1 Introduction

Governments use substantial resources to keep society safe and punish criminal activities. Annually, mass incarceration costs amount to approximately $182 billion in the United States (Wagner & Rabuy, 2017). The economic tradition to understand deviance and deterrence has its origins in the seminal work by Becker (1968), which stresses the importance of severity and certainty of punishments.Footnote 1 More recently, these concepts have been studied in the domain of uncertainty and ambiguity (see, e.g., DeAngelo & Charness 2012; for a recent review of economic research, see Chalfin & McCrary 2017, and for a cross-disciplinary discussion of experimental work, see Engel 2016).

Understanding the deterrence mechanisms of deviant behavior yields important policy implications. However, a particular institutional aspect that is implicitly factored in by Becker’s decision framework – the swiftness of punishment – has been under-researched in the economics literature. This concept is often referred to as celerity (see Bailey 1980; Howe & Brandau 1988; Yu 1994; Nagin & Pogarsky 2001; Nagin & Pogarsky2004).Footnote 2 Historically, this concept has its scholarly roots in the writings of Jeremy Bentham and Cesare Beccaria, and represents a fundamental component in deterrence theory that has been referred to as the ‘neglected middle child of the deterrence family’ (Pratt & Turanovic, 2018). However, despite theoretical advancements (see, e.g., Nagin 2013), little empirical work has been conducted regarding the celerity of punishment using observational data since there are very few environments where nearly instantaneous punishment or resolution occurs. For this reason, we capitalize on a stylized experimental setting in which we can systematically vary the structure of celerity.

The goal of this paper is to experimentally study the role of timing and uncertainty of formal punishment in deterring deviant behavior - aspects that have received surprisingly little attention in this area of research. Specifically, we are interested in how the timing of formal sanctions (be it conviction or sentencing) and the timing of the resolution of uncertainties surrounding these sanctioning mechanisms affect deterrence. We systematically vary the swiftness of a sanction within a new, stylized, experimental paradigm along the following two dimensions: first, we vary the delay between offense and detection; second, we vary the delay between offense and sanctioning. We also study the combination thereof. Our main objective is to test the behavioral assumption that swiftness matters, which we derived from existing theories (e.g., Loewenstein 1987; Frederick et al., 2002), and to advance our understanding of how to leverage swiftness and the dread of uncertainty as an effective third approach to deter deviance. We argue that swiftness can serve as a useful tool for policy makers to design more efficient and/or potentially also less expensive institutional deterrence mechanisms. Naturally, the concepts of decision-making under risk and uncertainty are not limited to the domain of criminal behavior. Most recently, this has been particularly apparent in the context of the COVID-19 pandemic where individual actions (e.g., to wear masks, to social distance, to obtain a vaccination) are guided by the same principles (Viscusi, 2020). The insights of celerity also find broader application beyond criminal behavior, such as finance and consumption (Palacios-Huerta, 1999; Chesson & Viscusi, 2003). For example, Kreps and Porteus (1978) and Kocher et al. (2014) show that preferences over temporal lotteries also depend on the point in time when the uncertainty is resolved: agents can show a preference for earlier or delayed resolution of uncertainty. Further evidence comes from consumer literature. Anticipatory emotions, compared with outcome-based emotions, are central in prospective consumption situations and the uncertainty associated with anticipatory emotions negatively affects intentions (Bee and Madrigal, 2013).

Given the high costs involved in increasing punishment’s certainty (e.g. costs for an executive body) or punishment’s severity (e.g. incarceration costs), we argue that the timing of punishment as well as the timing at which individuals are informed about the consequences, that is, their delay with respect to the transgression in question, can potentially serve as a powerful tool for deterrence. Our findings advocate for changes in institutional structures to increase the effectiveness of deterrence (Bigoni et al., 2015).

More generally, from an aggregate perspective, the speed of justice matters because it facilitates economic development and market efficiency (North, 1991; Djankov et al., 2008; Ponticelli & Alencar, 2016). Recent empirical results evaluating the impact of legal reform to increase judiciary adjudication in Senegal find an increase in procedural efficiency without any adverse effects on quality (Kondylis & Stein, 2018). From an individual perspective, existing research in various disciplines indicates that timing affects the association formed in one’s mind between the deviant act and the ultimate punishment: without proper swiftness, sanctions risk losing their bite, regardless of how certain or severe they are Chalfin and McCrary (2017) and Pratt and Turanovic (2018) – a fact already prominently argued a long time ago (Watson, 1924).

