## 1. Defining risk



In non-technical contexts, the word "risk" refers, often
rather vaguely, to situations in which it is possible but not certain
that some undesirable event will occur. In technical contexts, the
word has several more specialized uses and meanings. Five of these are
particularly important since they are widely used across
disciplines:


1. risk = an *unwanted event* that may or may not occur.



An example of this usage is: "Lung cancer is one of the major
risks that affect smokers." 


2. risk = the *cause* of an unwanted event that may or may not
occur.



An example of this usage is: "Smoking is by far the most
important health risk in industrialized countries." (The
unwanted event implicitly referred to here is a disease caused by
smoking.) Both (1) and (2) are qualitative senses of risk. The word
also has quantitative senses, of which the following is the oldest
one:


3. risk = the *probability* of an unwanted event that may or
may not occur.



This usage is exemplified by the following statement: "The risk
that a smoker's life is shortened by a smoking-related disease
is about 50%." 


4. risk = the statistical *expectation value* of an unwanted
event that may or may not occur.



The expectation value of a possible negative event is the product of
its probability and some measure of its severity. It is common to use
the number of killed persons as a measure of the severity of an
accident. With this measure of severity, the "risk" (in
sense 4) associated with a potential accident is equal to the
statistically expected number of deaths. Other measures of severity
give rise to other measures of risk.



Although expectation values have been calculated since the
17th century, the use of the term "risk" in
this sense is relatively new. It was introduced into risk analysis in
the influential Reactor Safety Study, WASH-1400 (Rasmussen et al.,
1975, Rechard 1999). Today it is the standard technical meaning of the
term "risk" in many disciplines. It is regarded by some
risk analysts as the only correct usage of the term.


5. risk = the fact that a decision is made under conditions of
*known probabilities* ("decision under risk" as
opposed to "decision under uncertainty")



In addition to these five common meanings of "risk" there
are several other more technical meanings, which are well-established
in specialized fields of inquiry. Some of the major definitions of
risk that are used in economic analysis will be introduced below in
section 7.1.



Although most of the above-mentioned meanings of "risk"
have been referred to by philosophers, a large part of the
philosophical literature on risk refers to risk in the more informal
sense that was mentioned at the beginning of this section, namely as a
state of affairs in which an undesirable event may or may not occur.
Several philosophers have criticized the technical definitions of risk
for being too limited and not covering all aspects that should be
included in risk assessments (Buchak 2014; Pritchard 2015;
Shrader-Frechette 1991). Linguistic evidence indicates that technical
definitions of risk have had virtually no impact on the non-technical
usage of the word (Boholm et al. 2016).



*Terminological note:* Some philosophers distinguish between
"subjective" and "objective" probabilities.
Others reserve the term "probability" for the subjective
notion. Here, the former terminology is used, i.e.
"probability" can refer either to subjective probability
or to objective chances.


## 2. Epistemology



When there is a risk, there must be something that is unknown or has
an unknown outcome. Therefore, knowledge about risk is knowledge about
lack of knowledge. This combination of knowledge and lack thereof
contributes to making issues of risk complicated from an
epistemological point of view.



In non-regimented usage, "risk" and
"uncertainty" differ along the subjective--objective
dimension. Whereas "uncertainty" seems to belong to the
subjective realm, "risk" has a strong objective component.
If a person does not know whether or not the grass snake is poisonous,
then she is in a state of uncertainty with respect to its ability to
poison her. However, since this species has no poison there is no risk
to be poisoned by it. The relationship between the two concepts
"risk" and "uncertainty" seems to be in part
analogous to that between "truth" and
"belief".



Regimented decision-theoretical usage differs from this. In decision
theory, a decision is said to be made "under risk" if the
relevant probabilities are available and "under
uncertainty" if they are unavailable or only partially
available. Partially determined probabilities are sometimes expressed
with probability intervals, e.g., "the probability of rain
tomorrow is between 0.1 and 0.4". (The term "decision
under ignorance" is sometimes used about the case when no
probabilistic information at all is available.) 



Although this distinction between risk and uncertainty is
decision-theoretically useful, from an epistemological point of view
it is in need of clarification. Only very rarely are probabilities
known with certainty. Strictly speaking, the only clear-cut cases of
"risk" (known probabilities) seem to be idealized textbook
cases that refer to devices such as dice or coins that are supposed to
be known with certainty to be fair. In real-life situations, even if
we act upon a determinate probability estimate, we are not fully
certain that this estimate is exactly correct, hence there is
uncertainty. It follows that almost all decisions are made
"under uncertainty". If a decision problem is treated as a
decision "under risk", then this does not mean that the
decision in question is made under conditions of completely known
probabilities. Rather, it means that a choice has been made to
simplify the description of this decision problem by treating it as a
case of known probabilities. This is often a highly useful
idealization in decision theory. However, in practical applications it
is important to distinguish between those probabilities that can be
treated as known and those that are uncertain and therefore much more
in need of continuous updating. Typical examples of the former are the
failure frequencies of a technical component that are inferred from
extensive and well-documented experience of its use. The latter case
is exemplified by experts' estimates of the expected failure
frequencies of a new type of component.



