Law enforcement and political misinformation

Abstract

Why is criminal law enforcement increasingly punitive, despite that the situation has improved for decades? This paper investigates this question from the perspective of political misinformation. To this end, we develop a law enforcement model with political competition and examine how political parties’ campaigns affect voters’ perceptions of crime and equilibrium law enforcement policy. In a political campaign stage, we show that one political party has an incentive to overstate the severity of crime, while the other party has an incentive to correct voters’ beliefs. However, although the two parties attempt to change voters’ beliefs in opposite directions, we find that the total effect of a political campaign is more likely to drive both parties’ policies in a harsh direction.

1. Introduction

For decades, developed countries have witnessed an increase in the punitiveness of law enforcement, as evidenced by both the incarceration rate and the severity of penalties (Lacey et al., 2018). One notable case is the phenomenon of mass incarceration in the United States, which denotes a substantial surge in the incarceration rate, defined as the number of inmates per 100,000 residents. Initially, the disparity between the United States and other developed countries in incarceration rates was not particularly pronounced. However, starting in the 1970s, the United States underwent a striking escalation in its incarceration rate, which peaked around 2010. While there has been a recent decline in the incarceration rate, attributed in part to the pandemic, decreasing property crime rates, and shifts in prosecution and sentencing practices (Pew Research Center, 2021), the United States still maintains the highest incarceration rate globally (see Table 1). ¹

Table 1.

Incarceration rate in developed countries (prison population rate per 100,000).

Year	United States	Canada	United Kingdom (England & Wales)	Germany	France	Australia	New Zealand
1970	161	104	80	–	57	82	83
1980	220	96	87	–	66	66	83
1990	457	121 (1990–1991)	90	–	78	84	116
2000	683	115 (2000–2001)	124	85	82	114	148
2010	731	117 (2011)	153	85	99 (2011)	135	198
2020	505	104 (2019)	133	72	119	160	186

Source: World Prison Brief.

Note: https://www.prisonstudies.org/world-prison-brief-data (accessed June 2023).

Scholars have long debated the reasons for this trend, including rising crime rates, race, illegal drug use, and inequality (Barker, 2009; Campbell et al., 2015; Enns, 2014, 2016; Gottschalk, 2006, 2016; Jacobs and Jackson, 2010). Among several reasons, Enns (2014, 2016) argues that mass incarceration largely reflects a political response to the public’s rising punitiveness. By using a longitudinal measure of the public’s support for being tough on crime and controlling for other factors, he shows that the public’s punitiveness has been a primary determinant of mass incarceration.

Why has the public become so punitive? One hypothesis is that the political discourse drives voters’ misperceptions about the crime situation. Several surveys show that many people believe that the crime rate is rising, despite that the situation has improved for decades. For example, a Gallup survey in 2022 shows that 78% of US adults believe that crime is up from the previous year, which does not match reality. ^{2 3} Additionally, Ramirez (2013a,b) shows that the presidential framing of crime, especially the punitive tone of presidential statements, increases the public’s punitive sentiment. Jacobs and Jackson (2010) argue that law-and-order campaign appeals by Republicans are the most plausible reason for the public punitive sentiment and the rapid increase in US imprisonment rates. Survey results may also support this idea. For example, in the 2016 presidential election, Donald Trump argued that the crime situation was becoming severe and blamed undocumented immigrants for committing crimes and causing violence. The survey results show that 78% of Trump supporters believed that crime had worsened but that only 37% of Clinton supporters did, which indicates that political discourse may affect voters’ perceptions of crime (Pew Research Center, 2016). Furthermore, the aforementioned Gallup survey also highlights a partisan disparity in the perception of the crime situation. Specifically, 73% of Republicans responded that there was an increase in crime compared to the previous year in their local area, whereas only 42% of Democrats shared the same perception.

However, even if voters’ perceptions and attitudes are malleable, it remains unclear why political campaigns enhance the public’s punitive attitude and lead politicians to implement harsh law enforcement. In the 2016 presidential election, for example, Trump emphasized the seriousness of the crime and the need to implement harsh enforcement, but Clinton instead emphasized the problem of mass incarceration and the need to reform the criminal justice system, including easing mandatory minimum sentences for offenders (Hill and Marion, 2018). If each party emphasizes the crime situation in different directions—one party overstates the situation, but another party attempts to correct it—it is unclear why the total effect of political campaigns leads to the public’s punitive attitude and punitive law enforcement policy. Are there any factors that foster punitiveness underlying the political process?

To address this question, this paper develops a law enforcement model incorporating political competition and examines the influence of political parties’ campaigns on voters’ perceptions of criminal harm and the resulting equilibrium law enforcement policy. Our model incorporates political parties, voters (i.e. victims), and criminals. Specifically, we assume that voters’ marginal disutility increases with respect to the harm caused by crime, which is the standard assumption in the law and economics literature (Becker, 1968; Stigler, 1970). In a political context, we employ two theories: the first is a probabilistic voting model (Lindbeck and Weibull, 1987; Persson and Tabellini, 2002). The second is an issue ownership theory that assumes that one party has accumulated reputation/expertise on a certain issue, enabling them to more efficiently implement a policy on this issue (Petrocik, 1996). Then, we allow parties to manipulate voters’ beliefs regarding the severity of crime harm through political campaigns. Consequently, uninformed voters, who lack sufficient information about the crime situation, adjust their beliefs based on the extent of each political party’s campaign.

We demonstrate that in equilibrium, the party with issue ownership on policing crime has an incentive to exaggerate the severity of criminal harm, while the opposing party has the incentive to rectify this misinformation. However, despite the parties’ contrasting strategies of misinformation and correction regarding the crime situation, we discover that the overall effect of political campaigns tends to steer both parties’ law enforcement policies toward harsher measures than in the absence of a political campaign.

The mechanism behind our main result comes from the combination of probabilistic voting and the assumption of increasing marginal disutility of voters regarding criminal harm. In an equilibrium, certain groups of voters become excessively fearful of harm through manipulation, while others become relatively less concerned about crime. Consequently, one might intuitively anticipate that these opposing effects of information manipulation cancel each other out. However, within this model, voters experience an increase in marginal disutility as the level of harm escalates. Therefore, the impact on the group of voters exhibiting greater fear outweighs that of the group with lesser fear. Under the probabilistic voting model, politicians respond to the average payoff of voters. As the total effect of political campaigns raises the average voter’s disutility due to crime, the resulting law enforcement tends to be punitive, even when the campaigns of both parties manipulate voters’ perceptions in opposing directions to the same extent. This mechanism can be more intuitively explained as follows: since voters are risk averse, they are more influenced by discourse suggesting that ‘crime rates are rising and growing more severe’ than by ‘crime rates are declining and becoming less severe.’

Adopting a theoretical perspective from the law and economics literature, this paper has a similar motivation to the Beckerian law enforcement model, for example, Becker (1968), Garoupa (1997), and Polinsky and Shavell (2000), and its extension to exploring effects of self-interested law enforcers, for example, Stigler (1970), Friedman (1999), Garoupa and Klerman (2002), and Yahagi (2018). In particular, this paper can be seen as an extension of law enforcement models with political competition such as in Langlais and Obidzinski (2017), Mungan (2017), Obidzinski (2019), Friehe and Mungan (2021), and Yahagi (2021). The most similar contribution is Langlais and Obidzinski (2017), who applies the Downs model with certainty to investigate the impact of political decision-making processes on law enforcement policies.

In the current paper, we extend these models in three ways. First, different from the Downs model with certainty used in Langlais and Obidzinski (2017), we employ a probabilistic voting model (Persson and Tabellini, 2002; Lindbeck and Weibull, 1987) to avoid a median voter equilibrium. ⁴ Second, we incorporate issue ownership by parties, which means that one party has a reputation for competence in handling a particular issue (Petrocik, 1996; Petrocik et al., 2003; Krasa and Polborn, 2010). In our setting, one party has a reputation for competence in policing crime, which appears to be well aligned with reality. ⁵ As a result of those settings, we obtain a divergence of equilibrium policy. Third, to investigate the effect of political misinformation, we incorporate a campaign competition stage, where political parties attempt to manipulate voters’ perceptions of crime. The idea is similar to the model of campaign finance (Baron, 1994; Grossman and Helpman, 1996; Meirowitz, 2008) and the model of priming (Amorós and Puy, 2013; Aragonès et al., 2015; Egorov, 2015; Dragu and Fan, 2016; Denter, 2020). In the campaign stage, we show that the two parties attempt to change voters’ perceptions in opposite directions.

