Deterrence of Unwanted Behavior: aTheoretical and Experimental Investigation1
Penélope Hernández2

Zvika Neeman3

September 1st, 2023

1 Acknowledgments to be added.
2 ERI-CES Universidad de Valencia
3 Economics, Tel Aviv University
4 Economics, Ben-Gurion University of the Negev

Ro’i Zultan4

Abstract
Suppose that spreading enforcement resources uniformly across time and space allows sanctioning anyone who engages in an unwanted activity with probability p.
However, by concentrating enforcement resources, it is possible to split the probability p into a higher probability of sanction p H > p in some targeted areas or times, atthe expense of a lower probability of sanction p L < p elsewhere. If the objective is tominimize the overall level of the socially unwanted activity, irrespective of its speciﬁc
location or time, does splitting the probability of sanction p help achieve this goal?
We present a theoretical model of this situation, and undertake an experimentthat allows us to answer this question empirically. Since the idea of beneﬁcial splittingof prior beliefs is central to Bayesian persuasion literature, our investigation presents
an experimental investigation into whether Bayesian persuasion can indeed yieldpractical beneﬁts in a realistic parametrized setting.

1 Introduction
Suppose that the chief of police in a certain town aims to deter crime, or some othersocially unwanted activity. Suppose that spreading enforcement resources uniformly
across time and space allows sanctioning anyone who engages in the unwanted activity with probability p. However, by concentrating enforcement resources, it is possible
to split the probability p into a higher probability of sanction p H > p in some targetedareas or times, at the expense of a lower probability of sanction p L < p elsewhere.1
If the objective is to minimize the overall level of the socially unwanted activity, irrespective of its speciﬁc location or time, does splitting the probability of sanction p
help achieve this goal?
We present a theoretical model that describes this situation, and undertake anexperiment that allows us to answer this question empirically. Since the idea of beneﬁcial splitting of prior beliefs is central to Bayesian persuasion literature, our investigation presents an experimental investigation into whether Bayesian persuasion canindeed yield practical beneﬁts in a realistic parametrized setting.
Speciﬁcally, we consider a model with a large number of individuals. Each individual faces a choice between a benign and a socially unwanted action. For example,
individuals may choose between parking legally and illegally. We assume that the
benign action generates a certain payoff for the individual. By contrast, the sociallyunwanted action induces a risky binary lottery, whose outcome depends on whether
the individual is sanctioned or not. Each individual is characterized by the threshold probability of sanction above which she prefers the benign action over the risky
lottery, which for simplicity we assume to be independent of time and place.
As mentioned above, we conduct an experiment to assess our theoretical model.
Since experimental results tend to be subject to noise, we enhance the realism of
our model by assuming that each individual’s threshold probability is normally distributed. As a result, individuals’ choices between the benign and unwanted actions
are also noisy in the theoretical model. Notably, each individual’s violation function, which relates the probability that the individual chooses the socially unwanted
action to the probability of sanction, is decreasing and S-shaped in the probabilityof sanction. This decreasing S-shape form is consistent with the intuition that, on
the one hand, very small probabilities of sanction should hardly affect individuals’
propensities to engage in the unwanted activity, and on the other hand, sufﬁciently
large probabilities of sanction should deter almost everyone from engaging in the un1 Of course, for splitting the probability of sanction to have any effect at all, it must be observable.

Namely, it must be known that in certain locations and times, enforcement is stricter.

wanted activity. Indeed, a famous experiment that was conducted in Kansas Cityin 1974 (Kelling et al., 1974) found that a doubling of police patrols had virtually nostatistically signiﬁcant effect on street crime.2
Summing up the individuals’ violation functions produces an aggregate violation function that relates the probability of sanction to the share of the population thatengages in the socially unwanted action, which is also decreasing and S-shaped.
If a violation function is decreasing and convex throughout its range, then splitting the sanction probability p would increase the overall likelihood that the individual would engage in the unwanted activity. However, if the violation function isconcave throughout, then splitting the sanction probability p would decrease the overall likelihood that the individual would engage in the unwanted activity. The fact that
individuals’, as well as the aggregate, violation functions are decreasing and S-shaped
implies that they are ﬁrst concave, and then convex. This suggests that small valuesof the sanction probability p may be split in a way that promotes overall deterrence,but large values cannot.
Moreover, for any ﬁxed probability of sanction, decreasing the magnitude of the
sanction, or increasing the reward from choosing the socially unwanted action without being sanctioned, increases the relative attractiveness of the socially unwanted action, and so shifts each individual’s as well as the aggregate violation function to theright. Accordingly, we say that decreasing the magnitude of the sanction, or increasing the reward from choosing the socially unwanted action, increases the temptation
to choose the socially unwanted action. Intuitively, when temptation is very low, theviolation function is shifted so much to the left that it becomes convex on the entire
relevant range of sanction probabilities, which in turn implies that splitting increases
the violation rate (hurts deterrence). By contrast, when temptation is very high, theviolation function is shifted so much to the right that it becomes concave on the entire
relevant range, which implies that splitting decreases the rate of violation (improvesdeterrence).
The main theoretical result of this paper formalizes this intuition. We show that
2 This ﬁnding had a big effect on the thinking on deterrence.

