the partially truthful equilibrium, however, the observed effect is smaller than the true value because the observer tempers their support after the business-as-usual message because the government may be concealing illegitimate violence. It becomes particularly important to know under what conditions the government will be truthful and the 14 The study also uses NGO-reported and self-reported exposure to violence. Using the latter, victimization by coalition security forces is associated with a reduction in support to the counterinsurgency in some treatments (Lyall, Shiraito, and Imai 2015, 845). 15 Our analysis would not change if we used average support, i.e., αs 1 þ 1−α ð Þs 2 for α ∈ 0, 1 ½ , but focusing on either initial or final support makes the exposition easier. 16 Civilian support is measured through frequency of informant "tips" in Shaver and Shapiro (2021), attitudes about counterinsurgency informant programs in Lyall, Shiraito, and Imai (2015), and attitudes about coalition forces in Lyall, Blair, and Imai (2013). difference between the true effect, γ, and its observed counterpart, Γ . Recall that by Lemma 2, when g is concave, there exists a κ ∈ R such that a truthful equilibrium exists if and only if κ ≥ κ . So when κ becomes larger, government disclosures are more likely (in the set inclusion sense) to provide incorrect estimates of γ in the case of the partially truthful equilibrium or infeasible estimates in the case of the never-admit-fault equilibrium.
Implication 2. Assume g s ð Þ = s. Then κ = γ 1þδ ð Þρ δλ −1 , and the truthful equilibrium becomes less likely in the set inclusion sense as the distaste for illegitimate violence γ increase-that is, ∂ κ ∂γ > 0. Moreover, in the partially truthful equilibrium σ, μ ð Þ, the difference between the observed and the true distaste for illegitimate violence Δ = γ−Γ σ ð Þis increasing in γ-that is, ∂Δ ∂γ > 0. A dilemma thus arises when estimating the effects of illegitimate violence on popular support using government reports. If the distaste of illegitimate violence, γ, is small, then the government is truthful. This means that the observed level of support will not be biased due to strategic reasons (though it might not be easily detectable in smaller samples). If this distaste is large, then the government is unlikely to be truthful, and the observed effect will be biased toward zero. The magnitude of this attenuation increases in the size of the true effect. Likewise, the bias emerges even though government disclosures and popular support are observed without measurement error. The result indicates that the estimates associated with the government-provided data in Lyall, Shiraito, and Imai (2015) could be interpreted as a lower bound on the degree to which civilians punish governments for exposure to violence.
Finally, the result also indicates that the truthful equilibrium is more likely to occur (in the set inclusion sense) when NGOs have small investigative costs, ρ, and the government cares about long-term support-δ is large. Thus, the observed effect Γ should correctly estimate the parameter γ in environments with wellfunded NGOs, strong transparency institutions, and governments that prioritize long-term support.
Implication 2 focuses on the relationship between illegitimate violence and observed equilibrium support.
Our model also includes a parameter β describing the government's baseline popularity absent illegitimate violence and cover-ups. We can therefore explore the relationship between popularity and the government's propensity to disclose illegitimate violence, which means government transparency can be assessed via government baseline popularity, a potentially observable quantity.
Implication 3. Assume g is strictly concave. As the government's baseline popularity, β, increases, the following hold: 1. The truthful equilibrium becomes less likely in the set inclusion sense-that is, ∂ κ ∂β > 0. 2. The never-admit-fault equilibrium becomes more likely in the set inclusion sense-that is, ∂κ ∂β > 0 , if and only if In other words, the government generally becomes less truthful as its baseline popularity increases if the decreasing marginal returns to support are sufficiently strong. Specifically, if g is strictly concave, then higher levels of baseline support imply a smaller set of parameters sustaining the truthful equilibrium (Implication 3.1). In addition, the left-hand side of Equation 6 is a measure of the strength of decreasing marginal returns.
The intuition for this is straightforward. With strong decreasing marginal returns to support, the loss of support that follows an exposed cover-up is more detrimental to a government with low baseline popularity. Thus, the government can more easily afford the costs of lying when it enjoys broad baseline support. Therefore, the truthful equilibrium becomes more difficult to sustain and the never-admit-fault equilibrium becomes easier to sustain as the baseline support increases. Thus, the result suggests that conflict researchers should have greater concerns about underreporting bias from government data when the government enjoys a high baseline popularity from the observer, all else equal.