In reality, the closest we can get to achieving maximal swiftness (celerity) of punishment is by catching deviants in the act and punishing them right away. For example, during the FIFA World Cup 2010 in South Africa, the local government agreed to establish 56 so-called ‘World Cup Courts’ across the country, assigning 1,500 dedicated personnel including magistrates, prosecutors, and public defenders. This was done to achieve speedy justice, in some cases leading to convictions on the same day.Footnote 3 Recently, the concept of celerity has entered the correctional arena through the project HOPE (Hawaii Opportunity Probation with Enforcement) as a new model for probation. “In 2004, First Circuit Judge Steven Alm launched a pilot program to reduce probation violations by drug offenders and others at high risk of recidivism. This high-intensity supervision program, called HOPE Probation [...], is the first and only of its kind in the nation. Probationers in HOPE Probation receive swift, predictable, and immediate sanctions – typically resulting in several days in jail – for each detected violation, such as detected drug use or missed appointments with a probation officer” (Alm, 2014). In a first pilot, the project was found to reduce drug use, crime, and incarceration. Simultaneously, it saved the government approximately $6,000 per participant per year through reduced incarceration (Hawken & Kleiman, 2009).Footnote 4

The existing line of research that has acknowledged the relevance of celerity has often resorted to observational studies with mixed approaches and insights. However, as also recognized by this stream of literature, the common absence of reliable observational data and the ability to account for potentially confounding influences (i.e., perceived or actual certainty or severity of punishment) render the study of celerity methodologically challenging (for a critical discussion, see Pratt & Turanovic 2018). A natural starting point to look for a clean effect is the laboratory environment in which institutional constraints are absent and where we can precisely control the incentives and relevant timings of resolution and punishment (Charness & DeAngelo, 2018). This is in the spirit of the recent surge of experimental economists studying related topics, such as corruption or tax evasion (e.g., Abbink 2018 ; Serra 2011) in controlled laboratory environments. This allows us to control the important elements of celerity: anticipation and revelation of information, timing and severity of punishment, and the opportunity to recidivate. Our experimental analysis is based on a novel cheating game where subjects may cheat in periodic instances to increase payoffs. These cheating periods are followed by an investigation such that cheaters will be detected and fined with a given probability. Across different treatments, we systematically vary the timing of both the resolution of uncertainty about whether one will be punished and actually being punished.

We analyze behavior alongside two dimensions: total propensity to cheat and recidivism (cheating conditional on having cheated at least once before). Our results show that delayed resolution of uncertainty has no direct systematic impact on behavior. With respect to the relation between the delay of punishment and deterrence, we observe a u-shaped relationship where deterrence is lowest for delayed punishment combined with no delay in resolution of uncertainty and for delayed resolution combined with no delay of punishment, whereas it is significantly higher for either no delay or a delay combined with delayed resolution of uncertainty. This result is at odds with discounted expected utility and theories of anticipatory utility. From a policy perspective, our results suggest that to improve deterrence mechanisms, punishment should either be swift or delayed and paired with the psychological dread of uncertainty.

Section 2 details our experimental design, procedures, and briefly discusses the existing literature on swiftness and deterrence. Results are presented and discussed in Section 3. We conclude in Section 4.

2 Experimental Design

Different theories come to different conclusions regarding the implications of the timing of a sanction for deterrence (see Section 3.1 for details). Clearly, this is an important point that has to be taken into consideration for the design of legal institutions. A systematic study of the role of celerity for deterrence poses a serious empirical challenge, because changing the celerity of an enforcement mechanism would most likely impact existing institutional structures on multiple levels. For that reason, isolating the impact of such an intervention is hardly possible in the field. In addition, it is unclear whether an actual or would-be offender is aware of this change or not, making identification almost impossible. Thus, a systematic study of celerity calls for a highly controlled environment that allows for the isolation of the direct effect of institutional changes varying celerity on behavior. Fortunately, the experimental laboratory provides such a controlled environment.