It convinced both academics andthe police itself that “police presence does not deter” (Sherman and Weisburd, 1995). Sherman andWeisburd (1995) famously criticized the Kansas City experiment by claiming that Kansas City is toolarge a unit of analysis for a doubling of patrols to produce an effect, or for a true reduction in crime tobe statistically signiﬁcant. Sherman and Weisbrud repeated the Kansas City experiment in Minneapolis
two decades after the Kansas City experiment, but restricted it to crime “hot spots,” which can be as
small as a street corner or a city block. They found that a doubling of police patrols in crime hot
spots produced reductions in total crime that ranged from 6 percent to 13 percent (however, “observed
disorder” decreased by one-half). Their ﬁndings are consistent with the prevailing view that “large
increases in dosage may be essential if any effect on crime is to be observed" (Sherman and Weisburd,1995).

2for any ﬁxed probability of sanction p, if it is possible to improve individual or aggregate deterrence by splitting p, then it is also possible to improve individual or aggregate deterrence, respectively, by splitting p under higher temptation. And conversely,
for any ﬁxed probability of sanction p, if it is impossible to improve individual or aggregate deterrence by splitting p, then it is also impossible to improve individual oraggregate deterrence, respectively, by splitting p under lower temptation.
We conﬁrm these theoretical predictions in a laboratory experiment, in which
subjects learn the probability of sanction from experience. In each round of the experiment, subjects can choose between a safe action, which pays 5 Experimental Currency
Units (ECU), and a binary lottery, which pays either a positive or a negative amount.3
A subject who chooses the safe action is said to be deterred. We implement splitting
by telling subjects to pay attention to a color that is ﬂashed in front of them, because itis related to the probability of receiving the positive payment in the risky binary lottery. The fact that the subjects in the experiment learn the sanction probability fromexperience, rather than being told what it is, supports the view that, for those parameter values when it is successful, splitting can also be useful in practice.
The importance of the experiment lies in that it allows us to quantify exactly
how high temptation needs to be in order for splitting to be effective in promoting
overall deterrence. In our experiment, the sanction probability p = 0.3 can be split
in a way that promotes overall deterrence if the binary lottery pays −10 and 50 ECU,
when the individual is and is not sanctioned, respectively. In such an environment,
not splitting the sanction probability p = 0.3 implies that 59% of participants’ choices
are for the socially unwanted action.4 If the binary lottery pays −30 and +30 ECU
upon sanction and no sanction, respectively, then splitting has no effect on overall
deterrence. In such an environment, not splitting the sanction probability p = 0.3
implies that 33% of participants’ choices are for the socially unwanted action. Finally,
if the binary lottery pays −50 and +10 ECU upon sanction and no sanction, respectively, then splitting hurts overall deterrence. In such an environment, not splitting
the sanction probability p = 0.3 implies that 12% of participants’ choices are for thesocially unwanted action.
The experiment thus both conﬁrms and quantiﬁes the observation that splitting
can be effective in promoting deterrence in environments in which the temptation to
3 At the conclusion of the experiment, the ECU is converted to euros so that the average payment to

the participants is held constant through the different sessions.
4 In choosing the parameters for our experiment, we have relied on the estimates produced by Erev

et al. (2017)’s Best Estimate and Sampling Tools (BEAST) model of choice under uncertainty. Accordingly,
our ﬁndings both rely on and validate the theoretical predictions produced by the BEAST model.

3commit the socially unwanted action is strong and the choice of this action is relatively
common, but not in environments in which the temptation to commit the sociallyunwanted action is weak and the choice of this action is relatively uncommon.
Estimates of the extent of illegal behavior are generally hard to get. But to put
the numbers above in perspective, it is noteworthy that between 25%-35% of the bus
passengers in Santiago, Chile, reportedly evaded payment of the required travel farebetween 2015 and 2019 (Cantillo, Raveau and Muñoz, 2022). According to the Internal Revenue Service (IRS), roughly one out of every six dollars owed in federal taxes
between 2008 and 2010 went unpaid.5 Recent research conducted in Barcelona and
New York City’s Murray Hill, Midtown Manhattan, revealed an average of 1.32 and0.28 illegally parked vehicles per 100 meters of road, respectively, suggesting that approximately 6% and 1.25% of vehicles in these areas were parked illegally (Morillo
and Campos, 2014). Lastly, the National Coalition Against Domestic Violence reportsthat over 10 million adults in the United States experience domestic violence annually, indicating that that approximately 3% of Americans are involved in perpetrating
domestic violence.6