Additionally, the current paper has a similar motivation to the literature investigating the effect of fake news/misinformation in the political arena (Hochschild and Einstein, 2015; Allcott and Gentzkow, 2017; Guess et al., 2018; Grossman and Helpman, 2023). From a theoretical perspective, the current paper can be seen as applying the model proposed by Grossman and Helpman (2023) to the law enforcement context. Grossman and Helpman (2023) develop a political competition model with imperfect information in which both parties can misinform voters and investigate when political behavior diverges. Unlike Grossman and Helpman (2023), we focus on the role of misinformation/information in law enforcement, especially to explain the cause of rising punitiveness. Furthermore, we explain the parties’ incentive to manipulate voters’ perceptions from a different perspective—issue ownership—and characterize the campaign competition between the parties.

Finally, strands of the literature investigate the cause of mass incarceration from historical and empirical perspectives (Barker, 2009; Campbell et al., 2015; Enns, 2014, 2016; Gottschalk, 2006, 2016; Jacobs and Jackson, 2010; Lacey et al., 2018). From a theoretical perspective, Mungan (2017) has a similar motivation to ours and investigates the cause of overincarceration from the perspective of disenfranchisement law. The current paper investigates a similar question to that of Mungan (2017) from a different perspective—political misinformation/information.

The remainder of this paper proceeds as follows. Section 2 explains the model and the behavior of each player. Section 3 shows the equilibrium of the game and explains how political misinformation affects the equilibrium law enforcement policy. Section 4 discusses the empirical implications of our model. Section 5 presents a conclusion and discusses directions for future research.

2. Model

Before presenting the details, we will provide an overview of the model. There are three types of players: political parties, voters, and criminals.

There are two political parties $P\in \{A,B\}$ . In stage 1, they propose a law enforcement policy $p_P\in [0,1]$ , which states expected sanctions for criminals. One interpretation is that $p$ is a probability of arrest, and the sanction is normalized to $1$ . In the following, we use the terms ‘law enforcement level’ and ‘probability of arrest’ interchangeably when referring to $p$ . The parties’ objective is to maximize the probability of winning. In stage 2, each candidate chooses the misinformation/information level and its direction regarding the perceived level of harm generated by crimes. The misinformation/information levels chosen by each party are represented as ${\sigma }_P\in [-\sigma ,\sigma ],\sigma >0$ . Uninformed voters are exposed to the party $A$ ’s political campaign with fixed probability $\alpha$ and $B$ ’s political campaign with probability $1-\alpha$ , where $\alpha \in (0,1)$ . ⁶

In stage 3, voters vote for the party that provides a higher utility. There are two types of voters: informed and uninformed voters. Informed voters have correct information about the harm of crime, and their beliefs are not affected by a political campaign. On the other hand, uninformed voters have a prior perception of crime with an upward bias over the actual value. ⁷ Additionally, their beliefs are malleable and affected by a political campaign.

In stage 4, the party that obtains more than half of the votes will win, and potential criminals choose whether they will commit a crime based on the implemented law enforcement policy.

The timing of the game can be summarized as follows:

Stage 1: Political parties announce a law enforcement policy $p_P\in [0,1]$ .

Stage 2: Political parties choose a misinformation/information level ${\sigma }_P\in [-\sigma ,\sigma ]$ . Uninformed voters’ perception of crimes is affected by $A$ ’s political campaign with probability $\alpha$ and $B$ ’s campaign with probability $1-\alpha$ .

Stage 3: Voters vote for the party that provides a higher utility.

Stage 4: The party that obtains half of the votes will win and implement the proposed policy. Potential criminals choose whether they will commit a crime under the implemented law enforcement.

The equilibrium concept is subgame perfect Nash equilibrium, and we focus on the pure strategy equilibrium. In the following subsections, we will explain each player’s behavior in detail.

2.1. Criminals

There are potential criminals who consider whether to commit a crime under the implemented law enforcement policy $p\in [0,1]$ . We assume that the number of potential criminals is normalized to $c\in [0,1]$ . Potential criminals are differentiated by their benefit from committing a crime $b$ , and we assume that the benefit is uniformly distributed on $b\in [0,1]$ . The utility function of criminals when committing an offense is defined as $u_c=b-p$ , where $p$ is the probability of arrest or expected sanction cost. If potential criminals do not commit a crime, $u_c=0$ . Therefore, a potential criminal will commit a crime if and only if $u_c\ge 0\iff b\ge p$ . Since $b$ is uniformly distributed on $[0,1]$ and the number of potential criminals is $c$ , the number of criminals who commit a crime under law enforcement level $p$ can be calculated as $q=c(1-p)$ .

Note that it is not obvious that harsh law enforcement (i.e. an increased $p$ ) leads to an increase in incarceration. While an increase in $p$ leads to criminals being arrested with high probability, it also discourages potential criminals from committing a crime. We require additional conditions to demonstrate that harsh law enforcement indeed leads to an increase in the incarceration rate. We will delve further into this matter in Section 3.5.

Additionally, unlike the combined law enforcement and political competition model proposed by Langlais and Obidzinski (2017), we assume that voters and criminals are different groups, that is, potential criminals do not have voting rights. ⁸ The examples of potential criminals in this setting are disenfranchised ex-convicts, undocumented immigrants, and interregional criminals. ⁹

2.2. Voters

There is a continuum of voters who are negatively affected by crimes. To simplify the following discussion, we assume that the pool of voters is normalized to one. The utility function of voters is written as

$$u=-\gamma (hq{)}^2-t$$

(1)

where $h>0$ is the marginal harm caused by a crime, $q$ is the number of potential criminals who commit a crime, and $t$ is the tax burden. Then, $hq$ can be interpreted as the total harm caused by criminals. $\gamma >0$ is the weight of the disutility from crimes relative to the tax burden $t$ , and we suppose that there is increasing marginal disutility of harm from crime, $hq$ . $t$ is the per capital tax burden and defined as $t=mp$ , where $m>0$ is a marginal monitoring cost for law enforcement policy. ¹⁰ Therefore, $t$ can be interpreted as the tax burden to finance monitoring activities to prevent crimes. To efficiently mitigate the harm from crimes, voters demand the appropriate law enforcement level $p$ . Although all voters have the same utility functional form, we will explain later that voters are differentiated by their idiosyncratic affinity for political parties.

Since the increasing marginal disutility of criminal harm is a crucial assumption for our results, we add some comments. The increasing marginal disutility assumption has a long tradition in Beckerian law enforcement models (Becker, 1968; Stigler, 1970). The assumption means that society (and individuals) will be more concerned with major crimes than minor crimes. For example, regarding crimes against property, the assumption means that the theft of $1,000 is more than twice as harmful as the theft of $500. This result is implied by the diminishing marginal utility of income (Stigler, 1970). The increasing marginal disutility assumption is also used in the immigration policy literature (Llavador and Solano-García, 2011). In this case, the assumption means that society believes that immigrants will generate social conflict, xenophobia, and insecurity, and the marginal negative social impact is assumed to be increasing in the number of immigrants.

By using the fact that $q=c(1-p)$ and $t=mp$ , the utility function of voters can be rewritten as

$$u=-\gamma h^2{c}^2(1-p{)}^2-mp$$

(2)

Following models of political competition with imperfect information such as those of Baron (1994), Grossman and Helpman (1996, 2023), and Strömberg (2004), we assume that voters are divided into two categories: informed and uninformed voters. In this paper, we suppose that informed voters have accurate information $h$ and are not affected by parties’ political campaigns regarding the severity of the harm from crime. On the other hand, uninformed voters have upward-biased prior beliefs about the crime situation. We denote the prior belief of an uninformed voter as $\tilde{h}=h+\mathrm{\Delta }$ , where $\mathrm{\Delta }>0$ . This assumption is based on the fact that many voters hold a misconception about the crime situation, perceiving it to be worse than it actually is, as we explained in the Introduction. Furthermore, we suppose that their belief about the severity of harm from crime is malleable, that is, their belief is affected by political parties’ campaigns, which we will explain below. We assume that the ratio of informed to uninformed voters is fixed—the percentage of uninformed voters is $\lambda \in (0,1)$ , and that of informed voters is $1-\lambda$ .