A major problem in the epistemology of risk is how to deal with the
limitations that characterize our knowledge of the behaviour of unique
complex systems that are essential for estimates of risk, such as the
climate system, ecosystems, the world economy, etc. Each of these
systems contains so many components and potential interactions that
important aspects of it are unpredictable. However, in spite of this
uncertainty, reasonably reliable statements about many aspects of
these systems can be made. Anthropogenic climate change is an example
of this. Although many details are unknown, the general picture is
clear, and there is no reasonable doubt about the existence of
anthropogenic climate change, about its major causes and mechanisms,
or the overall nature of the risks that it creates for our societies
(Mayer et al. 2017; Lewandowsky et al. 2018). The epistemological
status of such partial knowledge about complex systems, and the nature
of the uncertainty involved, are still in need of further
clarification (McKinney 1996, Shrader-Frechette 1997).



In the risk sciences, it is common to distinguish between
"objective risk" and "subjective risk". The
former concept is in principle fairly unproblematic since it refers to
a frequentist interpretation of probability. The latter concept is
more ambiguous. In the psychometric literature on risk from the 1970s,
subjective risk was often conceived as a subjective estimate of
objective risk. In more recent literature, a more complex picture has
emerged. Subjective appraisals of (the severity of) risk depend to a
large extent on factors that are not covered in traditional measures
of objective risk, such as control and tampering with nature. If the
terms are taken in this sense, subjective risk is influenced by the
subjective estimate of objective risk, but cannot be identified with
it. In the psychological literature, subjective risk is often
conceived as the individual's overall assessment of the
seriousness of a danger or alleged danger. Such individual assessments
are commonly called "risk perception", but strictly
speaking the term is misleading. This is not a matter of perception,
but rather a matter of attitudes and expectations. Subjective risk can
be studied with methods of attitude measurement and psychological
scaling (Sjoberg 2004).



The probabilistic approach to risk dominates in philosophy as well as
in other disciplines, but some philosophers have investigated an
alternative, modal account of risk. According to this account,
"[t]o say that a target event is risky is to say that (keeping
relevant initial conditions for that event fixed) it obtains in close
possible worlds" (Pritchard 2016, 562). The risk is smaller, the
more distant the nearest possible world is in which the target event
takes place. This approach has interesting connections with
psychological accounts of risk, but it is far from clear how distance
between possible worlds should be defined and determined.


## 3. Philosophy of science



The role of values in science has been particularly controversial in
issues of risk. Risk assessments have frequently been criticized for
containing "hidden" values that induce a too high
acceptance of risk (Cranor 2016; 2017; Intemann 2016; Heinzerling
2000). There is also a discussion on the need to strengthen the impact
of certain values in risk assessment, such as considerations of
justice (Shrader-Frechette 2005a), human rights (Shrader-Frechette
2005b), and the rights and welfare of future people (Caney 2009; Ng
2005). Issues of risk have also given rise to heated debates on what
levels of scientific evidence are needed for policy decisions. The
proof standards of science are apt to cause difficulties whenever
science is applied to practical problems that require standards of
proof or evidence different from those of science. 



A decision to accept or reject a scientific statement (for instance an
hypothesis) is in practice always subject to the possibility of error.
The chance of such an error is often called the inductive risk (Hempel
1965, 92). There are two major types of errors. The first of these
consists in concluding that there is a phenomenon or an effect when in
fact there is none. This is called an error of type I (false
positive). The second consists in missing an existing phenomenon or
effect. This is called an error of type II (false negative). In the
internal dealings of science, errors of type I are in general regarded
as more problematic than those of type II. The common scientific
standards of statistical significance substantially reduce the risk of
type I errors but do not protect against type II errors
(Shrader-Frechette 2008; John 2017).



Many controversies on risk assessment concern the balance between
risks of type I and type II errors. Whereas science gives higher
priority to avoiding type I errors than to avoiding type II errors,
the balance can shift when errors have practical consequences. This
can be seen from a case in which it is uncertain whether there is a
serious defect in an airplane engine. A type II error, i.e., acting as
if there were no such a defect when there is one, would in this case
be counted as more serious than a type I error, i.e., acting as if
there were such a defect when there is none. (The distinction between
type I and type II errors depends on the delimitation of the effect
under study. In discussions of risk, this delimitation is mostly
uncontroversial. Lemons et al. 1997; van den Belt and Gremmen
2002.)



In this particular case it is fairly uncontroversial that avoidance of
type II error should be given priority over avoidance of type I error.
In other words, it is better to delay the flight and then find out
that the engine was in good shape than to fly with an engine that
turns out to malfunction. However, in other cases the balance between
the two error types is more controversial. Controversies are common,
for instance, over what degree of evidence should be required for
actions against possible negative effects of chemical substances on
human health and the environment.



![figure 1](figure1.png)

Figure 1. The use of scientific data for policy purposes.




Such controversies can be clarified with the help of a simple but
illustrative model of how scientific data influence both scientific
judgments and practical decisions (Hansson 2008). Scientific knowledge
begins with data that originate in experiments and other observations.
(See Figure 1.) Through a process of critical assessment, these data
give rise to the scientific corpus (arrow 1). Roughly speaking, the
corpus consists of those statements that could, for the time being,
legitimately be made without reservation in a (sufficiently detailed)
textbook. When determining whether or not a scientific hypothesis
should be accepted, for the time being, as part of the corpus, the
onus of proof falls on its adherents. Similarly, those who claim the
existence of an as yet unproven phenomenon have the burden of proof.
These proof standards are essential for the integrity of science. 