Related Literature
The idea that, in a game with incomplete information, it may be possible to proﬁtably
manipulate players’ choices through the splitting of prior probabilities dates back at
least to work of Aumann and Maschler (1995), and is a key observation of the literature on Bayesian persuasion, which originated in Kamenica and Gentzkow (2011).
For a recent review of this literature, see Kamenica, Kim and Zapechelnyuk (2021).
We are aware of only three experimental studies of Bayesian persuasion. All
three papers have a very different focus from ours. Fréchette, Lizzeri and Perego(2022) study the role of commitment in communication and show that a form of commitment blindness leads some senders to overcommunicate when information is veriﬁable and undercommunicate when it is not. Au and Li (2018) perform an experimental study of the relationship between Bayesian persuasion and reciprocity, andNguyen (2017) studies experimentally whether subjects design their signals in a waythat maximizes their expected payoff.
The hotspots literature in criminology (see, e.g., Braga, Papachristos and Hureau,
5 See the IRS publication titled Federal Tax Compliance Research: Tax Gap Estimates for Tax Years

20082010 Publication 1415 (5-2016), https://www.irs.gov/pub/irs-soi/p1415.pdf.
6 See

the national intimate partner and sexual violence survey:(http://www.cdc.gov/violenceprevention/pdf/nisvs_report2010-a.pdf).

42010 summary report2014; and Braga et al., 2019) studies how to focus enforcement resources where they
make the most difference. For example, in a town with two neighborhoods and two
police cruisers, is it better to deploy these cruisers in the ﬁrst or second neighborhood,or to split them between the two neighborhoods? By contrast, we focus mainly on thequestion of whether it is possible to improve deterrence through resource allocation,
under the constraint that amount of resources is ﬁxed. Namely, in a town with twoneighborhoods as above, is it better to have the two cruisers patrol together, or separately?
Lando and Shavell (2004) and Eeckhout, Persico and Todd (2010) both considered the question of how to allocate enforcement resources, and argued that it may be
beneﬁcial to concentrate enforcement on a subset of the population. Eeckhout, Persico and Todd also demonstrated this idea empirically using trafﬁc data gathered by
the Belgian Police Department. More recently, Hernández and Neeman (2022) havegeneralized their theoretical results by considering any number of locations, adding
uncertainty, considering the question of how to further improve deterrence through
Bayesian persuasion, or communication.
The rest of the paper proceeds as follows ... All proofs are relegated the the
Appendix.

2 ModelWe consider the following choice problem. An individual faces a choice between
two actions. One is benign, and the other is socially unwanted but beneﬁcial for theindividual.
We assume that choice of the benign action generates a certain payment to
the individual, which we normalize to zero. We refer to the choice of this action
as “compliance.” Because choice of the socially unwanted action may by subject
to sanction, choice of this action induces a risky binary lottery L( p), which is parameterized by the probability of sanction p ∈ [0, 1]. With probability p the individual is sanctioned, and the lottery generates a loss L < 0 to the individual, andwith probability 1 − p the individual is not sanctioned, and the lottery generates areward R > 0. We refer to the choice of this action, which generates an expected
payment of E [ L( p)] = p · L + (1 − p) · R to the individual, as “committing a violation.” Accordingly, individuals who are induced to comply are said to be deterredfrom committing a violation.
The choice environment we consider is thus characterized by two parameters:
5

