Decision makers calibrate behavioral persistence on the basis
of time-interval experience
Joseph T. McGuire⇑, Joseph W. Kable
Department of Psychology, University of Pennsylvania, 3720 Walnut St., Philadelphia, PA 19104, USA
a r t i c l e i n f o
Received 21 October 2011
Revised 14 March 2012
Accepted 22 March 2012
Available online 23 April 2012
a b s t r a c t
A central question in intertemporal decision making is why people reverse their own past
choices. Someone who initially prefers a long-run outcome might fail to maintain that pref-
erence for long enough to see the outcome realized. Such behavior is usually understood as
reflecting preference instability or self-control failure. However, if a decision maker is
unsure exactly how long an awaited outcome will be delayed, a reversal can constitute
the rational, utility-maximizing course of action. In the present behavioral experiments,
we placed participants in timing environments where persistence toward delayed rewards
was either productive or counterproductive. Our results show that human decision makers
are responsive to statistical timing cues, modulating their level of persistence according to
the distribution of delay durations they encounter. We conclude that temporal expecta-
tions act as a powerful and adaptive influence on people’s tendency to sustain patient
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1.1. Failures of persistence
Intertemporal decision behavior can appear to be
dynamically inconsistent. As Ainslie (1975) framed the
problem, ‘‘people often change their preferences as time
passes, even though they have found out nothing new
about their situation’’ (p. 464). Reversals of choices in do-
mains as diverse and consequential as diet, addiction,
and financial planning create the impression that prefer-
ences are fundamentally unstable. Understanding the
cause of these reversals is important, since a tendency to
sustain the pursuit of delayed rewards correlates with
numerous positive life outcomes (Duckworth & Seligman,
2005; Mischel, Shoda, & Peake, 1988; Shoda, Mischel, &
The predominant theoretical explanations for such
reversals hold that multiple internal subsystems trade off
control over behavior. The relevant subsystems have been
variously characterized as cool vs. hot (Loewenstein,
1996; Metcalfe & Mischel, 1999), controlled vs. automatic
(Baumeister, Bratslavsky, Muraven, & Tice, 1998; Stanovich
& West, 2000), farsighted vs. myopic (Laibson, 1997;
McClure, Laibson, Loewenstein, & Cohen, 2004) or instru-
mental vs. Pavlovian (Dayan, Niv, Seymour, & Daw, 2006).
A related idea is that preference instability can arise from
non-exponential temporal discounting functions (Ainslie,
1975; Laibson, 1997; McClure et al., 2004; Strotz, 1955).
Previous theoretical enterprises have focused largely on
situations where decision makers hold full information
about the times at which future outcomes will occur.
However, the timing of real-world events is not always
so predictable. Decision makers routinely wait for buses,
job offers, weight loss, and other outcomes characterized
by significant temporal uncertainty. Timing uncertainty is
also a central feature of the well-known delay-of-gratifica-
tion paradigm (Mischel & Ebbesen, 1970), where young
children must decide how long to continue waiting for a
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⇑Corresponding author. Tel.: +1 215 746 4371; fax: +1 215 898 7301.
E-mail addresses: email@example.com (J.T. McGuire), kable@
psych.upenn.edu (J.W. Kable).
Cognition 124 (2012) 216–226
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preferred food reward, while lacking any information
about how long the delay will last. Even though persis-
tence is usually associated with successful self-control,
temporal uncertainty can create situations where limits
on persistence are appropriate (Dasgupta & Maskin,
2005; Rachlin, 2000). Our aim in the present paper is to
demonstrate that behavior resembling persistence failure
can arise as the rational response to uncertainty about an
awaited outcome’s timing.
1.2. Persistence under temporal uncertainty
A temporally uncertain outcome can be described in
terms of a probability distribution over its potential times
of arrival. Different timing distributions will apply to dif-
ferent categories of events, and the shape of the distribu-
tion determines how the expected remaining delay will
change as time passes. This general phenomenon has been
described previously in the contexts of survival and reli-
ability analysis (e.g., Elandt-Johnson & Johnson, 1980)
and Bayesian cognitive judgment (Griffiths & Tenenbaum,
2006). Here we present an overview focusing on the impli-
cations for intertemporal decision making (for quantitative
details see Fig. 2 and Section 2.3).
