ECOLEC-04208; No of Pages 8
Ecological Economics xxx (2012) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Ecological Economics
journal homepage: www.elsevier.com/locate/ecolecon
Valuing life: Experimental evidence using sensitivity to rare events
Olivier Chanel a,⁎, Graciela Chichilnisky b
a
b
Aix Marseille School of Economics (CNRS-GREQAM), 2 Rue de la Charité, F-13236 Marseille cedex 2, France
Departments of Economics and of Statistics, Columbia Consortium for Risk Management, Columbia University, New York 10027, USA
a r t i c l e
i n f o
Article history:
Received 31 March 2011
Received in revised form 24 October 2011
Accepted 5 March 2012
Available online xxxx
Keywords:
Decision under risk
Value of prevented fatality
Expected utility
Experiment
Catastrophic risk
a b s t r a c t
Global environmental phenomena like climate change, major extinction events or flutype pandemics can
have catastrophic consequences. By properly assessing the outcomes involved – especially those concerning
human life – economic theory of choice under uncertainty is expected to help people take the best decision.
However, the widely used expected utility theory values life in terms of the low probability of death someone
would be willing to accept in order to receive extra payment. Common sense and experimental evidence refute this way of valuing life, and here we provide experimental evidence of people's unwillingness to accept a
low probability of death, contrary to expected utility predictions. This work uses new axioms of choice, especially an axiom that allows extreme responses to extreme events, and the choice criterion that they imply.
The implied decision criteria are a combination of expected utility with extreme responses, and seem more
consistent with observations.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Global environmental phenomena like climate change, major extinction events or flu-type pandemics share two characteristics: potential catastrophic consequences and a high degree of uncertainty.
To determine the best decision to take in order to mitigate or avoid
their harmful consequences, decision theorists use the choice under
uncertainty framework, especially the widely-applied expected utility (EU). This then commonly assesses potential outcomes, including
those affecting human life when deaths are involved. In essence, EU
theory values life in terms of the low probability of death that
would be acceptable in return for a given amount of money.
However, Arrow (1966) provided the following illustration of how
people value their lives that puzzles decision economists: Most people would prefer 5 cents to 2 cents, and 2 cents to death. Does this
mean that they would prefer 5 cents and a very low probability of
death, to 2 cents? Kenneth Arrow famously commented that a positive response to this question would seem “outrageous at first
blush”. And yet the answer is ‘Yes’ according to the EU theory that
Von Neumann–Morgenstern, he and others pioneered (Arrow,
1971). For instance, if we take a value of a prevented fatality (VPF)
of $5.5 million (U.S. Environmental Protection Agency, 2004), the corresponding probability of death that would be acceptable for an extra
3 cents is 5.45 10 − 9 (i.e. 0.03/5.5.10 6) according to EU.
Consequently, Arrow's comment is fully relevant, although at first
glance it could be argued that the amounts at stake in his example are
⁎ Corresponding author. Tel.: + 33 4 91 14 07 80; fax: + 33 4 91 90 02 27.
E-mail addresses: olivier.chanel@univ-amu.fr (O. Chanel), chichilnisky1@gmail.com
(G. Chichilnisky).
too small to make sense. But Arrow's famous example can be
reworked within a simple experiment that provides much larger numerical values.
In February 1998, 64 subjects were invited to play a hypothetical
game in which they could choose whether or not to swallow one
pill among 1 billion (10 9) identical ones. Only one pill contained a lethal poison that was sure to kill, all the others being harmless. The
survivors (i.e. those who swallowed one of the 999,999,999 harmless
pills) received $220,000. We easily infer the value these subjects attribute to their own life according to EU predictions. Each of the 33 subjects who answered ‘No’, implicitly valued his/her own life at more
than $ 220 trillion (220,000/10 − 9). This VPF obviously contrasts
with the $1.7–$7 million range usually obtained in the literature.
The same game was played again by the same subjects as well as
new subjects in January 2009, providing similar results as well as motivations for their (possible change in) answers.
This article examines the results of this experiment, and takes another look at Arrow's comment. The theory we present reveals that
this puzzling result can be attributed to the failure of EU theory to
provide an appropriate value for catastrophic events such as death.
It is well known that EU theory has limitations and individuals have
been found to violate its axioms in a variety of settings since the
1950s (historical examples are Allais', 1953, and Ellsberg's, 1961 paradoxes). Chichilnisky (2000) showed that it underestimates our responses to rare events no matter how catastrophic they may be.
This insensitivity has unintended consequences. We argue that this
insensitivity, and the attendant inability to explain responses to
choices where catastrophic outcomes are possible, makes EU theory
less appropriate to properly express rationality in these situations. A
case in point is the experimental paradox presented above when
0921-8009/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolecon.2012.03.004
Please cite this article as: Chanel, O., Chichilnisky, G., Valuing life: Experimental evidence using sensitivity to rare events, Ecol. Econ. (2012),
doi:10.1016/j.ecolecon.2012.03.004
2
O. Chanel, G. Chichilnisky / Ecological Economics xxx (2012) xxx–xxx
In February 1998, the members of a Research Center in Quantitative Economics were asked by internal e-mail (in French): “Imagine
that you are offered the opportunity to play a game in which you
must choose and swallow one pill out of 1 billion (10 9) identical
pills. Only one contains a lethal poison that is sure to kill you, all the
other pills being ineffective. If you survive (i.e. you swallow one of
the 999,999,999 ineffective pills), you receive a tax-free amount of
€152,450. 1 Are you willing to choose one pill and to swallow it?”.
The value subjects attribute to their own life can be assessed using
the classic utility theory of choice under uncertainty. Indeed, statedependent models, simple single period models, life-cycle models
when the change in mortality lasts over an infinitesimally short
time (Johansson, 2003) as well as wage-risk trade-off models for marginal changes in risk (see Rosen, 1988; Viscusi, 1993) rely on the EU
theory and express the VPF as a marginal rate of substitution between
wealth and risk of death. What happens if this approach is crudely applied to the results of the above experiment?