a reward R > 0, and a loss L < 0. Obviously, for any ﬁxed probability of sanction p ∈ [0, 1], increasing the reward R, or decreasing the (absolute value of the) loss
| L| strengthens the individual’s temptation to commit a violation. In other words,
temptation introduces a binary relation over choice environments, which is deﬁned
as follows.
Deﬁnition 1 A choice environment ⟨ R, L⟩ induces a stronger temptation to commit a violation than the choice environment ⟨ R′ , L′ ⟩ if either R ≥ R′ or L ≥ L′ and at least one of theseinequalities is strict.
If a choice environment ⟨ R′ , L′ ⟩ induces a stronger temptation to commit a violation
than the choice environment ⟨ R, L⟩ then we say that “temptation increases” from
⟨ R, L⟩ to ⟨ R′ , L′ ⟩.
Denote by p̃ R,L the threshold probability of sanction above which the individual
prefers to comply in choice environment ⟨ R, L⟩. That is, when faced with choice problem ⟨ R, L, p⟩, where R and L denote the Reward and Loss, respectively, and p denotes
the probability of sanction, if p > p̃ R,L then the individual would comply; if p < p̃ R,L
the individual would commit a violation; and if p = p̃ R,L the individual would be
indifferent between compliance and the commitment of a violation. To simplify notation, as long as it does not cause confusion, we drop the subscript from the thresholdsanction and denote it as p̃.
As explained in the introduction, we test our theoretical model experimentally.
Because experimental results tend to be subject to noise, we enhance the realism of ourmodel by using a random preference model. Speciﬁcally, we assume that when facedwith a choice environment ⟨ R, L⟩, the individual’s threshold sanction p̃ is normally
distributed, with mean µ R,L and standard deviation σR,L .
Thus, when faced with a choice problem ⟨ R, L, p⟩, the probability that the individual would commit a violation is given by:
π R,L ( p) ≡ Pr ( p ≤ p̃ R,L )= 1 − Φ R,L ( p),where Φ R,L denotes the cumulative distribution function of a Normal distribution
with mean µ R,L and standard deviation σR,L . We refer to the function π R,L (·) as the“violation curve” for environment ⟨ R, L⟩. As before, to simplify notation, we drop thesubscript from the violation curve and denote it as π (·).
The function Φ R,L is an increasing S-shaped function, or a sigmoid function.
That is, Φ R,L is convex and then concave in its argument. Thus, the violation curve6π ( p) is a decreasing S-shaped function of the sanction probability p, which is concave
and then convex in p, as depicted in Figure 1a below.

π ( p)0π ( p)π ( p)p1(a) S-shaped violationp L =0ppH(b) Optimal splitting

0

1

p

1

(c) Changing ﬁne/reward

Figure 1: The Violation Curve

3 Splitting
Splitting the probability of a sanction p into two probabilities p L < p < p H may facilitate compliance while maintaining the same amount of enforcement resources. The
basic idea is the following. Instead of being faced with the choice problem ⟨ R, L, p⟩,
the individual would be faced with one of two choice problems. With probability λ,the individual would be faced with the choice problem ⟨ R, L, p L ⟩; and with probability
1 − λ, the individual would be faced with the choice problem ⟨ R, L, p H ⟩. The numbers p L , p H and λ are chosen such 0 ≤ p L < p < p H ≤ 1, and λp L + (1 − λ) p H = p.
This ensures that the mean probability of a sanction across the two choice problems⟨ R, L, p L ⟩ and ⟨ R, L, p H ⟩, λp L + (1 − λ) p H , remains ﬁxed at p. Being faced with oneof two choice problems instead of just with a single choice problem is called splitting
because the probability p used in the single choice problem ⟨ R, L, p⟩ is split into theprobabilities p L and p H used in the two choice problems ⟨ R, L, p L ⟩ and ⟨ R, L, p H ⟩ in a
way that preserves the overall probability that an individual who chooses to commit
a violation is sanctioned.
An individual who is faced with the choice problem ⟨ R, L, p L ⟩ chooses the lottery with probability π ( p L ); and an individual who is faced with the choice problem⟨ R, L, p H ⟩ chooses the lottery with probability π ( p H ). It follows that the expected
probability that an individual who is faced with one of the two choice problems

7⟨ R, L, p L ⟩ and ⟨ R, L, p H ⟩ as described above would commit a violation is equal to:λπ ( p L ) + (1 − λ)π ( p H ).
The next deﬁnition formalizes the sense in which a probability of a sanction p
can be split in a way that promotes social welfare.
Deﬁnition 2 Fix an environment ⟨ R, L⟩. A violation curve π (·) is said to be proﬁtably
convexiﬁable at p ∈ (0, 1) if there exist two probabilities p L < p < p H such thatλπ ( p L ) + (1 − λ)π ( p H ) < π ( p)for a probability λ ∈ (0, 1) that satisﬁes the equation λp L + (1 − λ) p H = p.
If a violation curve π (·) is proﬁtably convexiﬁable at p, then there exist two
probabilities p L < p < p H such that the straight line that connects the points ( p L , π ( p L ))and ( p H , π ( p H )) lies below π (·) on the interval ( p L , p H ). This is depicted in Figure
1b, where the value of p L is taken to be equal to zero.
Notably, while a function that is concave on an open interval is proﬁtably convexiﬁable at any point in this interval, a function can also be proﬁtably convexiﬁable
at points in which it is convex. Figure 1b depicts a violation curve that is both locallyconvex and proﬁtably convexiﬁable at points that are sufﬁciently close to p H frombelow.
If it is possible to split the probability of a sanction in a way that reduces theprobability of committing a violation, or that increases compliance, then we say thatsplitting is socially beneﬁcial. In principle, there could be many pairs of sanction
probabilities p L < p < p H , which make splitting socially beneﬁcial. The pair p L andp H that is depicted in Figure 1b is the optimal pair, which maximizes the probabilityof compliance.
Lemma 1 Fix an environment ⟨ R, L⟩. A probability of a sanction p is proﬁtably convexiﬁable
if and only if p < p∗R,L where p∗R,L is given by the unique solution of the problem
π (0) − π ( p ),pp∈[0,1]min(1)provided that p∗R,L ≤ 1. If p∗R,L > 1, then any sanction probability p < 1 is proﬁtably
convexiﬁable.
Increasing temptation makes non-compliance relatively more attractive for theindividual for every sanction probability. It is therefore natural to assume that in8