The most obvious way to use scientific information for policy-making
is to employ information from the corpus (arrow 2). For many purposes,
this is the only sensible thing to do. However, in risk management
decisions, exclusive reliance on the corpus may have unwanted
consequences. Suppose that toxicological investigations are performed
on a substance that has not previously been studied from a
toxicological point of view. These investigations turn out to be
inconclusive. They give rise to science-based suspicions that the
substance is dangerous to human health, but they do not amount to full
scientific proof in the matter. Since the evidence is not sufficient
to warrant an addition to the scientific corpus, this information
cannot influence policies in the standard way (via arrows 1 and 2).
There is a strict requirement to avoid type I errors in the process
represented by arrow 1, and this process filters out information that
might in this case have been practically relevant and justified
certain protective measures.



In cases like this, a direct road from data to policies is often taken
(arrow 3). This means that a balance between type I and type II errors
is determined in the particular case, based on practical
considerations, rather than relying on the standard scientific
procedure with its strong emphasis on the avoidance of type I
errors.



It is essential to distinguish here between two kinds of risk-related
decision processes. One consists in determining which statements about
risks should be included in the scientific corpus. The other consists
in determining how risk-related information should influence practical
measures to protect health and the environment. It would be a strange
coincidence if the criteria of evidence in these two types of
decisions were always the same. Strong reasons can be given for strict
standards of proof in science, i.e. high entry requirements for the
corpus. At the same time, there can be valid policy reasons to allow
risk management decisions to be influenced by scientifically plausible
indications of danger that are not yet sufficiently well-confirmed to
qualify for inclusion into the scientific corpus. 



The term *inductive risk* is usually reserved for the (type I
and type II) risks that follow directly from the acceptance or
rejection of an hypothesis. The term *epistemic risk* is used
for a wider category of risks in belief formation, such as risks taken
when choosing a methodology, accepting a background assumption, or
deciding how to interpret data (Biddle 2016).



Policy issues concerning risk have often been the targets of extensive
disinformation campaigns characterized by science denial and other
forms of pseudoscience (Oreskes 2010). Several philosophers have been
active in the repudiation of invalid claims and the defence of science
in risk-related issues (Cranor 2005; 2016; 2017; Goodwin 2009;
Prothero 2013; Shrader-Frechette 2014; Hansson 2017).


## 4. Philosophy of technology



Safety and the avoidance of risk are major concerns in practical
engineering. Safety engineering has also increasingly become the
subject of academic investigations. However, these discussions are
largely fragmented between different areas of technology. The same
basic ideas or "safety philosophies" are discussed under
different names for instance in chemical, nuclear, and electrical
engineering. Nevertheless, much of the basic thinking seems to be the
same in the different areas of safety engineering (Moller and
Hansson 2008). 



Simple safety principles, often expressible as rules of thumb, have a
central role in safety engineering. Three of the most important of
these are inherent safety, safety factors, and multiple barriers.



Inherent safety, also called primary prevention, consists in the
elimination of a hazard. It is contrasted with secondary prevention
that consists in reducing the risk associated with a hazard. For a
simple example, consider a process in which inflammable materials are
used. Inherent safety would consist in replacing them by
non-inflammable materials. Secondary prevention would consist in
removing or isolating sources of ignition and/or installing
fire-extinguishing equipment. As this example shows, secondary
prevention usually involves added-on safety equipment. The major
reason to prefer inherent safety to secondary prevention is that as
long as the hazard still exists, it can be realized by some
unanticipated triggering event. Even with the best of control
measures, if inflammable materials are present, some unforeseen chain
of events can start a fire. 



Safety factors are numerical factors employed in the design process
for our houses, bridges, vehicles, tools, etc., in order to ensure
that our constructions are stronger than the bare minimum expected to
be required for their functions. Elaborate systems of safety factors
have been specified in norms and standards. A safety factor most
commonly refers to the ratio between a measure of the maximal load not
leading to a specified type of failure and a corresponding measure of
the maximal expected load. It is common to make bridges and other
constructions strong enough to withstand twice or three times the
predicted maximal load. This means that a safety factor of two or
three is employed.



Safety factors are also used in regulatory toxicology and
ecotoxicology. For instance, in food toxicology, the highest dose
allowed for human exposure has traditionally been calculated as one
hundredth of the highest dose (per kilogram body weight) that gave no
observable negative effect in experimental animals. Today, higher
safety factors than 100 have become common (Dourson and Stara 1983;
Pressman et al. 2017).



Safety barriers are often arranged in chains. Ideally, each barrier is
independent of its predecessors so that if the first fails, then the
second is still intact, etc. For example, in an ancient fortress, if
the enemy managed to pass the first wall, then additional layers would
protect the defending forces. Some engineering safety barriers follow
the same principle of concentric physical barriers. Others are
arranged serially in a temporal or functional rather than a spatial
sense. One of the lessons that engineers learned from the
*Titanic* disaster is that improved construction of early
barriers is not of much help if it leads to neglect of the later
barriers (in that case lifeboats).



The major problem in the construction of safety barriers is how to
make them as independent of each other as possible. If two or more
barriers are sensitive to the same type of impact, then one and the
same destructive force can get rid of all of them in one swoop. For
instance, if three safety valves are installed in one and the same
factory hall, each with the probability 1/1,000 of failure, it does
not follow that the probability of all three failing is \(1 \times
10^{-9}\). The three valves may all be destroyed in the same fire, or
damaged by the same mistake in maintenance operations. This is a
common situation for many types of equipment. 



