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
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
where e is an error—a measurement error. To complete the story, we need to specify
the distribution of e. We do this in a way that is tractable and not unreasonable—we
assume that e has an Extreme Value distribution with parameters 0 and 1/s. Thus the
cumulative distribution function, F(.) of e is given by:
F e ð Þ ¼ exp ?exp ?es
It follows that the probability density function f(.) is given by:
f e ð Þ ¼ exp ?exp ?es
To summarise: we assume that the subjects each have a well-defined (Rank-
Dependent) preference function and that each subject is either naive, resolute or
sophisticated. For each subject we find the best-fitting preference function and the
best-fitting type (naive, resolute or sophisticated) to represent the subject’s responses
on the experiment. The responses are the bids (for the four trees in each of the three
sets) and the decisions (in four trees on each of the three sets)—a total of 24
observations for each subject. We note that the nature of the data is different—for the
bids we have a number, while for the decisions we have their choice. The former is
essentially a continuous variable while the latter is a discrete variable (taking one of
either two or three values). The analysis of the two kinds of data has to be different.
The details are given in the next section, where we are more specific about how we
have interpreted and applied our stochastic specification.
6 The stochastic and functional specifications
We have assumed that subjects evaluate lotteries with error. More specifically we
have assumed that the expressed valuation, VG, of a lottery G with true certainty
equivalent u−1(U(G)) is given by VG=u−1(U(G))+e, where e has an extreme value
distribution with parameters 0 and 1/s. When taking a decision between lottery A or
lottery B, we assume that the subject evaluates both A and B with error and chooses
the lottery with the highest evaluation. Given the properties of the extreme value
distribution, it follows that the probability that A is chosen is
probability that B is chosen is
VB the true valuation of lottery B. We note the importance of the parameter s: when s
is zero then both lotteries are chosen with probability one-half, and when s is infinite
then the lottery with the highest true valuation is chosen with certainty. We can
therefore interpret s as indicating the precision of the subject’s decision—the higher
is s the more precise is the subject, and the more likely he or she is to choose the
lottery with the truly highest value.
The above story can easily be extended to a choice between three lotteries A, B
and C: given our error story, lottery Z (=A, B or C) is chosen with probability
This error story also enables us to determine the probability density of the bids.
We assume that if a tree offers two choices—between lotteries A and B—then the bid
is put equal to the maximum of the evaluations of A and B. Using the properties of
the extreme value distribution, and assuming that the errors in the valuations are
eVAsþeVBs and the
eVAsþeVBswhere VA is the true valuation of lottery A and
16 J Risk Uncertain (2009) 38:1–25
independent, it follows that the cumulative distribution function of the bid x is given
exp ? exp ? x ? VA
ð Þ þ exp ? x ? VB
From equation 4 we can find the probability density of the bid x based on two
An obvious extension leads us to the cumulative distribution function of the bid
when the tree offers three lotteries A, B and C and the bid is put equal to the
maximum of the evaluations of these three lotteries:
exp ? exp ? x ? VA
From this we can find the probability density of the bid x based on three lotteries.
We now need to say something about how the different types are assumed to
behave in the four trees. Tree 1 is simple and all types do the same thing. The
decisions are based on the evaluations of M (Up) and O (Down), and the bids are
made on the basis of the maximum of these two evaluations.
In Tree 2 different types do different things. The naive subject takes the decision
at the decision node and this decision is based on the evaluations of K (Up) and L
(Down); however the bid is based on the evaluations as viewed from the beginning
of the tree—from which perspective the choice is between M and O—and hence the
bid is based on the maximum of the evaluations of M and O. The resolute subject,
however, bases both the decision and the bid as viewed from the beginning of the
tree—that is on the basis of the evaluations of M and O. The sophisticated works
backwards: he or she anticipates that the decision at the decision node will be based
on the relative evaluations of K and L; if K is chosen he evaluates the tree as being
worth the evaluation of M; if L is chosen the tree is worth the evaluation of O. He or
she can work out the probability of choosing K and L and hence work out the
expected evaluation of the tree—on which he or she bases his bid.