2.1 The Cheating Game

We use a simple card game, in which subjects have to guess which card was drawn from a deck of cards. The game is played repeatedly by our subjects for a total of 28 rounds. In certain rounds, subjects are presented with the option to “cheat.”Footnote 5 Cheating guarantees them the maximum possible payoff for that round. Our goal was to design a simple task where the option to cheat was auxiliary – that is, we wanted the task to be easy to understand – but meaningful, regardless of whether or not the option to cheat was presented. Specifically, we wanted to make sure that cheating was not considered part of the game, but a clear violation of the said game’s rules. In the game, a card is randomly drawn from a deck of 32 cards and subjects have to guess which card was drawn. A subject received 10 Experimental Currency Units (ECU) for a correct guess and 4 ECU for an incorrect guess. Given a 1 in 32 chance of being correct, the expected payoff of not cheating is about 4.2 ECU. In some rounds, participants are given the option to cheat. By cheating, participants are allowed to uncover the randomly drawn card before making one’s guess, ensuring a correct answer and the maximum payoff of 10 ECU less a possible fine if detected.Footnote 6 Participants were informed that each instance of cheating will be followed by an “investigation” that would detect cheating with a fixed probability of 25%, the occurrence and outcome of which is only revealed to the participant.Footnote 7

With that, cheating exposes them to the risk of being caught and fined. If caught, the consequences in the form of punishment are twofold. First, the subject has to pay a fine of 10 ECU. Second, the subject is suspended from the game for one round, is not allowed to make any decisions, and cannot earn any ECU. Hence, the total expected fine is about 14.2 ECU including the forfeited expected earnings of 4.2 ECU due to suspension. Consequently, cheating yields an expected payoff of about 6.5 ECU in that round, whereas the expected payoff of not cheating is about 4.2 ECU. Suspended participants are not allowed to play the game for one round and additionally are forced to wait 60 seconds before they are allowed to continue in the next period. We deliberately chose suspension combined with a short waiting period as part of the sanctioning mechanism to increase salience with regard to the timing of sanctions.Footnote 8 While one might argue that a delayed fine in a laboratory context where all “actual” payments are realized at the very end of the experiment decreases the result’s robustness, such concerns do not apply to the suspension as it is clearly linked to the particular round a subject is suspended.

In order to make the moral dimension of cheating more salient in our laboratory context, we introduce a third party, represented by a charity, that incurs a monetary damage as a result of cheating.Footnote 9 Specifically, for each experimental session, there is a charity pool of 250 ECU (worth $25) from which 50 ECU is deducted each time a particular subject decides to cheat. At the end of the experiment, one subject is randomly selected. This subject’s behavior is implemented towards the charity and the remainder is donated to “Doctors without Borders.”Footnote 10 The identity of the subject is not communicated and remains unknown to avoid reputation concerns.

2.2 Treatments

We vary the timing in a 2 × 2 design along the following two dimensions: the timing of punishment and the timing of the resolution of uncertainty. Resolution of uncertainty regarding whether cheating is detected (and hence whether there are sanctions) is either immediate (IR) or delayed by two periods (DR). Obviously, punishment cannot occur before a sentence is passed and, hence, necessarily occurs after the uncertainty is resolved. Relative to the timing of resolution of uncertainty, punishment is either immediate (no delay or ND) or delayed by 2 rounds (delayed or D). This gives us 4 treatments: IR-ND, IR-D, DR-ND, and DR-D. To ensure that decisions are not influenced by unresolved risk, each cheating decision was followed by 5 rounds without cheating. Footnote 11

All treatments consist of 28 rounds: 4 training rounds followed by 4 blocks of 6 rounds each. In the first 4 rounds participants play the card game without cheating to familiarize themselves with the game and the interface. In the first round of each block, subjects can cheat. In the remaining rounds of a block (rounds 2-6), they play the card game without the option to cheat. Using blocks of 6 rounds allows us to vary both the timing of the resolution of uncertainty, as well as the timing of punishment without an overlap with subsequent cheating decisions.