Before answering, it should be pointed out that studies aiming at
valuing life never ask the kind of direct question we use. They generally use either data from market choices that involve an implicit
trade-off between risk and money (labor or housing markets, transportation, self-protection or averting behaviors), or stated preferences elicited in more subtle ways and using unidentified victims. 2
Moreover, stated preferences suffer from limitations, both generally
and in this case: the actual behavior is not observed; due to incorrect
sensitivity to probabilities, smaller changes in risk tend to induce
higher VPF estimates (Beattie et al., 1998); a significant gap exists between willingness to pay and willingness to accept…
Finally, the lack of monetary incentives in this experiment may
puzzle the reader and is briefly justified below. A number of authors
(e.g. Smith, 1976; Harrison, 1994, or Smith and Walker, 1993) emphasize the importance of paying subjects in real cash and providing
appropriate monetary incentives in experiments, based on the principle that monetary incentives are needed to motivate people sufficiently when answering hypothetical questions and that this leads
to better performance. On the contrary, other authors, including social (and economic) psychologists (Loewenstein, 1999; Slovic, 1969;
Tversky and Kahneman, 1992), consider that subjects should be intrinsically motivated enough to answer truthfully in the experiment
and that social or affective incentives may be even better motivators
than monetary incentives.
This is a controversial issue among researchers, regularly raised by
new experiments or meta-analyses. A case in point is Camerer and
Hogarth (1999), who analyzed 74 experiments either known to
them (1953–1998) or published in famous US journals from 1990 to
98. These studies all varied incentives substantially. The authors
found no effect on mean performance in most of the studies (though
variance is usually reduced by higher payment) and noted that “no
replicated study has made rationality violations disappear purely by
raising incentives”. They conclude that apart from cases in which subjects are required to make a major cognitive effort and/or face an incitement to lie, monetary incentives are not mandatory.
Neither of these conditions applies to our experiment, which
moreover has several characteristics suggesting that subjects were intrinsically motivated to answer truthfully: they were volunteer colleagues, with a potential reciprocity concern vis-à-vis the
experimenter; they were told they would be provided with a summary of the experimental results; the topic can be considered entertaining and of intellectual interest; and the experiment was not timeconsuming at all (5 min). We are therefore confident that participants
answered seriously even without monetary incentives, which would
have been difficult to implement in this case.
All that being said, subjects face a choice between compensation
(€152,450) for accepting a change in risk of death (increase of
10 − 9) and a status quo alternative. Subjects who answer ‘Yes’ clearly
consider that €152,450 is enough to compensate for the increase in
death risk, whereas those who answer ‘No’ do not. Due to the
referendum-type elicitation question, the minimum amount at
which subjects would accept the increase in risk is unknown.
Among the 64 responses collected, 33 subjects answered ‘No’ and
31 answered ‘Yes’ (see the second column of Table 1 for details by answer type).
Do some subjects' characteristics explain such behavior? We look
for dependences between the answer given and individual characteristics with contingency chi-square tests (see the second column of
Table 2). No evidence of dependence is found: the p-values are far
from the usual significant levels in use. These results are confirmed
by performing an analysis of variance for main effects and crossed
1
Note that the original wording mentioned FRF 1,000,000. In 1998, the exchange
rate was 1 USD per 5.9 FRF.
2
However, in our experiment, the victim, although identified, is only exposed to an
(infinitesimal) risk change, not to certain death.
valuing human life, since EU theory does not “fit” with the stated behavior of most of the subjects in the experiment.
This paper provides a theoretical framework by considering death
as a ‘catastrophe’, namely a rare event with major consequences.
Using the new axioms of choice introduced in Chichilnisky (2000,
2002), we derive a choice criterion that is more consistent with the
experimental evidence on how people value catastrophic events
such as death. We show that EU theory underestimates rare events
and that this originates from the classic axiom of continuity (Monotone Continuity, defined in Arrow, 1971) which implies that rational
behavior involves insensitivity to rare events with major consequences
like death. We replace the axiom of continuity by an alternative
axiom of sensitivity to rare events, formalizing a theory of choice
under uncertainty where rare but catastrophic events (such as
death) are given a treatment in symmetry with the treatment of frequent events. As a consequence, a probability can be considered low
enough to make the lottery involving death acceptable; it all depends
on what the other outcomes are.
This implies a different way of valuing life, one that seems more in
tune with experimental evidence. First, this new way of valuing life is
in keeping with evidence provided by the experiment reported
below, given that age and family situation appear to affect the way
subjects change their decisions about whether or not to take action
impacting the value of their lives. More generally, it may explain
why in some experiments people appear to give unrealistically high
numerical values to life that are not consistent with the empirical evidence about how they choose occupations, for instance. Second, this
new way of valuing life is in keeping with evidence provided by experimental psychologists, who observe that the brain reacts differently when making a decision involving rare situations inspiring extreme
fear (LeDoux, 1996). Overall, the proposed framework suggests an alternative way to define rational behavior when catastrophic risks are
involved.
The remainder of the paper proceeds as follows. Section 2 presents
the experimental evidence. Section 3 recalls recent contributions in
the literature on modeling risk and catastrophic events, shows how
EU theory fails to appropriately value life and proposes a solution.
The final Section discusses the results and draws conclusions.
2. Experimental Evidence
We present the results of an experiment (referred to below as the
pill experiment) which twice asked a sample of subjects a question
implying a trade-off between the risk of dying and a fixed amount
of money, at an interval of 11 years.
2.1. The 1998 Initial Pill Experiment
Please cite this article as: Chanel, O., Chichilnisky, G., Valuing life: Experimental evidence using sensitivity to rare events, Ecol. Econ. (2012),
doi:10.1016/j.ecolecon.2012.03.004
O. Chanel, G. Chichilnisky / Ecological Economics xxx (2012) xxx–xxx
Table 1
Composition of the samples (in %).
Sample
1998 survey
(n = 64)
2009 survey
(n = 120)
1998 and 2009
survey (n = 57)
Answers
Answers
Answer changed?
