Naive, resolute or sophisticated? A study of dynamic
John D. Hey & Gianna Lotito
Published online: 10 January 2009
# Springer Science + Business Media, LLC 2009
Abstract Dynamically inconsistent decision makers have to decide, implicitly or
explicitly, what to do about their dynamic inconsistency. Economic theorists have
identified three possible responses—to act naively (thus ignoring the dynamic
inconsistency), to act resolutely (not letting their inconsistency affect their
behaviour) or to act sophisticatedly (hence taking into account their inconsistency).
We use data from a unique experiment (which observes both decisions and
evaluations) in order to distinguish these three possibilities. We find that the majority
of subjects are either naive or resolute (with slightly more being naive) but very few
are sophisticated. These results have important implications for predicting the
behaviour of people in dynamic situations.
JEL classifications D90.D80.C91
This paper is concerned with dynamic decision-making1. An important and recurring
issue in this analysis concerns the behaviour of dynamically inconsistent people. Do
they know that they are dynamically inconsistent, and, if so, what do they do about it?
Economic theory has identified three possible responses (though there are obviously
many more): that such decision makers act naively (ignoring their inconsistency); that
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NO9058; No of Pages
1See, amongst others, Cubitt et al. (2004), Machina (1989), McClennen (1990).
J. D. Hey
LUISS, Rome, Italy
J. D. Hey (*)
Department of Economics & Related Studies, University of York, Heslington, York YO10 5DD, UK
Università del Piemonte Orientale, Alessandria, Italy
they act resolutely (not letting their inconsistency affect their behaviour); that they act
sophisticatedly (taking their inconsistency into account). We report on an experiment
that lets us infer which of these responses describes behaviour better. We have
designed the experiment in such a way that we can not only observe choices in
dynamic decision problems but also we obtain subjects’ evaluations of such problems.
Combining these two types of data we can estimate the preferences of the decision
makers, and crucially infer whether they are naive, resolute or sophisticated.
Dynamic decision-making has two dimensions: the sequentiality of the decision
process, and the passage of real time. In both dimensions, the issue of dynamic
consistency arises: whether decision-makers implement plans that they made earlier.
A key element is whether the preferences of the decision maker change through the
decision-making process. If preferences do change, the potentiality of dynamic
inconsistency arises. This may happen in several ways depending upon the context.
In the context of a risk-free problem with no passage of real time, this can only occur
if preferences explicitly change during the decision process. In other contexts the
reasons are more subtle. In the context of a risky sequential problem, potential
dynamic inconsistency may occur if the preferences of the decision-maker are not
those of Expected Utility theory. In the context of a risk-free problem with the
passage of real time, potential dynamic inconsistency may occur if the preferences of
the decision-maker are not those of (exponentially) Discounted Utility theory. Our
analysis is relevant to all contexts, though the context which we examine is that of
sequential decision-making under risk.
The problem of inconsistent choice in a dynamic decision problem was initially
analysed in the literature in a context of certainty, and was related to the problem of
preferences changing either endogenously or exogenously through time (see, for
example, Strotz 1956). Hammond (1976 and 1977) generalised the analysis,
overcoming the distinction between exogenously and endogenously changing tastes,
and concentrating the analysis on the essential aspect of the problem—namely that
preferences change over time. However he kept it confined to a situation without risk
or uncertainty. This was introduced only later: Hammond (1988a, b, 1989), Raiffa
(1968), Machina (1989) and McClennen (1990).
The simple example of the drug addict in Hammond (1976) illustrates the
problem. Suppose that an individual is considering whether to start taking an
addictive drug. The individual2would prefer at most to take the drug without
consequences. However, he is certain that, if he starts, he will become an addict,
with serious consequences for his health. Of course, he can refuse to take the drug in
the first place. This agent is facing a simple dynamic decision problem with the
following structure (squares representing decision nodes):
2We presume that he is male to avoid expositional clumsiness.
2 J Risk Uncertain (2009) 38:1–25
Here three options are available to the agent, which lead to the following outcomes:
take the drug as long as it is harmless, then stop, which leads to outcome a; become an
addict, which leads to outcome b; not take the drug, which leads to outcome c.