In Tree 3 again the different types do different things. The actual decision of a
naive subject at the second decision node is based on the evaluations of L and K, but
the naive agent does not anticipate this when taking the decision at the first node and
in making the bid for the tree. These are determined by the evaluations of M, N and
O. If either the evaluations of M or O are bigger than the evaluation of N the naive
agent chooses Up at the first node; otherwise he or she chooses Down. The bid of a
naive is simply determined by the maximum of the evaluations of M, N and O. In
contrast a resolute subject bases both his decisions at both nodes on the evaluations
of M, N and O: if the evaluation of M is the highest he or she chooses Up and then
Down; if the evaluation of O is the highest he or she chooses Up and then Up; and if
the evaluation of N is the highest, he or she chooses Down; the bid of a resolute
subject is based on the maximum of the evaluations of M, N and O. The
sophisticated subject once again anticipates his or her decision at the second node—
this will be Up if L is evaluated more highly than K and Down otherwise. If L is
evaluated more highly than K then his or her decision at the first node (and the bid
for the tree) is based on the evaluations of O and N; if, however, K is evaluated more
highly than L then his or her decision at the first node (and the bid for the tree) is
based on the evaluations of M and N.
ð Þ þ exp ? x ? VB
ð Þ þ exp ? x ? VC
J Risk Uncertain (2009) 38:1–25 17
Tree 4 is simple and all do the same: the decisions and the bids are all based on
the evaluations of M, N and O.
There is one slight complication that we have so far ignored: and that concerns what
exactly a sophisticated person does when working backwards—when backwardly
inducting.We haveassumedsofar thatthisagent eliminatesthe branchesofthe treethat
he or she will not be following in the future, and then simplifies the tree by using
reduction—that is, reducing the remaining compound lottery to a simple lottery using
the usual probability rules. But there is an alternative—that this agent simplifies the
tree by substituting in certainty equivalents of the remaining bits of the tree. Segal
(1999) notes and discusses this distinction. Accordingly we should distinguish
between two types of sophisticated agents—those who work backwards using
reduction and those who work backwards using certainty equivalents. We refer to
these as Type 1 sophisticated and Type 2 sophisticated respectively. This distinction
does not affect the story we have told above about how sophisticated agents process a
dynamic tree, but it does affect the specification of the appropriate likelihood
We confine all the technical details to Web Technical Appendix 2 (with the
GAUSS program for one of the combinations in Web Technical Appendix 3).
Suffice it to say here that we estimate the parameter of the utility function, the
parameter of the weighting function and the precision parameter s using maximum
likelihood (implemented in GAUSS), subject by subject and type by type. To do this
we need to specify the likelihood of the observations. This is different for the
decision and for the bids: in essence the likelihood of a decision is the probability
that that decision is taken given the parameters and the stochastic specification; the
likelihood of a bid is the probability density at the bid given the parameters and the
There is one further problem that we need to consider—the specifications of the
utility function and of the weighting function. Ideally one would not specify functional
forms but estimate the functions at all possible values. The problem with this is, as we
have already noted, that there are too many values to make this procedure possible:
given the number of observations that we have (24 for each subject), we would lose
too many degrees of freedom. So we are forced to assume functional forms. The most
obvious ones are the CARA and CRRA specifications for the utility function
(u x ð Þ / ?exp ?rx
specification for the weighting function (w p
respectively). The Quiggin specification allows for an S-shaped weighting function
while the Power specification does not. In order to ensure the robustness of our results
we use all four possible combinations: CRRA with Power; CRRA with Quiggin;
CARA with Power; and CARA with Quiggin, and for each subject we choose the
ðÞ and u x ð Þ / xrrespectively) and the Quiggin (1982)22and Power
ð Þ ¼
ðÞ1=g and w p
ð Þ ¼ pg
21It should be noted that we do not consider reduction by substitution of certainty-equivalents for the
naive and resolute types. In this we follow McClennen (1990) which implies simplification by reduction to
hold for these models.
22To be strictly correct we should attribute this to Tversky and Kahneman (1992), who proposed this
variation on the original specification proposed by Quiggin, namely: w(p)=pg/[pg+(1-p)g]
18 J Risk Uncertain (2009) 38:1–25
combination which gives the highest maximised log-likelihood averaged over all four
(naive, resolute, Type 1 and Type 2 sophisticated) types23.