Inherent safety, safety factors, and multiple barriers have an
important common feature: They all aim at protecting us not only
against risks that can be assigned meaningful probability estimates,
but also against dangers that cannot be probabilized, such as the
possibility that some unanticipated type of event gives rise to an
accident. It remains, however, for philosophers of technology to
investigate the principles underlying safety engineering more in
detail and to clarify how they relate to other principles of
engineering design (Doorn and Hansson 2015). 



Many attempts have been made to predict the risks associated with
emerging and future technologies. The role of philosophers in these
endeavours has often been to point out the difficulties and
uncertainties involved in such predictions (Allhoff 2009; Gordijn
2005). Experience shows that even after extensive efforts to make a
new product safe, there is a need for post market surveillance (PMS)
in order to discover unexpected problems. For instance, before the
massive introduction of automobile air bags around 1990, safety
engineers performed laboratory tests of different crash scenarios with
dummies representing a variety of body weights and configurations
(including infants and pregnant women). But in spite of the
adjustments of the construction that these tests gave rise to,
inflated airbags caused a considerable number of (mostly minor)
injuries. By carefully analyzing experiences from actual accidents,
engineers were able to substantially reduce the frequency and severity
of such injuries (Wetmore 2008). For pharmaceutical drugs and some
medical devices, post market surveillance is legally required in many
jurisdictions.


## 5. Ethics


### 5.1 A difficulty for moral theories



Until recently, problems of risk have not been treated systematically
in moral philosophy. A possible defence of this limitation is that
moral philosophy can leave it to decision theory to analyse the
complexities that indeterminism and lack of knowledge give rise to in
real life. According to the conventional division of labour between
the two disciplines, moral philosophy provides assessments of human
behaviour in well-determined situations. Decision theory takes
assessments of these cases for given, adds the available probabilistic
information, and derives assessments for rational behaviour in an
uncertain and indeterministic world. On this view, no additional input
of moral values is needed to deal with indeterminism or lack of
knowledge, since decision theory operates exclusively with criteria of
rationality. 



Examples are easily found that exhibit the problematic nature of this
division between the two disciplines. Compare the act of throwing down
a brick on a person from a high building to the act of throwing down a
brick from a high building without first making sure that there is
nobody beneath who can be hit by the brick. The moral difference
between these two acts is not obviously expressible in a probability
calculus. An ethical analysis of the difference will have to refer to
the moral aspects of risk impositions as compared to intentional
ill-doing. More generally speaking, a reasonably complete account of
the ethics of risk must distinguish between intentional and
unintentional risk exposure and between voluntary risk-taking, risks
imposed on a person who accepts them, and risks imposed on a person
who does not accept them. This cannot be done in a framework that
treats risks as probabilistic mixtures of outcomes. In principle,
these outcomes can be so widely defined that they include all relevant
moral aspects, including rights infringements as well as
intentionality and other pertinent mental states. However, this would
still not cover the moral implications of risk taking *per se*,
since these are not inherent properties of any of the potential
outcomes. 



Methods of moral analysis are needed that can guide decisions on
risk-takings and risk impositions. A first step is to investigate how
standard moral theories can deal with problems of risk that are
presented in the same way as in decision theory, namely as the (moral)
evaluation of probabilistic mixtures of (deterministic) scenarios.


### 5.2 Utilitarianism



In utilitarian theory, there are two obvious approaches to such
problems. One is the *actualist* solution. It consists in
assigning to a (probabilistic) mixture of potential outcomes a utility
that is equal to the utility of the outcome that actually
materializes. To exemplify this approach, consider a decision whether
or not to reinforce a bridge before it is used for a single, very
heavy transport. There is a 50% risk that the bridge will fall down if
it is not reinforced. Suppose that a decision is made not to reinforce
the bridge and that everything goes well; the bridge is not damaged.
According to the actualist approach, the decision was right. This is,
of course, contrary to common moral intuitions.



The other established utilitarian approach is the maximization of
expected utility. This means that the utility of a mixture of
potential outcomes is defined as the probability-weighted average of
the utilities of these outcomes.



The expected utility criterion has been criticized along several
lines. One criticism is that it disallows a common form of
cautiousness, namely disproportionate avoidance of large disasters.
For example, provided that human deaths are valued equally and
additively, as most utilitarians are prone to do, this framework does
not allow that one prefers a probability of 1 in 1000 that one person
will die to a probability of 1 in 100000 that fifty persons will die.
The expected utility framework can also be criticized for disallowing
a common expression of strivings for fairness, namely disproportionate
avoidance of high-probability risks for particular individuals. Hence,
in the choice between exposing one person to a probability of 0.9 to
be killed and exposing each of one hundred persons to a probability of
0.01 of being killed, it requires that the former alternative be
chosen. In summary, expected utility maximization prohibits what seem
to be morally reasonable standpoints on risk-taking and risk
imposition.



However, it should be noted that the expected utility criterion does
not necessarily follow from utilitarianism. Utilitarianism in a wide
sense (Scanlon 1982) is compatible with other ways of evaluating
uncertain outcomes (most notably with actual consequence
utilitarianism, but in principle also for instance with a maximin
criterion). Therefore, criticism directed against expected utility
maximization does not necessarily show a defect in utilitarian
thinking.