At the initial decision node n0the agent has to decide whether to take the drug or
not, and his preferences are a ≻ c ≻ b. If he gets to the choice node n1he has become
an addict, and therefore the only relevant preferences are those concerning a and b,
and addiction itself means that b ≻ a. Thus, at n1his initial preferences between a
and b get reversed. The agent will choose b inconsistently with his previous
preference for reaching a.
The requirement of dynamic consistency is a requirement of consistency between
planned choice and actual choice. In the example, the dynamically inconsistent agent
decides initially to take the drug, then stop (option a), but chooses later not to stop
(thus actually choosing option b).
Hammond (1976) considers myopic (naive) and sophisticated choice as two
possibilities available to an agent in such a situation of dynamic inconsistency. When
acting according to a myopic approach, the agent selects at each point those
strategies which he judges acceptable from the perspective of that point. In the
example, the myopic agent ignores that his tastes are changing, and chooses at each
stage the option he considers as best at that moment. Therefore, he will choose
option a at n0, but change his mind and choose b at n1. The final outcome will be b.
When acting according to a sophisticated approach, the agent anticipates his
future choice and chooses the best plan among those he is ready to follow to the end:
he rejects those plans which imply a choice he anticipates he will not make. By
doing so, he always ends up choosing ex post according to his ex ante plans, and
avoids violating dynamic consistency. In the example, at n0he will forecast that by
taking the drug he will become an addict, and realises that his only options are b and
c. Therefore he will choose the most preferred option between the two, that is, c.
Hammond’s example of dynamically inconsistent choice allows us to consider
another possible model of behaviour, which was formalised only later in the
literature—McClennen (1990), Machina (1989)—in the context of dynamic
inconsistency under risk—resolute choice. The agent resolves to act according to a
plan judged best from an ex ante perspective, and intentionally acts on that resolve
when the plan imposes on him ex post to make a choice he does not prefer at that
point. By so choosing he manages to act in a dynamically consistent manner. In
Hammond’s example, at n0a resolute agent would have resolved to act according to
the plan leading to the most preferred outcome a—take the drug while it is harmless,
then stop; and he would have acted on that resolve when at n1the plan imposed on
him to choose the less preferred option—therefore going for a and not for b.
So far we have introduced the problem of dynamic inconsistencies in the context
of changing preferences in a world of certainty. As mentioned above, there are
however other contexts in which the same considerations apply. One is in a dynamic
certain world in which agents discount the future non-exponentially—for example,
using quasi-hyperbolic discounting, as in Harris and Laibson (2001). We note that
Harris and Laibson remark (p. 939) that “we model an individual as a sequence of
autonomous temporal selves”, explicitly making clear that preferences are changing
through time, in the sense that the relative evaluation of consumption in any two
periods varies depending upon when the evaluation is carried out. In this paper, and
J Risk Uncertain (2009) 38:1–253
indeed in much of the literature on quasi-hyperbolic consumers, the analysis assumes
that the decision maker is sophisticated in the sense that we have used it above.
Hence the decision maker takes into account his or her decisions in the future when
deciding in the present. In this way, the decision maker resolves his or her dynamic
inconsistency. However, it could also be the case that the decision maker acts either
naively or resolutely3—in which cases the predictions of the quasi-hyperbolic model
would be different.
A further dynamic context in which exactly the same considerations apply is that
of a risky world in which agents do not have Expected Utility preferences. This is
the context in which we carry out our experimental investigation. We begin to give
detail in Section 1.
However, we have exactly the same interest in all contexts: how does a
dynamically inconsistent agent respond to, and resolve, his or her dynamic
inconsistencies: naively, resolutely or sophisticatedly4? We attempt to answer this
question by adopting an experimental design where the subjects have to take a series
of decisions, some of which are dynamic in nature, and in which different kinds of
inconsistent agent respond in different ways to the decision problems posed.
Subjects are required both to take decisions and also to evaluate decision problems.
By observing their behaviour (both over decisions and evaluations) we can infer
whether they are (more likely to be) naive, resolute or sophisticated. Interestingly we
find that the majority of our subjects appeared to be either naive or resolute (with
roughly equal numbers of each), while very few appeared to be sophisticated. This
finding has important implications for the modelling of dynamically inconsistent
agents (for example, as in the literature on the behaviour of agents who
hyperbolically discount the future).