2.3. Political parties

There are two political parties, $P\in \{A,B\}$ . The parties are winning probability maximizers. In this model, we suppose that parties differ in their reputation/expertise in implementing law enforcement. Specifically, we assume that one party can implement policing activities with a more efficient cost and voters know this advantage. Without loss of generality, we suppose that party $A$ has an advantage in policing activities and the marginal monitoring cost of $A$ is smaller than $B$ . In the following, we assume that $m_A<m_B$ , where $m_P$ represents the marginal monitoring cost if $P$ wins the race and implements its law enforcement policy. In other words, this assumption means that party $A$ has issue ownership of law enforcement (Petrocik, 1996; Petrocik et al., 2003). As we will explain below, the policy divergence is caused by the issue ownership assumption. If there is no policy divergence, parties do not have an incentive to manipulate voters’ beliefs. Therefore, the assumption of issue ownership is crucial for our results. ¹¹

Note that in this paper, we assume political competition with a one-dimensional policy space. However, the literature regarding issue ownership theory (Amorós and Puy, 2013; Aragonès et al., 2015; Egorov, 2015; Dragu and Fan, 2016; Denter, 2020) considers a multidimensional policy space and investigates how parties’ issue ownership of different policies affects the incentives of political campaign spending and the electoral results. In Appendix A, we extend our model into a multidimensional setting and show that the main result of our model does not change under an additional condition.

Next, we will explain stage 2: parties’ political campaigns. The main idea is similar to Grossman and Helpman (2023). In this model, we assume that uninformed voters’ prior belief on harm from crime $\tilde{h}$ is pliable, and political parties can affect it through political campaigns. Specifically, we assume that parties can choose the exaggerated/correct level of harm by ${\sigma }_P\in [-\sigma ,\sigma ]$ , and if voters are exposed to the campaign rhetoric of $P$ , then their perceived harm from crime would change to $h_P=h+\mathrm{\Delta }+{\sigma }_P$ . To simplify the following discussion, we suppose that $\mathrm{\Delta }>\sigma$ so that $h_P>h$ always holds (i.e. the perception of criminal harm cannot be below the actual value). ¹² Note that the assumption of ${\sigma }_P\in [-\sigma ,\sigma ]$ means that the maximum degree of manipulation is symmetric in both exaggeration ( ${\sigma }_P>0$ ) and correction ( ${\sigma }_P<0$ ). Therefore, $\sigma >0$ denotes the impact of misinformation/information, and a large $\sigma$ means that the political parties’ campaign has a larger effect on uninformed voters’ perception of the harm from crime in both directions.

We assume that the probability of voters being exposed to party $P$ ’s political campaign is exogenously determined. Specifically, we assume that voters are exposed to party $A$ ’s political campaign with a probability of $\alpha \in (0,1)$ and to party $B$ ’s political campaign with a probability of $1-\alpha$ . One interpretation of $\alpha$ is the relative campaign budget of party $A$ compared to party $B$ . If party $A$ has a larger campaign budget than party $B$ , it can more effectively influence voters’ beliefs, resulting in a higher probability of exposure to party $A$ ’s campaign. In Appendix B, we demonstrate that the probability of exposure can be endogenized by incorporating the campaign contest between the two parties. We also show that even in this alternative setting, the main results of this paper remain unchanged.

Then, the payoff function of political party $P$ can be defined as

$$u_P=W_P(p_A,p_B,{\sigma }_A,{\sigma }_B)$$

(3)

where $W_P$ is the winning probability of party $P$ . Parties choose the direction of information ${\sigma }_P$ to align with their platform.

As a final remark, we would like to discuss the interpretation of the political competition in our model. In this paper, we consider a two-candidate presidential or congressional election. Within this interpretation, although political parties cannot directly implement law enforcement, we assume that candidates have the ability to influence the capacity for investigation, prosecution, and incarceration through budgetary appropriations. Furthermore, candidates can impact incarceration rates by defining crime criteria and imposing sentencing requirements through federal or state laws. Under this framework, we simplify the discussion by abstracting from how law enforcement agencies react to political decision-making. However, several papers reveal that political pressure can affect the implemented law enforcement policy (Levitt, 1997; Huber and Gordon, 2004; Dyke, 2007; Berdejó and Yuchtman, 2013; McCannon, 2013; Bandyopadhyay and McCannon, 2014, 2015; Nadel et al., 2017). Therefore, the notion that the two-candidate presidential or congressional election and the proposed platforms influence actual law enforcement policy does not appear unrealistic.

On the other hand, another interpretation of the electoral competition in this model could be a two-candidate election for sheriff or district attorney. In fact, several empirical studies have investigated how elections for sheriff and district attorney affect law enforcement (Thompson, 2020; Krumholz, 2020; Okafor, 2021). However, despite the potential applicability of this interpretation, we will adopt the interpretation of a two-candidate presidential or congressional election. This is because we are focusing on the scenario where candidates influence voters’ perceptions of crimes through political campaigns and discourse, which seems to align more closely with a two-candidate presidential or congressional election than an election for sheriff or district attorney.

3. Analysis

3.1. Law enforcement level that maximizes voters’ utility

As a benchmark, we calculate the ‘utility-maximizing’ law enforcement level in this subsection. Note first that optimal policy varies depending on the cost efficiency of each party. Intuitively, if $A$ implements a policy, the optimal policy is determined given that the cost of policing crime is $m_A$ . On the other hand, if $B$ implements a policy, the optimal policy is determined given that the cost of policing crime is $m_B$ . Therefore, the ‘utility-maximizing’ law enforcement level depends on the party, and we denote it as $p_P^{*},P\in \{A,B\}$ .

Furthermore, there are two possibilities for the definition of a utility-maximizing policy: (i) calculating the policy that maximizes the voters’ utility, including the uninformed voters’ upward bias $\mathrm{\Delta }$ , and (ii) calculating the policy that maximizes the voters’ utility without the uninformed voters’ upward bias $\mathrm{\Delta }$ . We choose the former definition because this helps us to explore how political campaigns/misinformation cause positive or negative effects compared to the utility-maximizing policy. If we calculate the utility-maximizing policy without considering prior bias, it is unclear whether the difference between the utility-maximizing policy and equilibrium arises from the prior upward bias $\mathrm{\Delta }$ or distortion caused by political misinformation.

Based on the definition, we obtain the following result by differentiating total utility with respect to $p$ .

Proposition 1.

The optimal law enforcement policy for voters, denoted as $p_P^{*}\in [0,1]$ , is determined as follows: (i) If ${\mathrm{\Omega }}^O\le m_P/2\gamma c^2$ , then $p_P^{*}=0$ . (ii) If ${\mathrm{\Omega }}^O>m_P/2\gamma c^2$ , then $p_P^{*}=({\mathrm{\Omega }}^O-m_P/2\gamma c^2)/{\mathrm{\Omega }}^O$ , where ${\mathrm{\Omega }}^O=\lambda {\tilde{h}}^2+(1-\lambda )h^2$ .

In the proposition, ${\mathrm{\Omega }}^O$ represents the squared value of the weighted average perception of criminal harm among voters. The superscript $O$ means the original perception without being affected by a political campaign.

Proof.