No
Yes
No
Yes
No
Yes
(n = 33) (n = 31) (n = 77) (n = 43) (n = 42) (n = 15)
Gender
Male
Female
69.70
30.30
Position
Ph.D. student
42.42
Adm. Staff
9.10
Ass. prof./Jr Res. 24.24
fellow
Professor/Sr Res. 24.24
fellow
Age
20–29 years old
30–39 years old
40–49 years old
Over 50 years old
36.37
30.30
24.24
9.09
Net individual income
b€1500/month
42.42
€1500–€2499/
21.21
month
€2500–€3499/
24.24
month
>€3500/month
12.12
Parenthood
No children
At least one child
a
60.61
39.39
67.74
32.26
63.63
36.37
74.42
25.58
76.19
23.81
33.33
66.66
35.48
22.58
19.35
29.88
15.58
16.88
51.16
6.98
23.26
33.33
9.52
30.96
46.66
33.33
6.66
22.58
29.88
18.60
26.19
13.33
41.94
29.03
16.13
12.90
36.36
24.68
27.27
11.69
53.49
23.25
11.63
11.63
30.96
35.71
26.19
7.14
53.33
26.66
6.66
13.33
54.84
16.13
41.56
18.18
58.14
11.63
40.47
21.43
66.66
41.40
9.68
15.58
13.95
23.81
0.00
19.35
24.68
16.28
14.29
35.19
61.29
38.71
29.87
70.13
37.21
62.79
62.38a
47.62a
26.67a
73.33a
Change in Parenthood between 1998 and 2009.
effects (interactions): no characteristic appears significantly discriminant in explaining the Yes/No answer. It is hence not surprising that
these characteristics fail to explain subject answers when used as explanatory variables in binomial discrete choice models (Probit and
Logit): the percentage of correct predictions obtained does not differ
from what would be obtained by chance!
Hence, according to EU predictions, each of the 33 subjects who answers ‘No’, implicitly values his/her own life at more than € (152,450/
109), that is more that € 152.4 trillion! This is seven times the world's
total GNP at the time of the survey (€20.8 trillion in 1998, World
Bank, 1999), far from the $1.7–$7 million range usually obtained in
the literature (see for instance Miller, 2000; Mrozek and Taylor, 2002;
Viscusi and Aldy, 2003; or U.S. Environmental Protection Agency,
2004). As a consequence, the overall self-assessed value of the members
of this research center amounts to at least €6.87 10 14!
What is going wrong? If we rule out the possibility that subjects
cannot correctly understand low probabilities (all belong to a Quantitative Economics research center and 84% of them have followed (or
given) graduate courses in Statistics), one plausible explanation is
that subjects who answer ‘No’ gave no consideration to what ‘one
Table 2
Contingency chi-square tests (p-values).
Answers in 2009 Changes in answers
Answers in
1998 → 2009 (n = 57)
1998 (n = 64) (n = 120)
Gender
Position
Age
Individual
(EUR)
Parenthood
0.866
0.513
0.832
income 0.344
0.822
0.176
0.150
0.344
0.003
0.047
0.261
0.158
0.955
0.413
0.081
3
chance in 1 billion’ means, but rather focussed on the frightening
event and disregarded the probability corresponding to this event.
We will show in Section 3 that it is because EU is insensitive to rare
events that it cannot handle the catastrophic dimension associated
with the rare event in this pill experiment.
2.2. The 2009 Follow-up Pill Experiment
In January 2009, the same question was again put by e-mail to the
initial 1998 sample as well as to new members of the same Research
Center. 3 Examining the motivations for their answer is crucial, and to
this end, they were questioned on the influence various factors had
on their answer. Subjects then gave a mark on a scale of 0 to 5
(where ‘0’ equates to ‘no influence at all’ and ‘5’ equates to ‘very
strong influence’) to the following changes in factors: marital/familial
status (Family), financial status (Financial), health status (Health), age
(Age), life expectancy (LifeExpec), perception of the probability (PercProba), opinion w.r.t. this type of issue (OpIssue), relation to chance
(Chance), relation to death (Death). An open question at the end
allowed subjects to state other factors (Other) or give open
comments.
Of the 64 initial members, 2 had unfortunately died, it was impossible to find a way of contacting 3 at the time of the study, and 2 did
not answer e-mails. The 2009 sample is thus composed of 57 out of 64
(89%) initial members and 63 new members, i.e. a total of 120 subjects. The answers to the pill question were as follows: 77 subjects answered ‘No’ and 43 answered ‘Yes’ (see third column of Table 1 for
descriptive statistics by answer type). Once again, dependences between the answer given (Yes/No) and individual characteristics
were tested with contingency chi-square tests (see the third column
of Table 2) and, as for the 1998 answers, no evidence of significant dependence was found.
Three interesting questions remain: what motivates 2009 subjects' answers, do subjects give different answers in 1998 and 2009
and why do some subjects answer differently in 1998 and 2009?
We first present in Table 3 sample statistics on the motivations
given by subjects, ranked by decreasing mean mark. Note that 17
out of 120 subjects (14.2%) gave a null mark to all motivations (including Other), thus considering that none of them influenced their
2009 answer. Table 3 shows that among the sample, opinion w.r.t.
this type of issue (OpIssue), perception of the probability (PercProba)
and marital/familial status (Family) seem to have the greatest weight
in explaining the answers.
We then test whether 1998 subjects answer differently from 2009
subjects. The standard test consists in comparing the proportion of
‘Yes’ (or indifferently ‘No’) in the two samples. However, we should
take into account that the samples overlap since 57 subjects belong
to both samples. We then use a test of equal proportion that accounts
for that (in particular for the fact that the variance of the two proportions is the same under the null hypothesis, see Bland and Butland).
The proportion of ‘No’ for the 2009 sample (n = 120) significantly exceeds that for the 1998 sample (n = 64, two-sample proportioncomparison test with “Bland and Butland” p-value = .0246). If we restrict to the overlapping subjects (n = 57), we obtain the same result
(one-sample proportion-comparison t test with “standard” pvalue = .0297). Let us consider now the two sub-samples that answer
the 2009 follow-up experiment: the 57 subjects that previously answered the 1998 experiment and the 63 new subjects. The proportions of ‘No’ for these two sub-samples do not significantly differ
(two-sample proportion-comparison test with “standard” pvalue = .1776).
Finally, let us focus on the 57 subjects that answered both 1998 and
2009 surveys. Moving from the aggregate level to the individual level,
3
The 1998 amount is about €182,000 in 2009 due to inflation (the exchange rate
was 1 USD per 0.77 € in 2009).
Please cite this article as: Chanel, O., Chichilnisky, G., Valuing life: Experimental evidence using sensitivity to rare events, Ecol. Econ. (2012),
doi:10.1016/j.ecolecon.2012.03.004
4
O. Chanel, G. Chichilnisky / Ecological Economics xxx (2012) xxx–xxx
Table 3
Descriptive statistics on marka by motivation (n = 120).
Motivation
Mean
Std.-Dev.