### 5.3 Rights-based moral theories



The problem of dealing with risk in rights-based moral theories was
formulated by Robert Nozick: "Imposing how slight a probability
of a harm that violates someone's rights also violates his
rights?" (Nozick 1974, 74).



An extension of a rights-based moral theory to indeterministic cases
can be obtained by prescribing that if *A* has a right that
*B* does not bring about a certain outcome, then *A*
also has a right that *B* does not perform any action that (at
all) increases the probability of that outcome. Unfortunately, such a
strict extension of rights is untenable in social practice.
Presumably, *A* has the right not to be killed by *B*,
but it would not be reasonable to extend this right to all actions by
*B* that give rise to a very small increase in the risk that
*A* dies -- such as driving a car in the town where
*A* lives. Such a strict interpretation would make human
society impossible. 



Hence, a right not to be risk-exposed will have to be defeasible so
that it can be overridden in some (but not necessarily all) cases when
the increase in probability is small. However, it remains to find a
credible criterion for when it should be overridden. As Nozick
observed, a probability limit is not credible in "a tradition
which holds that stealing a penny or a pin or anything from someone
violates his rights. That tradition does *not* select a
threshold measure of harm as a lower limit, in the case of harms
certain to occur" (Nozick 1974, 75).


### 5.4 Deontological moral theories



The problem of dealing with risks in deontological theories is similar
to the corresponding problem in rights-based theories. The duty not to
harm other people can be extended to a duty not to perform actions
that increase their risk of being harmed. However, society as we know
it is not possible without exceptions to this rule. The determination
of criteria for such exceptions is problematic in the same way as for
rights-based theories. All reasonable systems of moral obligations
will contain a general prohibition against actions that kill another
person. Such a prohibition can (and should) be extended to actions
that involve a large risk that a person is killed. However, it cannot
be extended to all actions that lead to a minuscule increase in the
risk that a person is killed, since that would exclude many actions
and behaviours that few of us would be willing to give up. A limit
must be drawn between reasonable and unreasonable impositions of risk.
It seems as if such delimitations will have to appeal to concepts,
such as probabilities and/or the size of the benefits obtained by
taking a risk, that are not part of the internal resources of
deontological theories.


### 5.5 Contract theories



Contract theories may appear somewhat more promising than the theories
discussed above. The criterion that they offer for the deterministic
case, namely consent among all those involved, can also be applied to
risky options. It could be claimed that risk impositions are
acceptable if and only if they are supported by a consensus. Such a
consensus, as conceived in contract theories, is either actual or
hypothetical. 



Actual consensus is unrealistic in a complex society where everyone
performs actions with marginal but additive effects on many
people's lives. According to the criterion of actual consensus,
any local citizen will have a veto against anyone else who wants to
drive a car in the town where she lives. In this way citizens can
block each other, creating a society of stalemates. 



Hypothetical consensus has been developed as a criterion in contract
theory in order to deal with inter-individual problems. We are invited
to consider a hypothetical initial situation in which the social order
of a future society has not yet been decided. When its future citizens
meet to choose a social order, each of them is ignorant of her or his
position in any of the social arrangements which they can choose
among. According to John Rawls's theory of justice, they will
then all opt for a maximin solution, i.e. a social order in which the
worst position that anyone can have in that society is as good as
possible. In arguing for that solution, Rawls relied heavily on the
assumption that none of the participants knows anything at all about
the probability that she will end up in one or other of the positions
in a future social order (Rawls 1957; 1971; 1974). John Harsanyi, who
discussed this problem prior to Rawls, assumed that the probability of
finding oneself in a particular social position is equal to the share
of the population that will have the position in question, and that
this is also known by all participants. Hence, if a fifth of the
population in a certain type of society will be migrant workers, then
each participant in Harsanyi's initial situation will assume
that she has a twenty per cent probability of becoming a migrant
worker, whereas none of the participants in Rawls's initial
situation will have a clue what that probability can be. In
Harsanyi's initial situation, the participants will choose the
social order with the highest expected utility (probability-weighted
utility), thus taking all potential future positions into account,
rather than only the least favourable one (Harsanyi 1953; 1955;
1975).



However, in discussions about various risks in our existing societies
we do not have much use for the hypothetical initial situations of
contract theory. The risks and uncertainties in real life are of quite
a different nature than the hypothetical uncertainty (or ignorance)
about one's own social position and conditions which is a
crucial requirement in the initial situation. The thought experiment
of an initial situation does not seem to provide us with any
intellectual tools for the moral appraisal of risks in addition to
those to which we have access even without trying to think away who we
are.