1 Dynamically inconsistent behaviour in a sequential risky context
In the context of risk and uncertainty, the problem of dynamic inconsistency is crucially
linked to the questionofwhether the decisionmaker has ExpectedUtilitypreferences or
not. As is well known, if the decision maker’s preferences satisfy Expected Utility
theory, then the problem of dynamic inconsistency does not arise. For a non-Expected-
Utility decision maker, however, the problem may arise5. We illustrate this in Fig. 1,
where three of the decision problems played in the experiment are shown. In this
figure, the squares (green in the experiment) represent decision nodes; and the circles
(red in the experiment) represent chance nodes (where Nature moves with the
probabilities indicated in the figure). The letters on the various choice branches denote
3Or in some other way—these three ways of reacting to the potential dynamic inconsistency are not the
4We should note that the issue could also be regarded as one of framing. As we shall see later when we
discuss the actual decision problems used in the experiment, different ways of framing the problem might
lead to different solutions, for a potentially dynamically-inconsistent person. Indeed, it could be argued
that dynamic-consistency and immunity-to-framing-effects are one and the same thing.
5See Machina (1989) for a detailed description of the “argument for the dynamic inconsistency of the
non-expected utility maximisers” (p. 1636). Other references include Raiffa (1968) and McClennen
4 J Risk Uncertain (2009) 38:1–25
the implied lottery: for example, in Problem 1, M indicates the lottery obtained
choosing Up and O the lottery obtained by choosing Down; at the decision node of
Problem 2, K is the certainty of £30, while L is the lottery which gives £50 with
probability 0.8 and £0 with probability 0.2. In Problem 36, O denotes the lottery
obtained by playing Up at both decision nodes; M denotes the lottery obtained by
6A tree with a similar structure is found in McClennen (1990).
a The violation of expected utility preferences
b The implications for dynamic choice
O (if L)
M (if K)
Fig. 1 The problem of dynamic
inconsistency. a The violation
of expected utility preferences.
b The implications for dynamic
J Risk Uncertain (2009) 38:1–255
playing Up at the first and Down at the second; while N denotes the lottery obtained
by playing Down at the first decision node.
Consider first Fig. 1a and consider an individual7who prefers O to M in Problem 1
and prefers K to L in Problem 2. These preferences imply a violation of Expected
Utility theory in the form of a Common Ratio Effect (see Starmer (2000) for
references concerning empirical evidence of this effect)8. Now consider how such an
individual would tackle Problem 3, in which there are two decision nodes D1and D2.
SupposethisindividualisatdecisionnodeD2. Then her preferences indicate that she
would choose to move Down at that node, because she prefers K to L. Now look at the
situation as viewed from node D1. As viewed from there, if she moves Up at that first
node, then she is faced with either getting O (by moving Up at D2) or getting M (by
moving Down at D2); if she moves Down she is faced with getting N. Assume that for
this individual O is preferred to N, which is preferred to M9. Then as viewed from
node D1she prefers O, and that, by assumption, is preferred to the lottery obtained by
moving Down at D1namely N. Hence she sets off by choosing Up at D1. A problem
arises, however, if she arrives at node D2. As we have already argued, at this node
Down is preferred, and so, if the preferences at that point are followed, the individual
will choose Down at node D2. Hence the individual, in arguing in this way, plans, at
node D1, to choose Up at node D2but, if she arrives there, actually chooses Down.
This is dynamic inconsistency: adopting a plan but then not implementing it at a later
node. Moreover, it is a problem to the individual when at node D1. If she is aware of
this dynamic inconsistency, then she realises that by choosing Up at D1and then
choosing Down at D2she therefore ends up with the lottery M which is dominated by
the lottery N—which she could get by choosing Down at D1.