creasing temptation increases the rate of violation π (·) for every sanction probabilityp ∈ (0, 1).
Lemma 2 Suppose that increasing temptation increases the rate of violation π (·) for every
sanction probability p ∈ (0, 1). Then if temptation increases from choice environment ⟨ R, L⟩
to ⟨ R′ , L′ ⟩, then µ R′ ,L′ > µ R,L and σR′ ,L′ < σR,LIntuitively, because a larger reward R and a smaller (absolute value of) loss | L|make any lottery L( p) more attractive, the mean µ R,L is weakly increasing in both R
and L and the standard deviation σR,L is weakly decreasing in both R and L. In otherwords, higher temptation, or a larger reward and a smaller loss, implies that individuals are both more likely to choose the socially unwanted action, and the varianceassociated with their choice of the socially unwanted action is smaller.
The next proposition describes the main theoretical result of the paper.
Proposition 1 Increasing temptation shifts the violation curve to the right. Speciﬁcally, suppose that the choice environment ⟨ R, L⟩ induces a stronger temptation to commit a violation
than choice environment ⟨ R′ , L′ ⟩. Then, as long as the standard deviation σR,L does not decrease too fast in R and L, if the probability of sanction p is proﬁtably convexiﬁable in choice environment ⟨ R′ , L′ ⟩, then it is also proﬁtably convexiﬁable in choice environment ⟨ R, L⟩. And,
if the probability of sanction p is not proﬁtably convexiﬁable in the choice environment ⟨ R, L⟩,then it is also not proﬁtably convexiﬁable in the choice environment ⟨ R′ , L′ ⟩.
Because the violation curve is decreasing and S-shaped, Proposition 1 implies
that increasing temptation shifts the violation curve to the right. It follows that theviolation curve becomes more concave as temptation increases, and more convex as
temptation decreases. By moving the violation curve sufﬁciently to the right, it canbe made concave over the entire range of sanction probabilities, and by moving theviolation curve sufﬁciently to the left, it can be made convex over the entire range ofsanction probabilities.
Proposition 1 is formulated for the case of a single individual. By aggregating
individuals’ violation functions, it is possible to obtain an aggregate analog of Proposition 1 as follows.
Suppose that there are n different individuals. Let p̃iR,L denote the thresholdprobability of sanction above which individual i prefers to comply in choice environment ⟨ R, L⟩. Suppose that individuals compliance decisions are stochastically independent. That is, when faced with choice problem ⟨ R, L, p⟩, if p > p̃iR,L then individual i complies; if p < p̃iR,L then individual i commits a violation; and if p = p̃iR,L thenindividual i is indifferent between compliance and the commitment of a violation,9independently of whether other individuals’ comply or not. As before, to simplify
notation, we drop the subscript from the threshold sanction and denote it as p̃i .
Each individual i’s threshold sanction p̃ is normally distributed, with mean µiR,L
i . Thus, when faced with a choice problem ⟨ R, L, p ⟩, theand standard deviation σR,Lmean fraction of individuals who would commit a violation is given by:Σ( p) ≡
π R,L

(
)
1 n
Pr p ≤ p̃iR,L∑n i =1= 1 − ΦΣR,L ( p),where ΦΣR,L denotes the cumulative distribution function of a Normal distribution
√
n
1 n
1
Σ
i
Σ
with mean µ R,L = n ∑i=1 µ R,L and standard deviation σR,L =(σi )2 . Wen2 ∑i =1 R,LΣ (·) as the “aggregate violation curve” for environment ⟨ R, L ⟩.
refer to the function π R,LAs before, to simplify notation, we drop the subscript from the violation curve anddenote it as π Σ (·). Because it is equal to a sum of decreasing S-shaped functions,the aggregate violation curve π Σ ( p) is a also a decreasing S-shaped function of the
sanction probability p, which is concave and then convex in p, as depicted in Figure
1a above.
We thus have the following proposition, which describes the aggregate analog
to Proposition 1.