Minimum
Maximum
# non null
OpIssue
PercProba
Family
Death
Financial
Chance
Age
LifeExpec
Health
Otherb
2.15
2.01
1.83
1.75
1.63
1.21
1.09
1.01
.82
4.62
2.14
2.21
2.13
2.07
1.85
1.84
1.70
1.62
1.43
0.74
0
0
0
0
0
0
0
0
0
3
5
5
5
5
5
5
5
5
5
5
68
60
56
58
64
43
43
40
37
17
a
b
Mark on a scale of 0 (no influence at all) to 5 (very strong influence).
17 subjects express another motivation and state it.
we observe that 15 subjects changed their mind between 1998 and
2009: 12 by switching from ‘Yes’ to ‘No’ and 3 from ‘No’ to ‘Yes’. Five
of them gave open comments to explain what motivates their change
“I take much bigger risks in everyday life without such a high reward,
so I have decided to change from ‘No’ to ‘Yes’”, “I now have two children
and do not want to add any additional risk – however tiny it may be –
that may have painful implications for their life” (‘Yes’ to ‘No’), “My position on the consequences my death would have for my relatives has
changed” (‘Yes’ to ‘No’), “I am now married with twins, fully happy
and I want nothing more” (‘Yes’ to ‘No’) and “the 1998 ratio of gain variation over risk variation expressed in French Francs (i.e., 106/10− 9)
was more attractive than the current ratio expressed in euros
(182, 000/10 − 9) w.r.t. the probability perception, even though the
monetary gains are similar in terms of purchasing power” (Yes to No).
Note that in 9 changes out of 15, the subject had had one (or several)
child(ren) since 1998.
Among these 57 subjects, descriptive statistics by change in answer
are shown in the fourth column of Table 1 (with ‘Yes’ for a change and
‘No’ for no change). Dependences between a change in answer and individual characteristics were also tested with contingency chi-square
tests (see last column of Table 2) and significant dependences were
found for gender (p-value= 0.003), position (p-value= 0.047) and
change in parenthood status (p-value= 0.081). We now turn to a conditional analysis of the determinants of a change in answer by estimating a Logit model. Table 4 presents the best model (Huber/White robust
estimator of variance is used) explaining the probability of a change in
answer. The overall quality of the model is good, as shown by the strong
rejection of the joint nullity of the estimated coefficient (p-valueb0.01),
the high Pseudo R2 (0.364) and the percentage of correct predictions
(48 out of 57, i.e. 84.2%).
Considering now the significant variables, we found that being a female (p-value= 0.008), being over 50 years old (p-value= 0.03) and
giving a high mark to the motivation “Family” (p-value= 0.003) and
“PercProba” (p-value= 0.019) significantly increase the probability of
a change. The marginal effect of the explanatory variables on the probability of a change is computed at the sample means and given in the
last column of Table 4, as well as the corresponding p-values. Hence,
Table 4
Estimation of the probability of change (n = 57).
Variable
Estimate
Robust p-value
Marg. effect
Robust p-value
Intercept
Female (= 1)
Older than 50 (= 1)
Family (0–5)
PercProba (0–5)
− 3.954
2.386
2.815
0.524
0.431
.000
.008
.030
0.003
.019
–
.396
.583
.070
.058
–
.007
.018
.022
.026
LRI/Pseudo R2: 0.3640
Wald test of joint nullity (p-value): 15.41 (.0039)
Percentage of correct predictions: 84.2%
ceteris paribus, being a female instead of a male increases the probability of change by 40%, being older than 50 by 58%, and one additional
point in the mark given to Family (resp. PercProba) increases the probability of a change by 7% (resp 5.8%).
Overall, the pill experiments lead to the following results. First, 52%
of the 1998 sample and 64% of the 2009 sample implicitly value their
own life at more than €152.4 trillion according to EU predictions. Second, subjects' characteristics are not significant in explaining 1998 or
2009 answers, but being a female and being over 50 years old increase
the likelihood of a change in answer between 1998 and 2009 in the
sample that answered both surveys. The marital/familial status and
the perception of the probability motivations also explain change in answers. Third, because in 9 changes out of 15 the subject had had one (or
several) child(ren) since 1998, and because a change in parenthood status is significant in explaining changes in answers, familial status could
be seen as the main driver of the answers. However, the perception of
(low) probability is certainly also relevant in explaining answers because the 2009 answers of subjects already surveyed in 1998 and
those of new subjects do not differ significantly.
Section 3 presents recent contributions to the modeling of catastrophic risks and shows how well these experimental results fit the
axiomatic approach introduced by Chichilnisky (2000, 2002).
3. Why EU Theory Fails, and a Solution
A close examination of EU theory appears worthwhile, to identify
the source of its inability to rationalize about half of subjects' choices
among catastrophic and rare outcomes.
3.1. Notations and Expected Utility
Uncertainty is described by a system that is in one of several states,
indexed by the real numbers with a standard Lebesgue measure. In
each state a monotonically increasing continuous utility function u :
R n → R ranks the outcomes, which are described by vectors in R n.
When the probability associated with each state is given, a description of the utility achieved in each state is called a lottery: a function
f : R → R. Choice under uncertainty means the ranking of lotteries.
Bounded utilities are required by Arrow (1971) and many others to
avoid the St. Petersburg Paradox (see Chichilnisky, 2000, 2009). An
event E is a set of states, and E c is the set of states of the world not
in E (i.e. the complement of the set E).
Axioms for choice under uncertainty describe natural and selfevident properties of choice, like ordering, independence and continuity. These classic axioms were developed half a century ago by
von Neumann, Morgenstern, Arrow, Hernstein and Milnor. Continuity
is a standard requirement that captures the notion that nearby stimuli give rise to nearby responses, which is reasonable enough. Arrow
(1971) calls it Monotone Continuity (MC) and Hernstein and Milnor
(1953) call it Axiom 2. 4 However continuity depends on the notion
of ‘closeness’ that is used. A monotone decreasing sequence of events
{E i}i∞= 1, is a sequence for which for all i, E i + 1 ⊂ E i. If there is no state
in the world common to all members of the sequence, ∩ i∞= 1E i = ∅
and {E i} is called a vanishing sequence. For example, in the case of
the real line, the sequence {(n, ∞)}, n = 1, 2, 3 …, is a vanishing sequence of sets.
In Arrow (1971), two lotteries 5 are close to each other when they
have different consequences in small events, which he defines as “An
4
Note that Arrow (1971), p. 257, introduces the axiom of Monotone Continuity attributing it to Villegas (1964), p. 1789. It requires that modifying an action in events
of small probabilities should lead to similar rankings. At the same time Hernstein
and Milnor (1953) require a form of continuity in their Axiom 2 that is similar to Arrow's Monotone Continuity and leads to their Continuity Theorem on p. 293.