Table 1 summarizes the four treatments where the timing of punishment is always considered relative to the timing of resolution of uncertainty. In treatment IR-ND, subjects experience immediate resolution of uncertainty – instant feedback within the same round about whether cheating was detected. The subjects also experience no delay in punishment. That is, the fine (if due) and the potential suspension are implemented immediately for the next period. In treatment IR-D, resolution of uncertainty is again immediate, but now there is a delay in punishment of two periods; when cheating in period t, the uncertainty will be resolved immediately, but the potential fine and suspension are executed only in period t + 3 (as opposed to t + 1 in IR-ND). We will also refer to IR-D as immediate resolution of uncertainty and delay of punishment. In treatment DR-ND, the investigation into cheating does not conclude immediately, but lasts for two additional periods. Only after that is the participant informed about whether his cheating was detected or not. As in IR-ND, there is no delay of punishment once uncertainty is resolved. We hence refer to DR-ND as delayed resolution of uncertainty and no delay of punishment. Finally, in treatment DR-D, resolution is again delayed, but now punishment is delayed for another two periods after resolution of uncertainty. That is, cheating in period t results in resolution of uncertainty in period t + 2, followed by the actual punishment (if due) in period t + 5.

Table 1 Overview of timing of resolution of uncertainty and punishment in the different treatments
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2.3 Experimental Procedures

We conducted 32 experimental sessions at the Decision Science Lab at Harvard University. Participants were recruited vie e-mail invitation from the laboratory’s database which contains students as well as non-students. A total of 296 subjects (46.6% male) participated in the experiment split between treatments as follows: 66 subjects in IR-ND, 85 subjects in IR-D, 69 subjects in DR-ND and 76 subjects in DR-D. The experiment was programmed and run using z-Tree (Fischbacher, 2007).Footnote 12 Within each session, participants were randomly assigned to a computer booth in which they would participate in the experiment anonymously. The consent forms and instructions for the corresponding treatment were distributed (see instructions in the Online Appendix). Following an informed consent, the participants were given sufficient time to read the instructions carefully. Before the start of the experiment, subjects had to answer a series of comprehension questions in order to check their understanding of the game and its payoff structure.Footnote 13 Subjects then played 28 periods after which they were informed of their total earnings via a detailed summary screen. One subject was randomly drawn to determine the charity pool and all participants were informed about the final amount left in the pool to be donated to “Doctors without Borders.”Footnote 14 At the end of the experiment, subjects completed a questionnaire containing questions on personal characteristics (demographics, education, income, age), risk-attitudes (SOEP; see Dohmen et al.2011)Footnote 15, consideration of future consequences (Strathman et al., 1994) and self-control (Tangney et al., 2004).

Sessions lasted approximately 45 minutes excluding the time for payment. A participant’s payoff was determined by the sum of his earnings over all 28 rounds. The total payoff in ECU was then converted to dollars at a rate of 10 ECU = $1. The average payment was $14.29, which included a show-up payment of $2.50.

2.4 The Role of Swiftness in Deterrence

Standard theories produce conflicting results with respect to the role of timing on deterrence. The classic theoretical approach towards the deterrence of criminal activity (e.g. Becker 1968) is based on the assumption that potential offenders mainly weigh the expected gains against the expected adverse consequences of an offense. This poses an inter-temporal decision problem under uncertainty: the benefits of criminal behavior are usually immediate, whereas any proceeding detection, conviction, and implementation of legal consequences are generally delayed and stochastic. In the standard framework of discounted expected utility, delayed punishment should reduce deterrence due to a discounting effect, whereas the timing of resolution of uncertainty should have no effect on behavior. However, it has been argued that the uncertainty associated with anticipatory emotions negatively affects intentions (Bee & Madrigal, 2013). Psychological learning theories (Skinner, 1963; Tversky & Kahneman, 1986; Ehrlich, 1996; Hackenberg, 2009) second the argument that the time between a transgression and the punishment as well as the uncertainty that is associated with the punishment are driving forces for effective behavioral changes. If this is indeed the case, then the classical interpretation of celerity as the time between committing an offense and the actual punishment (e.g. fine or imprisonment) should be complemented by the time the uncertainty is resolved, that is, the time of sentencing.