7 The results
To summarise: we have, for each subject, fitted the naive, resolute and (the two
versions of) sophisticated types to the 24 observations for each subject for each of
the four combinations of utility and weighting function. The detailed estimates are
available on request. An illustration of some of the results for one of the four
combinations (CARA with Quiggin) is provided in Web Technical Appendix 4. For
each type and for each combination we have the following information from our
The value of the maximised log-likelihood;
Information as to whether the maximum likelihood converged correctly;
The estimates and standard errors of (the transformed values24of) the parameters—
r, g and s.
(Hence) The estimates of the parameters.
We should note that for some of the specifications there were occasional problems
with the convergence of the maximum likelihood process, but there were
combinations (particularly CARA with Quiggin) that seemed to be particularly
robust. We should note that there is no guarantee that the likelihood function is
smoothly concave everywhere; accordingly we tried several starting values for the
maximum likelihood software. We are reasonably convinced that we have found the
overall maximum in essentially all cases.
The bottom line is the categorisation of subjects as to whether they are naive,
resolute or sophisticated. As we have already noted, we begin by choosing, for
each subject, the combination of utility function and weighting function that best
explains the data on the subject. In some cases this was simple: irrespective of the
type (naive, resolute or sophisticated) the same combination yielded the maximum
of the maximised log-likelihoods—there were 25 subjects for whom the data had
this property. For the rest we had to select the best combination in some manner;
our criterion was to select the combination for which the average maximised log-
likelihood (across all types) was maximised. This may seem somewhat arbitrary
but we have carried out an independent check by seeing if the chosen type might
have been different if we had chosen an alternative (but possible) combination. We
found that, for all except nine of the 50 subjects, the maximised log-likelihood
23We should note that for 25 subjects there was one combination which fitted best on all four specifications;
there were 20 subjects for whom one combination fitted best on three of the four specifications; and there were
five subjects for whom one combination fitted best on two of the four specifications. The conclusion seems to
be clear—for virtually all subjects, the data seems to be telling us that one combination (of utility function and
weighting function) fits best independently of the specification.
24To avoid problems with the maximum likelihood algorithm we transformed our parameters to restrain
their range (for example to stop the s parameter becoming negative). The returned estimates and variance-
covariance matrix are thus those of the transformed parameters—and they need to be transformed back
before they can be interpreted.
J Risk Uncertain (2009) 38:1–25 19
with our chosen ‘best’ combination and chosen ‘best’ type was, in fact, the highest
of all the 16 maximised log-likelihoods that we computed; for seven of these nine,
the maximised log-likelihood of our chosen ‘best’ type was higher than for any
other type in all the other combinations. Just in two cases (subjects 11 and 30)
would our chosen combination change our chosen type—subject 11, whom we
have classified as more likely to be naive, could be classified as being
sophisticated; and subject 30, whom we have also classified as being naive, could
possibly be reclassified as resolute. We feel that our classification of our types is
We give some examples of what we have done. Consider Tables 5, 6 and 7 in
which we give examples of three subjects which show different characteristics.
All the entries in the tables are maximised log-likelihoods. The rows indicate
the type (naive, resolute or the two types of sophisticated) and the columns indicate
the combination (of utility function and weighting function) used in the
Table 5 shows the log-likelihoods for a subject for whom the best type is the
same irrespective of the combination. (As we have already noted 25 of the 50
subjects had this property.) Indeed, for this subject the CARA/Power combination
performs better than the CARA/Quiggin combination and those two significantly
better than either of the combinations involving the CRRA utility function.
Moreover, independently of the combination, the naive type fits the data better
than the others. We therefore select the CARA/Power combination as representing
the subject’s preferences and conclude that this subject is more likely to be naive
than any of the other types.
Table 6 Examples of log-likelihoods: subject for whom best combination is the same for three of the types
Subject number 3 aqaprqrpBest Highest log-likelihood
Winner is resolute. For abbreviations, please refer to Table 5 legend
Table 5 Examples of log-likelihoods: subject for whom best combination is the same irrespective of the type
Subject number 8aq aprq rp BestHighest log-likelihood
Winner is naive
n naïve, r resolute, s1 Type 1 sophisticated, s2 Type 2 sophisticated, aq CARA and Quiggin, ap CARA
and Power, rq CRRA and Quiggin, rp CRRA and Power
20J Risk Uncertain (2009) 38:1–25
Subject 3, shown in Table 6, is somewhat less clear cut. For this subject the best
fitting combination depends upon the type; only by averaging across the types
are we able to declare the CARA/Quiggin combination ‘best’ for this subject.