5
The equivalent to the notion of “lotteries” in our framework is the notion of “actions” in Arrow (1971).
Please cite this article as: Chanel, O., Chichilnisky, G., Valuing life: Experimental evidence using sensitivity to rare events, Ecol. Econ. (2012),
doi:10.1016/j.ecolecon.2012.03.004
O. Chanel, G. Chichilnisky / Ecological Economics xxx (2012) xxx–xxx
event that is far out on a vanishing sequence is ‘small’ by any reasonable standards” and more formally, as follows:
Axiom of Monotone Continuity (MC) Given a and b, where a ≻ b, a
consequence c, and a vanishing sequence {E i}, suppose the sequences
of actions {a i}, {b i} satisfy the conditions that (a i, s) yields the same
consequences as (a, s) for all s in (E i) c and the consequence c for all
s in E i, while (b i, s) yields the same consequences as (b, s) for all s in
(E i) c and the consequence c for all s in E i. Then for all i sufficiently
large, a i ≻ b and a ≻ b i (Arrow, 1971, p. 48).
In Arrow's framework, two lotteries that differ in sets of small
enough Lebesgue measure are very close to each other.
On the basis of the standard axioms of choice (including MC), a
crucial result established that individuals optimize the ranking of lotteries WEU(f) according to an expected utility function. The expected
utility of a lottery f is a ranking defined by WEU(f) = ∫ x∈R f(x)dμ(x)
where μ is a measure with an integrable density function ϕ(.) that belongs to the space of all measurable and integrable functions on R so
μ(A) = ∫ A ϕ(x)dx, where dx is the standard Lebesgue measure on R.
The ranking WEU(.) is a continuous linear function that is defined by
a countably additive measure μ. 6
3.2. Recent Contributions to the Modeling of Catastrophic Risks
In recent work, a catastrophic risk is described as an event that has
“a very low probability of materializing, but that if it does materialize
will produce a harm so great and sudden as to seem discontinuous
with the flow of events that preceded it” (Posner, 2004, p. 6). This interpretation of catastrophic risks is entirely consistent with ours.
However, Posner (2004) does not model decisions with catastrophic
risks — he refers to EU analysis and points out that this analysis is inadequate to explain the decisions that people make when confronted
with catastrophic risks.
The modeling of catastrophic risks in Weitzman (2009) is based
on EU and assumes that there are “heavy tails” (defined as distributions that have an infinite moment generating function). He seeks
to explain behavioral discrepancies by attributing them to these
unexplained “heavy tails”. It should be noted that, “heavy tails”
being inconsistent with the main axioms of EU, this leads to the
non-existence of a robust solution and unacceptable conclusions,
like using all the current resources to mitigate future catastrophes.
As noted by Buchholz and Schymura (2010), a model assuming both
EU theory and utility function unbounded below leads to catastrophic
events playing a dominant role in the decision-making process. In
contrast, the choice of utility functions that are bounded below
leads to implausibly low degrees of relative risk aversion and catastrophic events playing no role in the decision-making process.
To avoid infinite values, Weitzman (2011) suggests thinning or
truncating the probability distribution, or putting a cap on utility.
Other authors try to reconcile EU and “heavy tails” by using specific
utility functions other than power Constant Relative Risk Aversion
utility. For instance, Ikefuji et al. (2010) propose the two-parameter
Burr function or exponential utility function and Millner (2011) proposes the (bounded) Harmonic Absolute Risk Aversion function to
model individual preferences. However, this approach yields results
driven by subjective choices like functional forms or parameter
values, which is not fully satisfactory, as Weitzman (2011) admits.
Chichilnisky's (1996, 2000, 2002, 2009) approach, presented hereafter, differs from the above in proposing a systematic axiomatic
foundation for modeling catastrophic risks or for making decisions
when risks are catastrophic. She shows why EU theory fails to explain
half the answers in the experiment and identifies the MC axiom as the
source of the problem, proposing an alternative set of axioms that appears to fit the experimental evidence.
6
A ‘countably additive’ measure is defined in Appendix.
5
3.3. The Failure of EU and a Solution
The failure of EU appears to be due to the MC axiom which implicitly postulates that rational behavior should be ‘insensitive’ to rare
events with major consequences. More specifically, the culprit is the
underlying definition of proximity that is used in the MC axiom,
where two events are close to each other when they differ on a set
of small measure no matter how great the difference in their outcomes. Chichilnisky (2000, 2002, 2009) used a L∞ sup norm that is
based on extreme events to define closeness: two lotteries f and g
are close when they are uniformly close almost everywhere (a.e.),
i.e. when supR|f(t) − g(t)| b a.e. for a suitable small > 0. 7 As a consequence, some catastrophic events are small under Arrow's definition
but not necessarily under Chichilnisky's.
The core here is that her definition of closeness is more sensitive
to rare events than Arrow's. It implies that a probability can be considered as low enough to make the lottery involving death acceptable,
depending on what the other outcomes are. This higher sensitivity
constitutes the second of her three axioms, which must be satisfied
by a ranking W to evaluate lotteries:
Axiom 1. The ranking W : L∞ → R is linear and continuous on lotteries.
The ranking W is called continuous and linear when it defines a linear function on the utility of lotteries that is continuous with respect
to the norm in L∞.
Axiom 2. The ranking W : L∞ → R is sensitive to rare events.
A ranking function W : L∞ → R is called insensitive to rare events
when it neglects low probability events; formally if given two lotteries (f, g) there exists ε = ε (f, g) > 0, such that W(f) > W(g) if and only if
W(f′) > W(g′) for all f′, g′ satisfying f′ = f and g′ = g a.e. on A ⊂ R when
μ(A c) b ε. We say that W is sensitive to rare events, when W is not insensitive to low probability events.
Axiom 3. The ranking W : L∞ → R is sensitive to frequent events.
Similarly, W : L∞ → R is said to be insensitive to frequent events
when for every two lotteries f, g there exists ε = ε (f, g) > 0 such that
W(f) > W(g) if and only if W(f′) > W(g′) for all f′, g′ such that f′ = f
and g′ = g a.e. on A ⊂ R : μ(A c) > 1 − ε. We say that W is sensitive to frequent events when W is not insensitive to frequent events.
Our notion of ‘nearby’ is stricter and requires that the lotteries be
close almost everywhere, which implies sensitivity to rare events.