Starting with the seminal paper of Loewenstein (1987), several theories propose that anticipation of future events is an important determinant of intertemporal utility (see e.g., Wu 1999; Lovallo and Kahneman 2000; Caplin & Leahy 2001; Dillenberger 2010; Strzalecki 2013; Golman et al., 2019). These models are based on the idea that a non-negligible proportion of the overall consequences from future consumption (be it negative or positive) is already consumed in the form of so-called anticipatory utility before actual consumption takes place. As a consequence, resolution of uncertainty may affect deterrence and the effect of delayed punishment may be reverted, leading to an increase in deterrence (for recent experimental evidence studying preferences for different forms of resolution of uncertainty see e.g. Zimmermann 2014 and Nielsen 2018). While there is growing theoretical literature supporting anticipatory utility theory and its implications, little empirical, and especially experimental, work exists. Footnote 16

In light of these conflicting predictions, it remains an empirical question to show how the timing of punishment, the timing of resolution of uncertainty, and the interaction thereof affects both deviance and recidivism (learning from experiencing punishment and uncertainty). Our experiment is set out to test these important open questions by systematically varying the structure of the timings. In a last step, we will use our insights to point to future research directions.

3 Results

Here, we present our results using parametric and non-parametric comparisonsFootnote 17, as well as various regression models to analyze differences in cheating behavior. Please note that not only were the number of cheating opportunities (4) the same in all treatments, but also their position within the block was constant (in the first round of each block). Hence, any difference in behavior can only result from our systematic variation in the timing of punishment and the timing of resolution of uncertainty.

There are a number of factors that might affect the propensity to cheat. First and foremost, risk attitudes might affect the individual propensity to engage in cheating. Hence, it is important to note that we find no differences in risk attitudes across treatment groups (chi-square test of independence, N = 296, χ2(30) = 30.422, p = 0.296). It is also conceivable that subjects who were detected cheating in a previous round, are more likely to cheat in future rounds to make up for the missed income opportunity. Consequently, differences in punishment frequencies across treatments, especially in early blocks, might introduce a confound. To rule out this possibility, we ran chi-square tests of independence comparing total punishment frequencies and punishment frequencies in the first block across treatments. Both tests show no significant difference across treatments (Overall: N = 1184, χ2(3) = 0.6806, p = 0.878; first block: N = 293, χ2(3) = 1.6887, p = 0.639) indicating that the frequency of (randomly) received punishments are comparable across treatments.

First, we look at the mean differences in total cheating across all treatments. Total cheating is defined as the proportion of all individual cheating incidences across all four blocks. We present a graphical illustration in Fig. 1 and regression results in Table 2. We find no significant difference between IR-D and DR-ND. However, our results illustrate an interesting inverted u-shaped relationship between total cheating and delay of punishment: compared to the IR-D case (immediate resolution, delayed punishment), cheating is significantly lower in both IR-ND (BSM, 53.2% vs. 38.2%, p = 0.03) and DR-D (BSM, 53.2% vs. 37.5%, p = 0.02). This result alone indicates that the most deterrent form of punishment is either swift or delayed and coupled with a period of unresolved uncertainty.Footnote 18

In order to check for robustness, we capitalize on the panel structure of our data by performing a random effects Logit regression (coefficients represent odds ratios). Our dependent variable in Table 2 is a dummy variable indicating the decision to cheat in each block. For ease of comparison and to improve readability of the results, we use IR-D (immediate resolution, delayed punishment) as our reference category. The regressions include a battery of relevant covariates such as gender, age, block number, number of correct card guesses, and risk preferences.Footnote 19 In model 2, we also control for previous cheating decisions (L.Cheat) and experience with punishment from past cheating (L.Punishment).Footnote 20

Our analysis in Table 2 on the full sample (model 1) suggests that, relative to immediate resolution and delayed punishment (IR-D), both swifter punishment (IR-ND) as well as delayed punishment coupled with delayed resolution (DR-D) are significantly less likely to render individual cheating decisions. The introduction of delayed uncertainty resolution, holding the timing of punishment constant (DR-ND), does not significantly affect cheating behavior. A direct comparison of our treatments mirrors this finding, indicating that higher deterrence can be achieved by either implementing swift punishment (IR-ND) or through the combination of delayed uncertainty resolution and delayed punishment (DR-D). Post-estimation tests yield a marginally significant difference between DR-ND and DR-D (p = 0.071), and no significant difference between the coefficients of IR-ND and DR-D (p = 0.823), suggesting that the effectiveness of deterrence is of a similar magnitude in both cases. All of these results corroborate our previously obtained non-parametric comparisons.Footnote 21