Note that the perception of criminal harm for uninformed voters is $\tilde{h}=h+\mathrm{\Delta }$ , while the perception for informed voters is $h$ . Suppose that law enforcement policy $p$ is implemented by party $P$ . Then, the total utility with the prior biased beliefs of uninformed voters can be written as

$$U=\lambda [-\gamma {\tilde{h}}^2{c}^2(1-p{)}^2-m_{P}p]+(1-\lambda )[-\gamma h^2{c}^2(1-p{)}^2-m_{P}p]$$

(4)

By differentiating $U$ with respect to $p$ , we have

$$\frac{\mathrm{\partial }U}{\mathrm{\partial }p}=\lambda [2\gamma {\tilde{h}}^2{c}^2-2\gamma {\tilde{h}}^2{c}^{2}p-m_P]+(1-\lambda )[2\gamma h^2{c}^2-2\gamma h^2{c}^{2}p-m_P]$$

(5)

Since ${\mathrm{\partial }}^{2}U/\mathrm{\partial }{p}^2=-\lambda (2\gamma {\tilde{h}}^2{c}^2)-(1-\lambda )(2\gamma h^2{c}^2)<0$ , a second-order condition is satisfied. If $(\mathrm{\partial }U/\mathrm{\partial }p){|}_{p=0}\le 0$ , then voters do not have an incentive to increase $p$ . This condition can be written as

$$\begin{array}{rl}& \lambda [2\gamma {\tilde{h}}^2{c}^2-m_P]+(1-\lambda )[2\gamma h^2{c}^2-m_P]\le 0\\ & \iff 2\gamma c^2[\lambda {\tilde{h}}^2+(1-\lambda )h^2]\le m_P\\ & \iff {\mathrm{\Omega }}^O\le \frac{m_P}{2\gamma c^2}\end{array}$$

(6)

Then, we obtain (i). If ${\mathrm{\Omega }}^O>m_P/2\gamma c^2$ , $p_P^{*}$ must satisfy the first-order condition. The first-order condition can be written as

$$\begin{array}{rl}& \frac{\mathrm{\partial }U}{\mathrm{\partial }p}=\lambda [2\gamma {\tilde{h}}^2{c}^2-2\gamma {\tilde{h}}^2{c}^{2}p-m_P]+(1-\lambda )[2\gamma h^2{c}^2-2\gamma h^2{c}^{2}p-m_P]=0\\ & \iff 2\gamma c^2{\mathrm{\Omega }}^O-2\gamma c^{2}p{\mathrm{\Omega }}^O-m_P=0\\ & \iff p_P^{*}=\frac{{\mathrm{\Omega }}^O-m_P/2\gamma c^2}{{\mathrm{\Omega }}^O}\end{array}$$

(7)

3.2. Probabilistic voting

Next, we consider voting behavior. As noted above, we use a probabilistic voting model such as those in Lindbeck and Weibull (1987) and Persson and Tabellini (2002). Suppose that $A$ proposes $p_A$ and $B$ proposes $p_B$ . Then, voter $i$ votes for $A$ if and only if

$$u(p_A,m_A,h)>u(p_B,m_B,h)+{\eta }_i$$

(8)

where ${\eta }_i$ is an idiosyncratic affinity for party $B$ and ${\eta }_i$ is uniformly distributed on $[\eta -(1/2\psi ),\eta +(1/2\psi )]$ . Voters have the same preference for law enforcement policy but are heterogeneous in their idiosyncratic affinities. For example, if ${\eta }_i<0$ , then voter $i$ has an affinity for party $A$ and vice versa. We suppose that parties know the distribution of ${\eta }_i$ but do not know the actual value. Additionally, we assume that voters’ idiosyncratic affinities are affected by aggregate uncertainty. Here, $\eta$ denotes aggregate uncertainty, and we suppose that $\eta$ is uniformly distributed on $[-1/2\varphi ,1/2\varphi ]$ . One interpretation of aggregate uncertainty is that parties are uncertain about what type of event will happen after the policy announcement and before voting. Examples are sudden changes in the economic state and political scandals, which may happen during elections and affect voters’ affinities for political parties.

In the following discussion, we suppose that $m_A=\delta m<m=m_B$ , where $0<\delta <1$ . Therefore, $\delta$ represents a relative cost advantage of party $A$ . Then, equation (8) can be rewritten as

$$\begin{array}{c}-\gamma h^2{c}^2(1-p_A{)}^2-\delta m{p}_A>-\gamma h^2{c}^2(1-p_B{)}^2-m{p}_B+{\eta }_i\\ \iff 2\gamma h^2{c}^2(p_A-p_B)-\gamma h^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)>{\eta }_i\end{array}$$

(9)

Since ${\eta }_i$ is uniformly distributed on $[\eta -(1/2\psi ),\eta +(1/2\psi )]$ , the probability that voter $i$ votes for $A$ can be rewritten as ¹³

$$\psi [2\gamma h^2{c}^2(p_A-p_B)-\gamma h^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\psi }-\eta ]$$

(10)

Since uninformed voters’ perception of the harm from crime can be affected by political campaigns, these voters’ perceptions might differ from the actual value $h$ . As explained above, we denote the uninformed voters’ perceived harm from crime by $h_P$ if they are exposed to party $P$ ’s political campaign. Note that uninformed voters are exposed to party $A$ ’s campaign with probability $\alpha$ and party $B$ ’s campaign with probability $1-\alpha$ . Furthermore, note that the proportion of uninformed voters is $\lambda$ and that of informed voters is $1-\lambda$ . Then, the expected total vote share for party $A$ can be calculated as

$$\begin{array}{rl}& \underset{\mathrm{informed}\mathrm{voters}}{\underbrace{(1-\lambda )\psi [2\gamma h^2{c}^2(p_A-p_B)-\gamma h^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\psi }-\eta ]}}\\ & +\underset{\mathrm{uninformed}\mathrm{voters}\mathrm{exposed}\mathrm{to}{\mathrm{A}}^{\prime }\mathrm{s}\mathrm{campaign}}{\underbrace{\lambda \alpha \psi [2\gamma h_A^2{c}^2(p_A-p_B)-\gamma h_A^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\psi }-\eta ]}}\\ & +\underset{\mathrm{uninformed}\mathrm{voters}\mathrm{exposed}\mathrm{to}{\mathrm{B}}^{\prime }\mathrm{s}\mathrm{campaign}}{\underbrace{\lambda (1-\alpha )\psi [2\gamma h_B^2{c}^2(p_A-p_B)-\gamma h_B^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\psi }-\eta ]}}\end{array}$$

(11)

If the total vote share is larger than one-half, party $A$ will win the race. Therefore, the probability of winning for party $A$ can be defined as

$$\begin{array}{rl}{W}_A& =\mathrm{Prob}\{(1-\lambda )\psi [2\gamma h^2{c}^2(p_A-p_B)-\gamma h^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\psi }-\eta ]\\ & +\lambda \alpha \psi [2\gamma h_A^2{c}^2(p_A-p_B)-\gamma h_A^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\psi }-\eta ]\\ & +\lambda (1-\alpha )\psi [2\gamma h_B^2{c}^2(p_A-p_B)-\gamma h_B^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\psi }-\eta ]>\frac{1}{2}\}\end{array}$$

(12)

Since $\eta$ is uniformly distributed on $[-1/2\varphi ,1/2\varphi ]$ , the winning probability of $A$ can be calculated as ¹⁴

$$\begin{array}{rl}{W}_A& =(1-\lambda )\varphi [2\gamma h^2{c}^2(p_A-p_B)-\gamma h^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\varphi }]\\ & +\lambda \alpha \varphi [2\gamma h_A^2{c}^2(p_A-p_B)-\gamma h_A^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\varphi }]\\ & +\lambda (1-\alpha )\varphi [2\gamma h_B^2{c}^2(p_A-p_B)-\gamma h_B^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\varphi }]\end{array}$$

(13)

Following the same logic, the winning probability of $B$ can be written as

$$\begin{array}{rl}{W}_B& =(1-\lambda )\varphi [\frac{1}{2\varphi }-2\gamma h^2{c}^2(p_A-p_B)+\gamma h^2{c}^2(p_A^2-p_B^2)+m(\delta p_A-p_B)]\\ & +\lambda \alpha \varphi [\frac{1}{2\varphi }-2\gamma h_A^2{c}^2(p_A-p_B)+\gamma h_A^2{c}^2(p_A^2-p_B^2)+m(\delta p_A-p_B)]\\ & +\lambda (1-\alpha )\varphi [\frac{1}{2\varphi }-2\gamma h_B^2{c}^2(p_A-p_B)+\gamma h_B^2{c}^2(p_A^2-p_B^2)+m(\delta p_A-p_B)]\end{array}$$

(14)

3.3. Equilibrium in campaign contest

Next, we will show the subgame perfect Nash equilibrium of this game by using backward induction. Since we already analyzed stage 4 (criminal behavior) and stage 3 (voting behavior) in the previous subsections, we will begin the analysis from stage 2. In stage 2, each political party chooses the direction of misinformation/information ${\sigma }_P$ . In the following discussion, we assume preliminarily that $p_A>p_B$ holds in equilibrium. In the later analysis, we will show that this must be true in equilibrium. Intuitively, the divergence of law enforcement policy stems from the fact that party $A$ has a cost advantage in policing activities and can propose a higher law enforcement level $p$ than party $B$ . The result can be summarized as follows:

Proposition 2.