Proposition 2 Suppose that the choice environment ⟨ R, L⟩ induces a stronger temptation tocommit a violation than choice environment ⟨ R′ , L′ ⟩. Then, as long as the standard deviaΣ does not decrease too fast in R and L, if the probability of sanction p is proﬁtablytion σL,Hconvexiﬁable in choice environment ⟨ R′ , L′ ⟩, then it is also proﬁtably convexiﬁable in choiceenvironment ⟨ R, L⟩. And, if the probability of sanction p is not proﬁtably convexiﬁable in the
choice environment ⟨ R, L⟩, then it is also not proﬁtably convexiﬁable in the choice environment ⟨ R′ , L′ ⟩.

4 Experiment
4.1 Design and Procedure
We ran an experiment in which we manipulated splitting within-subjects and temptation between-subjects to test the beneﬁt from splitting under different levels of temptation. The experiment consisted of 200 trials in two blocks, which were divided into
one block of 100 splitting trials and another block of 100 pooling trials. The orderof the blocks was randomized at the participant level. In each trial, each participant
10Table 1: Experimental Treatments
Temptation

Safe

RewardLoss

Weak
Medium-weakMediumMedium-strong
Strong

5
5
5
5
5

10
10
30
30
50

−50
−30
−30
−10
−10

observed a signal in the shape of a colored circle and chose between a safe option(comply) and a risky option (violate). The safe option always yielded a payoff of 5
ECU (Experimental Currency Units). The possible payoffs obtained from the risky
option varied depending on the treatment and color of the circle. Each treatment was
associated with two possible payments, one positive and one negative. The positive
payment captured the beneﬁt from committing a violation without being sanctioned,
and the negative payment captures the loss from committing the violation and being
sanctioned.
In the pooling trials, the circle was always yellow and the probability of sanction
upon choosing to commit a violation was ﬁxed at 53 . In the splitting trials, the circle
was either red with a probability of .56 or blue with a probability of .44. The color redcorresponded to a high rate of enforcement, or probability of sanction, and the blue
color corresponded to a low rate of enforcement, or probability of sanction. Accordingly, a red circle indicated that the probability of a sanction was 1, that is, when thecircle was red, a subject who chose to commit a violation was sanctioned with probability 1. The blue circle indicated a small probability of sanction. When the circle was1
blue, a subject who chose to commit a violation was sanctioned with probability 12.
1
The mean probability of a sanction was thus .56 · 1 + .44 · 12 = .597, slightly less thanin the pooling trials.
Table 1 presents the between-subjects treatments. The treatments manipulated
temptation by gradually varying the negative or loss payments and the positive or reward payments. Starting from weak temptation, with a low reward of 10 ECU and ahigh absolute loss of −50 ECU, temptation increased by ﬁrst decreasing the loss payment to −30 in the Medium-weak temptation treatment, and then also increasing the
reward payment to 30 in the Medium temptation treatment. This was followed by further decreasing the loss payment to −10 in the Medium-strong temptation treatment
and ﬁnally by further increasing the reward payment to 50 in the Strong temptation
treatment.
For each treatment, we ran one session with 50 participants. The total number11

of participants in the experiment was thus 250. The experiment was carried out at
the Laboratory for Research in Experimental Economics (LINEEX) in the Universityof Valencia in December, 2022, and March, 2023. The instructions, available in the
appendix, were read aloud at the beginning of each session/treatment. Each session
lasted approximately 50 minutes. The average payment to participants was 13.8 Euros.

4.2 Hypotheses
Because participants in the experiment faced three different probabilities of sanction:
1 3
12 , 5 , and 1, the experiment generated three points on the participants’ violation
curve. We refer to the piecewise linear curve resulting from connecting these threepoints as the experimental violation curve. The beneﬁt from splitting is directly tied to1
the curvature of this curve. If the line that connects the points with p = 12
and p = 1
lies below the point with p = 35 , then the sanction probability p = 35 can be beneﬁciallysplit. Otherwise, it probably cannot.7 As explained above, Proposition 1 implies thatthe violation curve should become more concave as temptation increases, and suggests that the violation curve may be strictly convex and strictly concave at very low
and very high levels of temptation, respectively. Accordingly, our ﬁrst hypothesis
concerns the concavity of the experimental violation curve.
Hypothesis 1 The experimental violation curve is convex under ”low” temptation. It becomes less convex/more concave as temptation increases and it is strictly concave under ”high”
temptation.
The second hypothesis tests the predictions of Proposition 1 directly.
Hypothesis 2 The beneﬁt of splitting increases as temptation increases. It is strictly positive
under high temptation and strictly negative under low temptation.
The next two hypotheses relate the share of individuals who comply to thestrength of temptation.
Hypothesis 3 As temptation increases, the violation curves of more individuals become concave.
Hypothesis 4 As temptation increases, the share of individuals who comply increases too.
7 Only ”probably" because the fact that p = 3 cannot be proﬁtably split into p = 1 and p = 1 does512

not necessarily imply that it cannot be proﬁtably split into two other sanction probabilities. However,
the monotonicity and curvature of the experimental violation curve suggests that proﬁtable splitting is
unlikely in this case.

