# A game theoretic approach to lowering incentives to violate speed limits in
Finland

Mika Sutela University of Eastern Finland: Law School. Contact by email:
<EMAIL_ADDRESS>Nino Lindström University of Helsinki: Institute of
Criminology and Legal Policy / Helsinki Graduate School of Economics. Contact
by email<EMAIL_ADDRESS>
###### Abstract

We expand on earlier research on the topic by discussing an infinitely-
repeated-game model with a subgame perfect equilibrium strategy profile (SPE)
as a solution concept that diminishes incentives to violate speed limits in a
_carrot-and-stick_ fashion. In attempts to construct an SPE strategy profile,
the initial state is chosen such that the drivers are playing a mixed strategy
whereas the police is not enforcing with certainty. We also postulate a
“short-period” version of the repeated game with generalized stage-game
payoffs. For this game, we construct a multi-stage strategy profile that is a
Nash equilibrium but not an SPE. Some solution candidates are excluded by
showing that they do not satisfy a one-shot-deviation property that is a
necessary condition for an SPE profile in a repeated game of perfect
information.

INTRODUCTION

As already Ross (1960) 57 wrote, traffic law violations are a costly and
widespread form of criminal activity, illegal acts which are not stigmatized
as criminal by the public. In public opinion, it is not considered ”real”
criminality, and it has generally not been treated as a subject for
criminological study. (Ross, 1960) 57. It is not obvious why many injurious
acts are regarded as crimes when others are not. Some illegal acts, such as
many traffic violations, can be victimless (see Dnes, 2000: 70.)16. Traffic
offences are crimes _mala quia prohibita_ , and therefore, are not in
themselves morally objectionable like traditional crimes. Every road user is a
potential criminal (Tolvanen, 1999: 61.) 70. At least in England, ”anti-social
traffic behaviour” has been used as a target of policing (see Corbett, 2008:
135; 11 Huotari, 2020: 18, 130; 32). Also speeding has been socially
constructed as almost a ”non-crime” or as not ”real” crime (Corbett, 2000)10.
The speeding behaviour is pervasive, and arguably socially acceptable (Fleiter
$\&$ Watson, 2005). Speeding is a socio-cultural problem and cannot be located
solely with the individual driver (see Corbett et al., 1998;13). Unfortunately
speeding has not been viewed as an anti-social activity as many drivers do not
perceive speeding as having negative consequences or being a serious offence
(Webster $\&$ Wells, 2000.)76. Road traffic is a rich phenomena which can be
analysed from many disciplines point of view. It includes e.g. educational,
legal, medical, psychological, sociological and technical aspects. Driver’s
behaviour are influenced by several factors, such as traffic rules, different
costs (the costs of breaking them, the costs of safety measures decided by the
motorist himself, as well as the costs of crashes) as well as external safety
factors (Tammi, 2013)68. Heuristics have an important role in the field of
behavioural economics. They influence people’s decision-making in everyday
situations. One heuristic, relevant considering traffic environment, is the
availability heuristic. It describes the fact that people assess probabilities
according to the mental availability of the corresponding events. For example,
if severe traffic offences and fatal road traffic crashes are reported in
media more frequently, people become familiar with the topic. Some dramatic
event will temporarily alter your feelings about the traffic safety. Fatal
crashes are able to bring that topic faster and easier to mind, and therefore,
it is estimated to be more likely to occur. The availability heuristic also
considers that the assessment of the probabilities of threats (crash or
sanction for a traffic crime) depends on individual experiences and
observations. Persons who have already got a sanction or involved in a crash,
estimate a higher likelihood of becoming a victim of crash or getting a
sanction than persons who have been spared so far. The reason for that,
according to the availability heuristic, is the ease with which the memories
about the crash or crime are mentally available. In contrast, persons who have
not been involved in crashes or captured for a traffic crime in the past, only
have an abstract idea of being a victim of a crash or a suspect of criminal
act in traffic. Personal experiences, pictures, and vivid examples are more
available than incidents that happened to others, or mere words, or
statistics. The availability heuristic can lead to probability estimation
errors. It is typical for these errors that the probabilities of the
occurrence of harmful events, such as sanctions for speeding or traffic
crashes, are underestimated (see Tversky $\&$ Kahneman, 1982;71 Kahneman,
2012;34 Schulan, 2019;60).

DRIVERS’ TRAFFIC BEHAVIOUR AND SPEED ENFORCEMENT OF THE POLICE

Traffic environment is social by nature, a special kind of social situation,
in which road users are interacting. Interaction between drivers, who
typically are anonymous, is brief and non-recurring. In road traffic,
individuals are not united by any traditional group identity, and road users
do not know each other by default. Communications are limited in content and
are mediated through mechanical aids, such as lights and horns. Driving is
inherently a coordinative activity among individuals. It is plausible that
trust and norms of trustworthiness would affect the nature of interactions on
the roads. (Ross, 1960;57 Fleiter et al., 2010;22 Tammi, 2013;68 Nagler,
2013;47). Moreover, traffic policing draws police officers into interactions
with citizens who may not otherwise receive the attention of the authorities
(Schafer and Mastrofski 2005)59. The police officer acts as a gatekeeper to
the criminal justice system. It is assumed that the officer will act as a
’common sense filter’ (Wells 2008, p. 813-814.) 77. Since its early days,
traffic enforcement has undermined the relationship between citizens and the
police. When the police response to traffic behaviour of a road user and it is
perceived as unfair, it can weaken the image of the policing in general,
weaken the legitimacy of the police, reduce the citizens’ willingness to
cooperate and thus weaken the police’s chances of succeeding in other tasks
that belong to it, which rely on the help, cooperation and active support of
the citizens (Huotari, 2020: 140.) 32.

Speed limits are one common rule of the road traffic but not a science for all
people or everyday motorists. Why this or that limit? To answer this, you
should probably know the statistics, the history of the speed limits on the
road, the (serious) crashes that have occurred, the pressure from local
residents to lower the limit and the measures taken for one reason or another.
The unpopularity of the theory of speed limits and their practice lead to
difficulties to follow speed limits rationally from time to time (see Hänti
$\&$ Huotari, 2019)29 Driving faster than the speed limit is globally
relatively commonplace (Glendon, 2007;25 Holocher $\&$ Holte, 2019;31).
Speeding is more socially accepted than other traffic violations (see European
Commission, 2021:8)19. Though, according to the a Finnish survey speeding
drivers arouse the most fear among road users (Liikenneturva, 2023). Many
different people are speeders and a majority of drivers admit to speeding at
some times (Webster $\&$ Wells, 2000). In a Finnish study, most of the drivers
reported that they drive at high speed because of a hurry (Mäkinen, 1990)44.
Being in a hurry largely contribute to drivers exceeding the speed limits, on
a general level (Wallén Warner, 2006;74 see also Bogdan et al., 2014;5).

Drivers’ speed is influenced e.g. by road factors, other road users and
enforcement (Haglund, 2001). Driving at excessive or inappropriate speed is a
major threat to safety on the road. It is estimated that up to 15$\%$ of all
crashes and 30$\%$ of all fatal crashes are the direct result of speeding or
inappropriate speed (European Commission, 2021) 19. Road users are aware of
speeding being one main cause of a road crash. Many road users think that
accidents due to speeding apply to others, but not to themselves (Holocher
$\&$ Holte, 2019) 31. Too high driving speed has been associated with about 40
percent of fatal road accidents investigated by road accident investigation
teams in Finland, according to the Crash Data Institute OTI’s accident report
2021 (Sihvola 2022)62.

Criminals are affected by the police, as they avoid committing crimes if the
chances of capture are high (Mukhopadhyay et al., 2016)45. Much police-public
contact happens on roads which are central to modern lives (Corbett $\&$
Grayson, 2010) 12. The police follow traffic rules as part of their work
(Hänti $\&$ Huotari, 2019)29. The police role in traffic has become more
extensive and complex as the number of cars on roads and associated traffic
offences have very strongly increased, and more complex traffic systems in
general (Mäkinen et al., 2003: 53) 43. The primary role of speed enforcement
conducted by the police is deterrence. The police officers, in general, are
more interested in preventing speeding than in catching speeders although some
drivers believe that the police are out there to get them. As the police
develop ever more sophisticated means of apprehending traffic violators, the
motoring public develops ever more sophisticated means of countering these
efforts. This engages the police and the motoring public in a cat-and-mouse
game. If the majority of drivers violate the speed limits, they are very hard
to enforce (Shinar, 2017) 61. To keep drivers from speeding, regular police
checks are essential. Drivers really often consider it rather unlikely to get
checked by the police for speeding on a typical journey (Holocher $\&$ Holte,
2019)31. Jørgensen and Pedersen (2022)33 found that drivers who committed
serious speeding offences overestimated the probability of being caught
speeding. Older drivers had less knowledge about the threshold level for
serious speeding but more knowledge about the detection rate than younger
drivers do.