In the equilibrium of stage 2, party $A$ proposes ${\sigma }_A=\sigma$ and $B$ proposes ${\sigma }_B=-\sigma$ .

Proof.

Note that $h_A=\tilde{h}+{\sigma }_A$ and ${\sigma }_A\in [-\sigma ,\sigma ]$ . By differentiating $W_A$ with respect to $h_A$ , we have

$$\frac{\mathrm{\partial }{W}_A}{\mathrm{\partial }{h}_A}=\lambda \alpha \varphi [4\gamma h_A{c}^2(p_A-p_B)-2\gamma h_A{c}^2(p_A^2-p_B^2)]$$

(15)

$\mathrm{\partial }{W}_A/\mathrm{\partial }{h}_A$ must be positive because

$$\begin{array}{rl}& 4\gamma h_A{c}^2(p_A-p_B)-2\gamma h_A{c}^2(p_A^2-p_B^2)>0\\ & \iff 2>p_A+p_B\end{array}$$

(16)

Since $(\mathrm{\partial }{W}_A/\mathrm{\partial }{h}_A)>0$ holds, party $A$ has an incentive to increase $h_A$ , and ${\sigma }_A=\sigma$ holds. The same logic can be applied to party $B$ , and ${\sigma }_B=-\sigma$ holds.

Before proceeding to the next step, we would like to emphasize that the underlying mechanism driving this result shares similarities with the fundamental findings of studies that investigate the manipulation of issue salience in political campaigns (Amorós and Puy, 2013; Aragonès et al., 2015; Egorov, 2015; Dragu and Fan, 2016; Denter, 2020). According to these papers, a party that possesses issue ownership in a particular area is interested in amplifying that issue’s salience (i.e. the weight of importance) through political campaigning. In our model, instead of directly manipulating the weight of interest in a specific issue, political parties manipulate the perception of criminal harm. When voters perceive criminal harm to be substantial, they believe that addressing crime becomes more crucial. Conversely, if voters perceive criminal harm as less significant than anticipated, they assign greater importance to another political issue. Consequently, if party $A$ holds a greater advantage in addressing crime, it is incentivized to exaggerate the severity of harm (i.e. increase the salience of this issue). Conversely, party $B$ has the incentive to downplay crime harm to divert voters’ interest away from addressing crime.

However, one may question how issue ownership can be modeled without considering multiple dimensions, as several theory papers that discuss issue competition (Amorós and Puy, 2013; Aragonès et al., 2015; Egorov, 2015; Dragu and Fan, 2016; Denter, 2020) adopt multidimensional policies. For instance, in those papers, one party possesses issue ownership in a specific policy area, while the other party has ownership in a different policy area. On the other hand, in our model, issue competition works within a one-dimensional context as follows: party $A$ is incentivized to exaggerate the severity of crime, thereby increasing the salience of the crime issue, over which they have issue ownership. Conversely, party $B$ is motivated to understate/correct voters’ beliefs, decreasing the salience of the crime issue, where it lacks an advantage. Consequently, party $B$ ’s campaign makes voters base their ballot choices more on their idiosyncratic affinity ${\eta }_i$ than on law enforcement policy, thereby increasing the winning probability of party $B$ .

To ensure the robustness of our findings, we will also demonstrate in Appendix A that our results remain unchanged even if we expand our model to incorporate a multidimensional context.

3.4. Equilibrium in political competition

Next, we will investigate the equilibrium in stage 1. The winning probability of $A$ can be written as

$$\begin{array}{rl}{W}_A& =(1-\lambda )\varphi [2\gamma h^2{c}^2(p_A-p_B)-\gamma h^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\varphi }]\\ & +\lambda \alpha \varphi [2\gamma h_A^2{c}^2(p_A-p_B)-\gamma h_A^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\varphi }]\\ & +\lambda (1-\alpha )\varphi [2\gamma h_B^2{c}^2(p_A-p_B)-\gamma h_B^2{c}^2(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\varphi }]\\ & =\varphi [2\gamma c^2\mathrm{\Omega }(p_A-p_B)-\gamma c^2\mathrm{\Omega }(p_A^2-p_B^2)-m(\delta p_A-p_B)+\frac{1}{2\varphi }]\end{array}$$

(17)

where $\mathrm{\Omega }$ is defined as

$$\mathrm{\Omega }=\underset{(\mathrm{square}\mathrm{of})\mathrm{uninformed}\mathrm{voter}{\mathrm{s}}^{\prime }\mathrm{perception}\mathrm{of}\mathrm{harm}}{\underbrace{\lambda [\alpha (\tilde{h}+\sigma {)}^2+(1-\alpha )(\tilde{h}-\sigma {)}^2]}}+\underset{(\mathrm{square}\mathrm{of})\mathrm{informed}\mathrm{voter}{\mathrm{s}}^{\prime }\mathrm{perception}\mathrm{of}\mathrm{harm}}{\underbrace{(1-\lambda )h^2.}}$$

(18)

Note that $\tilde{h}+\sigma$ represents the perception of criminal harm for voters exposed to $A$ ’s campaign, $\tilde{h}-\sigma$ represents the perception of criminal harm for voters exposed to $B$ ’s campaign, and $h$ represents the perception of criminal harm for informed voters. Therefore, $\mathrm{\Omega }$ corresponds to the squared value of the weighted average perceptions of criminal harm. By differentiating (17) with respect to $p$ and combining this with the results in the previous subsections, we obtain the subgame perfect Nash equilibrium of the model.

Proposition 3.

In a political competition (stage 1), party $P\in \{A,B\}$ proposes the following platform: (i) if $\mathrm{\Omega }\le (m_P/2\gamma c^2)$ , then $p_P=0$ . (ii) if $\mathrm{\Omega }>(m_P/2\gamma c^2)$ , then $p_P=(\mathrm{\Omega }-m_P/2\gamma c^2)/\mathrm{\Omega }$ . In stage 2, party $A$ proposes ${\sigma }_A=\sigma$ and $B$ proposes ${\sigma }_B=-\sigma$ .

Since $m_A=\delta m<m=m_B$ , if $\mathrm{\Omega }>\delta m/2\gamma c^2$ , $p_A>p_B$ holds. In other words, the party with issue ownership (party $A$ ) proposes a harsher law enforcement policy than the party that does not have an advantage (party $B$ ). In the campaign contest (stage 2), the party that has issue ownership (party $A$ ) attempts to overstate the seriousness of the harm from crime, and the party that does not have issue ownership (party $B$ ) attempts to correct voters’ beliefs.

Proof.

Let us consider the case of party $A$ . By differentiating the winning probability with respect to $p_A$ , we have

$$\frac{\mathrm{\partial }{W}_A}{\mathrm{\partial }{p}_A}=\varphi [2\gamma c^2\mathrm{\Omega }-2\gamma c^2\mathrm{\Omega }{p}_A-\delta m]$$

(19)

Then, if $(\mathrm{\partial }{W}_A/\mathrm{\partial }{p}_A){|}_{p_A=0}\le 0$ , the party has no incentive to increase $p_A$ . This condition can be written as $\varphi [2\gamma c^2\mathrm{\Omega }-\delta m]\le 0\iff \mathrm{\Omega }\le \delta m/2\gamma c^2$ , so we obtain (i).