### 5.6 Summary and outlook



In summary, the problem of appraising risks from a moral point of view
does not seem to have any satisfactory solution in the common versions
of the above-mentioned types of moral theories. The following are
three possible elements of a solution:


1. It may be useful to shift the focus from risks, described
two-dimensionally in terms of probability and severity (or
one-dimensionally as the product of these), to actions of risk-taking
and risk-imposing. Such actions have many morally relevant properties
in addition to the two dimensions mentioned, such as who contributes
causally to the risk and in what ways and with what intentions, and
how the risk and its associated benefits are distributed.
2. Important moral intuitions are accounted for by assuming that each
person has a prima facie moral right not to be exposed to risk of
negative impact, such as damage to her health or her property, through
the actions of others. However, this is a prima facie right that has
to be overridden in quite a few cases, in order to make social life at
all possible. Therefore, the recognition of this right gives rise to
what can be called an *exemption problem*, namely the problem
of determining when it is rightfully overridden.
3. Part of the solution to the exemption problem may be obtained by
allowing for reciprocal exchanges of risks and benefits. Hence, if
*A* is allowed to drive a car, exposing *B* to certain
risks, then in exchange *B* is allowed to drive a car, exposing
*A* to the corresponding risks. In order to deal with the
complexities of modern society, this principle must also be applied to
exchanges of different types of risks and benefits. Exposure of a
person to a risk can then be regarded as acceptable if it is part of
an equitable social system of risk-taking that works to her advantage.
Such a system can be required to contain mechanisms that eliminate, or
compensate for, social inequalities that are caused by disease and
disability. (Hansson 2003; 2013)



Discussions on the overall issue of risk acceptance can be found in
Macpherson (2008), Hansson (2013) and Oberdiek (2014). Justice in risk
impositions is discussed in Ferretti 2010 and Heyward & Roser
2016. Issues of rights and risks are discussed in Thomson 1986 and,
with a particular emphasis on responsibilities, in Kermisch 2012 and
van de Poel, et al. 2012.


## 6. Decision theory



Decision theory is concerned with determining the best way to achieve
as valuable an outcome as possible, given the values that we have. In
decision theory, our values and goals are taken as given, and the
analysis concerns how best to achieve them to an as high degree as
possible. Decision-making under risk and uncertainty is one of the
major topics in decision theory. It is usually assumed that if the
values of a set of potential outcomes are known (for instance from
moral philosophy), then purely instrumental considerations are
sufficient for determining how best to act under risk or uncertainty
in order to achieve the best possible result. (For a critical
discussion of that presumption, see Hansson 2013, 49-51.) The
values taken for given in decision-theoretical analysis can, but need
not, be moral values of the types that are developed and analyzed in
moral philosophy.


### 6.1 Decision weights



Decision theory has traditionally had a predilection for
consequentialism, whose structure is suitable for most formal models of
decision-making. The standard decision-theoretical approach to risk is
maximization of expected utility, which can be seen as a smooth
extension of (act) utilitarianism. In expected utility theory, the value associated with an uncertain situation is equal to the sum of the probability-weighted values of all its possible outcomes. Let \(p\) be a function that assigns probabilities to outcomes, and \(u\) a function that assigns values to them. Then the value associated with a situation with three possible outcomes \(x\_1\), \(x\_2\) and \(x\_3\), is equal to



\[p(x\_1) \cdot u(x\_1) + p(x\_2) \cdot u(x\_2)+ p(x\_3) \cdot u(x\_3).\]


However, influential proposals have been made for alternative decision
rules. There are two major types of justification for such endeavours.
First, examples have been put forward in which it would seem
implausible to claim that expected utility maximization is the only
normatively reasonable decision rule (Allais 1953; Ellsberg 1961).
Secondly, numerous psychological experiments have shown that human
decision-makers tend to deviate substantially from expected utility
maximization. The first type of justification puts the normative
soundness of expected utility in question, where the second exposes
its shortcomings as a descriptive model. 



In an important class of alternative decision rules, the probabilities
used in expected utility calculations are replaced by some other
numbers ("decision weights"). This approach was proposed
by William Fellner (1961). In most of these constructions, all
probabilities are transformed by some transformation function
*r*. Instead maximizing the standard expected utility

\[p(x) \cdot u(x)\]
 the agent will then maximize 
\[r(p(x)) \cdot
u(x)\]




Several decision rules with this structure have been proposed. One of
the earliest was Handa (1977). Currently, the best known proposal in
this tradition is prospect theory (Kahneman and Tversky 1979; Tversky
and Kahneman 1986), which was developed in order to describe
observations from psychological decision experiments more accurately
than in expected utility theory. Prospect theory a is a fairly complex
theory that also deviates in other ways from expected utility theory.
The traditional focus on outcomes is replaced by a focus on losses and
gains, which are treated asymmetrically. (See also Section 7.3.)



A problem with the function \(r\), as defined above, is that the
transformed probabilities which it gives rise to will not add up to 1
except in the trivial case when *r* is the identity function
(Fishburn 1978). To solve this problem, Quiggin (1982) introduced the
rule of maximizing *anticipated utility* (also called utility
with rank dependent probabilities). Instead of replacing \(p(x)\) by a
function of the individual probability, \(r(p(x))\), he replaced it by
a function that also depends on the other probabilities and utilities
involved in the problem. The outcomes are first ordered from worst to
best, which results in a vector \(\langle x\_1, x\_2 ,\ldots ,
x\_n\rangle\) of outcomes, such that \(u(x\_1) \leq u(x\_2) \leq \ldots
\leq u(x\_n)\). A decision weight can then be assigned to each outcome,
taking into account both its probability and its position in the
ranked sequence of outcomes. Since the decision-weight can be
different for outcomes with the same probability, Fishburn's
trivialization result does not apply here. There is evidence
indicating that rank-dependent utility models may be more empirically
adequate than prospect theory (Harrison and Ross 2017).



Other models have been proposed that replace the probabilities in
expected utility maximization by some other type of decision weight
(Gilboa and Schmeidler 1994; Buchak 2014).