Chichilnisky (2009) proved that EU theory fails to explain the behavior of individuals facing catastrophic events since:
Theorem 1. A ranking of lotteries W(f) : L∞ → R satisfies the Monotone
Continuity Axiom if and only if it is insensitive to rare events (see
proof in Chichilnisky, 2009).
A formal statement of the theorem is hence MC ⇔ ¬Axiom 2. The
simple example below shows why the axiom MC leads to insensitivity
to rare events.
Example. Assume that the Axiom MC is satisfied. By definition, this
implies for every two lotteries f ≻ g, every outcome c and every vanishing sequence of events {E i} there exists N such that arbitrarily altering the outcomes of lotteries f and g on event E i, where i > N,
does not alter the ranking, namely f′ ≻ g′, where f′ and g′ are the altered versions of lotteries f and g respectively. 8 In particular since,
for any given f and g, Axiom MC applies to every vanishing sequence
of events {E i}, we can choose a sequence of events consisting of
open intervals I = {I i}i∞= 1 such that I i = {x ∈ R : x > i} and another J =
7
A similar topology was used in Debreu's (1953) formulation of Adam Smith's Invisible Hand theorem.
8
For simplicity, we consider alterations in those lotteries that involve the ‘worst’
outcome c = infR|f(x), g(x)|, which exists because f and g are bounded a.e. on R by
assumption.
Please cite this article as: Chanel, O., Chichilnisky, G., Valuing life: Experimental evidence using sensitivity to rare events, Ecol. Econ. (2012),
doi:10.1016/j.ecolecon.2012.03.004
6
O. Chanel, G. Chichilnisky / Ecological Economics xxx (2012) xxx–xxx
{J i}i∞= 1 such that J i = {x ∈ R : x b − i}. Consider the sequence K = {K i}
where K i = I i ∪ J i. The sequence K is a vanishing sequence by construction. Therefore there exists an i > 0 such that for all N > i, any alterations of lotteries f and g over K N, denoted f N and g N respectively,
leave the ranking unchanged i.e. f N ≻ g N. Therefore Axiom MC implies
insensitivity of ranking W in unbounded sets of events such as {E i}.
Hence, because EU (and more generally approaches relying on the
EU axiomatic) relies on the MC Axiom that considers two lotteries to
be close when they differ in events of small measure, it is insensitive
to rare events. 9 This leads directly to the way of valuing life we describe above: measured by the low probability of death acceptable
in exchange for an extra amount of money. This classic axiom postulates that rational behavior should be ‘insensitive’ to rare events with
major consequences such as death, as proved in Theorem 1. This insensitivity underestimates our true responses to catastrophes, creating an impression of irrationality from observed behavior that is not
fully justified. Taking a family of subsets of events containing no
rare events, for example when the Lebesgue measure of all events
contemplated is bounded below, EU satisfies all three Axioms. Indeed,
in the absence of rare events, Axiom 2 is an empty requirement and
Axioms 1 and 3 are consistent with EU theory.
The MC axiom is strong and somewhat counterintuitive, so it is not
surprising that the experimental results contradict it. It requires, for
example, that the measures of a nested sequence of intervals of events
{x : x > n} for n = 1, 2,...decrease all the way to zero. This zero limit is required even though the intervals themselves {x : x > n} are all essentially
identical and could, for example, be expected to have the same measure,
at least in some cases. No explanation is provided for this somewhat unusual and strong axiom, in which essentially a sequence of identical sets
is assumed always to have measures converging to zero.
In contrast, the axioms for decision-making with catastrophic
risks utilized here require sensitivity to rare events, a new axiom
which applies in some cases and not in others. This new axiom allows
for the measures in such sequences converging to zero in some cases
and not in others, a logical negation of MC. This flexibility seems more
intuitive and plausible. While the universal applicability of MC is
ruled out, a different type of continuity is required in the new axioms.
Experimental evidence contradicts EU theory when catastrophes are
involved (as Posner, 2004, explains himself), and therefore it contradicts the universal applicability of its underlying axiom, MC. The experimental results presented in this article are consistent with the
new axiom of sensitivity to rare events. The difference between MC
and sensitivity to rare events is at the basis of the new form of
decision-making under uncertainty involving catastrophic risks introduced and developed in Chichilnisky (1996, 2000, 2002, 2009), and
can be viewed as the axiomatic foundation for the experimental results presented in this article.
In the experiments presented here, the value of life can be considered as clearly defined if a subject is willing to accept the same probability of death in exchange of the same amount of money and the
value of life is therefore contingent on the amount itself. Our findings
show, however, that for the same amount of money, the subject may
or may not be willing to accept the same small probability of death,
which indicates that subjects are taking into consideration factors
other than the amount of money and the probability of death when
making a decision. This appears to conflict with existing theory
based on the MC axiom, where the probability of death a subject is
prepared to accept can be clearly defined for any given payment. In
particular we establish the following result:
Theorem 2. A ranking of lotteries W : L∞ → R that satisfies Axiom 2 (i.e.
that is sensitive to rare events) determines a value of life that changes
depending on outcomes other than the amount of money.
9
See also Chichilnisky (1996, 2000, 2002) for a general proof that EU is insensitive
to rare events.
Proof. Theorem 1 showed that sensitivity to rare events is the negation of the MC Axiom. This implies that for two given lotteries f ≻ g,
every outcome c and every vanishing sequence of events {E i} there
exists N such that arbitrarily altering the outcomes of lotteries f and
g on event E i, where i > N, does not alter the ranking, namely f′ ≻ g′,
where f′ and g′ are the altered versions of lotteries f and g respectively, while for some other lotteries f ≻ g, every outcome c and some vanishing sequence of events {E i} there exists N such that arbitrarily
altering the outcomes of lotteries f and g on event E i, where i > N,
does alter the ranking, namely g′ ≻ f′, where f′ and g′ are the altered
versions of lotteries f and g respectively. Recall that – as in
Theorem 1 above – we have considered alterations in the lotteries
that involve the ‘worst’ outcome c = infR|f(x), g(x)|. The worst outcome can be identified with “death”, and is common to the two sets
of lotteries under comparison. The alterations in both cases are the
same, representing small enough probabilities of death. This implies
that, for a small enough probability of death (determined by N) in
the first two lotteries, the subject will accept the risk of death when
offered the small payment, while in the second lottery the same probability of death (represented by N) is not small enough. In other
words, depending on other outcomes of the lotteries, the subject
will accept a small probability of death, or will not accept that probability. Therefore the value of life depends on other outcomes of the
lottery, as we wished to establish.