Different traffic violations are not independent of each other but in many
cases are associated with each other. Speeding increases the likelihood of
other traffic violations, especially of those associated with overtaking (see
e.g. Mäkinen et al., 2003) 43. Research shows a high cross-over between minor
and major traffic offending, offending on the roads, and mainstream crime (see
Corbett, 2008)11. According to Crosetta et al. (2021)14, first-time serious
traffic offenders are more likely to have a previous or future initial non-
traffic offence. Thus, first-time serious traffic offences can act as
trigger/strategic offences (Crosetta et al., 2021) 14. Traffic enforcement is
a key way for the police to influence traffic safety and crime in traffic.
During enforcement, the police must actively try to uncover non-traffic crimes
as well. Crime that occurs in traffic is typically hidden crime that becomes
apparent during traffic enforcement (Poliisi, 2022: 14.)52

Legitimacy, as trust towards the police, influences behaviour strongly. People
with higher levels of trust and support for the (law enforcement) authorities
are less likely to engage in behaviour against the system (Tyler, 2006: 33.)
72. In traffic, driver beliefs and attitudes are among the strongest
predictors of risky driving (Fernandes et al., 2010) 20. The lack of police
enforcement has been found to be a significant predictor of aggressive and
risky driving (Stanojević et al., 2018). Favourable attitudes to speeding and
experiences of punishment avoidance are among the factors that significantly
predict the frequency of speeding (Fleiter $\&$ Watson, 2005)21. However,
attitudes and driving behaviour do not always match (Corbett $\&$ Simon, 1999:
82). The attitude and responsibility of each road user contributes to the
entire traffic culture. Selfish and aggressive traffic behaviour means more
dangerous traffic (Tolvanen, 2018: 44) 70. Many young drivers believe that
avoiding detection by law enforcement is a breeze (Hoekstra and Wegman, 2011)
30. Speed enforcement does not influence the general respect for speed limits
(Rintamäki et al., 2022: 73) 56. The effectiveness of enforcement is directly
related to its perceived threat. To maximize enforcement the driver has to see
many police vehicles (that actually seem to control the driving speeds). One
effective method to increase the perceived presence is to use a mix of marked
and unmarked police vehicles, and make sure through communications that the
motoring public is aware of the presence of them (Shinar, 2017) 61. Traffic
enforcement is, to a considerable extent, communication on control and
campaign related to that. The communication serves educational purposes and
recalls police traffic enforcement and the risk of capture (Huotari, 2020:
135.) 32. Visible enforcement influences traffic behaviour and non-visible
traffic enforcement increases the experience of the possibility of capture
everywhere and at all times (Huotari, 2020: 27) 32. Speeds should be
controlled visibly, so that the largest possible number of road users can
notice the control. If the police implementate a massive high visibility
enforcement operation and announce that, drivers’ awareness of the operation
has the potential to decrease aggressive driving behaviour. Communication
together with high visibility get people to drive more slowly and responsibly
to avoid sanctions. (Sonduru Pantangi et al., 2020;64 Bauernschuster $\&$
Rekers, 2022.3) However, invisible and completely surprising control is also
needed to keep the perceived risk of capture as high as possible (Poliisi,
2022: 40) 50. It is possible to influence road users with different attitudes
with wide-ranging and different forms of enforcement (Sisäministeriö, 2021)
63. To influence and to tackle speeding behaviour, police enforcement is one
tool but there is a need for multiple intervention tools (see Hankonen,
2019;28 Hoekstra and Wegman, 2011;30 Sutela $\&$ Aaltonen, 2021;67). It has
been suggested that high levels of enforcement (about three times the normal
level) are needed to obtain effects on drivers’ perceived risk of detection
(Haglund 2001;27 Østvik, 198949).

The effects of speed enforcement on driving behaviour are easily local and
short-term, even if capture and being punished quickly follow breaking the
law. It would be essential to raise the real and perceived risk of capture.
This requires coverage and continuity from the speed enforcement. Generally,
there are no resources for this due to the frequency of speed limit
violations. The risk of capture is perceived as small indeed (Huotari, 2020:
28) 32.

Ryeng (2012) 58 found that the largest speed reducing effects on individual
speed choice were found by either making most other drivers on the road reduce
their speed, or by substantially increasing enforcement. Visible police
presence in traffic is important, as a measure for reducing average speeds, as
well as for the enforcing of minor offences (Vadeby et al., 2018;73 see Bogdan
et al., 2014;5 Stanojevic et al., 2018;66). Traffic enforcement creates an
objective risk of capture. Control experiences (detection of surveillance and
traffic violations) and information on the enforcement in turn create
subjective risk of punishment (Mäkinen, 1990)44. In order to strengthen the
effectiveness of traffic enforcement, the available resources must be
alloctaed more intelligently and the enforcement must be automated (Huotari,
2020: 26-27) 32. Speed enforcement relies increasingly on automatic traffic
enforcement of the police where driving speed are controlled by traffic safety
cameras, the advantage of which compared to traditional, manned speed
enforfcement is cost efficiency (see Rintamäki et al., 2022: 89)56. The idea
that merely seeing the camera poles have an effect on driving behaviour. When
drivers see a camera pole, the fear of capture and also other deterrents guide
them to drive within legal limits (Sutela $\&$ Aaltonen 2021)67.

The police has to prioritise their resources and duties. In helping determine
the allocation of police resources, mapping of crime ”hot spots” has become an
essential tool. It is more useful to consider how police officers are deployed
than just increasing numbers of them (see Ratcliffe, 2016: 105, 139)55. Police
effectiveness is a crucial component of policing (Pryce $\&$ Time, 2023)54.
Traffic enforcement is a traditional, relatively resource-intensive branch of
policing (Huotari, 2020: 35) 32. Because the police organisation has a
multiple goal structure, resource deployment is a complex task, since
effectiveness in one area may mean ineffectiveness in another (Chatterton
1987, 81) 7. In the police organisations, traffic safety work easily appears
as a secondary task, which can generally be moved aside without necessarily
influencing anyone, but the resources used or attached to it are always away
from other (actual or proper) police work (Huotari, 2020: 131) 32. The
resource deployment strategies of the police are often decided in an ad-hoc
manner (Samanta et al., 2022). Police departments decide quite independently
how traffic enforcement is organized in the area and how its resources are
determined and distributed among different functions of the unit (Karnaranta,
2019: 51) 36.

Also traffic enforcement needs prioritisation. Forms of misconduct in traffic
behaviour are numerous and focusing on them all systematically in enforcement
is simply impossible. Relatively few drivers may be influenced, only a few
aspects of driving behaviour may be focused on. There are also reasons other
than resources why the police need to set priorities for enforcement. These
are associated with the capabilities of the police to measure and detect
various violations (Mäkinen et al., 2003: 48.) 44. As a response to crime
citizens strongly support aggressive traffic enforcement practices focusing on
”hot spots”. However, it has been suggested that law enforcement officials
often develop and implement strategies without public input (Chermak,
McGarrell $\&$ Weiss, 2001)8. Communication of enforcement strategies and
themes should be regarded as an important part of the effectiveness of
enforcement. The citizens should be involved in target setting and actively
asked to contribute to the achievement of the traffic safety objectives (see
Mäkinen et al., 2003: 79.)43.

Game theory deals with any situation in which a strategy i.e. is a plan for
acting that responds to the reactions of others is important. To characterize
a game, we must specify three things: players, strategies of each player, and
payoffs to each player for each strategy (Cooter $\&$ Ulen, 2012: 33.)9. In
Bjørnskau and Elvik (1992)17 it is suggested that game theory has been shown
to be _most fruitful_ in analysing police and road user interaction with
respect to traffic law violations. Essentially, this line of argumentation is
based on the idea that the traffic can be seen as a game where the drivers
gain utility from speeding and the police from possibility to allocate
resources to some other policing operations than traffic enforcement. Given
that road traffic is a very rich phenomenon, as already established in this
article, it’s very clear that no mathematical model of ours will be able to
encompass everything that factors into the road user dynamics. Thus, we will
focus our attention to incentives to violate the speed limits and incentives
to prevent such violations with an abstraction of the strategic interaction
between the police and the drivers insipred by earlier research and the more
complicated structure that can be generated from that basis mathematically.
Everything beyond the game theoretic model consists of _externalities_ that
are maily discussed only if they bear any relevance (real-life or theoretical)
to what emerges from that analysis.

A central and famous concept of game theory is _Nash equilibrium_ \- named
after American mathematician John Nash (1928-2015) - for which there exist
many refinements. In this article, we wish to examine whether the game played
by the drivers and the police has such an equilibrium. We will do this mainly
by following and elaborating on Elvik (2015)18. Elvik explored the
relationship between changes in the risk of apprehension for speeding in
Norway and changes in the amount of speeding. Unlike Elvik, we do not attempt
to test the game-theoretic model empirically. The Norwegian basis of our paper
can be justified by the fact that Finland and Norway are very similar
countries, for example in terms of main characteristics of mobility (Pöllänen
et al., 2023)53. Norway is one of the best-performing countries for road
safety. For the improvement for road safety, Finland can learn from Norway by
e.g. lowering speeds and enhancing compliance with speed limits (Pöllänen et
al., 2023)53. As Elvik, we focus too on speeding violations, due to speeding
has one of the highest priority in traffic policing (Mäkinen et al., 2003:
56)44 and because, as stated above, speeding is such a traffic violation
drivers gain utility from.