### 6.2 Pessimism, cautiousness and the precautionary principle



In decision-theoretical analysis of risk, it is important to
distinguish between two pairs of concepts that are both related to
risk: Pessimism vs. optimism and cautiousness vs. risk-taking.



Pessimism and optimism concerns a person's patterns of
value-related beliefs about future events. A person is pessimistic
about some future development to the extent that she assigns a higher
probability to undesired outcomes, and a lower probability to desired
ones, than the most plausible assignment. She is optimistic to the
extent that she assigns higher probabilities to desired outcomes, and
lower probabilities to undesired ones, than the most plausible
assignment.



Cautiousness (risk aversion) and risk-seeking (risk affinity) concern
a person's decisions and actions, rather than her beliefs. When
making a choice between different action alternatives, a cautious
(risk-averse) person tends to choose an alternative with as small
probabilities as possible for the worst outcomes. She does so even if
it prevents her from maximizing the expected utility of her choice. In
contrast, a risk-seeking person looks for options with a high
probability for the best outcomes, and she does this even at the
expense of not maximizing the expected utility of her choice. (See
also section 7.2.)



The *precautionary principle* is often believed to be a general
instruction to be cautious. However, this is not what is usually meant
by the term "the precautionary principle". This is a
principle that is defined in international treaties and in European
legislation. According to these definitions, the precautionary
principle concerns how we should act when it is scientifically
reasonable to suspect a risk to health or the environment, but the
evidence is not strong enough to show conclusively that the risk
exists. The precautionary principle says that in such cases we may,
and often should, take measures against the potential danger. This
definition has been the standard at least since 1992, when the
precautionary principle was included both in the United Nations'
Rio Declaration and in the European Union's Treaty of Rome,
which is now called the Treaty on the Functioning of the European
Union (Pyhala et al., 2010). Its practical application
depends on the interpretation of scientific evidence that indicates
the presence of a danger, but is not strong enough for scientists to
treat the existence of that danger as known. The evaluation of such
evidence raises methodological and epistemological issues that are in
need of further clarification, not least with conceptual tools
developed in the philosophy of science.



There is also a still ongoing philosophical discussion about other
definitions of a precautionary principle. Many of these proposals are
closer to general (decision-theoretical) cautiousness than the more
narrowly defined legal concept (Munthe 2011).


## 7. Risk in economic analysis



Risks have a central role in economic activities. In capitalist market
economies, taking economic risks is an essential part of the role of
the entrepreneur. Decisions on investments and activities on financial
markets can only be understood against the background of the risks
involved. Therefore it is no surprise that modern economic theory,
with its emphasis on mathematical models of economic activities, has
developed several formal models of risk-taking.


### 7.1 Measures of economic risks



Portfolio analysis, which was developed in the 1950s by Harry
Markowitz (1952), James Tobin (1958) and others, was an important step
forward in the economic analysis of risk. These authors employed a
simple statistical measure, namely the standard deviation (or
alternatively the variance, that is the square of the standard
deviation) as a measure of riskiness. Hence, in a comparison between
two investment alternatives, the one whose economic outcome is
calculated to have the largest standard deviation is regarded as the
most risky one. In a comparison between different such alternatives,
each of them can be characterized by two numbers, namely its
expectation value and its standard deviation or riskiness. Investors
typically prefer investments with as high expectation values and as
low riskiness as possible. However, investors differ in the relative
weight that they assign to expectations respectively risk avoidance.
Given these decision weights, an individual's optimal portfolio
can be determined.



Since the late 1960s, alternative measures of risk have been
developed. Perhaps the most influential of these was provided by
Michael Rothschild and Joseph Stiglitz (1970): If we move probability
mass from the centre to the tails of a probability distribution, while
keeping its mean unchanged, then we increase the risk associated with
the distribution. A measure based on this principle (mean preserving
spread) can be constructed that has more attractive mathematical
properties than those of the older standard deviation measure.


### 7.2 Measures of attitudes to risks



We differ in our attitudes to risk. Some of us willingly take risks
which others deem much too large. The notion of certainty-equivalent
outcomes can be used to specify such differences. Consider a risky
outcome *X*. Another outcome *Y* is a
certainty-equivalent outcome for *X* if and only if (1)
*Y* involves no uncertainty and (2) *X* and *Y*
are considered by the agent to be equally good. For instance, let
*X* be a lottery ticket with a 50 per cent chance of winning
something to which you assign 10 utiles (utility units) and a 50 per
cent chance of winning nothing. The expected utility of this ticket is
5 utiles. Now suppose that you are indifferent between receiving 3
utiles for sure and receiving *X*. Then your attitude is risk
averse. The general criterion for risk aversion (risk avoidance,
cautiousness) is that \(CE(X) < EU(X)\). Similarly, you are risk
neutral concerning *X* if \(CE(X) = EU(X)\) and you are risk
affine (risk seeking, risk loving) if \(EU(X) < CE(X)\).