Do some decision criteria satisfy all three Axioms in the presence
of rare events? Yes, if we modify EU by adding another component
called ‘purely finitely additive’ elements of L∞∗ 10 that embodies the notion of sensitivity for rare events. The only acceptable rankings W
under the three axioms above are a convex combination of L1 function
plus a purely finitely additive measure putting all weight on extreme
or rare events, as stated in the Theorem below:
Theorem 3. A ranking of lotteries W : L∞ → R satisfies all three Axioms 1,
2 and 3, if and only if there exist two continuous linear functions on L∞,
ϕ1 and ϕ2 and a real number λ, 0 b λ b 1, such that:
W TFA ðf Þ ¼ λ∫xR f ðxÞϕ1 ðxÞdx þ ð1−λÞbf ; ϕ2 >
ð1Þ
where ∫ R ϕ1(x)dx = 1, while ϕ2 is a purely finitely additive measure (see
proof in Chichilnisky, 1996, 2000, 2002).
The intuition behind this Theorem is that the first term of the utility in (1) is akin to EU, and therefore introduces a measure of sensitivity to normal or relatively frequent events. The density ϕ1(x) defines a
countably additive measure that is absolutely continuous with respect to the Lebesgue measure. 11
The second term of the utility in (1) is inconsistent with the MC
axiom, and satisfies a different type of continuity, under a topology
called “The Topology of Fear” (Chichilnisky, 2009). This second term
is very sensitive to rare events and balances out the first term in the
characterization, which is only sensitive to normal or frequent events.
The operator b f, ϕ2 > represents the action of a measure ϕ2∈L⁎∞ that
differs from the Lebesgue measure in placing full weight on rare
events. Remember that ϕ2 cannot be represented by an L1 function.
The two terms together therefore satisfy both ‘sensitivity to rare
events’, and ‘sensitivity to frequent events’, as is required by the
new axiomatic treatment of decision making under uncertainty
10
∗
The space L∞
is called the ‘dual space’ of L∞, and is known to contain two different
types of rankings W(.), (i) integrable functions in L1(R) that can be represented by
countably additive measures on R, and (ii) ‘purely finitely additive measures’ which
are not representable by functions in L1 (Chichilnisky, 2000), and are not continuous
with respect to the Lebesgue measure of R. See Appendix for a definition of a ‘finitely
additive’ measure.
11
A measure is called absolutely continuous with respect to the Lebesgue measure
when it assigns zero measure to any set of Lebesgue measure zero; otherwise the measure is called singular.
Please cite this article as: Chanel, O., Chichilnisky, G., Valuing life: Experimental evidence using sensitivity to rare events, Ecol. Econ. (2012),
doi:10.1016/j.ecolecon.2012.03.004
O. Chanel, G. Chichilnisky / Ecological Economics xxx (2012) xxx–xxx
with catastrophic risks used here. The implied decision criteria that
emerge from the new axioms are a combination of EU with extreme
responses (to extreme events like death), and seem more in line
with experimental evidence.
Indeed, it seems that purely finitely additive measures could play
an important role in explaining how our brains respond to extreme
risks. When the number of choices is finite there is a simpler way to
explain the criterion of choice: it is similar to a convex combination
of EU and a maximin. EU is optimized while at the same time avoiding
those choices that involve catastrophic outcomes, such as death. This
rule is inconsistent with EU and will rank a choice that involves death
much lower than EU would. Therefore any observer that anticipates
EU optimization will be disappointed, and will believe that there is irrationality. 12 But this is not true, as the rule becomes rational once we
take into account rational responses to extreme events. It is consistent with what people do on an everyday basis, with what is observed
in the experiments presented here and also with what Arrow's famous comment implies.
4. Discussion and Concluding Remarks
How the Axiom MC creates insensitivity to rare events can be illustrated by the following situation used by Arrow (1966) to show
how people value their lives, along the same lines as the discussion
in the Introduction. If a is an action that involves receiving one cent,
b is another that involves receiving zero cents, and c is a third action
involving receiving one cent and facing a low probability of death,
Arrow's Monotone Continuity requires that the third action involving
death and one cent should be preferred to the second involving zero
cents when the probability of death is low enough. Even Arrow says
of his requirement ‘this may sound outrageous at first blush…’
(Arrow, 1966, p. 256, l. 28–29). Outrageous or not, we saw in
Theorem 1 that MC leads to the neglect of rare events with major consequences like death.
Theorem 1 shows that Axiom 2 eliminates those examples that
Arrow calls outrageous. We can also see how Axiom 2 provides a reasonable solution to the problem, as follows. Axiom 2 implies that
there are catastrophic outcomes, such as the risk of death, so terrible
that people are unwilling to accept a low probability of death to obtain one cent versus 0 cents, no matter how low that probability
may be. Indeed, according to the sensitivity Axiom 2, no probability
of death is acceptable when one cent and 0 cents are involved. However according to Axiom 2, in some cases, the probability can be low
enough to make the lottery involving death acceptable. As shown in
Theorem 2, it all depends on what the other outcomes are. This becomes clear in our approach and seems a reasonable solution to the
problem that Arrow raises.
For example, if in the above example “one cent were replaced by
one billion dollars” – as Arrow (1966 p. 256, lines 31–32) suggests –
under certain conditions we may be willing to choose the lottery
that involves a low probability of death and one billion dollars over
the lottery that offers 0 cents. 13 Indeed, some of the subjects in the
pill experiment state that they might have chosen to take the pill
for a larger amount (“the amount is not big enough” (3 subjects),
“the amount is too small to dramatically change my life”, “I would
have answered ‘Yes’ if I was almost in absolute poverty”).
More to the point, consider the same type of death risk: a low
probability of death caused by a medicine that can cure an otherwise
12
Note that EU could be used in certain cases to rationalize answers like the one we
obtained, without providing a consistent set of axioms that create a well defined theory, by assuming that some subjects are infinitely averse to risk (unbounded below
utility function). However, this is a somewhat ad hoc treatment not satisfactory theoretically since it brings back in the St Petersburg Paradox.
13
Or if the death event had been replaced by a less frightening event, say a €152,450
loss, we are willing to bet that most of the subjects in the pill experiment would have
accepted the 10− 9 probability of loss.