A SIMPLE SEQUENTIAL GAME MODEL AND NASH EQUILIBRIA

First of all, it’s important to note that a Nash equilibrium is not ”good” or
”bad” by its plain definition. Essentially it just implies that all players’
stragies are best responses to each other and no party has an incentive to
deviate by changing one’s strategy. Every finite game will always have at
least one equilibrium, which was shown by Nash in 1950***This was proven by
Nash in his paper _Equilibrium points in n-person games_ (January 1950) when
working at Princeton University.. This equilibrium may be in _pure_ or in
_mixed_ strategies. Here, ”pure” refers to strategies or actions being played
with a probability $1$ or $0$, whereas ”mixed” refers to strategies or actions
being played with probabilities from the open interval $(0,1)$. A mixed
strategy Nash equilibrium, however, may be a bit abstract and difficult to
interpret in our setting of drivers and traffic enforcement of the police. The
best interpretation seems that drivers should violate and certain _frequency_
and the police should enforce at another frequency.

|  | Drivers
---|---|---
|  | $S$ | $DS$
Police | $E$ | $-10000,-300$ | $-10000,-50$
$DE$ | $-20000,50$ | $0,-50$
Table 1: Normal form game matrix from Elvik (2015) [18].

What may also seem particularly problematic at first is the idea that the
drivers should make choices collectively, as a kind of a rational ”hive mind”.
However, earlier research on the topic provided by Bjørnskau and Elvik
(1992)17 argues that drivers should not be seen as

_passive (parametric) receivers of safety measures_

but rather as

_strategic actors who take advantage of whatever measure available to attain_
their _goals_

(with emphasis on ”their”) regardless of whether it makes the roads safer or
not. In the paper it is further argued that considering the drivers as
parametric receivers is in fact a _fatal mistake_ potentially resulting in
_no-safety effect of potential safety measures_. Watson et al. (2015) 75 has
suggested speeding offenders are not a homogeneous group: important
differences exist between different types of offenders, based on the extent
and severity of their offending behaviour.

Certainly it always makes sense to suggest that every individual driver seeks
to act according to his own interests and that ultimately a driver may be (and
even should be), concerned about the utility of the group to which he belongs.
In all scenarios the driver’s utility with respect to his or her speeding
behaviour is dependent on the enforcement strategy of the police. Rather than
trying to predict the outcome of this strategic interaction from some
equilibria, the key approach here is to examine whether, and under which
circumstances, an individual driver has an _incentive_ to violate speed limits
and if there is a way to _diminish_ such incentives.

Note that simply having incentives to perform certain actions does not imply
that an individual automatically acts according to those incentives. The
individual agent may not even be aware of such incentives in the first place.
Therefore, rather than relying on every driver computing his or her expected
utility every time upon entering the vehicle to find a mathematically optimal
strategy, the focus is ultimately on controlling and restricting the road
users’ potentially utility-maximizing behaviour by an optimal set of rules (or
a ”mechanism”) that has a minimal amount of societally harmful loopholes that
calculative law-breakers might take advantage of.

Traffic enforcement of the police is largerly revealing activity. The more the
police control, the larger number of traffic crimes and violations are
uncovered (Kautto, 2021) 37. Periodic traffic enforcement i.e. intermittently
enforcement throughout a specific time period has been found to be an
effective in reducing average speed and the proportion of drivers exceeding
the speed limit. The existence of such law enforcement could potentially
modify drivers’ behaviour. (Karimpour et al., 2021)35. Elvik (2015)18 points
out that the police adapts to changes in violation periodically. The existence
of such periods implies that a model consisting of a _repeated game_ may be
appropriate. On the other hand, even the notion of the police adapting to
something the drivers are doing implies that the game has to be _sequential_
and not simultaneous. Potential offenders, including traffic offenders, are
rather aware of changes in policing. There is considerable and robust evidence
that crime, also traffic crime, is responsive to police operations, increases
in police manpower and to many varieties of police redeployments. Individuals
respond to the incentives that are the most immediate and salient (see Chalfin
$\&$ McCrary 2017)6. The police would move first and annually (or quarterly or
even monthly) after learning information about the traffic violations of the
past year (or some previous weeks or months). So rather than starting by
postulating a simultanous-move normal form game that is played once, perhaps a
sequential, infinitely***Clearly, ”infinitely” here is just an abstaction
referring to the idea that a game is repeated for a very long time or even
that it’s is only played for a short or medium duration but consists of a
countless number of periods. We will discuss period durations later in this
paper in more detail. and periodically repeated game would model better the
dynamics at hand. We’ll start by modeling an extensive form game where the
police moves first and the drivers second†††What if, hypthetically, we
reversed the setting and let the drivers move first? The result is that we get
a game with a different pure strategy SPE through backward induction. In fact,
in this SPE the police are not enforcing and the drivers are not speeding -
exactly the kind of equilibrium the police would prefer. If the police were to
mix in the off-path node, the drivers would have an incentive to switch their
strategy given that $\mathds{U}_{D}(S)>\mathds{U}_{D}(DS)\Leftrightarrow
b_{P}(E)<1/4$ where $\mathds{U}$ denotes expected utility (ommitting the steps
due to the similarity to calculations on p. 2).Would it, however, make sense
to let the drivers move first when modeling a real-life scenario? In the game-
theoretic model the advantages and disadvantages of speeding stem solely from
the enforcing strategy of the police. Thus, it’s not too far fetched to
postulate that a driver would not be prevented from speeding by the idea that
his criminal behaviour would have a minor influence to the enforcement
strategy of the police next year. In order to directly appeal to his
incentives, we should have an enforcement strategy in place while the person
is making his speeding decision. This is why we choose to stick to the model
where the police is moving first, or at least _not later than the drivers_ ,
every period..

PoliceDrivers$(-2\cdot 10^{4},50)$_Speed_ $(0,-50)$_Don’t Speed_ _Don’t
Enforce_ Drivers$(-10^{4},-300)$_Speed_ $(-10^{4},-50)$_Don’t Speed_ _Enforce_
Figure 1: Extensive form for the sequential game with the police moving first.

Now, this extensive form game is different from the original normal form game
(as seen in Table 1) and it has a Nash equilibrium in _pure strategies_ where
the police are enforcing traffic and the drivers are not speeding on the path
of play. This equilibrium can be found through a process called _backward
induction_. If the driver finds himself in the left decision node, he would
like to play ”Speed”. In the right decision node, he would like to play ”Don’t
Speed”. Proceeding backwards, the police would then like to play ”Enforce” for
a bigger payoff. Therefore, we have found a pure strategy‡‡‡A common style of
denoting player $i$’s pure strategy is $s_{i}$ which is a member of that
player’s set of pure strategies $S_{i}$, i.e. $s_{i}\in S_{i}$. In this case,
given that P is short for Police and D short for Driver(s), the SPE profile
(1) is $(s_{P},s_{D})$, where $s_{P}$ equals Enforce and $s_{D}$ equals (Speed
(in the left node), Don’t speed (in the right node)).  Nash equilibrium of the
game

$\centering(\textrm{Enforce},(\textrm{Speed (in the left node)},\textrm{ Don't
speed (in the right node)})\@add@centering$ (1)

This Nash equilibrium is in fact called a _subgame perfect equilibrium_ (SPE),
which means that this strategy profile corresponds to a Nash equilibrium in
every _subgame_ of the full game. This game has three subgames, the two
starting at the driver’s decision nodes and the full game (a game is always a
subgame of itself). Now that we’ve found the SPE of the game, we know that the
driver will not have an incentive to deviate from not speeding when the police
is controlling and also the police will not have an incentive to switch from
control to not control when the drivers are not speeding.

Even though this is an SPE, it is not an outcome the police would like. In
fact, they would naturally prefer the outcome where ”Don’t Enforce” and ”Don’t
Speed” are played on the path of play, which is the outcome we would have in a
world without speeding. Since this outcome would yield the drivers exactly
what they obtain in the SPE outcome, this outcome is in fact a _Pareto
improvement_ , i.e. nobody’s utility is taken away but at least someone is
better off. Nevertheless, no pure strategy equilibrium where the police plays
”Don’t Enforce” and the drivers play ”Don’t Speed” exists since knowing that
the police are not enforcing the drivers will have an incentive to change
their strategy to speeding on the path of play.

We should examine off-path behaviour that involves _mixed strategies_. A mixed
strategy is simply a probability distribution over the set of pure strategies.
This means that in every node the sum of the probabilites of the possible
actions is always one, i.e. $100\%$. The drivers are rational to mix only if
they are indifferent between their choices of pure strategies or in decision
nodes that are off the path of play. Notice that, as derived in Bjørnskau and
Elvik (1992)17, there exists a mixed strategy equilibrium in the simultaneous-
move game as given in Table 1. In this equilibrium, the drivers are speeding
50 percent of the time and the police is enforcing a bit less than 30 percent
of the time. However, the authors do not find this equilibrium “stable” due to
the enforcer reducing the level of enforcement whenever the number of traffic
law violations is decreasing. This implies there clearly exist some
externalities to the game as given by Table 1 since the police is clearly
expecting the drivers to change their strategy soon in the future - something
that isn’t captured by the simple $2\times 2$ game matrix alone.