What might the individual do about this dynamic inconsistency? Well, she might
simply be unaware of it, or ignore it, acting naively, and choose at each point the
strategy most preferred from the perspective of that point—Up at D1and then Down
at D2. However, if she is aware of this dynamic inconsistency, she might want to do
something about it. One possibility is that the individual, anticipating the fact that
she will choose Down at D2, if she reaches it, realises that it would be better to
choose Down at node D1. This, in the literature, is termed acting sophisticatedly—
see, for example, McClennen (1990) and Machina (1989)10. This sophisticated
individual anticipates her future behaviour, and avoids the inconsistency by making
a choice at the initial decision node which is constrained by her anticipated choice at
each following node.
However, as viewed from the individual at node D1this behaviour implies the
lottery N whereas, as viewed from D1, a better lottery is O which is achievable by
choosing Up at D1and Up at D2. The individual at D1might accordingly decide to
10Machina though does not use directly the term sophisticated choice for this kind of approach. Also
Karni and Safra’s (1989) model of “behavioural consistency” represents a way of implementing the
sophisticated choice approach, and represents a solution to the problem of dynamic inconsistency with
9Note that lottery N stochastically dominates lottery M.
8The Independence Axiom of Expected Utility theory implies that £30 for sure is preferred to a 80:20
lottery over £50 and £0 if and only if a 25:75 lottery over £30 and £0 is preferred to a 20:80 lottery over
£50 and £0.
7Whom we assume here is female to avoid expositional clumsiness.
6 J Risk Uncertain (2009) 38:1–25
be resolute and choose Up at D1and Up at D2. This resolute individual avoids
inconsistency by making the choice of a plan most preferred at the initial node to
constrain future behaviour. She resolves to implement the plan originally adopted,
despite the fact that this implies, at some future node, making a choice that she
would not have liked to make once arrived at that node. Again a discussion of this
kind of behaviour can be found in Machina (1989)11and McClennen (1990)12.
Such problems do not arise with Expected Utility individuals. In the context of
the above decision problem, EU preferences could be such that K is preferred to L
and M to O, or the reverse. In the first case, the individual chooses Down at node D1
and in the reverse case chooses Up at D1and Up at D2.
2 Related experiments
The purpose ofthe studyreportedinthispaper istotryandfindout whether subjectsare
naive,resoluteorsophisticated.To the bestofour knowledge,this isthe first suchstudy.
However, there are some related experiments, which also study behaviour in dynamic
contexts. We shall confine ourselves to those in economics which have an appropriate
incentive mechanism13. One such paper is that by Cubitt et al. (1998), which is
primarily concerned with trying to discover, in the context of dynamic decision
making, which of a set of dynamic choice principles is apparently violated by
individuals with non-EU preferences. Cubitt et al. presented (to different groups of
subjects) different decision trees and studied the behaviour of subjects in such trees.
These trees show some similarities with those in Problems 1, 2 and 3 shown in Fig. 1.
A sub-set of the trees in Cubitt et al. was used in Hey and Paradiso (2006) who,
instead of looking at the decisions of subjects, collected data on subjects’ evaluations
of decision problems. The two studies are closely related though they had different
objectives. These three sets of trees are relevant to the objectives of this present paper
since naive, resolute and sophisticated subjects could have different behaviours in the
trees, as well as having different evaluations of them. Cubitt et al. found that subjects
behave differently in the different trees, while Hey and Paradiso find that the temporal
frame also affects the evaluations the subjects make of the different trees.
Our experiment differs from both these previous studies, most particularly
because this present paper has a different agenda. However, we build on the
framework set by these two papers by using similar decision trees to those used in
Cubitt et al. and in Hey and Paradiso. We also add a fourth tree which helps us in our
task of distinguishing between naive, resolute and sophisticated subjects. In addition,
our experiment differs from both these other two in that we observe not only choices
but also gather data on valuations14. Therefore in our experiment we combine these
11Machina’s model of choice is equivalent even if differently formalised to McClennen’s model of
resolute choice. According to Machina, resolute choice represents one of the “antecedents of the formal
approach” presented in his paper.
12An interesting point is how this person can force him- or herself to behave resolutely.
13Busemeyer and his associates in psychology have carried out some related experiments (see, for
example, Busemeyer et al. 2000) but without such incentives.
14We note that valuation data is potentially more informative than choice data, as the latter only tells us
which choice is preferred and nothing about the strength of preference.