Next, let us consider case (ii). Since $({\mathrm{\partial }}^2{W}_A/\mathrm{\partial }{p}_A^2)=-2\varphi \gamma c^2\mathrm{\Omega }<0$ , the second-order condition is satisfied. By the first-order condition, we obtain

$$\begin{array}{rl}\frac{\mathrm{\partial }{W}_A}{\mathrm{\partial }{p}_A}& =\varphi [2\gamma c^2\mathrm{\Omega }-2\gamma c^2\mathrm{\Omega }{p}_A-\delta m]=0\\ \iff p_A& =\frac{\mathrm{\Omega }-\delta m/2\gamma c^2}{\mathrm{\Omega }}\end{array}$$

(20)

Following the same logic, if $\mathrm{\Omega }\le m/2\gamma c^2$ , then $p_B=0$ . Moreover, if $\mathrm{\Omega }>m/2\gamma c^2$ , then $p_B$ can be calculated as

$$\begin{array}{rl}\frac{\mathrm{\partial }{W}_B}{\mathrm{\partial }{p}_B}& =\varphi [2\gamma c^2\mathrm{\Omega }-2\gamma c^2\mathrm{\Omega }{p}_B-m]=0\\ \iff p_B& =\frac{\mathrm{\Omega }-m/2\gamma c^2}{\mathrm{\Omega }}\end{array}$$

(21)

Since $0<\delta <1$ , $p_A>p_B$ holds if $\mathrm{\Omega }>\delta m/2\gamma c^2$ .

Before discussing the implications of this result, we would like to discuss the difference between Grossman and Helpman (2023) (hereafter GH) and our findings. GH also discovers policy divergence along with misinformation without assuming issue ownership. In GH, policy divergence arises from the interaction between sequential moves of political parties and the characteristics of misinformation, where parties can announce false states of the world to uninformed voters. Specifically, in GH, the incumbent proposes the platform first, followed by the challenger. Then, if the challenger has an advantage in influencing uninformed voters’ beliefs, they are motivated to shift their policy position away from the incumbent. Due to the challenger’s greater impact on uninformed voters, it announces a policy opposite to the incumbent’s and promotes a false state of the world aligned with its policy, seeking to divert votes from the incumbent. This setting allows policy divergence without issue ownership, but it does not have a pure-strategy equilibrium when parties simultaneously announce their policies.

In contrast to GH, our model introduces policy divergence through issue ownership. One party always proposes a higher probability of arrest (harsh law enforcement) due to its cost advantage in policing crimes. This policy divergence leads to the polarization of political misinformation, similar to GH. Our model offers two advantages over GH. First, policy divergence and the polarization of political misinformation occur even without the assumption of sequential policy proposals as in GH. Second, our model makes an empirical prediction regarding which party has an incentive to overstate/understate the state of the world (e.g. severity of criminal harm) during a political campaign, which is not evident in GH (see Proposition 3 in GH).

3.5. The effect of political misinformation/information

Next, we investigate the impact of political misinformation/information on the equilibrium platforms. By differentiating $p_A$ and $p_B$ with respect to $\sigma$ , we have

$$\frac{\mathrm{\partial }{p}_A}{\mathrm{\partial }\sigma }=\frac{\mathrm{\partial }\mathrm{\Omega }/\mathrm{\partial }\sigma \cdot \mathrm{\Omega }-(\mathrm{\Omega }-\delta m/2\gamma c^2)\cdot \mathrm{\partial }\mathrm{\Omega }/\mathrm{\partial }\sigma }{{\mathrm{\Omega }}^2}=\frac{\delta m/2\gamma c^2}{{\mathrm{\Omega }}^2}\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }\sigma }$$

(22)

$$\frac{\mathrm{\partial }{p}_B}{\mathrm{\partial }\sigma }=\frac{\mathrm{\partial }\mathrm{\Omega }/\mathrm{\partial }\sigma \cdot \mathrm{\Omega }-(\mathrm{\Omega }-m/2\gamma c^2)\cdot \mathrm{\partial }\mathrm{\Omega }/\mathrm{\partial }\sigma }{{\mathrm{\Omega }}^2}=\frac{m/2\gamma c^2}{{\mathrm{\Omega }}^2}\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }\sigma }$$

(23)

Therefore, $\mathrm{sign}(\mathrm{\partial }{p}_A/\mathrm{\partial }\sigma )=\mathrm{sign}(\mathrm{\partial }{p}_B/\mathrm{\partial }\sigma )=\mathrm{sign}(\mathrm{\partial }\mathrm{\Omega }/\mathrm{\partial }\sigma )$ holds. Then, if $(\mathrm{\partial }\mathrm{\Omega }/\mathrm{\partial }\sigma )>0$ , the political campaign will make both parties’ proposals harsher. The conditions can be calculated as follows:

$$\begin{array}{rl}\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }\sigma }>0& \iff \lambda [2\sigma +2\tilde{h}(2\alpha -1)]>0\\ & \iff \alpha >\frac{1}{2}-\frac{\sigma }{2\tilde{h}}\end{array}$$

(24)

This condition means that both parties are more likely to propose a harsher policy as a result of the political campaign, even if the relative political campaign effectiveness of $A$ is low, that is, $\alpha <(1/2)$ . In other words, the effect of misinformation by party $A$ is more likely to exceed the impact of correction by $B$ even if $A$ has a disadvantage in campaign effectiveness ( $\alpha$ ). As a result, the total effect of the political campaign is more likely to lead to harsher law enforcement despite the fact that the two parties misinform/inform about the crime situation in opposite directions and to the same degree ( $\sigma$ ). The result can be summarized as follows.

Proposition 4.

Suppose that $\alpha >(1/2)-(\sigma /2\tilde{h})$ holds. Then, the political campaign will change both parties’ law enforcement policy in a harsher direction.

The mechanism behind this result stems from the combination of the probabilistic voting model and the assumption of voters’ increasing marginal disutility concerning the harm caused by crime, a standard assumption in the law and economics literature, as explained in Section 2.2.

To illustrate this, consider a simple example. Suppose that there are only uninformed voters, and half of them are exposed to party $A$ ’s political campaign (referred to as group $A$ ), while the other half are exposed to party $B$ ’s political campaign (referred to as group $B$ ). Both parties can manipulate the uninformed voters’ beliefs about the harm from crime by $\sigma >0$ . Consequently, voters in group $A$ believe that the harm is $\tilde{h}+\sigma$ , while voters in group $B$ believe that the harm is $\tilde{h}-\sigma$ . One might intuitively expect that this would cancel out the effect of information manipulation since each group of voters believes the harm is higher/lower by the same amount $\sigma$ . However, this is not the case.

In a probabilistic voting model, political parties respond to the average voter’s utility. In this simplified scenario, the average voter’s utility can be calculated as:

$$\underset{\mathrm{group}\mathrm{A}}{\underbrace{\frac{1}{2}[-\gamma (\tilde{h}+\sigma {)}^2{q}^2-t]}}+\underset{groupB}{\underbrace{\frac{1}{2}[-\gamma (\tilde{h}-\sigma {)}^2{q}^2-t]}}$$

(25)

Note that to the extent that $p$ remains unchanged, political misinformation/information does not impact the number of criminals, denoted by $q=c(1-p)$ , since potential criminals’ decision-making is unaffected by the harm from crime. In other words, voters believe that potential criminals do not care about the harm caused by crime when choosing to violate a law. Consequently, the tax burden $t=mp$ also remains unchanged due to political misinformation/information as long as $p$ remains unchanged. Thus, we can observe that political campaigns exacerbate the average voter’s disutility due to the increasing marginal disutility assumption:

$$\begin{array}{rl}& \frac{1}{2}[-\gamma (\tilde{h}+\sigma {)}^2{q}^2-t]+\frac{1}{2}[-\gamma (\tilde{h}-\sigma {)}^2{q}^2-t]\\ & \iff -\gamma {\tilde{h}}^2{q}^2-t\underset{campaigneffect}{\underbrace{-\gamma {\sigma }^2{q}^2}}\end{array}$$

(26)

The first term represents the average voter’s utility without political campaigns, while the second term represents the campaign effect. As political campaigns exacerbate the average voter’s utility, he or she demands harsh law enforcement ( $p$ ).

In this simple example, we assume that exactly half of the uninformed voters are exposed to each party’s political campaign. However, as long as the number of uninformed voters exposed to party $B$ ’s campaign is not significantly larger than that of party $A$ , the misinformation effect of party $A$ still outweighs the information effect of party $B$ .