If delay durations in a given environment follow a uni-
form or Gaussian distribution, the expected remaining de-
lay will become steadily shorter as time elapses. Gaussian
distributions characterize delimited events, such as movies
or human lifetimes (Griffiths & Tenenbaum, 2006). Con-
sider, for example, the case of waiting for a talk to end. If
it has gone on longer than expected, one might be inclined
to assume that only a small amount of time still remains.
Fig. 1 illustrates this phenomenon for a Gaussian distribu-
tion (specifically, a truncated Gaussian with a lower bound
corresponding to the current time).
Under the standard assumption that rewards are
subjectively discounted as a function of their delay
(Samuelson, 1937), rewards with Gaussian timing will
tend to increase in present subjective value over time
while they are being awaited. If a delayed reward is ini-
tially preferred relative to other alternatives that are avail-
able immediately, this preference should strengthen as
time passes. All else equal, the initial patient choice should
A very different situation can occur if the reward’s tim-
ing follows a heavy-tailed distribution (e.g., a power func-
tion; see Fig. 1). In this case, the expected remaining delay
can increase with the passage of time. Heavy-tailed distri-
butions describe open-ended events, where some delays
are short but others are indefinitely long. Consider the
example of waiting for a reply to an email (Barabási,
2005). One might initially expect a reply to come quickly,
but if it does not, one might conclude that the remaining
delay will be longer than initially expected.
If a reward is characterized by a heavy-tailed timing
distribution, its expected delivery time grows more distant
with time elapsed, implying that its present subjective va-
lue progressively deteriorates. Even if the delayed reward
were initially preferred, it might eventually become so re-
mote that it no longer outcompeted immediately available
alternatives. Under these circumstances, decision makers
could produce reversing sequences of choices, equivalent
to the patterns often attributed to self-control failure: they
might choose a delayed reward, wait for a period of time,
and then shift to an immediate outcome instead. Such a
decision maker would not be dynamically inconsistent,
but would instead be responding rationally to new infor-
mation gained from observing the passage of time. There
is precedent for the idea that mere time passage may be
informative in this way, warranting reassessments of both
the delay and the degree of risk associated with future
events (Dasgupta & Maskin, 2005; Fawcett, McNamara, &
Houston, 2012; Rachlin, 2000; Sozou, 1998).
Heavy-tailed distributions characterize timing in a vari-
ety of real-life situations where intervals are open-ended.
Distributions with this form have been empirically docu-
mented in examinations of the time between emails
(Barabási, 2005), the length of hospital stays (Harrison &
Millard, 1991), and time between retrievals of the same
memory (Anderson & Schooler, 1991). Heavy-tailed distri-
bution is unknown (Gott, 1993, 1994; Jeffreys, 1983). It
seems plausible that decision makers routinely encounter
environments characterized by heavy-tailed timing statis-
tics, in which they must continually reassess whether a for-
merly preferred delayed outcome remains worth pursuing.
Decision makers are also likely to encounter situations
where timing is uncertain but delimited. For example,
endogenous variability in time-interval perception and
memory can produce a Gaussian pattern of subjective
uncertainty (i.e., scalar variability; Gallistel & Gibbon,
2000; Gibbon, 1977). This kind of situation would call for
persistence: if a delayed reward was worth pursuing in
the first place, it should be pursued until it is obtained.
The above observations lead to a hypothesis: a person’s
willingness to continue waiting ought to depend on a
Time (min)Time (min)
Fig. 1. Schematic illustration of how time passage may change a reward’s
expected time of arrival. The left and right columns represent different
kinds of beliefs one might hold about an awaited outcome’s timing. The
solid line represents the current time (shown at 0, 2, and 4 min). The
dashed line represents the outcome’s expected arrival time, defined as the
mean of the area to the right of the current time. For Gaussian beliefs
(mean = 3, SD = 1), the expected delay starts at 3 min and grows shorter
with time. For heavy-tailed beliefs (generalized Pareto distribution (see
Eq. (2)), k = 0.5, r = 1.5), the delay starts at 3 min and rises with time.
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