To discuss mixing further with respect to the sequential form game, assume
that the drivers are mixing between speeding and not speeding in their left
decision node. The police then has an incentive to switch from enforcing to
not enforcing given that their _expected utility_ , simply the expected value
of their utility, from not enforcing with respect to the drivers’ mixed
strategy is greater than from enforcing. We denote the drivers mixed strategy

$(b_{D}(\textrm{Speed (in the left node)}),b_{D}(\textrm{Don't speed (in the
left node)})),$

where ”b” refers to a _behaviour strategy_ (simply a name for a mixed strategy
in a sequential game). To simplify our notation a little bit, we now denote
the action of speeding in the left node by $S_{L}$ and not speeding in the
left node by $DS_{L}$. We also shorten the notation for other strategies
similarly: enforce will be $E$ and don’t enforce $DE$, with an appropriate
subscript (”$L$” for an action played in the left subame and ”$R$” in the
right) if needed. Then it’s possible to write the aforementioned behaviour
strategy as $(b_{D}(S_{L}),b_{D}(DS_{L}))=(b_{D}(S_{L}),1-b_{D}(S_{L}))$. Let
us denote the function for expected utility with the symbol $\mathds{U}$ with
an appropriate subscript. In the case of the drivers using a mixed strategy in
the left node, the expected utility of the police is

$\displaystyle\mathds{U}_{P}(DE)$ $\displaystyle=-2\cdot
10^{4}b_{D}(S_{L})+0\cdot b_{D}(DS_{L})$ (2) $\displaystyle=-2\cdot
10^{4}b_{D}(S_{L})$ (3) Therefore the police plays ”Enforce” as long as
$\displaystyle\vspace{1em}\mathds{U}_{P}(DE)$
$\displaystyle<\mathds{U}_{P}(E)$ (6) $\displaystyle\Leftrightarrow-2\cdot
10^{4}b_{D}(S_{L})$ $\displaystyle<-10^{4}$ (7)
$\displaystyle\Leftrightarrow\hskip 10.00002ptb_{D}(S_{L})$
$\displaystyle>1/2$ (8)

Then

$\centering\\{(E,((b_{D}(S_{L}),1-b_{D}(S_{L})),DS_{R}):b_{D}(S_{L})\in(1/2,1)\\}\@add@centering$
(9)

is a Nash equilibrium (or a spectrum of equilibria given by varying
$b_{D}(S_{L})$) of the game. However, this is not an SPE since mixing in the
left subgame is not a Nash equilibrium of that subgame.

How would this translate into a real-life setting, if at all? Exceeding the
speed limit may often be a deliberate choice. Speeding can be defined as a
type of driver error, deliberate error i.e. violation (Haglund, 2001: 13.)27.
People drive deliberately at excessive speed out of habit (De Pelsmacker and
Janssens, 2007) 15. Over half (57$\%$) of drivers who responded to the survey
by the Finnish Road Safety Council (Liikenneturva, 2020)41 told that slight
excess of speed limits has become a habit. In 2020, on Finland’s main roads,
about 43 percent of drivers drove over the speed limits in summer period and
54 percent in winter period (Kiiskilä, Mäki $\&$ Saastamoinen 2021) 38. From
the perspective of the police, drivers speeding with a probability of less
than half could be understood as less than half of the large population of
drivers speeding and the rest not speeding. Knowing that the drivers are
speeding with a probability greater than 50 percent, the police would not have
an incentive to deviate from controlling. Knowing that the police are
controlling, the drivers as a group would not have an incentive to speed.

Deterrence is an old idea and has been discussed in academic literature at
least from eighteenth-century. There are three core concepts embedded in
theories of deterrence. According to them, individuals respond to changes in
the certainty, severity, and celerity (or immediacy) of punishment (see
Chalfin $\&$ McCrary, 2017) 6. The deterrence hypothesis suggests that
individuals balance off the costs and benefits of crime in deciding whether to
engage in criminal activity. According to the the deterrence approach,
following Becker (1968), increasing the probability of capture (and
punishment), for example by increasing the number of police officers, should
deter criminal behaviour (Dnes, 2000: 72–73.16). Control experiences and
information on the enforcement should create subjective risk of capture.
Influencing and increasing drivers´ risk perception i.e. subjective risk of
capture is one of the most significant factors to prevent traffic violations,
such as speeding. However, it has quite a marginal effect to drivers who
repeatedly are committing traffic violations (e.g. Mäkinen 1990 44; Tolvanen
1999;70). Police enforce road rules through strategies that aim to create a
credible risk of punishment for wrong-doing. However, more recent theorising
has proposed that informal deterrence, instead of formal one, is most likely
to influence behaviour. Researchers have claimed that those who have
significant relations and social bonds to would-be offenders aremore likely to
have success in deterring their non-compliance because they are more likely to
elicit internal feelings of shame over wrong-doing among the recipient. For
example, parents have to be found to deter young drivers non-compliance more
likely than police (Allen et al., 2017;2 Murphy $\&$ Helmer, 2013;46 Grasmick
and Bursik, 1990;26).

However, if $b_{D}(S_{L})\leq 1/2$, i.e. the drivers are speeding with a
probability no more than 50 percent, the police would have an incentive to
deviate to not control. Notice that (7) tells us that the only way to alter
the treshold of $b_{D}(S_{L})$ this NE is to alter the ratio

$\frac{u_{P}(E,DS_{R})}{u_{P}(DE,S_{L})}=-2\cdot 10^{4}/10^{4}$

which is independent of the gains and penalties of the drivers. This could
mean, for example, that the police are not going to change their enforcement
strategy before the drivers are speeding with the probability of less than 80
percent or 30 percent or 15 percent… all depending on the ratio of the utility
of controlling while the drivers are not speeding and not enforcing while the
drivers are speeding.

THE PHENOMENON OF ALTERNATING STRATEGIES

According to Elvik (2015)18 the violations will start to drop when the police
are controlling traffic and rise when it’s not controlling. We discuss and try
to model this ”oscillating” behaviour mathematically by using our simple
sequential game model as a stage game for an indefinitely (not necessarily
infinitely) repeated game. We also try capture and include the behaviour of
the drivers as a group, which is ultimately to which the police responds with
a choice of strategy. In a repeated game this aggregate behaviour could be
modelled with an appropriately decreasing or increasing sequence
$\\{b_{D}^{t}(S_{L}\vline E)\\}_{t\in\mathds{N}}$, where the superscript $t$
is the number of the period. Here, $S_{L}$ and $E$ separated by a vertical
line means _the probability of $S_{L}$ conditional on $E$_. However, we have
to careful here. Since we are addressing speeding in the left subgame, we are
talking about the drivers’ choice of action as a response to the police not
enforcing while on the path of play the police are still enforcing and the
drivers not speeding. Game theoretically this is to think of what the police
think and know that the drivers currently _would do_ if the police _were not_
controlling. Since on the path of play the drivers are not speeding, this
means they are playing ”don’t speed” with probability one (also called a
_degenerate_ mixed strategy). Again, rather than thinking about this strategy
as ”most” drivers not speeding, it’s more appealing to think of it as a pure
strategy of an individual driver. The behaviour strategy in the left node,
however, can be understood as a probability distribution over an individual
driver’s pure strategies and more interestingly, as the statistical
distribution of the actions taken by the population of drivers. The police are
not going to switch from enforcing to not enforcing unless the population’s
response to that is to not speed with the probability of one or close enough
to one. The phenomenon of the violations decreasing during the police enfocing
traffic is therefore consistent with an individual driver varying his or her
behaviour strategy in the left subgame such that $b_{D}^{t}(S_{L})$ is drawn
from a decreasing sequence. Once $b_{D}^{t}(S_{L})$ reaches $1/2$, the police
will switch to not enforcing. As the violations will begin to rise once the
police is not enforcing, $\\{b_{D}^{t}(S_{L})\\}_{t\in\mathds{N}}$ will
continue as an increasing sequence $\\{b_{D}^{t}(S_{L}\vline
DE)\\}_{t\in\mathds{N}}$.

If $b_{D}^{t}(S_{L})$ drops to $1/2$§§§This means $b_{D}^{t}(S_{L})\geq 1/2$
and $b_{D}^{t}(S_{L})\downarrow 1/2$ while $t$ is increasing., the drivers are
acting against their best interest because they don’t have a rational
incentive to change their Nash equlibrium strategy in our stage game. This
means the drivers are acting irrationally as a group in our game theoretic
model. On the other hand, they are acting not only with lawful intentions, but
also increasingly many drivers are acting rationally with respect to their own
utility in a _myopic_ sense (for each driver it’s within his or her best
interest to not to speed while the police are enforcing, even if the police
would change their enforcement strategy some time in the future). In this
scenario, the drivers changing their mixed strategy in the left subgame can be
seen as an ”overcorrection” to their behaviour with respect to the current
traffic enforcement. Generally speaking, there is no ”wrong” or ”right” way to
model irrational behaviour, but in our case we can draw the pattern of this
behaviour from what has been observed. This is also consistent information
incompleteness of a real-life setting: the drivers would not know what the
exact limits of the probability (or frequency) interval that keeps the
strategy of the police unchanged. Rather, the drivers are getting a ”feel’ for
the limits. Also, people tend to follow the herd behaviour in traffic. Drivers
tend to influence one another which is nicely reflected in the finding that
fast drivers, as compared with slow drivers, thought a higher number of other
drivers were speeding. It is much easier to see the consequences of an action,
such as speeding, if you watch someone else do it first. Sometimes the
instinct to follow the crowd, can get us into trouble (Orrell, 2021: 130;48
Haglund, 2001: 4227).