Figure 1 shows how political misinformation/information affects the equilibrium platform. Here, we set $\alpha =0.4$ , meaning that party $B$ (i.e. the party that attempts to understate/correct the severity of criminal harm) affects uninformed voters’ perception with a higher probability. The solid lines show the equilibrium platforms when $\sigma =2$ (with political misinformation), and the dotted lines show the equilibrium platforms when $\sigma =0$ (without political misinformation). Since $\alpha >(1/2)-(\sigma /2\tilde{h})$ holds under those parameters, the political campaign leads to harsh law enforcement. The interesting point is that although the two parties’ campaign directions are the opposite (overstate/correct) and the impact is the same ( $\sigma$ ), the total effect of a political campaign makes both parties’ law enforcement policies harsher. The mechanism behind this phenomenon is simple. In equilibrium, the voters exposed to $A$ ’s campaign demand harsher law enforcement, and those exposed to $B$ ’s campaign demand a more lenient policy. However, since the marginal disutility from crime increases at an increasing rate, the effect of overstatement is more significant than that of correction. As a result, the average voter demands harsher policy despite that the two parties attempt to change voters’ beliefs in opposite directions.

Figure 1.

The effect of misinformation on the equilibrium platforms. Note: In this graph, we set $\alpha =0.4,\lambda =0.7,\delta =0.8,m=0.5,\gamma =1,c=0.2,\mathrm{\Delta }=2$ .

Figure 2 shows how the distortion of law enforcement happens to depend on the misinformation/information level $\sigma$ . In this figure, we compare the proposed law enforcement policy $p_P$ and utility-maximizing policy $p_P^{*}$ .

Figure 2.

Utility-maximizing policy and equilibrium platform. Note: In this graph, we set $\alpha =0.4,\lambda =0.7,\delta =0.8,m=0.5,\gamma =1,c=0.2,\mathrm{\Delta }=2,h=1$ .

We would like to highlight three points. First, if $\sigma =0$ , $p_P=p_P^{*}$ holds, meaning that when there is no political misinformation/information, both parties propose the policy that maximizes utility. This is evident from the definitions of $p_P$ and $p_P^{*}$ . According to Proposition 1, the utility-maximizing law enforcement policy implemented by party $P$ can be expressed as:

$$p_P^{*}=\frac{{\mathrm{\Omega }}^O-m_P/2\gamma c^2}{{\mathrm{\Omega }}^O}$$

(27)

On the other hand, according to Proposition 3, $p_P$ can be calculated as:

$$p_P=\frac{\mathrm{\Omega }-m_P/2\gamma c^2}{\mathrm{\Omega }}$$

(28)

It is clear that by definition, if $\sigma =0$ , then $\mathrm{\Omega }={\mathrm{\Omega }}^O$ . Therefore, if $\sigma =0$ , then $p_P=p_P^{*}$ .

Second, if $\sigma$ is sufficiently small such that $\alpha <(1/2)-(\sigma /2\tilde{h})$ , political misinformation leads to lenient law enforcement, which falls below the utility-maximizing level. It is important to recognize that this only occurs when party $B$ holds a relative advantage in campaign effectiveness since $\alpha <(1/2)-(\sigma /2\tilde{h})$ holds only when $\alpha <1/2$ . Intuitively, this implies that the political campaign of party $B$ (understating/correcting the severity of criminal harm) outweighs that of party $A$ only when uninformed voters are more likely to be exposed to the information provided by $B$ and the level of misinformation $\sigma$ is not substantial.

Third, and most important, if $\sigma$ becomes large and $\alpha >(1/2)-(\sigma /2\tilde{h})$ holds, the impact of party $A$ ’s political campaign (overstating the severity of crimes) surpasses that of correcting voters’ beliefs. Consequently, the average voter begins to demand harsh law enforcement policies. As a result, if $\sigma$ becomes large, $p_P$ exceeds the utility-maximizing level. This example demonstrates that even when party $B$ can manipulate uninformed voters’ beliefs with higher probability and attempts to correct voters’ beliefs, an increase in the level of misinformation/information ( $\sigma$ ) enables party $A$ to misinform voters more intensely, leading to harsh law enforcement. In other words, if voters’ beliefs are malleable and politicians possess the means to manipulate them, it can be challenging to avoid excessive law enforcement.

Finally, we would like to discuss the condition that harsh law enforcement will increase the incarceration rate. An increase in the probability of arrest, denoted by $p$ , leads to more apprehension of individuals who have violated the law. However, this increase in $p$ also acts as a deterrent, discouraging potential criminals from attempting crimes. Therefore, an increase in $p$ does not necessarily result in a corresponding increase in the incarceration rate.

Let us define the incarceration rate in our model. Note that the number of potential criminals violating the law is $q$ , and they are arrested with a probability of $p$ . Therefore, the number of arrested criminals is calculated as $qp$ . If the potential criminal’s benefit from committing a crime, denoted as $b$ , is higher than the expected cost, that is, $b-p\ge 0$ , he or she would violate the law. Since $b$ is uniformly distributed on $[0,1]$ , in equilibrium, $q=c(1-p)$ holds.

Since the total population is $1+c$ , the incarceration rate of the society is calculated as $qp/(1+c)=(c(1-p)p)/(1+c)$ . Note that in this definition, the effect of increasing $p$ depends on its level: an increase in the probability of arrest leads to the apprehension of individuals who have violated the law, but it also acts as a deterrent, discouraging potential criminals from attempting crimes. By differentiating the incarceration rate with respect to $p$ , we obtain:

$$\begin{array}{rl}& \frac{\mathrm{\partial }(\frac{qp}{(1+c)})}{\mathrm{\partial }p}>0\iff \frac{c-2cp}{1+c}>0\\ & \iff \frac{1}{2}>p\end{array}$$

(29)

Based on the above, we obtain the following corollary:

Corollary 1.

Suppose that $\alpha >(1/2)-(\sigma /2\tilde{h})$ . Then, a political campaign will lead to changes in both parties’ law enforcement policies toward harsher measures, resulting in an increase in the incarceration rate as long as $p_P<1/2$ holds.

In equilibrium, for example, when the marginal cost of law enforcement $m_P$ is large, we have $p_P<1/2$ .

Although the actual probability of arrest varies depending on the type of crime, it does not appear to be as high as one might think. For instance, in the US in 2018, Baughman (2020) estimated the true arrest rates (known crimes compared to the arrest rates for those crimes): 80.95% for murder, 37.41% for aggravated assault, 15.38% for robbery, and 6.77% for burglary.

Furthermore, Baughman (2020) demonstrates that the overall true arrest rate in the US in 2018 was 10.57%, which is lower than expected. Additionally, the conviction rate would be lower than the arrest rate since not all arrests result in conviction (Baughman, 2020).

Corollary 1 states that the impact of harsh law enforcement on the incarceration rate varies depending on the type of crime and its arrest/conviction rates. However, it seems reasonable to argue that harsh law enforcement leads to higher total incarceration rates because the probability of arrest does not appear to be very high for most crimes and satisfies the condition of Corollary 1. ¹⁵

3.6. Other comparative statics

Finally, we summarize how other parameters affect the equilibrium platforms. First, the proportion of uninformed voters $\lambda$ increases the equilibrium law enforcement level $p_A,p_B$ . This can be checked as follows. Note that $\mathrm{sign}(\mathrm{\partial }{p}_A/\mathrm{\partial }\lambda )=\mathrm{sign}(\mathrm{\partial }{p}_B/\mathrm{\partial }\lambda )=\mathrm{sign}(\mathrm{\partial }\mathrm{\Omega }/\mathrm{\partial }\lambda )$ . Then, we have

$$\begin{array}{rl}& \frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }\lambda }={\tilde{h}}^2+{\sigma }^2+2\sigma \tilde{h}(2\alpha -1)-h^2>0\\ & \iff (\tilde{h}-\sigma {)}^2+4\sigma \tilde{h}\alpha -h^2>0\end{array}$$

(30)

This is true because $\tilde{h}-\sigma >h$ holds by definition. As we explained in Proposition 3, uninformed voters’ perception is affected by a political campaign, and the total effect makes equilibrium platforms harsher. Therefore, the result that having more uninformed voters leads to harsh law enforcement is straightforward.

Second, the size of the potential criminal group $c$ is positively related to the law enforcement level, that is, $(\mathrm{\partial }{p}_A/\mathrm{\partial }c)>0$ , and $(\mathrm{\partial }{p}_B/\mathrm{\partial }c)>0$ , which is evident by the definition of equilibrium policies. The interpretation is also straightforward because the size of the population of potential criminals directly increases external harm for voters, so they demand harsher law enforcement policy.