There were 20 subjects for whom the estimates had this property. However, we
do note that, whatever the combination, the log-likelihood for the resolute type is
higher than for the other types. We therefore conclude that this subject is
Subject 46, shown in Table 7, is even less clear cut. Only five of the 50 subjects
were similar to this one. It will be seen that for two of the types (naive and resolute)
the CRRA/Power combination is best, whereas for the two sophisticated types the
CARA/Power combination is best. However, both sophisticated types are best for
three of the combinations and it is only for the CRRA/Power combination does the
sophisticated type come out worse than the other two types. To decide on the best
type for this subject, we have averaged the log-likelihoods over all types for each
combination and have chosen the combination for which this average is highest—
this leads us to the choice of the CARA/Power combination as best representing this
subject’s preferences. Given that, we note that the sophisticated likelihoods are the
largest in the CARA/Power combination—and hence we declare ‘sophisticated’ as
the winner. Note too, that of all 16 maximised log-likelihoods in the table (excluding
the means) those for the sophisticated with the CARA/Power combination are the
At this point we have chosen the best combination for each subject. At the
same time we have also chosen the best type for each subject (that is, the type
for which the maximised log-likelihood for the chosen combination is greatest).
However we also want to give an indication of how much better the best-fitting
type is relative to the others. To this end we proceeded as follows. For each
subject we have three maximised log-likelihoods (where we take just the better
of the two sophisticated specifications—that with the highest maximised log-
likelihood). Let us denote these by ll(n), ll(r) and ll(s) for the naive, resolute and
sophisticated types respectively. If we adopt a Bayesian interpretation of the
results, and if we start with equal priors on the three types, then the posterior
probabilities of the naive, resolute and sophisticated types being the correct ones
P i ð Þ ¼
exp ll i ð Þ
Þ þ exp ll r ð Þ
Þ þ exp ll s ð Þ
exp ll n ð ÞððÞi ¼ n;r;s
Table 7 Examples of log-likelihoods: subject for whom best combination is the same for two of the types
Subject number 46aq aprqrp BestHighest log-likelihood
Winner is sophisticated (either Type). For abbreviations, please refer to Table 5 legend
J Risk Uncertain (2009) 38:1–2521
Hence, for example, for Subject 125on the CRRA plus Quiggin combination we
ll n ð Þ ¼ ?11:78101;ll r ð Þ ¼ ?14:15611 and ll s ð Þ ¼ ?12:49711
and hence, applying formula 6, we have that the posterior probabilities for the three
P n ð Þ ¼ 0:632;P r ð Þ ¼ 0:059 and P s ð Þ ¼ 0:309:
Hence, for this subject the naive type is over twice as likely as the sophisticated
type, and the resolute type is relatively very unlikely.
We have applied this analysis to each of the subjects (using the best fitting
combination of utility function and weighting function for each subject individually)
and present the results graphically in Fig. 3. In this triangle we represent the
probability of the naive type being correct on the horizontal axis, and the probability
of the sophisticated type being correct on the vertical axis. The probability of the
resolute type being correct is the residual. In the triangle subjects are indicated by a
number. The triangle is divided into three areas—the one to the top being where the
sophisticated type is most probable, the one to the right being where the naive type is
most probable and the one nearest the origin being where the resolute is most
probable. We note that there are 25 subjects in the “naive most likely area”, 20
subjects in the “resolute most likely area” and just five subjects in the “sophisticated
most likely” area.
Of course, for subjects whose preferences are those of Expected Utility, there is
no possibility of dynamic inconsistency and hence no meaning to the distinction
between the different types (naive, resolute and sophisticated). In principle,
therefore, we should exclude such subjects from our analysis. However, given the
stochastic nature of our data we cannot be sure whether a subject is EU or not,
though we do have estimates of the parameter g of the weighting function. If g is
equal to 1, the weighting function reduces to w(p)=p and the Rank Dependent model
reduces to Expected Utility. We can carry out formal tests (based on our estimates)
of the hypothesis that the g parameter is significantly different from 1. Out of the 50
subjects there are 28 subjects for whom the g parameter is significantly different
from 1 at the 1% level—and hence probably not subjects with Expected Utility
preferences. If we restrict our analysis to these 28 subjects we get Fig. 4. It will be
seen that there are 15 in the “naive most likely” area, 12 in the “resolute most likely”
and just 1 in the “sophisticated most likely”. These results strengthen our comments
We conclude that the sophisticated type performs consistently worse than the
other two, and that the naive type performs marginally better than the resolute type.