Nevertheless, this behaviour gives the police an opportunity to get out of the
SPE where they are enforcing while the drivers are rational not to speed.
Interestingly enough, once $b_{D}^{t}(S_{L})=1/2$, the police would have the
chance to meet the drivers at the unique mixed strategy equilibrium with the
best response of enforcing around 30 percent of the time¶¶¶The exact mixed
strategy equilibrium is
$((\frac{1}{2},\frac{1}{2}),(\frac{2}{7},\frac{5}{7}))$ as derived in
Bjørnskau $\&$ Elvik (1992) [17] .. But because the drivers are speeding
less and less - to which the police are also getting a “feel” by the same
figure of speech as before - the police anticipates that drivers speeding
probability is not going to remain fixed at $50$ percent. Thus, there is no
stable equilibrium at hand, as argued by Bjørnskau $\&$ Elvik (1992)17. Once
the police stops enforcing, the each driver is immediately rational to play
the pure strategy ”Speed” to which the police would again be rational to
respond ”Enforce”. However, even though an individual driver gains the
incentive to start speeding, in the aggregate the drivers can’t be expected to
change their strategy immediately but rather follow the sequence of behaviour
strategies where $b_{D}^{t}(S_{L}\vline DE)$ is increasing.

DriversPolice$(-50,-10^{4})$_Enforce_ $(-50,0)$_Don’t Enf.__Don’t Speed_
Police$(-300,-10^{4})$_Enforce_ $(50,-2\cdot 10^{4})$_Don’t Enf.__Speed_
Figure 2: Extensive form for the sequential game with the drivers moving first
(see f.n. on p. 7)

So far we haven’t discussed too much about information. The reason we have
obtained such an SPE in the game-theoretic model we have developed so far, is
that we have assumed the game to be of _perfect information_. This means that
all players know each others’ stragegies, utilites and past and future play.
They know what each player does (or ”would do”) in each subgame. Notice that
the notion of drivers having incentives and being aware of them relies on this
kind of information assumptions.

How realistic is this assumption with respect to a real-life setting? The
police obtains statistical information about drivers’ violations from previous
periods, which probably is in line with this perfect information assumption
farely well. However, it cannot be completely clear to the police what happens
_in the future_ after they make their choice of strategy. When it comes to the
drivers, this assumption loses its credibility even more. It is not too
unrealistic to postulate that in the aggregate the drivers are not completely
certain of whether the police are enforcing traffic or not. On the other hand,
the idea of an incentive to deviate from not speeding to speeding hinges on
the notion of being aware of the ”opponent’s” current strategy. To simplify
things, however, we choose to take into account only the drivers who are well
aware of the actions of the police and model them as having perfect
information. Better yet, we can equivalently view the situation from the point
of view of an individual driver who has perfect information and plays the game
alone against the law enforcement.

What happens if this stage game is repeated infinitely? Would playing (9) or
(1) infinitely also be an SPE of this game? To answer this question, we first
recall that (9) itself isn’t an SPE profile, which directly rules it out.
Profile (1), however, is an SPE for the game in a repeated form given any
number of periods∥∥∥Suppose the game is played for $T$ periods. If the profile
isn’t an SPE, there would exist some $t\leq T$ such that on $t$ one of the
players has an incentive to deviate from following the profile. But this is
impossible since a NE is played on $t$.. This begs the question: how is this
information _useful_ in any way? Clearly, it would serve as an odd prediction
of the play in most societies. It’s also a very undesirable and unrealistic
outcome for both parties in most scenarios. The actions (or _action profile_)
played on the path of play in (1) (and in (9)), first _Enforce_ and then
_Don’t Speed_ , do not constitute a NE of the plain simultaneous-move stage
game, which mostly takes away the predictive power of the outcome . The
equilibrium only arises once the game becomes sequential and the off-path
actions for both players common knowledge. This shows how abundant information
is absolutely crucial for the existence and relevance of this kind of
equilibria and how an action profile that would not be an equilibrium in
itself can be sustained as an _equilibrium outcome_ given the structure of the
entire game******An archetypical example of this is the _Prisoner’s Dilemma_
where a profile consisting of repeated play of actions otherwise strictly
dominated in the stage-game become a supportable outcome only in the case of
infinite periods. See Mailath (2018)42, chapter 7.. As we move on to repeated-
game-framework, we’ll see how the game being endowed with new structure could
in principle allow us to sustain a wide variety of outcomes in the equilibrium
- some of which may be worthwhile to consider with respect to a real-world
setting.

INFINITELY REPEATED GAME AND ITS FORMALISM

We move on to examine whether the repeated game framework allows us to
construct feasible SPE strategy profiles with respect to our game theoric
setting derived from Elvik (2015) 18. We are especially interested in SPEa
that allow the police to escape enforcing every period that was implied in the
earlier analysis with the sequential form stage game. Also, our focus shifts
from just modeling the phenomena game theoretically to trying to solve optimal
policing strategies in a _carrot-and-stick_ fashion. Doing this, we avoid
going too deep into the general theory of repeated games - for which a great
resource is Fudenberg and Tirole (1991)24. In repeated games, one must
consider an economic factor called the _discount factor_ , commonly denoted by
$\delta$, which represents the agent’s devaluation of the good after one
perior. For example, if $\delta=1$ the agent’s valuation of the good is
unchanged by a period change, and if $\delta=0$, the good has no value at all
to the agent on the following period. In this article we make no assumption
that the discount factor of the drivers should differ from the one of the
police.

Consider constructing an automaton with stage-game utilites given by the
2$\times$2 game matrix as given in Elvik (2015)18. Denoting an action profile
for period $t$ as $a(t)=(a_{D}(t),a_{P}(t))$, the utility for repeating an
infinite sequence $\\{a(t)\\}_{t\in\mathds{N}}$ for a player is

$(1-\delta)\sum_{t\in\mathds{N}}\delta^{t-1}u_{i}(a(t))$ (10)

where $i\in\\{D,P\\}$. Since we are discussing _infinitely_ repeated games,
certain interesting dynamics come into play that set them apart from finitely
repeated games. Clearly, a backwards induction cannot be applied to these
games since there is no final period. Nevertheless, it’s possible to find SPEa
using _automata_. An automaton is a tuple $(\mathcal{W},w^{0},f,\tau)$ such
that $\mathcal{W}$ is the set of _states_ with an initial state $\omega^{0}$,
$f:\mathcal{W}\rightarrow A$ is an output function and $\tau:\mathcal{W}\times
A\rightarrow\mathcal{W}$ is the transition rule. For a given a state
$w\in\mathcal{W}$, we have that $f(w)$ is the set of actions the players are
expected to play in that state given that they’re following the automaton and
$\tau(\omega,a)$ is the state the players are going to transfer to after
whichever action profile $a$ they play.

For any state $\omega\in\mathcal{W}$ in the automaton and for any player
$i\in\\{D,P\\}$ there is an associated value $V^{i}_{\omega}$. Intuitively,
this is player $i$’s ”value for being in the state $\omega$”. These state-wise
values become critical when investigating whether an automaton describes an
SPE of the infinitely repeated game. A necessary condition of an automaton
describing an SPE is that _no player_ has an incentive to deviate (that is,
playing actions other than prescribed by the automaton) from _any_ state of
the automaton. Specifically, a deviation refers to a _one-shot deviation_
where we only assume a deviation from the automaton that takes place on a
single period.

###### Theorem 1

A strategy profile given by an automaton $(\mathcal{W},w^{0},f,\tau)$ is an
SPE if and only if no player has profitable one-shot deviations from any state
$\omega\in\mathcal{W}$.

A proof can be found in Mailath (2018)42, chapter 7. With respect to the
notion of a one-shot deviation, we must define a suitable definition for one
period. With respect to the period duration, we start by following Elvik
(2015)18 where it is stated that the police collects the data from violation
annually and chooses their strategy accordingly.††††††Ultimately, however, it
becomes very obvious that the period duration does not really make a
difference within the model when it comes to optimal strategies, even though
it obviously would in reality due to the gains and losses for both parties
getting higher. This is because the utilities for the players will away be
multiples of the stage-game utilities such that the multipliers will always
cancel out of the inequalities for comparing total utilities. To illustrate,
when the drivers not speeding and the police are enforcing for one year, the
resultant utility the utility vector ($-10000,-50$) multiplied by some
$\alpha\in\mathds{R}_{+}\setminus\\{0\\}$, i.e. $\alpha(-10000,-50)$ such that
$\alpha$ will cancel out. This will be discussed in detail later in the
article.