Rows denote all possible message-violence pairs in the truthful (top) and partially truthful (bottom) equilibria, and NA denotes message-violence pairs that do not emerge in equilibrium. Columns denote the values used to compute the observed effect of illegitimate violence on support Γ. The value Γ is not defined in the never-admit-fault equilibrium.

How and When NGOs Benefit Governments
Scholars have sought to explain why governments create transparency institutions at all given the benefits of controlling information about its behavior (Grigorescu 2003). Several study the variation in FOI laws (Berliner 2014), but transparency institutions include broader legal and regulatory frameworks that facilitate civil society's access to information (as in Egorov, Guriev, and Sonin 2009;Lorentzen 2014). Understanding the drivers and effects of these institutions is especially important in this domain of national security, as governments are especially reticent to disclose information (Colaresi 2012). 17 In the Naxalite case, for example, the government imposed significant costs on NGOs investigating the conflict via the Chhattisgarh Special Public Security Act of 2005. Those convicted of contacting suspected Naxalite rebels faced six years in prison. This policy and others that determine press protections affect investigative costs, ρ, and the equilibrium strategies capture the transparency behavior of the government. Thus, we use the model to study the effects of transparency institutions (via smaller ρ) on government truthfulness and the conditions under which the government would have incentives to manipulate NGO investigative costs through changes to transparency institutions.
Lemma 3. In the partially truthful equilibrium, the government becomes less likely to disclose illegitimate violence as NGO investigative costs, ρ , increase-that is, In the other equilibria, the government's strategy and thus beliefs are constant in ρ.
Because transparency institutions reduce investigative costs, they encourage government disclosures in the partially truthful equilibrium. This is illustrated in Figure 3's left panel. Notice that transparency institutions are not necessary for government truthfulness, however. Even in their absence, as long as ρ < ∞, NGO investigations still expose cover-ups in equilibrium, which means the government could truthfully disclose illegitimate violence when the importance of long-term support, δ, and the distaste of cover-ups, κ, are large.
Recall that uninformed support after message m = 0 depends on the probability that the government lied, μ 0 . By increasing the equilibrium probability that governments disclose illegitimate violence, efficient NGOs create positive belief spillover effects to governments after legitimate violence via enhanced uninformed support. The next implication states when this effect can increase the government's ex ante expected utility. Implication 4. If g is strictly concave, then the following hold: 1. In the partially truthful equilibrium, the government's ex ante expected utility is strictly decreasing in NGO investigative costs, ρ, if λ ≥ ρ 1þδ ð Þ−2qδ δ 1−2q ð Þ . This inequality always holds if q ≤ 1 2 . 2. In the never-admit-fault equilibrium, the government's ex ante expected utility is strictly increasing in ρ. 3. In the truthful equilibrium, the government's ex ante expected utility is constant in ρ.
In other words, when g is strictly concave, governments can benefit ex ante from more efficient NGOs-that is, smaller ρ produced from transparency institutions-only in the partially truthful equilibrium. Implication 4 states two sufficient conditions for this to happen: the probability of illegitimate violence is small or the NGO is sufficiently motivated to investigate events regardless of the surprise dividend. To see the intuition, notice that, in the partially truthful equilibrium, the government wants to commit to telling the truth ex ante. After illegitimate violence, the government is mixing and thus indifferent between lying and telling the truth, the latter entails a payoff of 1 þ δ ð Þg β−γ ð Þ, which is independent of ρ. After legitimate violence, the government sends message m = 0and would like the message to be believed with certainty, and Lemma 3 shows that the message becomes more believable with more efficient NGOs. Thus, increasing the efficiency of NGOs via transparency institutions can weakly increase the expected utility of both types of government as the business-as-usual message m = 0 becomes more believable.
In contrast, in the never-admit-fault equilibrium, more efficient NGOs decrease the government's expected payoffs. Here, the government expects three levels of final support: uninformed s 2 = β− γ þ κ ð Þq with probability 1−σ N 0 ð Þ, informed after legitimate violence s 2 = β with probability σ N 0 ð Þ 1−q ð Þ, and informed after illegitimate violence s 2 = β−γ−κ with probability σ N 0 ð Þq. As the NGO faces higher investigative costs, there is greater probability that final support will be uninformed, reducing uncertainty from an ex ante perspective. When the government is risk averse (g is strictly concave), this increases the government's expected utility. In the truthful equilibrium, government disclosures remove uncertainty about the type of violence, so the NGO's report and thus the NGO's cost of effort does not affect its payoffs.