The results (model 2) also suggest that cheating behavior increases with both past rounds’ cheating behavior and having been caught and punished for cheating. This finding indicates that individuals try to make up for incurred losses by increasing the frequency of cheating and taking larger risks, thus being more risk-seeking in losses. Further, a participant’s age is inversely and significantly correlated with one’s probability to cheat.Footnote 22 This indicates that potential wealth effects cannot explain cheating behavior. It is worth noting that we observe substantial gender heterogeneity, indicating that males cheat significantly more than females.Footnote 23 To further explore potential heterogeneous effects for males and females, we ran additional regressions with restricted subsamples using only male or female subjects, respectively (model 3 and 4).Footnote 24 All this suggests that swift punishment or delayed punishment in combination with delayed resolution of uncertainty significantly increases the deterrence of deviant behavior. The delay of uncertainty alone remains ineffective.

We conclude that both very efficient (no delay of punishment) and very inefficient (delayed punishment combined with delayed resolution of uncertainty about the status of discovery) punishment institutions are equally effective in deterring deviant behavior. What is more, one could also plausibly assume the presence of learning effects. A large body of existing literature suggests that the learning effects that emerge through experience are shaped by the timing of rewards and punishments. Due to this, they affect subsequent behavior (cf. Camp et al.1967; Parke & Deur 1972). This is of particular importance in the punishment context, because such learning effects would directly speak to the occurrence of recidivism among former felons. Following this logic, the experience of uncertainty and punishment following transgressive behavior could lead to differences in subsequent transgressions. We refer to this as recidivism. Recidivism in this context is defined as the propensity to cheat conditional on at least one previous transgression (which can occur at the beginning of any of the first three blocks). The idea behind this measure is to understand whether experiencing the drain of uncertainty of punishment following the first cheating decision will affect the individual’s subsequent propensity to cheat. Our results indicate that cheating behavior following the experience of uncertainty and punishment is congruent to our previous findings on unconditional cheating behavior. We present a graphical illustration in Fig. 2.

Fig. 2
figure 2

Recidivism. Percentage of individual cheating attempts relative to the maximum of possible cheating attempts following their first cheating decision

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In order to shed light on this mechanism, we employ the same estimation approach as before, but now restrict our sample to observations where participants have cheated at least once before. Our dependent variable of interest continues to be the propensity to cheat in any given period. Through these regressions, reported in Table 3, we look to analyze the propensity to recidivate and continue to cheat.

Table 3 Propensity to recidivate (Random Effects Logit Regression)
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Our results for recidivism are consistent with our previous findings, suggesting that the recidivism of individuals is lowest when punishment is either immediate or late and paired with uncertainty. The delay of uncertainty alone is ineffective. We again find an inverted u-shaped relationship; when combining delayed punishment with uncertainty of resolution, recidivism rates return to levels similar to those found when immediate punishment is paired with immediate uncertainty resolution (IR-ND). In support of this, post estimation tests show that the drop in recidivism rates in DR-D is marginally significant compared to cheating in DR-ND (p = 0.077). In contrast to total cheating behavior, we do not observe gender heterogeneity for recidivism, except in DR-ND, where males are significantly more likely to recidivate (interaction not shown in the table). Age continues to be inversely related to cheating propensity. We observe a particularly interesting finding for the lagged cheating and lagged punishment variables. The results suggest that following one’s previous decision to cheat, subsequent cheating decisions become less likely, unless previous cheating was accompanied with punishment. In the latter case, participants are substantially more likely to recidivate, which suggests recidivism is driven by one’s desire to make up for previous losses. To further investigate whether previous punishment has a differential treatment effect, we ran an additional regression that interacts lagged punishment with the treatment dummies. We find all interactions to be non-significant. Post-estimation tests show that the overall effects identified above remain unchanged.Footnote 25

In summary, we can conclude that the same institutional settings that are capable of reducing recidivism are also the ones deterring deviant behavior in the first place. Our results demonstrate that swift or delayed punishment combined with an extensive dread of uncertainty regarding one’s detection reduces future criminal behavior.