In our model, the police has no other _endogeneous_ way of influencing the
drivers’ incentives to speed other than to alter the level of traffic
enforcement. This can be viewed as _rewarding_ and _punishing_ the drivers -
that is - the police punishes the drivers due to their past speeding by
enforcing and rewards them by not enforcing. An essential research question
is: if the drivers should and can be punished, how long should the punisment
last and will the police have an incentive to carry out the punishment.

It seems the only sensical way of the police to punish the drivers for
speeding is to use one period, i.e. one year, for enforcement in case the
drivers have been speeding excessively. This means that in the case of too
much traffic violations, the automaton should be such that the drivers are
taken on a _punishment path_ where they are punished for one period (year) and
one period only. We try to construct a two-state automaton describing an SPE
such that the initial state $\omega_{0}\in\mathcal{W}$ is the societally
optimal one with no player having an incentive to deviate from playing
$f(\omega_{0})$ due to threat of being taken on a punishment path. Given the
$2\times 2$-stage game matrix from Elvik (2015)18, it’s tempting to try to
construct an SPE automaton such that the initial state is $\omega_{(NE,S)}$
and the punishment state is $\omega_{(E,DS)}$. Obviously, the punishment state
cannot be $\omega_{(E,S)}$ since - assuming perfect information and that the
police moves first on each period - the drivers will not speed knowing that
the police is controlling. However, in this automaton the drivers are never
rewarded with being able to speed and the stage-game utility with respect to
the punishment state is equal to the initial state (i.e. there is no
”punishment” at all) and the drivers will always have an incentive to play
$S$. It’s then clear that an infinite repetition of the play $(DE,DS)$ cannot
be sustained.

$\hskip 1.99997pt\omega_{\tiny{DE,DS}}\hskip 1.99997pt$start$\hskip
5.0pt\omega_{\scriptsize E,DS}\hskip 5.0pt$(DE,DS)not (DE,DS)(E,DS)not (E,DS)
Figure 3: Automaton i This fails to be an SPE.

Even though this failure of an SPE is quite obvious, we can also show this
formally with calculations and more importantly introduce mathematical
apparatus generally needed for this kind of analysis. For this, consider the
following payoff function (see e.g. Mailath (2018)42):

###### Definition 1

(Payoff function) Given a player $i\in I$ and a transition function
$\tau:\mathcal{W}\times A\rightarrow\mathcal{W}$ we define a _payoff function_
$g_{i}^{\omega}:A\rightarrow\mathds{R}$ as

$g_{i}^{\omega}(a)=(1-\delta)u_{i}(a)+\delta V^{i}_{\tau(\omega,a)}.$

Player $i$’s payoff function $g^{\omega}_{i}$ thus takes an action profile
$a\in A$ and a state $\omega\in\mathcal{W}$ as parameters. It returns the sum
of the utility of the action profile $a$ being played for one round and the
discounted value of the state that the players are being transferred to by the
automaton according to the transfer rule $\tau$ given that the players are
initially in state $\omega$ and an action profile $a$ is played. To then show
that the automaton in Figure 3 fails to be an SPE, we note that in order for
the drivers to have no incentive to deviate from playing $(DE,DS)$, we should
have

$\displaystyle V_{\omega_{(DE,DS)}}^{D}>(1-\delta)u_{D}(DE,S)+\delta
V_{\omega_{(E,DS)}}^{D}$ (11)

where the right-hand-side of the inequality is the utility from deviating to
playing $S$ for one period and then transitioning to the state
$V_{\omega_{E,DS}}$. Here, the value for the inital state is simply
$V_{\omega_{DE,DS}}^{D}=-50$, i.e. the drivers’ stage game utility for
$(DE,DS)$, since following the automaton the action profile $(DE,DS)$ is
repeated infinitely implying (10) becomes $(1-\delta)\cdot(-50)\cdot
1/(1-\delta)=-50$. To calculate $V_{\omega_{(E,DS)}}$, we use the payoff
function and follow the automaton:

$\displaystyle V_{\omega_{(E,DS)}}^{D}$
$\displaystyle=(1-\delta)u_{D}(E,DS)+\delta V_{\omega_{(DE,DS)}}^{D}$ (12)
$\displaystyle=-50\cdot(1-\delta)+(-50)\delta$ (13) $\displaystyle=-50.$ (14)

If inequality (11) held, the drivers would never have an incentive to speed
since being taken on the punishment path - here simply being punished with
$(E,DS)$ for one round (year) - would be too costly. However,

$\displaystyle V_{\omega_{(NE,DS)}}^{D}$
$\displaystyle>(1-\delta)u_{D}(DE,S)+\delta V_{\omega_{(E,DS)}}^{D}$ (15)
$\displaystyle\Leftrightarrow-50$ $\displaystyle>(1-\delta)\cdot
50+\delta\cdot(-50)$ (16) $\displaystyle\Leftrightarrow-50$
$\displaystyle>50-100\delta$ (17) $\displaystyle\Leftrightarrow\delta$
$\displaystyle>1,$ (18)

so this is shown to be impossible due to $\delta\in[0,1]$. We reach the same
(fairly obvious) conclusion as before: given the stage-game utilities given as
in Elvik (2015)18, it’s not possible to take away the drivers’ incentive to
speed with the threat of punishing them with traffic enforcement the next year
for the duration of one year only.

A more reasonable attempt to construct a two-state automaton describing an SPE
is to postulate the initial state as a mixed strategy state where the police
are enforcing with a certain probability (or frequency) and the drivers
speeding with another probability. As noted earlier, there exists a mixed
strategy equilibrium for the stage game where the drives are speeding 50 $\%$
of the time. Then repeating this equilibrium infinitely gives us a very simple
example of the kind of subgame equilibria we’re looking for. Remember,
however, that this equilibrium is viewed as unstable by Bjørnskau $\&$ Elvik
(1992)17. This implies that any stable equilibrium outcome we postulate should
at least have the property that the drivers are speeding clearly less than 50
$\%$ of the time.

Let us denote the mixed-strategy-state (which is also the initial state) by
$\omega_{ms}$ and the probability of the drivers speeding by $b_{D}(S)$ as
before. The punishment state remains $\omega_{(E,DS)}$. We next ask the
following question: if the police was not enforcing _at all_ (i.e.
$b_{P}(E)=0$), meaning speeding becomes as attractive as possible and the
intial state as valuable as possible for the drivers, how large could
$b_{D}(S)$ be in an SPE? For the remainder of this article, we will make the
assumption of $b_{P}(E)=0$ which, if supportable, would be the optimal
equilibrium outcome for the police.

For there to be no incentive for the drivers to deviate from the state
$V^{D}_{\omega_{ms}}$, the following has to hold:

$\displaystyle V^{D}_{\omega_{ms}}$
$\displaystyle>(1-\delta)u_{D}(DE,S)+\delta V^{D}_{\omega_{(E,DS)}}$ (19)
$\displaystyle=(1-\delta)u_{D}(DE,S)+\delta((1-\delta)u_{D}(E,DS)+\delta
V_{\omega_{ms}}^{D})$ (20)
$\displaystyle\Leftrightarrow(1-\delta^{2})V_{\omega_{ms}}^{D}$
$\displaystyle>50(1-\delta)-50\delta(1-\delta)$ (21) $\displaystyle
V_{\omega_{ms}}^{D}$ $\displaystyle>50\frac{1-\delta}{1+\delta}$ (22)

where we have used the payoff function twice to get (20). Since
$V_{\omega_{ms}}^{D}=50b_{D}(S)+(-50)(1-b_{D}(S))=100b_{D}(S)-50$ when the
police is playing pure strategy $DE$ and the drivers are mixing, the criterion
for deviation becomes

$\displaystyle 100b_{D}(S)-50$ $\displaystyle>50\frac{1-\delta}{1+\delta}$
(23) $\displaystyle\Leftrightarrow b_{D}(S)$
$\displaystyle>\frac{1}{1+\delta}$ (24)

This implies $1/2<b_{D}(S)<1$, so we’d have to accept quite a high value for
$b_{D}(S)$ to achieve an SPE strategy profile. However, the police can
quarantee a payoff of at least $-10000$ by just playing $E$ every period, so
at the very least we’d need to have

$\displaystyle\centering V_{\omega_{ms}}^{P}\@add@centering$
$\displaystyle>-10^{4}$ (25) $\displaystyle\Leftrightarrow-2\cdot
10^{4}b_{D}(S)+0\cdot(1-b_{D}(S))$ $\displaystyle>-10^{4}$ (26)
$\displaystyle\Leftrightarrow b_{D}(S)$ $\displaystyle<\frac{1}{2}.$ (27)

But this contracicts (24) and shows it’s not possible to have the automaton in
Figure 4 depicting an SPE in this case.

$\hskip 10.00002pt\omega_{\textrm{ms}\hskip 7.00002pt}$start$\hskip
5.0pt\omega_{E,DS}\hskip 5.0pt$$(DS,(b_{D}(S),1-b_{D}(S)))$$b_{D}(S)\geq
1/2$(E,DS)not (E,DS) Figure 4: Automaton ii. $\omega_{ms}$ is the initial
state. This profile also fails to be an SPE.