Overall, whether the government benefits from transparency institutions and more efficient NGOs depends on exactly how truthful the government becomes after their adoption. If the government does not become very truthful, then the NGO will better expose cover-ups, meaning the government is worse off. If the government becomes very truthful, then uninformed support increases, benefiting the government and minimizing the chances of being caught in a cover-up. These competing effects produce the nonmonotonic relationship between investigative costs ρ and the government's expected payoffs in Figure 3's right panel. Here, the government is worse off with moderately efficient NGOs, ρ ≈ 6 , and would prefer either more or less efficient investigating NGOs. 17 Absent the capacity to manipulate transparency institutions, officials could use other means of manipulating NGO costs such as attacks against journalists (Carey and Gohdes 2021;Davenport 2009).

CONNECTIONS TO OUR CASES
Several implications are born out in our cases. First, recall that NGO and government data exist for the Naxalite conflict, specifically in Chhattisgarh during 2005-07 and that substantial differences arise between lists. Implication 1 suggests that one list is likely to be a more accurate depiction of the illegitimate violence in conflict than the other. In the Naxalite case, this is because at least one of the NGOs was especially interested in catching the government in a lie given the competition for attention and resources among NGOs. 18 This places the Naxalite context along the light-gray line in Figure 2, suggesting that the NGO list is likely to exhibit less underreporting bias than the SATP list of encounters. Regarding the number of civilians killed, we would expect the NGO list to more accurately represent the nature of the conflict.
We can also observe the relationship between baseline popularity and truth telling. India's long-running National Congress Party was in power and widely popular between 2004 and 2008, although it was in a coalition government. Marginal returns to additional support were minimal. Consequently, the resulting loss of support from an exposed cover-up would impose a smaller cost than it would have for a less popular governing party, which can explain the differences between the government and NGO accounts.
Implication 3 helps explain the emergence of the Chhattisgarh Special Public Security Act in 2005, which drastically increased the cost of NGO investigations, ρ, by introducing harsh penalties for affiliating or communicating with suspected Naxalites. Figure 3's left panel illustrates that such a change in ρ can move behavior into a never-admit-fault equilibrium from a partially truthful equilibrium. Figure 3's right panel illustrates that if the initial ρ was not too small and the subsequent increase in ρ was large, then the government is strictly better off by having passed the 2005 Act.
In the US targeted-killings program, we can think of this case as exhibiting two different periods: before the release of any data by the administration and after the release of the first report in 2016. In the first period, only NGO data are available and the government's list is effectively zero, with the administration operating in the never-admit-fault equilibrium. In the later period, it is likely operating in the partially truthful equilibrium, releasing reports that convey only a portion of noncombatant deaths resulting from the drone strikes. This difference in equilibrium could be explained by a change in Obama's time preferences over support, δ. In 2011, δ is arguably small, as Obama was running for reelection and immediate political support was crucial. In the second period, δ increases drastically as Obama becomes more concerned with his legacy than immediate electoral support. Consider Figure 1 to see that an increase in δ could move the government away from the never-admit-fault equilibrium into the partially truthful one.
The effect of baseline popularity is more difficult to interpret in this case. Obama faced lower than average approval ratings throughout most of his presidency; his NGO cost of effort, ρ G's ex ante expect utility Note: Left panel graphs the government's equilibrium probability of truthfully reporting illegitimate violence, σ G 1 ð Þ, as a function of the NGO's cost of effort, ρ. Right panel graphs the government's ex ante expected utility as a function of ρ. Dashed vertical lines demarcate the three types of equilibrium behavior: truthful (small ρ), partially truthful (moderate ρ), and never admit fault (large ρ). Graphs generated assuming g s ð Þ = log s, β = 2, κ = 1, γ = 0:95, δ = 2, and q = 0:25.