3.1 Discussion

In this section, we discuss our results in light of existing theories of intertemporal decision making. Specifically, we seek to illustrate how deterrence should be affected by our two main treatment variables, the delay of punishment and the timing of resolution of uncertainty, given a certain theoretical framework. We consider two prominent theories that seem particularly important for our setting, discounted expected utility theory (DEU) and utility from anticipated emotions (UAE). For brevity, we will focus on the intuition behind derived implications and delegate the formal derivation to the 1.

Intuitively, most theories of temporal discounting suggest that future costs or benefits receive a lower weight than immediate ones; this weight decreases as one moves further into the future (Frederick et al., 2002). First, we consider a standard DEU model with discount factor 0 < δ < 1, and for simplicity, restrict attention to a single block of six periods involving one cheating opportunity only. Note that in this model, the timing of the resolution of uncertainty does not affect utility. Furthermore, note that only the utility generated from cheating differs across treatments and that this utility is larger the more the punishment is delayed. Hence, if a potential offender discounts delayed legal consequences, then deterrence decreases as the delay extends. As a consequence, higher celerity (less delay) would increase the efficiency of legal sanctions, which is the classical hypothesis in criminology literature (Nagin & Pogarsky, 2004; Paternoster, 2010). Thus, according to DEU, the efficacy of deterrence decreases as the delay of punishment increases, independent of the timing of resolution of uncertainty. In particular, the point in time where the uncertainty is resolved has no effect on deterrence.

Our results provide only limited support for this implication. When resolution of uncertainty is immediate, we find the expected effect of an additional delay of punishment. However, if resolution of uncertainty is delayed this effect is reversed.

We remark that empirically the above implication of DEU has to be taken with some caution. Gains are usually discounted with a positive rate, making earlier gains more attractive than delayed ones. However, fines are losses, and the evidence for discounting of losses is rather mixed. While studies consistently show that losses are discounted less than gains, it is unclear whether the discount rate for losses is positive, equal to zero, or even negative (Thaler, 1981; Loewenstein, 1987; MacKeigan et al., 1993).

Second, we turn to theories of anticipatory utility, which incorporate anticipatory feelings such as excitement, fear or dread into classical expected utility theory. DEU fails to take such considerations into account. These theories suggest that one might want to bring forward an unpleasant event to shorten the period of dread (or delay a positive event to enjoy the excitement for a longer period of time). The idea is that future events influence current utility. More precisely, negative future events cause more negative utility today the further away the event is (at least up to a certain point). If this effect outweighs the discounting effect described above, then a further delay increases deterrence due to the negative utility from anticipation. Consequently, in UAE, deterrence increases in the delay of punishment if the effect of anticipation is sufficiently high, for a fixed timing of resolution of uncertainty.

Anticipated emotions in Loewenstein’s model refer to future consumption under certainty. Caplin and Leahy (2001) extend this model by allowing for uncertainty and point toward the importance of anticipatory feelings prior to the resolution of uncertainty. Anticipatory emotions, such as anxiety, are often predicated on an uncertain future and thus, are mainly relevant prior to the resolution of uncertainty. As a consequence, the point in time at which uncertainty is resolved matters. For example, Kreps and Porteus (1978) and Kocher et al. (2014) show that preferences over temporal lotteries also depend on the point in time when the uncertainty is resolved. That is, agents can show a preference for earlier or delayed resolution of uncertainty. Caplin and Leahy (2001) argue that anxiety experienced one period before the resolution of uncertainty should depend on both the detection probability and the severity of the punishment. Typically, this anxiety, if large enough, leads to a preference for early resolution of uncertainty for negative outcomes. That is, if the effect of anxiety experienced just before the resolution of uncertainty is large enough, a decision-maker discounting future events will prefer to resolve this uncertainty as late as possible to minimize the impact of anxiety. Thus, under UAE, delayed resolution of uncertainty increases deterrence if anxiety prior to the resolution of uncertainty is large enough and the timing of punishment is held fixed.

Given that the effect of anticipation and anxiety are large enough, the combined effect of delayed resolution and delayed punishment can be grasped by comparing, for example, DR-ND to IR-ND. Taken together the combination of both effects predicted by UAE imply an increase in deterrence when the delays of both punishment and resolution are increased. To summarize, UAE implies a positive effect of delayed punishment on cheating, in contrast to DEU. Delayed resolution of uncertainty may matter under UAE, potentially further increasing deterrence.