This is a meaningful finding in itself. We have discovered that in addition to
the drivers having an incentive to speed, the police has _no incentive_ to
take away the drivers’ incentive to speed by threatning to punishment them
with enforcement for one year. One-shot deviations always exist for both
parties given the stage-game matrix as in Elvik (2015)18.

To try to construct an SPE strategy profile, we could investigate the
possibility of having a more elaborate punishment path - the drivers being
punished for more than one period - but if a period corresponds to one year,
it wouldn’t seem feasible to punish the drivers for multiple years. Given that
the police is able to change their strategy quicker (as discussed earlier), we
are going to suggest the possibility of shortening the period duration. This
would imply assessing the data multiple times a year. Traffic safety work of
the police has both strategic and operational dimensions (Poliisi, 2022:
24)52. The police can use monthly reports on traffic enforcement and/or
traffic offences and crashes in their traffic analysis at operational (or
tactical) level but one year can be more relevant for strategic level. Thus,
the police are able to change their strategy somewhat easily to some extent.
Now, if there are multiple ”checkpoints” a year, the stage-game utilities
should also vary. In addition to decreasing the magnitude off the stage-game
utilities, we - more importantly - also suggest generalizing their
proportions. For this, suppose that in the new stage-game the negative
utilites for the police are replaced with functions of $b_{D}(S)$. After all,
the more speeding takes place, the greater should be the extent of enforcement
and the greater the potential damage done by speeding. We therefore replace
$-10000$ with $-\beta(b_{D}(S))$ and $-20000$ with $\alpha(b_{D}(S))$
($\alpha$ and $\beta$ taking only positive values so that the minus sign exits
to remind us the utilites for the police are non-negative). When the duration
of one period is shortened to one $N$th of of a year
($N\in\mathds{N}\setminus\\{1\\}$), the drivers’ utilites are divided by $N$.
Also, the discount factor should be $N$-dependent such that it it’s an
increasing function of $\delta_{N}$ of $N$.

|  | Drivers
---|---|---
|  | $S$ | $DS$
Police | $E$ | $-\beta(b_{D}(S)),-300/N$ | $-\beta(b_{D}(S)),-50/N$
$NE$ | $-\alpha(b_{D}(S)),50/N$ | $0,-50/N$
Table 2: Stage-game matrix for short-period repeated game

Let $n<N$ be the number of periods the driver is going to be punished for
deviating from $\omega_{ms}$. The punishment stage-game action profile remains
as before, $(E,DS)$. The police then values being in the state $\omega_{ms}$
as

$V^{P}_{\omega_{ms}}=-\alpha(b_{D}(S))b_{D}(S)+0\cdot(1-b_{D}(S))=-\alpha(b_{D}(S))b_{D}(S)$

since the police is playing a pure strategy and the drivers are mixing. For
the drivers, this value is

$V^{D}_{\omega_{ms}}=50\frac{1}{N}\cdot
b_{D}(S)+(-50\frac{1}{N})\cdot(1-b_{D}(S))=\frac{1}{N}(100b_{D}(S)-50).$

Since the new punishment path has $n$ states, $i$th state denoted
$\omega_{(E,DS)_{i}}$, the value for the first state on that path for the
drivers is

$\displaystyle V_{\omega_{(E,DS)_{1}}}^{D}$
$\displaystyle=(1-\delta_{N})((-50\frac{1}{N})+(-50\frac{1}{N})\delta_{N}+(-50\frac{1}{N})\delta_{N}^{2}+$
(28) $\displaystyle\hskip
10.00002pt\cdots+(-50\frac{1}{N})\delta_{N}^{n-1})+\delta_{N}^{n}V_{\omega_{ms}}^{D}$
(29)
$\displaystyle=-(50\frac{1}{N})(1-\delta_{N}^{n})+\delta_{N}^{n}(100\frac{1}{N}b_{D}(S)-50\frac{1}{N})$
(30) $\displaystyle=\frac{1}{N}(100\delta_{N}^{n}b_{D}(S)-50)$ (31)

When it comes to the drivers, we in fact only need this value with respect to
the states on the punishment path to check for one-shot deviations. This is
because the drivers will not have an incentive to deviate from playing $DS$
whenever the police plays $E$. As for the requirement for the drivers to have
no incentive to deviate from $\omega_{ms}$, the following must hold:

$\displaystyle\frac{1}{N}(100b_{D}(S)-50)$
$\displaystyle>(1-\delta_{N})u_{D}(DE,S)+\delta_{N}V_{(E,DS)_{1}}^{D}$ (32)
$\displaystyle=\frac{1}{N}(50(1-\delta_{N})+\delta_{N}(100\delta_{N}^{n}b_{D}(S)-50))$
(33) $\displaystyle=\frac{1}{N}(100\delta_{N}^{n+1}b_{D}(S)-100\delta_{N}+50)$
(34) $\displaystyle\Leftrightarrow(100-100\delta_{N}^{n+1})b_{D}(S)$
$\displaystyle>-100\delta_{N}+100$ (35) $\displaystyle\Leftrightarrow
b_{D}(S)$ $\displaystyle>\frac{1-\delta_{N}}{1-\delta_{N}^{n+1}}.$ (36)

So we observe that $b_{D}(S)$ is dependent on both the discount factor and the
number of periods. We can study***An illustrative way would be to do this
graphically, but we leave it for the reader. this dependence by varying $n$.
Clearly we’d like $n$ to be as small as possible to maximize the feasibility
of the punishment. If $n=1$, (36) becomes $\frac{1}{2}<b_{D}(S)<1$, so in this
case the drivers would have to be speeding at least half of the time, which is
clearly suboptimal. If $n=2$ and $\delta$ is close to $1$, it’s possible to
have $b_{D}(S)$ nearly as small as $0,33$ (this is because
$\lim_{\delta_{N}\rightarrow
1}(1-\delta_{N})/(1-\delta_{N}^{3})=\lim_{\delta_{N}\rightarrow
1}1/(\delta_{N}^{2}+\delta_{N}+1)=\frac{1}{3}$). If $n=3$, the lower limit
becomes as low as $0,20$. The implication here is that the police could allow
the probability of speeding only somewhere near this lower limit and use the
punishment path whenever the drivers speed at a higher probability.

How should $N$ be chosen in the light of these values? If we fix $N=365$, the
punishment lasts for $n$ days. If $N=12$, the punishment lasts for $n$ months.
With respect to our model of perfect information, we could make $N$ enourmosly
large with the following consequences: $n$ becomes miniscule resulting in a
punishment path of extremely short duration (e.g. one nanosecond), the drivers
become extremely patient ($\delta_{N}\approx 1$) and the drivers still have no
incentive to deviate from $\omega_{ms}$. Then again, if the drivers had an
incentive to deviate even for a nanosecond, they would do this continuously,
and nanoseconds would eventually turn into minutes, hours, days, weeks and
months and so on. We therefore make the observation that, when it comes to the
drivers’ incentives, the duration of a period is not relevant (within the
model). However, from the perspective of the police there are at least two
externalities that become limiting factors here, namely the time it takes for
the police to react to the drivers’ deviation and the resources the police
requires in order to carry out the punisment for the deviation.

To investigate the incentives of the police, we first calculate how the police
values being in $\omega_{(E,DS)_{1}}$:

$\displaystyle V_{\omega_{(E,DS)_{1}}}^{P}$
$\displaystyle=(1-\delta_{N})(-\beta(b_{D}(S))+(-\beta(b_{D}(S)))\delta_{N}+(-\beta(b_{D}(S)))\delta_{N}^{2}$
(37) $\displaystyle+$
$\displaystyle\cdots+(-\beta(b_{D}(S)))\delta_{N}^{n-1})+\delta_{N}^{n}V^{P}_{ms}$
(38)
$\displaystyle=-\beta(b_{D}(S))(1-\delta_{N}^{n})+\delta_{N}^{n}V^{P}_{\omega_{ms}}$
(39)
$\displaystyle=-\beta(b_{D}(S))(1-\delta_{N}^{n})+\delta_{N}^{n}(-\alpha(b_{D}(S))b_{D}(S)),$
(40)

This means the police has no incentive to deviate from the initial state
$\omega_{ms}$ given that

$\displaystyle-\alpha(b_{D}(S))b_{D}(S)$
$\displaystyle>(1-\delta_{N})(-\beta(b_{D}(S)))+\delta_{N}V_{\omega_{(E,DS)_{1}}}^{P}$
(41) $\displaystyle=(1-\delta_{N})(-\beta(b_{D}(S))$ (42) $\displaystyle\hskip
20.00003pt+\delta_{N}(-\beta(b_{D}(S))(1-\delta_{N}^{n})$ (43)
$\displaystyle\hskip 20.00003pt+\delta_{N}^{n}(-\alpha(b_{D}(S))b_{D}(S)))$
(44)
$\displaystyle\Leftrightarrow-\alpha(b_{D}(S))b_{D}(S)(1-\delta_{N}^{n+1})$
$\displaystyle>(\delta_{N}(1-\delta^{n})+(1-\delta_{N}))(-\beta(b_{D}(S)))$
(45) $\displaystyle\Leftrightarrow-\alpha(b_{D}(S))b_{D}(S)$
$\displaystyle>\frac{(\delta_{N}(1-\delta_{N}^{n})+(1-\delta_{N}))(-\beta(b_{D}(S)))}{(1-\delta_{N}^{n+1})}$
(46) $\displaystyle\Leftrightarrow b_{D}(S)$
$\displaystyle<\frac{(\delta_{N}(1-\delta_{N}^{n})+(1-\delta_{N}))}{1-\delta_{N}^{n+1}}\frac{\beta(b_{D}(S))}{\alpha(b_{D}(S))}$
(47) $\displaystyle=\frac{\beta(b_{D}(S))}{\alpha(b_{D}(S))}$ (48)

As this condition holds if and only if the probability of speeding $b_{D}(S)$
must be lower than the ratio $\beta/\alpha$ at $b_{D}(S)$, there is an
incentice to bring the cost of enforcement $\beta$ as low as possible with
respect to the cost of damages from not enforcing $\alpha$, quite intuitively.