12

1
0

.2

Violation rates
.4
.6

.8

Temptation
Weak
MediumStrong

0 1/12

3/5
Enforcement rate

1

Figure 2: Violation rates across treatments.

4.3 Results
We ﬁrst analyze the effects of temptation on the concavity of the experimental violation curve and the effectiveness of splitting at the aggregate level. We proceed withanalyses and tests at the individual level.
4.3.1 Aggregate Violation
Figure 2 depicts the aggregate violation rates by treatment and probability of sanction.
The qualitative pattern depicted in Figure 2 is consistent with the prediction stated in
Hypothesis 1. Namely, the violation curve changes gradually from convex to linear
to concave as temptation increases from weak to medium to strong.
To formally measure the curvature of the experimental violation curves, we calculated for each participant the violation rate for each enforcement rate. We thenestimated the following regression with robust standard errors clustered on individ-

13

Table 2: Curvature of the violation curves.

Weak temptation
Medium temptation
Strong temptation

Term

Convexity

t-valuep-value

β2
β2 + β5
β2 + β80.357−0.051−0.843

2.98
−0.36
−4.43

0.003
0.722
0.000

Notes: Marginal self interactions of enforcement rate based on an OLS regression of subject-level mean
violation rate by enforcement rate with robust standard errors clustered on participants.

uals:
VRip = β 0 + β 1 · p + β 2 · p × p+ β 3 · Mi + β 4 · Mi × p + β 5 · Mi × p × p(2)+ β 6 · Si + β 7 · Si × p + β 8 · Si × p × p,
where VRip is the violation rate of participant i across the periods in which the enforcement rate was p, and Mi and Si are dummy variables that indicate whether theparticipant participated in the Medium or Strong) temptation treatment, respectively.8
The sum of the coefﬁcients of the all the terms that include the interaction of the enforcement rate with itself, p × p, captures the convexity (if positive) and concavity (ifnegative) of the violation curve. Under weak temptation, the term p × p is multipliedby β 2 ; under medium temptation, the term p × p is multiplied by β 2 + β 5 ; and understrong temptation, the term p × p is multiplied by β 2 + β 8 . The results are presented
in Table 2 and are consistent with Hypothesis 1. The experimental violation curve issigniﬁcantly convex under weak temptation. Under medium temptation, the experimental violation curve is indistinguishable from linear. And under strong temptation,
the experimental violation curve is signiﬁcantly concave.
As explained above, a concave violation curve implies that splitting improvesmean compliance, and a convex violation curve implies that splitting hurts mean compliance. However, we also test the effectiveness of splitting directly. We ﬁrst compare
the number of violations (out of 100 trials) between the split and pooled environmentsby treatment. Wilcoxon signed-rank tests show that splitting of the sanction proba1
bility 35 into the sanction probabilities 12
and 1 signiﬁcantly reduces the violation rate
under strong temptation (z = 3.70, p < .001) and signiﬁcantly increases the violation
rate under weak temptation (z = 3.72, p < .001). Splitting has no signiﬁcant effect on
8 Each of the 150 participants was faced with three different enforcement rates. This gave us threetreatments, and 450 observations overall.

14

Table 3: Regression on violation rates.

MediumWeak
Split
Medium × Split
Weak × Split
Constant

Coefﬁcient

Robust std.
error

z-statistic

p-value-1.079-2.404-0.673
0.645
1.297
0.377

0.219
0.240
0.160
0.211
0.254
0.171

-4.94
-10.03
-4.21
3.06
5.10
2.20

.000
.000
.000
.002
.000
.028

Notes: Logistic regression of violation rates on treatment interacted with splitting based on 30, 000
observations with robust standard errors clustered on participants.