Finally, we want to briefly remark that, recently, Baucells and Bellezza (2017) proposed a new theory of inter-temporal decision making. They extend the existing models of anticipatory utility by a reference point which adjusts. It does so during the anticipation phase by altering a utility of recall in the periods succeeding the consumption and changing the magnitude effect in discounting. In this theory it is possible that the utility maximizing timing of an unpleasant event is somewhere in the middle of the time horizon, i.e. fines in earlier or later periods hurt more and should, therefore, lead to higher deterrence. While our experiment was not designed to test this theory, it is the only theory which is compatible with the findings of our experiment.

4 Conclusion

We investigate along two dimensions how timing can impact the effectiveness of sanctions. We use a controlled laboratory experiment designed to study the effect of delayed punishment and delayed resolution of uncertainty on deterrence. Our experimental findings show that the timing of resolution of uncertainty has no effect on deterrence. For the delay of punishment, we observe the following inverted u-shaped relationship: deterrence is highest for no delay or delayed punishment combined with delayed resolution, whereas it is lowest when there is delayed resolution but no delay in punishment or when there is delayed punishment but no delay in resolution of uncertainty. The observed inverted u-shape is at odds with both discounted expected utility theory and anticipatory utility theory. According to the first theory, deterrence should decrease monotonically with the delay of punishment. According to the second, there should also be a monotonic relation between deterrence and delay, which would be the inverse of that in the previous case if the effect of anticipation is sufficiently high.

It is important to note that the effects of treatments on the total cheating behavior can be obtained by two different, possibly simultaneous operating processes. First, the variations in the experimental treatments could have affected the participants’ anticipatory reasoning about how a possible punishment would impact them. If the impact is anticipated to be severe, this could lead to delayed or no cheating. Second, learning processes may have affected cheaters who (at least once) underwent the respective treatments differently by experiencing the (non)waiting for a resolution of uncertainty and the potential execution of an immediate or delayed punishment. This may have influenced their likelihood to cheat again in the future. Inspecting the results for recidivism (i.e. future cheating upon having cheated before) shows that they closely mirror the results of the total cheating behavior. Even if some experience for the treatments to become effective would be needed, basic learning theories (e.g. Azrin1956; Banks & Vogel-Sprott 1965) are at odds with the inverted u-shaped relation between deterrence and delay of punishment also observed for recidivism.

Arguably, the highly effective deterrence of deviant behavior in DR-D could be interpreted in one of the following two ways: firstly, only a delay of punishment, and not the existence of uncertainty resolution, is responsible for the decrease in cheating; secondly, it is the combination of both the delay in punishment and the existence of uncertainty that imposes additional dread and, thus, the interaction of both is driving the strength of deterrence. Our results and theoretical foundation suggest that it is most likely the former. We consider this as a promising venue for future research.

Our findings yield important insights for optimally designing sanctioning schemes in legal systems. Existing deterrence literature has almost exclusively focused on the role of severity and certainty of legal consequences in deterring proscribed actions. Our study shows that celerity, the timing of sanctions through sentencing, may also be a crucial component of an effective legal system. Our results imply that if deterrence is to be maximized, then the punishment should either follow the criminal act quickly or be delayed and combined with uncertainty. As immediate punishment may be relatively costly, an optimally delayed punishment along with a certain degree of uncertainty could be the most efficient solution to reduce both deviance and recidivism.

Our study provides a first step into analyzing the effects of deterrence in a sanctioning system. In order to draw conclusions for an optimal policy in the real world, future research needs to tackle several limitations of our study. In particular, it seems necessary to study celerity when the delay of punishment extends to the real payout of subjects. While the concept of celerity has already been backed theoretically in the scholarly literature for an extended period of time, only recently has the concept of celerity entered the correctional arena through the project HOPE (Hawaii Opportunity Probation with Enforcement) as a new model for sanctions and probation (Hawken & Kleiman, 2009; Kilmer et al., 2013; Alm, 2014; Nicosia et al., 2016). More research should follow and consider additional aspects such as the role of optimal delay, which can be very sensitive to the type of punishment (e.g. the optimal delay may be rather different for monetary fines than for imprisonment). With this in mind, we are confident that our study highlights the role of celerity in designing optimal sanctioning systems and points to fruitful avenues for future research.