In economics, risk aversion is usually related to money. Let
*X* represent a lottery ticket with a 50 per cent chance of
winning EUR 100, and suppose that you consider this ticket to be
worth EUR 30. We then have \(EU(X) = u(100)/2\), and \(CE(X) =
u(30)\). This means that \(u(30) = u(100)/2\). If this is a consistent
pattern, then the utility of money, as illustrated in a diagram with
amounts *x* of money on the x axis and their utilities \(u(x)\)
on the y axis, will be represented by a concave (hill-like) curve.
Similarly, if a risk affine behaviour will be represented by a convex
(valley-like curve). Provided that the utility function *u* is
twice continuously differentiable, this can be expressed more
precisely in the form of the Arrow-Pratt measure of risk
aversion/affinity, according to which the agent's risk aversion
at any point *x* is equal to \(-u''(x)/u'(x)\). Hence, a person
with the utility function \(u\_1\) is more risk averse at a point
*x* than one with utility function \(u\_2\) if and only if

\[-{u\_1}''(x)/{u\_1}'(x) > -{u\_2}''(x)/{u\_2}'(x)\]
 (Arrow 1965; Pratt 1964). The Arrow-Pratt measure has the
advantage of being invariant under transformations of the utility
function that preserve the preference relation that it represents
(i.e. it is invariant under multiplication of the utility with a
positive constant and addition of an arbitrary constant). On the other
hand, it can be questioned on philosophical grounds whether risk
attitudes can be adequately represented by variations in the utility
of money. It can be argued that cautiousness and the utility of money
are two separate issues and that they should therefore have
independent representations.


### 7.3 Experimental economics



Studies in experimental economics reveal that actual agents often do
not conform with theoretically derived rationality criteria. One of
the most popular descriptive theories that tries to capture actual
behaviour under risk is prospect theory, which was developed by Daniel
Kahneman and Amos Tversky around 1980 (Kahneman & Tversky 1979;
Tversky & Kahneman 1986). It distinguishes between two stages in a
decision process. In the first phase, the *editing* phase,
gains and losses in the different options are identified. They are
defined relative to some neutral reference point that is usually the
current asset position. In the second phase, the *evaluation*
phase, the options are evaluated in a way that resembles expected
utility analysis, but both utilities and probabilities are replaced by
other, similar measures. Utility is replaced by a measure that is
asymmetrical between gains and losses. Objective probabilities are
transformed by a function that gives more weight to probability
differences close to the ends than to those near the centre of the
distribution. Thus it makes a greater difference to decrease the
probability of a negative outcome from 2 to 1 per cent than to
decrease it from 51 to 50 percent. 



Prospect theory can explain some of the ways in which actual behaviour
deviates from standard economic models of rational behaviour under
risk. Hence the overweighting of probability changes close to zero or
unity can be used to explain why people both buy insurance and buy
lottery tickets. However, prospect theory is not plausible as a
normative theory for rational behaviour under risk. Probably,
normative and descriptive theories of risk will have to go in
different directions.


### 7.4 Risk-benefit analysis



Risk-benefit analysis (RBA), also called cost-benefit analysis (CBA)
is a collection of decision-aiding techniques that weigh advantages
against disadvantages in numerical terms. In a typical risk-benefit
analysis, multi-dimensional problems are reduced to a single
dimension. This is achieved by assigning monetary values to all
potential outcomes. Usually, uncertain outcomes are evaluated
according to the expected utility model. This means that the disvalue
of a risk is obtained by multiplying the probability of the undesired
event by a monetary value representing its severity (Sen 2000;
Sunstein 2005).



In order to make overall alternatives comparable in simple numerical
terms, all consequences, including losses of human lives, have to be
assigned monetary values. The discussion about assigning monetary
value to human lives seems to have had its origin in criticism by the
American Air Force against a report from the RAND Corporation in 1950.
The report recommended a military strategy that would result in
mortality rates for crew members that the military considered to be
unacceptably high. This was because loss of crew members was not
included in the calculations. This omission had the same effect as
setting the value of a soldier's life at zero. In response to
the criticism, the RAND Corporation began to include a life value in
their analyses, but they had considerable difficulties in deriving
this value (Banzhof 2014). Their employee Thomas Schelling proposed
that life values should be derived from a marginal analysis of
individuals' own valuation of risks of death (Schelling 1968).
If a person would accept a death risk of 1/10,000 against a
compensation of $400, then that is taken to express a life value of
10,000 \(\times\) $400 or $4,000,000.



Most of the philosophical discussion on risk-benefit analysis has been
focused on the assignment of a monetary value to the loss of a human
life (MacLean 1994; Heinzerling 2000; 2002). It has been claimed that
lives and money are incommensurable, and that the determination and
use of such "live values" express a lack of respect for
human lives. Defenders of risk-benefit analysis have countered that
these values are just technical constructs representing what society
tends to pay (alternative: ought to pay) in order to save a human
life. Risk-benefit analysis can help decision-makers to save as many
lives as possible, given that they have a fixed amount of resources
that they can assign to life-saving policies (Sunstein 2005).



Many of the value assignments used in cost-benefit analysis are based
on estimates or measurements of (hypothetical) willingness to pay.
Such estimates will give more influence to affluent people since they
can pay more than others to have it their way. This can be corrected
with income-based adjustments of the reported willingness to pay.
However, there are considerable problems involved in the performance
and interpretation of willingness-to-pay studies (Grune-Yanoff
2009).



Risk-benefit analysis gives rise to several other philosophical
problems of considerable philosophical interest (Hansson 2007). Due to
its quantitative nature, it tends to leave out problems that are
difficult to quantify, such as risks of cultural impoverishment,
social isolation, and increased tensions between social strata.
Furthermore, due to its aggregative structure, risk-benefit analysis
often leaves out social justice and other distributional aspects,
although these are in fact accessible to quantitative treatment.