We comment further on these findings in the next and concluding section.
25In this case the maximised log-likelihoods for the Type 1 and Type 2 sophisticated specifications
were −13.32295 and −12.49711 and we accordingly prefer Type 2 to represent sophistication.
22 J Risk Uncertain (2009) 38:1–25
Dynamically inconsistent economic agents provide a serious challenge to economic
theory. In order to predict their future behaviour, one needs to know how they are
resolving any dynamic decision problem—in particular, for example, whether they
are naive, resolute or sophisticated. In conducting experiments to try to determine
whether subjects are naive, resolute or sophisticated, experimenters have a problem
in designing the experiment in such a way that they can infer what subjects are doing
when processing the dynamic decision problem. Essentially, the issue is concerned
Fig. 4 The ex post probabilities
for those subjects who appear to
have non-EU preferences
Fig. 3 The ex post probabilities of the various types for all subjects. Large number—g parameter
significantly different from 1 at 1%. Medium sized number—g parameter significantly different from 1 at
5%. Small number—g parameter insignificantly different from 1. Very small number—g parameter
insignificantly different from 1 but GAUSS unable to calculate standard errors
J Risk Uncertain (2009) 38:1–2523
with whether subjects are planning their future behaviour and anticipating any
possible future inconsistencies. Ideally, one would like to design an experiment
where any plans that the subject is making are revealed. However, there are serious
(possibly insurmountable26) difficulties in observing plans and hence in seeing
whether they are implemented. If one simply asks the subject what he or she is
planning to do, then, unless one forces the subject to implement that plan, there is no
incentive for the subject to report any plans honestly; moreover, asking them such a
question raises in the subject’s mind the idea that they possibly ought to plan.
Furthermore, if one insists that the subject implements the plan that he or she has
stated, then one changes the nature of the problem from a dynamic problem to a pre-
The experiment surmounts these difficulties with a unique design in which not
only behaviour is observed but also reported evaluations of different dynamic
decision problems are obtained. Moreover, we have constructed the decision
problems in such a way that we can distinguish, using data on both decisions and
reported evaluations, the decision process actually followed.
In particular, we can distinguish between naive decision makers (those who
ignore any possible future inconsistencies), resolute decision makers (those who are
resolute in implementing their a priori plans) and sophisticated decision makers
(who anticipate their future inconsistencies and who are not sufficiently resolute to
overcome them). These different types will have different behaviour in our trees as
well as different evaluations of the trees. We have used the data to infer the type of
each subject. The picture is somewhat clouded as we do not know ex ante the
preference functional of each individual, but we have investigated four different
combinations for each subject and chosen the best. While the final picture is not
totally clear, it seems to be the case that around 50% of our 50 subjects are naive,
40% are resolute and just 10% sophisticated.
The large number of resolute subjects and the small number of sophisticated
subjects in our experiment surprised us, as we thought ex ante that it would be
difficult for subjects to be resolute. However, taking into account that we obliged the
subjects to spend 15 min evaluating the trees (and practising playing them out), it
seems to have been the case that subjects used this time to work out an ex ante
strategy and to realise that it was better for them to implement that rather than
behaving in a sophisticated way: remember that sophisticated behaviour is not
optimal as viewed by the decision maker at the beginning of the tree. It would be
interesting to explore whether this finding is sensitive to the length of the time spent
contemplating the trees.
The implications for economic theory are significant. If we look at models which
incorporate dynamically inconsistent behaviour (such as the literature on quasi-
hyperbolic discounting in the context of a life-cycle saving model27), it will be seen
that most of these models assume sophisticated behaviour. Our results suggest that
this might be descriptively implausible. If subjects are indeed naive or resolute rather
than sophisticated, then the predictions of these models need to be modified
27See, for example, Harris and Laibson (2001).
26But see Busemeyer et al. (2000) for the psychologists’ way round the problem.
24J Risk Uncertain (2009) 38:1–25
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