Third, if $\alpha >(1/2)-(\sigma /2\tilde{h})$ (i.e. political campaigns lead to harsh enforcement), increases in $\sigma$ lead to convergence between the political parties’ platforms, that is, $(\mathrm{\partial }{p}_B/\mathrm{\partial }\sigma )>(\mathrm{\partial }{p}_A/\mathrm{\partial }\sigma )$ . This can be checked as follows: by differentiating $p_A$ and $p_B$ with respect to $\sigma$ , we have

$$\frac{\mathrm{\partial }{p}_A}{\mathrm{\partial }\sigma }=\frac{\delta m/2\gamma c^2}{{\mathrm{\Omega }}^2}\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }\sigma },\frac{\mathrm{\partial }{p}_B}{\mathrm{\partial }\sigma }=\frac{m/2\gamma c^2}{{\mathrm{\Omega }}^2}\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }\sigma }$$

(31)

If $\alpha >(1/2)-(\sigma /2\tilde{h})$ , we have $(\mathrm{\partial }\mathrm{\Omega }/\mathrm{\partial }\sigma )>0$ . Additionally, since $0<\delta <1$ holds, we have $(\mathrm{\partial }{p}_B/\mathrm{\partial }\sigma )>(\mathrm{\partial }{p}_A/\mathrm{\partial }\sigma )$ . Similarly, the proportion of uninformed voters $\lambda$ also leads to convergence between the political parties’ platforms, that is, $(\mathrm{\partial }{p}_B/\mathrm{\partial }\lambda )>(\mathrm{\partial }{p}_A/\mathrm{\partial }\lambda )$ following the same calculation. The interpretation is as follows: As noted above, an increase in $\sigma$ and $\lambda$ leads voters to demand harsher policy because they believe that the crime situation is severe. However, since the marginal detection effect of $p$ is decreasing, the gap between $p_A$ (higher platform) and $p_B$ (lower platform) would shrink.

4. Empirical implications

In this section, we would like to discuss the empirical implications of our model by referring to empirical research on the related topics.

First, our model predicts that a political party with issue ownership on policing crime has an incentive to overstate the situation to align with its advantage. This appears to be an intuitive argument—if voters believe that the crime situation is dire, they will be more inclined to depend on the party that has competence in handling the situation. Therefore, if voters’ beliefs and perceptions are malleable, the party with issue ownership has a strong incentive to incite voters’ punitive attitudes toward crime. Although it is not stable (Holian, 2004), the conventional view is that Republicans have issued ownership of policing crime in the United States (Petrocik, 1996; Petrocik et al., 2003). Thus, Republican presidents often seem to overstate the crime situation. The most extreme example may be the false arguments advanced by Donald Trump. For example, in 2016, Donald Trump claimed that ‘the US murder rate was the highest it’s been in 45 to 47 years,’ but this was not true. ¹⁶ Although potentially not as extreme as in this example, politicians may mislead voters’ perceptions of crime by emphasizing the agenda even when the crime rate is decreasing. For example, Republican presidents are often said to incite the voters’ punitive attitude by combining law-and-order appeals with racial problems (Mendelberg, 2001; Weaver, 2007; Jacobs and Jackson, 2010). However, this is anecdotal evidence. A more serious empirical investigation is needed to check the plausibility of the hypothesis.

Second, our model predicts that issue ownership leads to divergence in law enforcement policy. Specifically, our model predicts that the party with a reputation for handling crime tends to propose harsher law enforcement than the other party. Several studies have examined the relationship between harsh law enforcement and the partisan effects of political parties. For instance, Gerber and Hopkins (2011) reports that cities electing Democratic mayors allocate a smaller proportion of their budget to public safety, an area with significant local discretion, than cities with similar characteristics electing Republican or independent mayors. Furthermore, several studies show that policy divergence may exist even in the election of district attorneys. For example, Krumholz (2020) demonstrates that the election of a Republican district attorney, as opposed to a Democratic district attorney, resulted in a 6% to 8% increase in total sentenced months and new prison admissions per capita during the four years following an election where a majority of district attorneys ran unopposed. Additionally, Okafor (2021) finds that being in a district attorney election year contributed to higher total prison admissions per capita and total months sentenced per capita, with larger effects observed in Republican counties. On the other hand, there is also research that presents different viewpoints. For example, Thompson (2020) investigates elections for sheriffs and demonstrates that Democratic and Republican law enforcers make similar choices regarding immigration enforcement. Therefore, further empirical investigations are necessary to comprehensively understand situations where partisan differences arise in law enforcement.

Third, our model predicts that political discourse regarding the crime situation leads to harsh enforcement, even when the same amount of effort is devoted to correcting voters’ perceptions of misinforming them. There is some evidence that political discourse may lead to misperceptions by citizens. For example, using a survey on the US Social Security program, Jerit and Barabas (2006) show that citizens mistakenly believe that Social Security will run out of money because politicians frequently refer to pessimistic assessments of its financial future. Esberg and Mummolo (2018) demonstrate that elite partisan cues enhance citizen misperceptions of crime and diminish confidence in the official data. On the other hand, there is some empirical research that investigates the effect of corrective information. Larsen and Olsen (2020) report that if citizens are continually supplied with correct information about burglary rates, this can reduce the misperception of the prevalence of burglaries. On the other hand, Nyhan and Reifler (2010) show that corrective information sometimes fails to reduce misperceptions about politics, especially when people are ideologically biased. However, to the best of our knowledge, the aggregate effect of political misinformation/information about crime situations remains unclear. To assess the plausibility of our prediction, more rigorous empirical research is needed.

Finally, our model predicts that it is challenging to avoid harsh law enforcement to the extent that voters’ beliefs about the crime situation are pliable. If this prediction is correct, our results may suggest the importance of improving voters’ prior knowledge of the facts. As Esberg and Mummolo (2018) show, if people have sufficient knowledge of the facts, political discourse is less likely to manipulate their beliefs. However, if they do not, it is difficult to avoid the distortion of law enforcement policy. In the current paper, this corresponds to a decrease in the percentage of uninformed voters $\lambda$ .

5. Conclusion

This paper provides a formal framework to explain why law enforcement policies have become harsher despite that the crime situation has improved for decades. Our model provides the following explanation: in political competition regarding the issue of criminality, the party that has an advantage in policing crime has the incentive to overstate the severity of the crime, and the other party has the incentive to correct voters’ beliefs. This political misinformation/information makes some voters believe that the crime situation is worse than reality, while other voters’ belief approaches reality. However, voters’ punitive attitudes are more likely to be incited rather than appeased; the former outweighs the latter effect, leading the average voter to demand harsher law enforcement.

There are several potential research topics that are relevant to the current paper. First, our model might be applied to various law enforcement settings, including the regulation of companies’ activities. In the context of political competition, climate change emerges as a critical issue, and political parties consider regulating companies’ actions accordingly. One party may manipulate the issue by providing misinformation about climate change to suit its advantage, while the other party may argue for more lenient regulation and attempt to manipulate voters’ beliefs in line with its intentions. Exploring how these political campaigns influence resulting regulations for companies could be an intriguing area for future research. Second, our model assumes that voters and criminals are different groups. Although this assumption may be valid when law enforcement applies to major crimes, it does not address minor criminal cases, such as speeding tickets and double-parking. In those cases, offenders (criminals) can affect law enforcement through voting, so the equilibrium platforms may change. To investigate those minor crime cases in our model, we would have to modify them to address more general settings. Third, our model assumes that the probability of voters being exposed to a party’s campaign is independent of the voters’ partisanship. However, in reality, campaign effects and partisanship may be correlated. To investigate that case, future research will need to generalize the model. Fourth, in the campaign competition stage, we do not explicitly consider media outlets’ behavior. However, in reality, media outlets’ behavior may influence the effectiveness of a political campaign. Additionally, media outlets have a crucial effect on voters’ perceptions. Future research will require a model that explicitly incorporates media outlets’ behavior. ¹⁷

Finally, in this paper, we do not intend to argue that the mechanism we provide is the only explanation for the increasing punitiveness of law enforcement. However, we believe that the model sheds light on a new perspective to explain modern penal society.