7
incurable cancer may be preferable to no cure. A sick person may
evaluate a cure – no matter how risky – higher than a healthy person
would, and may be willing to take risks that a healthy person would
not. In the same spirit, as shown in the pill experiment, the same individual may change his/her mind depending on factors exogenous to
the outcomes. Here, among the two reasons that significantly
explained such a change in the subjects that answered both 1998
and 2009 surveys, were “change in marital/familial status” and
“change in the perception of the probability”.
The former has to do with the painful implications the death
would have for relatives. The latter has to do with the subjective perception of the probability 10 − 9. One reason suggests itself: the 9/11
attacks, which occurred between the two experiments. Indeed, although subjects may have considered the outcome “simultaneous
crashes of two commercial flights into the World Trade Center within
the same hour” of tiny probability, the fact that it actually occurred
may have led them to change their perception of tiny probability.
No catastrophe had ever been so widely covered worldwide. This
may explain that, in 2009 answers, new subjects do not significantly
differ from previously 1998 surveyed subjects when surveyed in
2009. Indeed, Sunstein (2003) also provides evidence that individuals
show unusually strong reactions to low-probability catastrophes especially when their emotions are intensely engaged. This “probability
neglect” is also explored in Sunstein and Zeckhauser (2011) regarding both fearsome risks and the resulting damaging overreactions
shown in individual behavior and government regulations.
Recently, Chanel and Chichilnisky (2009) report experimental results from a study of the predictions of the standard EU framework
under catastrophic risks. Subjects faced choices among events involving “being locked up in a room with no chance of escaping, being
freed or communicating (with relatives, friends…), with nothing interesting to do”. The events differed as to the duration of detention,
and the catastrophic event was created by making the period of detention 40 years. Interestingly, the results obtained are in line with
that obtained here: more than half the subjects did not behave
according to EU theory whereas the remaining half answered according to EU theory; however all behaved according to the approach proposed in this article.
To conclude, the alternative approach proposed in this article furthers the treatment of catastrophic outcomes in two ways.
First, it provides a new measure for the value of life with two important characteristics: it values life more highly than under the EU criterion, and this value is shown to depend on other factors, not only on
the numerical value of what is being offered. This is because catastrophes are worst-case events whose weight depends on what else is
going on in people's lives.
Second, the alternative approach challenges the belief that EU properly expresses rationality in situations involving catastrophic outcomes.
Indeed, EU theory is found to perform poorly in explaining the actual
behavior of most of the subjects in the experiment, even though these
subjects are fully familiar with the logic behind EU theory, while the
alternative approach performs quite well. We do not reject the MC
axiom outright — nor do we reject EU outright. Rather, we find that
it is more realistic and satisfactory that MC be satisfied in some
cases and not in others — EU axioms therefore being satisfied in
some cases and not in others. Requiring that MC be satisfied in all
cases and thus that EU utility axioms hold in all cases is problematic,
since it implies insensitivity to rare and potentially catastrophic
events. Hence, our approach stands as an alternative proposal for defining rational behavior in the face of catastrophes such as death. In
any case, in the absence of rare events with major consequences,
our theory of choice under uncertainty is consistent with and mirrors
the standard EU theory (our Axiom 2 is indeed void of meaning), and
can therefore be viewed as an extension of classic decision theory.
Finally, an interesting avenue for future research might be to explore how the brain works while considering outcomes involving
Please cite this article as: Chanel, O., Chichilnisky, G., Valuing life: Experimental evidence using sensitivity to rare events, Ecol. Econ. (2012),
doi:10.1016/j.ecolecon.2012.03.004
8
O. Chanel, G. Chichilnisky / Ecological Economics xxx (2012) xxx–xxx
catastrophic and non-catastrophic events. Are the same zones activated? In the same order? For the same length of time? Functional magnetic resonance imaging or positron emission tomography should
help answer these questions, since neuroeconomic decision science
is no longer a utopian concept (see for instance Smith et al., 2002;
or Knutson and Peterson, 2005).
Acknowledgments
This research is part of Columbia's Program on Information and
Resources, and the Columbia Consortium for Risk Management
(CCRM). It was motivated by Olivier Chanel's experimental research
and is based on earlier results in “The Topology of Fear”
(Chichilnisky, 2000, 2002, 2009). It is partly supported by program
Riskemotion (ANR-08-RISKNAT-007-01), which is gratefully acknowledged. We thank two anonymous referees, Marjorie Sweetko
and Jean-Christophe Vergnaud for helpful comments and suggestions,
congress participants at Montréal 2010 WCERE for helpful discussions, and the 127 former and current Greqam members for their
kind participation.
Appendix. Countably and Purely Finitely Additive Measures
The space of continuous linear functions on L∞ is a well known
space called the “dual” of L∞, and is denoted L∞∗ . This dual space has
been fully characterized e.g. in Yosida and Hewitt (1952) or Yosida
(1974). Its elements are defined by integration with respect to measures on R. The dual space L∞∗ consists of (i) L1 functions g that define
countably additive measures μ on R by the rule μ(A) = ∫ A g(x)dx
where ∫ R|g(x)|dx b ∞ and therefore μ is absolutely continuous with respect to the Lebesgue measure, namely it gives measure zero to any
set with Lebesgue measure zero, and (ii) a ‘non-L1 part’ consisting
of purely finitely additive measures ρ that are ‘singular’ with respect
to the Lebesgue measure and give positive measure to sets of Lebesgue measure zero; these measures ρ are finitely additive but they
are not countably additive. A measure η is called finitely additive
when for any family of pairwise disjoint measurable sets {Ai}i = 1, … N
η(∪iN= 1 Ai) = ∑ iN= 1 η(Ai). The measure η is called countably additive
when for any family of pairwise disjoint measurable sets {Ai}i = 1, … ∞
η ∪∞i¼1 Ai ¼ ∑∞i¼1 ηðAi Þ. The countably additive measures are in a
one-to-one correspondence with the elements of the space L1(R) of
integrable functions on R. However, purely finitely additive measures
cannot be identified by such functions. Yet purely finitely additive
measures play an important role, since they ensure that the ranking
criteria are ‘sensitive to rare events’ (Axiom 2). These measures define continuous linear real valued functions on L∞, thus belonging to
the dual space of L∞ (Yosida, 1974), but cannot be represented by
functions in L1.
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Please cite this article as: Chanel, O., Chichilnisky, G., Valuing life: Experimental evidence using sensitivity to rare events, Ecol. Econ. (2012),
doi:10.1016/j.ecolecon.2012.03.004