J Risk Uncertain (2009) 38:1–257
8 J Risk Uncertain (2009) 38:1–25
two other approaches but with a differently directed research agenda—that of
detecting whether subjects are naive, resolute or sophisticated.
Our experiment is built around the four trees shown in Fig. 2 (which is a screen
shot from our experiment—the colour version of which can be found in the Web
Technical Appendix). Trees 1, 2 and 3 are identical to Problems 1, 2 and 3 shown in
Fig. 1. Tree 4 is the static version of Tree 3, where the decision problem is reduced
to that of a single decision—rather than that of a sequence of decisions. We note that
Expected Utility theory has a strong prediction about relative behaviour in Trees 1
and 2; given that, as we have discussed above, we have predictions about the
behaviour in Tree 3 of naive, resolute or sophisticated subjects whose preferences do
not respect Expected Utility theory.
In essence, observing behaviour in these trees should enable us to detect whether
people are naive, resolute or sophisticated. However, there are two problems. The
first is that we can only hope to detect differences in type if the agent has non-EU
preferences—since the distinction only has sense for non-EU agents. Moreover, it
may be the case that a subject has non-EU preferences but our trees do not enable us
to detect this fact: a subject who chooses M in Problem 1 and K in Problem 2 (or O
in Problem 1 and L in Problem 2) is not necessarily a person with EU preferences. A
second problem is that, inevitably in experiments, there is noise in the subjects’
responses (see Hey 2005). It is well known from many previous experiments that
this noise can be quite large; to give some idea of this, we note that, when asked the
same question on two separate occasions, subjects give different answers roughly
25–35% of the time. Accordingly we cannot be sure that any stated decision (or any
stated valuation) is exactly in accordance with the subject’s preferences.
To help us to get over these two problems, we used three sets of four trees—all
with the same structure as the four trees in Fig. 2 but with different probabilities and
payoffs. We give detail in the next section (Section 3). Moreover, in analysing the
data from the experiment, after a descriptive analysis (Section 4), we explicitly
assumed the existence of noise in the responses of the subjects and we use all the
data on each subject to analyse the preferences of that subject. We give detail in
Section 5. For all the technical material we refer to our Web Technical Appendices at
3 The experimental design and implementation
The experiment was built on the four decision problems introduced in the previous
section. We used three sets each with four trees, with different values for the payoffs
and the probabilities, giving a total of 12 decision trees. Table 1 gives the details; we
note that the possible payoff varies from a minimum of £0 to a maximum of £150.
We designed the experiment in such a way that we could observe subjects’ choices
in all 12 trees as well as their stated evaluations of the 12 trees. The amounts of
money and the probabilities in the three sets of trees were chosen to satisfy various
criteria. First, we wanted one set of trees (Set 1) that was the same as used in
RFig. 2 A screen shot from the experiment—squares labelled C (green in the experiment) are choice
nodes; squares labelled N (red in the experiment) are chance nodes
J Risk Uncertain (2009) 38:1–259
previous experiments (Cubitt et al. 1998 and Hey and Paradiso 2006). Second, we
wanted different amounts of money, and different probabilities in the different sets,
but with the same properties. Hence the structure of the payoffs and probabilities are
identical in the three sets—they are always such that we should be able to detect
non-EU behaviour from the choices, and then to distinguish the naive, resolute and
sophisticated types. Finally we wanted payoffs and probabilities that gave an
incentive to the subjects that was roughly similar in the three different sets (the
maximum expected payoff ranges from £7.50 to £10 in the different sets) as well as
being reasonable. Here there is a problem in that we preferred that subjects would
not leave the experiment with less money than that with which they came. Obviously
there is no way to guarantee this, but giving a relatively high participation fee
reduces this possibility—and also increases the amount of money that they can bid
for the various trees. For a risk-neutral subject, the value of a randomly chosen tree
in the different sets is £8.83, so while the value of bidding for a tree was not high, it
was also not negligible. Moreover subjects were encouraged to think seriously about
their bids by having to spend 15 min deciding on them, which turned out to be a
more than sufficient length of time.