the violation rate under medium temptation (z = 0.15, p = .880).
Table 3 presents the results of a logistic regression of the violation rate on treatment interacted with splitting with robust standard errors clustered on participants.
Figure 3 plots the results from the regression, with the violation rate presented in theleft panel and the estimated marginal effect of splitting presented in the right panel.
The regression conﬁrms the results from the non-parametric tests. Splitting of the1
sanction probability 35 into the sanction probabilities 12
and 1 signiﬁcantly reduces
violations under strong temptation (z = 4.32, p < .001), signiﬁcantly increases violations under weak temptation (z = 3.13, p = .002), and has no signiﬁcant effectunder medium temptation (z = 0.20, p = .838). Overall, the results strongly support
Hyopthesis 2. Namely, splitting of the sanction probability 53 into the sanction proba1
bilities 12
and 1 leads to (i) an overall signiﬁcant reduction in violations under strong
temptation; (ii) an overall signiﬁcant increase in violations under weak temptation;and (iii) no signiﬁcant effect on violation under medium temptation.
4.3.2 Individual Heterogeneity
Individual participants may of course differ in their response to deterrence. In orderto capture this heterogeneity in individual participants’ responses, we also estimateda separate regression for each participant in each treatment. The speciﬁcation is as inEquation (2), with violation as the dependent variable, except we excluded the treatment terms β 3 –β 8 . Figure 4 summarizes the results. The ﬁgure provides a breakdownof the participants in each treatment, distinguishing between those with statisticallysigniﬁcant convex and concave responses to the enforcement rate. The ﬁgure also
shows the share of participants whose response deviates from linearity in a statisti-

15

.8

Pool

Split

Medium.2

Temptation

Violation rates.4.6Strong

0

Weak

Weak

MediumTemptation

Strong-.3(a) Violation rates.

-.2

-.1
0
.1
Effect of Splitting on violation.2.3(b) Marginal effects.

Figure 3: Effects of splitting enforcement on violation rates.

cally signiﬁcant way.9 The ﬁgure clearly shows that the share of participants witha concave experimental violation curve increases as temptation increases from weak
to medium to strong. The comparison between the distributions across the ﬁve categories is highly signiﬁcant (χ2 (8) = 59.392, p < .001), as is the comparison acrossthree categories ignoring signiﬁcance (χ2 (4) = 34.728, p < .001).

9 Under weak temptation, there are seven participants (14%) for which the regression is not es-

timable because they never violated under any sanction probability. These participants appear under
the linear category.

16

Temptation

Strong20%

10%Medium

10%

40%

Weak

60%

4%

36%

0%

30%

26%

30%

20%

40%

Convex (sig)Concave (sig)

14%

60%

12%

8%

80%

100%

Convex (ns)Concave (ns)linearFigure 4: Distribution of curvatures across participants.

Temptation

1

2

3

0%

12%

20%

24%

10%

58%

24%

32%

30%

34%

20%

40%

22%

14%

60%

Negative (sig)
Positive (sig)

Negative (ns)Positive (ns)

14%

80%

6%

100%
Zero

Figure 5: Distribution of beneﬁt from splitting across participants.

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Appendix
Proof of Lemma 1
The proof consists of three steps.
1. any probability in which violation curve is concave can be proﬁtably convexiﬁable. this means that open interval from 0 to mu is proﬁtably convexiﬁable.
2. if p is proﬁtably convexiﬁable by a split to pL pH, then moving pL to left and
pH to right both increases the range of proﬁtably convexiﬁable sanction probabilites,
and increases social beneﬁt from convexiﬁcation.
3. pL is optimally set as small as possible at 0, and pH is set as large as possible at point in which the slope of the line that connects violation curve at PL withviolation curve at pH is minimized. Setting it larger implies connecting line intersects
violation curve, which implies it is not proﬁtably convexiﬁable for all points after theintersection.

Proof of Lemma 2
TBA

Proof of Proposition 1
The proof consists of two steps. 1. increasing R or L (or both) increases mu, whichshifts violation curve to the left in a parallel way. Prop 1 implies that optimal splitting
before change is between 0 and p*. A parallel shift to left of violation curve allows toconnect point p* + change in mu to 0 so that connecting line lies below violation curveon the open interval (0, p*+ change).
2. increasing R or L (or both) decreases sigma. The ﬁrst order condition of the
optimization problem yields
π ( p ∗ ) = π (0) + π ′ ( p ∗ ) p ∗ .
19(3)

The solution of Equation (3) describes the unique probability of a sanction p∗ that has
the property that the tangent of the violation curve π ( p) at p∗ is such that: (i) the tangent lies below π ( p) on the interval (0, p∗ ); and (ii) the tangent intersects the violation
curve at the point (0, π (0)). It follows that all the probabilities p < p∗ are proﬁtably
convexiﬁable, and no probability p > p∗ is proﬁtably convexiﬁable. because smaller
sigma implies distribution is more concentrated around mu, change increases pi(0)and decreases pi(p*). And, in order to decrease pi’, necessary to decrease p*. so in order to maintain equation, p* must decrease. (to show this, need to show that p pi’(p)is decreasing in p above mu ). so, as long as decrease in sigma, which decreases p*,
is small relative to increase in mu, which increases p*, increased temptation implies alarger p*.

Proof of Proposition 2
The proof is similar to the proof of Proposition 1.

20