18 Author first-hand experience.
favorability surpassed 50% only during his fourth and seventh years in office. During the second peak in popularity, the administration acknowledges the targeted-killings program and begins reporting associated civilian casualties. This pattern is inconsistent with the comparative statics in Implication 3, which suggests that the government's truthfulness should decrease after an increase in baseline support (assuming g is sufficiently concave). Nonetheless, at the end of his second term, Obama may have prioritized his progressive legacy, in which case truthfulness would have increased due to an increase in δ. Implication 3 helps us understand the effect of the BIJ, which began systematic data collection in 2010, publishing its first list of casualties in 2011. It was a well-funded actor that selectively chose reporting projects. As such, it constitutes a highly efficient NGO (i.e., it has a small ρ). As illustrated in the left panel of Figure 3, when the costs of NGO investigations decrease, the government's likelihood of disclosing illegitimate violence weakly increases. Furthermore, it is possible that such a change could shift the behavior from the never-admit-fault equilibrium to the partially truthful equilibrium. The overall effect of this shift on the government's expected utility is ambiguous. If such a change sufficiently commits the government to the truth, then it is better off after the BIJ begins its investigations.

CONCLUSION
We explore the mechanisms that lead governments to strategically disclose illegitimate violence and establish implications for the production and analysis of conflict data. We find that both government and NGO reports of illegitimate violence are likely to suffer from underreporting bias, and these biases should be positively correlated across cases. When NGOs face higher investigative costs, they invest less effort in reporting and are therefore less likely to expose cover-ups. At the same time, however, governments will have larger incentives to conceal illegitimate violence.
In addition, we illustrate a dilemma that arises when estimating the effects of collateral damage on civilian support using government reports: if this effect is small and unimportant, then government disclosures are likely to be truthful and the effect can be correctly estimated using standard research designs. If this effect is large and substantial, however, then government disclosures will understate the amount of illegitimate violence, leading to attenuation bias even when reports and support are observed without error. Finally, our analysis suggests that governments will have nonmonotonic preferences over the strength of transparency institutions, where moderately strong institutions leave the government the worst off.
Future research might evaluate how the model applies to domains outside the production and analysis of conflict data. For these applications, we highlight three core assumptions. First, governments release information that is not immediately verified, an assumption best characterizing the national security context where governments cannot reveal hard information without risking a security threat. This might also be true of particular kinds of financial information, the revelation of which might cause significant market volatility. Second, the watchdog NGO or media must be able to produce hard, verifiable information (e.g., pictures of mass graves or videos of noncombatant casualties) that is released to the observer. Our model is therefore not applicable to the study of partisan news organizations if they produce false or unverifiable reports or selectively choose what to report. Third, the government has sufficient certainty about the true state of the world. If, alternatively, the government sees sufficiently noisy signals of the state of the world, then it may hedge against the potential cost of a cover-up by admitting wrongdoing even after seeing a signal that suggests appropriate behavior. These incentives are particularly strong when the distaste of cover-ups and the prior probability of wrongdoing are large.
We note that, although some of our modeling parameters are unlikely to change over the course of a conflict, other parameters-for example, baseline popularity-may vary significantly during a conflict. Variation in these parameters might suggest that some periods may exhibit more or less underreporting bias than others within the same conflict and same dataset. We do not study how forward-looking governments disclose violence today to influence the evolution of their popularity throughout a conflict, although these dynamics could be explored in future research. Likewise, the occurrence of illegitimate violence is exogenous in our model in part because conflict is messy and violence against noncombatants often occurs unintentionally. Nonetheless, future research could endogenize the government's use of illegitimate violence to study how this changes the conditions under which government transparency arises.
Finally, understanding the incentives of warring parties to report the true nature of conflict events has implications for postconflict reconciliation and transitional justice. Recent work has demonstrated that transitional justice initiatives are often ineffective at promoting postconflict peace and reconciliation (Loyle 2018;Loyle and Davenport 2016). This may be because even genuine attempts to implement transitional justice institutions (e.g., tribunals, reintegration policies) are often built on a shared record of violence among warring parties. This shared record is often a compilation of violent conflict events provided by NGOs and the government. As such, it may fail to map accurately onto individual experiences. Understanding the biases that exist in this record may help explain why so many people feel left out of the transitional justice process and why such a record may serve as an ineffective tool for promoting transitional justice.

SUPPLEMENTARY MATERIALS
To view supplementary material for this article, please visit http://doi.org/10.1017/S0003055422001162.