The experiment was conducted at EXEC, the Centre for Experimental Economics
at the University of York. A total of 50 students, both graduate and undergraduate,
took part in the experiment. They were given written instructions (see Web Technical
Appendix 5). When all participants had finished reading the instructions, a
PowerPoint presentation was played at a predetermined speed on their individual
screens. After this, they could ask questions. The experiment then started, using a
Visual Basic program (available on request). The computer screen showing the four
decision trees is that shown in Fig. 2. In order to elicit the subjects’ evaluations for
each of the 12 trees, we used the second-price sealed-bid auction method as in Hey
and Paradiso (2006). This was implemented as follows. Subjects performed the
experiment in groups of five. They were sat at individual computer terminals and
were not allowed to communicate with each other. They individually made bids for
each of the three sets of four trees (12 trees overall) and were given 15 min to bid for
each set of four trees in the set. During the bidding period the subjects were allowed
to practice playing out the decision trees as much as they wanted in the time allowed.
It was made clear that the outcomes of the practice did not affect their payments in
any way. The number of seconds left for the practice and bidding was shown in the
box at the top left-hand side of each decision tree. When the bidding time was over,
the subjects played out all the 12 problems for real. We displayed on each subject’s
screen the results of his or her playing out plus the bids of all the five subjects in the
Table 1 The parameters used in the experiment
ParameterSet 1Set 2 Set 3
10 J Risk Uncertain (2009) 38:1–25
group for each of the 12 trees. Then we invited one of the five subjects to select a
ball at random from a bag containing 12 balls numbered from 1 to 12. This
determined the problem on which the auction was held. We then consulted the bids
made by the group members, and the subject with the highest bid for the problem
paid us the bid of the second-highest bidder. As all subjects were given a £20
participation fee, four of the five members earned £20, while the fifth earned £20
minus the bid of the second highest bidder plus the outcome. Both in the instructions
and in the presentation, it had been emphasised, through different examples, that the
highest bidder (in the auction corresponding to the randomly selected decision
problem) would have to pay the bid of the second-highest bidder, and it was made
clear that the bid for each of the 12 problems should be equal to the willingness to
pay for the decision problem. It was emphasised that in the case that the subject’s
bids were all £0, he or she was unlikely to be sold one of the decision problems and
thus will definitely end up with no less than the participation fee. Moreover, in the
case that the bid was higher than £0 and the subject’s bid was the highest for the
chosen tree, he or she would end up with the participation fee, minus the bid of
the second-highest bidder, plus the outcome. This could—depending on the bid—be
less than the participation fee and the subject could end up losing money. Subjects
were warned in advance that they had to bring enough cash to the experiment to
allow for this possibility, which however never occurred15.
Before proceeding to the analysis of the data from the experiment, we should
comment on one feature of the design: the (implicit) use of the random lottery
incentive mechanism: while subjects took decisions with respect to, and placed bids
on, 12 trees, the payment mechanism implied that there was an average 1/60
probability16that the decisions and bids on any tree would actually influence their
payment. This kind of random lottery incentive mechanism has a long tradition in
experimental economics, but implies that the subjects treat each tree as independent
from the other trees—that is, that they separate the trees one from the other when
they are taking decisions. In the context of dynamic choice, this assumption might
appear to be unrealistic, but the opposite assumption—that the subjects take into
account all 12 trees when taking their decisions on any one—might be considered as
even more unrealistic. However, there is considerable evidence that this random
lottery mechanism does have the desired property: a direct test was first carried out
by Starmer and Sugden (1991) with just two choice problems; an indirect test with
100 choice problems was implemented by Hey and Lee (2005a); a further test of the
proposition that subjects may not take into account all the questions when answering
any one, but instead take into account the recently answered questions was carried
out by Hey and Lee (2005b). All three of these studies seem to confirm that subjects
do separate and hence that they answer to each question (each tree in our context) as
if that were the only question (tree) in the experiment.
16Recall that there were five subjects in each group, implying an average probability of 1/5 that any
subject would be paid from playing out a particular tree, and that there were 12 trees, each chosen with
15We take this as a sign that the subjects understood the second-price sealed-bid auction method. Besides,
the raw data on the subjects’ bids (see Web Technical Appendix 1) show that no case of under or over-
J Risk Uncertain (2009) 38:1–25 11
4 Description of the data from the experiment
We recall from Section 1 that the experiment design allows us to classify subjects
according to whether their preferences obey EU or not by observing their behaviour
in trees T1and T2. Four patterns of behaviour can be observed on these two trees:
K preferred to L, and M to O
L preferred to K, and O to M
K preferred to L, and O to M
L preferred to K, and M to O
The first two patterns are subjects consistent with EU preferences. The last two
classify subjects who violate EU; pattern (3) represents the more common violation.