$\hskip 10.00002pt\omega_{\textrm{ms}}\hskip 10.00002pt$start$\hskip
1.99997pt\omega_{(\textrm{\tiny E,DS})_{1}}\hskip 1.99997pt$$\hskip
1.99997pt\omega_{(\textrm{\tiny E,DS})_{2}}\hskip 1.99997pt$$\hskip
1.99997pt\omega_{(\textrm{\tiny E,DS})_{n}}\hskip
1.99997pt$$(DS,(b_{D}(S),1-b_{D}(S)))$$b_{D}(S)>\frac{1-\delta_{N}}{1-\delta_{N}^{n}}+\epsilon$not
(E,DS)(E,DS)(E,DS)not (E,DS)not (E,DS) Figure 5: Automaton iii. $n$-state
punishment path. This profile is a NE but not an SPE.

Aside from externalities, there exists a bigger problem within the model when
it comes to the incentives of the police. Turns out that even though the
police has no incentive to deviate from $\omega_{ms}$ (all the other stage
game pure strategy outcomes yield a lower utility for the police), it has an
incentive to deviate from _any_ state on the punishment path. Take the first
state on the punishment path for example. The police values being in
$\omega_{(E,DS)_{1}}$ as
$-\beta(1-\delta^{n})+\delta_{N}^{n}(-\alpha(b_{D}(S))b_{D}(S))$ (see
(37)-(40)) which is always non-positive given the definitions of $\alpha$ and
$\beta$. In order for the police to not have an incentive to deviate from
playing $(E,DS)$ when in $\omega_{(E,DS)_{1}}$ would mean that the police
prefers $(DE,DS)$ to be played and being taken back to $\omega_{(E,DS)_{1}}$
by the transfer rule to following the automaton. We thus should have

$\displaystyle V_{\omega_{(E,DS)_{1}}}^{P}$
$\displaystyle>(1-\delta_{N})u_{P}(DE,DS)+\delta_{N}V_{\omega_{(E,DS)_{1}}}^{P}$
(49) $\displaystyle=(1-\delta_{N})\cdot
0+\delta_{N}V_{\omega_{(E,DS)_{1}}}^{P}$ (50)
$\displaystyle\Leftrightarrow(1-\delta_{N})V_{\omega_{(E,DS)_{1}}}^{P}$
$\displaystyle>0$ (51) $\displaystyle\Leftrightarrow
V_{\omega_{(E,DS)_{1}}}^{P}$ $\displaystyle>0,$ (52)

which is not possible given (37)-(40). We have shown that the police has a
profitable deviation from a state of the automaton and therefore it cannot
depict an SPE (we could have used any state on the punishment path instead of
$\omega_{(E,DS)_{1}}$ to similarly show this).

Because neither the police or the drivers have an incentive to deviate from
the inital state but there are states on the punishment path that one of the
players has an incentive to deviate from, the automaton actually depicts a
Nash equilibrium but not a subgame perfect equilibrium. The implication is
that if the drivers act irrationally (assuming they are more prone to doing
so) and deviate from the initial state by speeding with too high a probability
(frequency), the police would need to be funded to have an incentive to carry
out the punishment or they would have to be legally obligated to carry out the
punishment regardless of the expenses. Certain externalities, perhaps
especially the concept of deterrence as discussed earlier, may come to the aid
of the police here. Even if the drivers had an incentive to deviate from ”not
speeding” on the punishment path in a very technical sense, the may be
psychologically discouraged to do so. Better yet, deterrence together with the
game theoretic disincentive should, in general, contribute to keeping the
drivers in the inital state $\omega_{ms}$ in the first place.

Out of interest, we examine the possibility of there existing a third party
that gives financial aid to the police whenever the drivers behave
irrationally and deviate from $V_{\omega_{ms}}$. We denote the smallest sum
that is needed to fund the police with to prevent them from having an
incentive to deviate from any state on the punishment path with $\gamma$.
Notice here that the police is funded with $\gamma$ _only if_ the drivers
deviate from the intial state and there is no other way for the police to get
this funding. It should also be forbidden to use $\gamma$ for anything other
than covering the expenses for traffic enforcement. Then the payoff for the
police from $(E,S)$ becomes $-\beta(b_{D}(S))+\gamma/n$ \- that is, one $n$th
of $\gamma$ is added to the stage game utility.

Given this funding, the police has no incentive to deviate from playing $E$
given that

$(1-\delta_{N})(-\beta_{D}(S)+\gamma/n)(1+\delta_{N}+\delta_{N}^{2}+\cdots+\delta_{N}^{n-m})+\delta^{n-m+1}V_{\omega_{ms}}^{P}\geq
0$ (53)

for all $m\in{1,...,n}$. After rearranging the terms, we can easily see that
this condition is equivalent to

$\displaystyle\gamma/n$ $\displaystyle\geq\sup_{m\leq
n}\biggl{\\{}\Bigl{(}1-\frac{\delta_{N}^{n-m+1}}{1-\delta_{N}^{n-m+1}}\Bigr{)}\beta(b_{D}(S))\biggl{\\}}$
(54)
$\displaystyle=\Bigl{(}1-\frac{\delta_{N}^{n}}{1-\delta_{N}^{n}}\Bigr{)}\beta(b_{D}(S)),$
(55)

and thus at the optimum

$\displaystyle\gamma=\frac{1}{n}\Bigl{(}1-\frac{\delta_{N}^{n}}{1-\delta_{N}^{n}}\Bigr{)}\beta(b_{D}(S)).$
(56)

CONCLUSIONS

Even though subgame perfect strategy profiles always exist in infinitely
repeated games, constructing an SPE profile that regulates the drivers’
incentive to speed in the most desirable way is not a trivial task. We saw
that it is impossible to have a stable solution with a strategy profile having
only one state on its punishment path given that the probability of speeding
is at least 50 percent while the police is controlling. A strategy profile of
an infinitely repeated game with an initial state where the police is also
mixing - i.e. not enforcing some of the time and enforcing rest of the time -
was not discussed in this paper (except the repetition of the stage game mixed
strategy NE), which leaves room for further analysis and possibly discovery of
game theoretically stable solution that is also societally feasible. An
outcome in a “short period repeated game” where the drivers are speeding with
a probability $b_{D}(S)$ bounded by the ratio
$\beta(b_{D}(S))/\alpha(b_{D}(S))$, the functions $\beta$ and $\alpha$
generalizing the stage game payoffs, was shown to be supportable as a Nash
equilibrium but not an SPE. This has the implication that when a player enters
a certain subgame (or subgames), perhaps willingly - yet irrationally - or by
a mistake, the player no longer has an incentive to follow the strategy
profile, which means the player’s behaviour drifts farther from the social
planner’s control. Certain externalities, such as deterrence, would be likely
to help to maintain a stable outcome in a real-life setting.

It is clear that if the drivers are to be punished in some way or sense for
violating the traffic laws, the punishment policy and its extent can’t be just
arbitrarly guessed and implemented. Rather, it should stem from a carefully
premeditated analysis that follows rationally solid argumentation which, of
course, should further be evaluated and refined in the light of empirical
evidence. In this article we have demonstrated how game theoretic modeling and
the solution concept of SPE is one way to find basis for such logical
arguments.

Our infinitely repeated game model was restricted to the case of a driver
having perfect information. Even though it is very unrealistic that a driver
would calculate his or her expected utility whenever entering a vehicle in a
very complex fashion as we have done in this article, the repeated game
framework is able to show that _even if_ the driver did this with perfect
information, she or he could not do any better in an SPE than by following the
equilibrium strategy profile. Any deviation from the driver makes the
situation worse for the driver, which is a very appealing feature of the Nash
equilibrium concept. For further research, one might want to consider
introducing elements of uncertainty to the repeated game. Another good
question is how to properly define the utility generalizing functions $\alpha$
and $\beta$. They could be made state-dependent such that the state is
charaterized by weather, region, time of the year, traffic jams, traffic
culture etc. The same can be said about the discount factor $\delta$ that
needn’t be common among the players. The model could also be made dependent of
available traffic data. For example, the behaviour strategy of the driver(s)
could follow a period-wise stochastic process that fits real-life data. This
could help in predicting the next point of time when the drivers are about to
irrationally exit the inital mixed strategy state in our short period repeated
game.

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