Table 2 gives the number of subjects who behaved according to the above patterns in
the different sets.
For all the four patterns we can predict the subjects’ behaviour in T3and T4. In
cases (1), (2) and (4) there should be no differences in behaviour for the
sophisticated, naive and resolute subjects, while in case (3) there should be a
difference. Therefore, for pattern (3) we can also classify how many subjects are
consistent with the predictions of the different dynamic choice models (assuming no
error). Table 3 gives the number of subjects who behave consistently with the
predictions in T3and T4in each set.
We note that the rate of consistency to predicted behaviour—particularly of non-
Expected-Utility subjects—is low. It is interesting to note also that nearly all
inconsistent behaviour in case (3) followed a common pattern: subjects chose
prospect N in both T3 and T4. Choice of the N gamble is consistent with
sophistication in T3, but not in T4, where all subjects with this non-EU preference
pattern should have preferred gamble O.
The experiment design allows also the prediction of how subjects with different
preference patterns should evaluate the different trees, by assigning higher bids to
the trees which they value more. Table 4 gives—for each set and for each different
preference pattern—the predictions in terms of bids and the number of subjects who
were consistent with such predictions. Again we note that also consistency to the
predictions on subjects’ evaluations appears to be very low. This is almost certainly
because of error in the subjects’ responses, which this descriptive analysis does not
take into account.
Table 2 Number of subjects with EU and non-EU preferences in each set
Set 1Set 2 Set 3
Expected utilityK preferred to L, hence M to O
L preferred to K, hence O to M
K preferred to L, but O to M (common violation)
L preferred to K, but M to O
12 J Risk Uncertain (2009) 38:1–25
5 A formal analysis
There are two problems with the above description of the data. First, it is partial and
does not use all the data from each subject in a systematic fashion. Second, as we
have already noted, it ignores the existence of noise in the subjects’ responses.
However, there is noise in experimental data. This creates an immediate problem in
that subjects’ behaviour might suggest that they were naive on Set 1, resolute on Set
2 and sophisticated on Set 317. What would we conclude then? More seriously,
because of their noise, we cannot be sure whether behaviour in Trees 1 and 2
actually does reveal EU or non-EU preferences. They may be EU but choose Up in
Tree 1 and Down in Tree 2 because of this error; similarly they may be non-EU and
yet choose Up in both Trees 1 and 2 (either with or without error). We need to take
into account of this noise. What we do then, rather than carry out a series of
individual tests on the data which require the assumption of zero noise, we use all
the data to estimate models of behaviour. Doing so enables us to use all the data on
each subject (24 observations), and, in particular, to try and discover the type (naive,
resolute or sophisticated) of each subject. Methodologically, it seems better to use all
the data on each subject to try and fit the various types rather than to carry out a
series of individual tests on bits of the data.
At this point it is necessary to note that we deliberately fixed the random number
generator in the real playing-out of the Trees 2 and 3, so that we could have
observations on all the subjects’ choices later in the tree. This is deception, and we
feel that we should comment on our use of it. Although we are, like all experimental
economists, averse to practicing deception, we note two things18: first, that the
Table 3 Number of subjects with consistent behaviour, and total number, in each set
Set 1 Set 2 Set 3
K and M
L and O
K but O
L but M
17Recall that subjects were presented with three sets of trees with the structure as in Fig. 2.
18We should also note that one of the referees asked us to record that he or she “judged that this feature of
the design was both unethical and unnecessary”. We agree with the ‘unethical’ but would argue that it was
necessary to increase the number of useful observations and save expense on paying the subjects: if we
had not used this deception, we would have had to have many more subjects in the experiment.
J Risk Uncertain (2009) 38:1–2513
deception was in the subjects’ interests in that they were no worse off with the
deception than without it; second, that without this deception we would have had
significantly fewer observations, crucially those which enable us to distinguish
between the various types (naive, resolute and sophisticated). We did not, however,
fix the generator during the practice playing out.
Our data analysis strategy is the following. We assume that subjects are different,
both in terms of their preferences and in terms of their type (naive, resolute or
sophisticated). We fit types to the data from each subject individually. We use all the
data (24 observations) on each subject with our ultimate goal of telling whether the
subject is (more likely19to be) naive, resolute or sophisticated.
While the analysis of the data on decisions only requires us to know preferences
over a relatively small number of lotteries, the analysis of the data on evaluations of
the trees requires us to know preferences not only over these lotteries but also over
all the evaluations of these lotteries (in monetary terms by the subject). For each
subject, there are 12 of these evaluations (the bids made for each of the 12 trees). In
Table 4 Predictedbids andnumber of subjects consistent with the predictions,andtotal number, in each set
Set 1 Set 2 Set 3
consistent totalconsistent total consistenttotal
K and M T1=T2
L and O
K but O
L but M
0 4 06
19We note that the presence of this error means that inevitably our inferences can not be certain.
14 J Risk Uncertain (2009) 38:1–25
each set of four trees, there are ten lotteries, involving five amounts of money. We Download full-text
neither know which type the subject is, nor the preference functional of the subject.
This has to be estimated. Given that we have just 24 observations per subject, it is
clear that we cannot estimate the evaluation of each of the 30 lotteries (ten lotteries
in each set of trees) involved in the experiment; we have to make some restrictions.
Accordingly we assume a particular form of the preference functional—to be precise
that of the Rank Dependent Expected Utility model. This seems to be well accepted
in the literature as the empirically most-valid generalisation of EU; moreover, it
contains EU as a special case.
The Rank Dependent functional is composed of a utility function and a
probability weighting function; we denote the former by u(.) and the latter by w(.).
The Rank Dependent Expected Utility, U(G), of a gamble G=(x1, x2,…,xI; p1, p2, …,
pI), where the prospects are indexed in order from the worst x1to the best xI, is given
ð Þ ¼ u x1
ð Þ þ
ð Þ ? u xi?1
ðÞ½?w piþ piþ1þ ::: þ pI
We note that Rank Dependent Expected Utility preferences reduce to Expected
Utility preferences when the weighting function is given by w(p)=p.
To fully characterise the preferences of a subject obeying the Rank Dependent
Expected Utility model, we need to know the utility function u(.) and the weighting
function w(.). As we describe later, we assume particular functional forms for these
two functions and estimate the parameters of the functions. We choose the functional
forms which best fit the responses of each subject—in a way that we will describe
We now need to have a story about the noise in the subjects’ responses—more
technically, we need to specify the stochastic structure of the data. There are various
stories that one can use and we choose to follow a Fechnerian measurement error
story20. To be precise, we assume that, when evaluating any lottery (whether certain
or risky), the subject makes a measurement error. More specifically, if u(.) is the
utility function of the individual, and u−1(.) its inverse, then we assume that the
evaluated certainty equivalent of any gamble G is given by u−1(U(G))+e, where
U(G) is the Rank Dependent Expected Utility of the gamble (using equation 1) and
20Before arriving at this particular specification, we tried several others, of varying degrees of
sophistication. One simple alternative was that subjects made all evaluations with error but then
‘trembled’ (see Moffatt and Peters 2001) when taking decisions and when making bids; the trouble with
this story (in addition to the fact that it does not seem empirically valid) is that, while the tremble story is
simple to apply to decisions, it is not obvious how to interpret it with respect to bids. There are also other
variations that we have tried on the basic story that we report in this paper; in particular, we explored the
hypothesis that subjects made no mistakes when evaluating certainties—this performed worse than the
variant reported in the paper; and also a variant that takes into account that the extreme value distribution
incorporates a bias (the expected value of a variable with an extreme value distribution with parameters m
and 1/s is not m but rather m+γs where γ is the Euler-Mascheroni constant 0.5772156649), by exploring
the notion that the subjects corrected for this bias when making their bids; this also performed worse than
the variant we have used in the paper.
J Risk Uncertain (2009) 38:1–25 15