Tempospect Theory of Intertemporal Choice

doi:10.4236/psych.2012.38082

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Psychology

2012. Vol.3, No.8, 555-557

Published Online August 2012 in SciRes (http://www.SciRP.org/journal/psych) http://dx.doi.org/10.4236/psych.2012.38082

Tempospect Theory of Intertemporal Choice

Taiki Takahashi, Ruokang Han

Department of Behavioral Science, Center for Experimental Research in Social Sciences,

Hokkaido University, Hokkaido, Japan

Email: [email protected]

Received May 3rd, 2012; revised June 5th, 2012; accepted July 6th, 2012

Anomalies in intertemporal choice (e.g. hyperbolic discounting, subadditive discounting, a sign effect, a

magnitude effect, and a delay-speedup asymmetry) have been investigated in neuroeconomics and be-

havioral neuroeconomics. In this study we propose a “tempospect” theory of intertemporal choice which

can account for these anomalies in intertemporal choice. The key features of the present theory are: 1) de-

cision over time is made with psychological time; and 2) psychological time is determined by a change in

delay until receipt (i.e., positive or negative time-interval between options); 3) psychological time is less

sensitive to a decrease in delay in comparison to an increase in delay; and 4) psychological time is influ-

enced by the sign and magnitude of the delayed outcomes. Implications of the present theory for neu-

roeconomics are discussed.

Keywords: Time Discounting; Psychological Time; Psychophysics; Behavioral Economics;

Neuroeconomics

Introduction

People tend to devalue the subjective value of outcomes as

delay until its receipt increases, which is referred to as temporal

discounting (intertemporal choice). Studies in behavioral eco-

nomics (Loewenstein & Prelec, 1992; Frederick et al., 2002;

Read & Roelofsma, 2003; Scholten & Read, 2010) and neu-

roeconomics (McClure et al., 2004; Kable & Glimcher, 2007;

Takahashi, 2009) demonstrated several anomalies in intertem-

poral choice (temporal discounting). Neuropsychopharmacolo-

gical studies indicate that these anomalies are related to addic-

tion (Bickel & Marsch, 2001). Since its introduction by

Samuelson (1937), an exponential discounting model for in-

tertemporal choice has dominated in economic theory. In ex-

ponential discounting, people are assumed to be time-consistent,

because time-discount rate is constant over time. However, later

empirical evidence suggests that several anomalies exist in

human and animal intertemporal choice behavior (Thaler, 1981;

Loewenstein & Prelec, 1992; Frederick et al., 2002). The im-

portant anomalies are 1) hyperbolic discounting; 2) subadditive

discounting; 3) a sign effect; 4) a magnitude effect; and 5) a

delay-speedup asymmetry (see Loewenstein & Prelec, 1992;

Frederick et al., 2002; Scholten & Read, 2010, for a review).

We briefly explain these anomalies below.

Hyperbolic Discounting

Time preferences between two delayed outcomes often

switch when both delays are increased by a given constant

time-interval. For instance, a person might prefer one apple

today to two apples tomorrow, but at the same time, prefer two

apples in 101 days to one apple in 100 day. This behavioral

tendency gives rise to time-inconsistent behavior (Strotz, 1956).

In other words, people tend to make patient plans in the distant

future, but make impulsive (impatient) actions in the near future.

It is to be noted that in exponential discounting, this type of

time-inconsistency does not exist because the time-discount

rate is constant over time. Recently, McClure et al. (2004) and

Kable and Glimcher (2007) examined the neural correlates of

hyperbolic discounting.

Subadditi ve D iscounting

Similar to human probability judgment, temporal discounting

is subadditive. Consider someone judging the present value of

an outcome to be received in one month. He or she can sepa-

rately discount for each of the four weeks in the month, or dis-

count once for the unbroken one month. Subadditive discount-

ing (Read, 2001) means that the total discounting is greater

when the month is divided into weeks. This anomaly is a sharp

evidence against Samuelson’s discounted utility theory (Schol-

ten & Read, 2010).

Sign Effect

Empirical studies reported that loss is less steeply time-dis-

counted than gain (Thaler, 1981). In Thaler (1981)’s study,

time-discount rates for gains were three to ten times greater

than those for losses. A more recent study also found the sign

effect in temporal discounting (Estle et al., 2006).

Magnitude Effect

Behavioral economic studies revealed that larger gains are

less steeply time-discounted than smaller ones (Thaler, 1981;

Estle et al., 2006). Similar to the sign effect in intertemporal

choice, this anomaly challenges the assumptions in Samuelson’s

discounted utility model (Loewenstein & Prelec, 1992; Freder-

ick et al., 2002).

Delay-Speedup Asymmetry

Loewenstein (1988) demonstrated that time-discount rates

can be dramatically affected by whether the change in delivery

T. TAKAHASHI, R. HAN

time of an outcome is framed as an acceleration (“speedup”) or

a “delay” from a certain temporal reference point. For instance,

people who didn’t expect to receive a product for another year

would pay an average of $54 to receive it immediately, but

those who thought they would receive it immediately demanded

an average of $126 to delay its receipt by a year. In other words,

time-discount rate is larger for “delay” than for “speedup” of

receiving the delayed reward. This anomaly also contradicts

with the assumptions in discounted utility theory (Loewenstein

& Prelec, 1992; Frederick et al., 2002).

Taken together, there has been accumulating evidence

against the standard exponentially discounted utility model. To

date, however, no theory is capable of accounting for these

important anomalies, although some behavioral economists

proposed descriptive models of intertemporal choice (Frederick

et al., 2010; Scholten & Read, 2010). In this study, we propose

a model which can explain all of the anomalies in intertemporal

choice by incorporating nonlinear and delay-speedup asymmet-

rical subjective time in decision over time.

Tempospect Theory for Intertemporal

Decision-Making

Temporal Cognition in Intertemporal Choice

In order to account for the anomalies in intertemporal choice,

various theoretical models have been proposed (Loewenstein &

Prelec, 1992; Takahashi, 2005; Takahashi, 2006; Takahashi,

2009; Scholten & Read, 2006; Scholten & Read, 2010). Gener-

ally speaking, Loewenstein and Prelec’s theory is a value-based

account of the anomalies in intertemporal choice, in contrast,

Scholten and Read’s group and Takahashi’s theories are time-

based accounts. In Takahashi’s theory, both hyperbolic (Taka-

hashi, 2005) and subadditive discounting is due to nonlinearity

in psychological perception (weighting) of delay or time-in-

terval between options. Specifically, time-interval in the distant

future is perceived as a shorter psychological time in compari-

son to that in the near future, which can explain why people are

patient regarding the distant future but impatient regarding the

near future (hyperbolic discounting). Zauberman et al. (2009)

experimentally confirmed the predictions from Takahashi’s

nonlinear time-perception theory of hyperbolic and subadditive

discounting. However, neither Loewenstein-Prelec theory,

Scholten-Read theory, nor Takahashi’s theory totally explains

all the anomalies mentioned earlier. In order to explain all the

anomalies mentioned, the present study generalizes the time-

based accounts proposed by Scholten-Read and Takahashi. In

generalizing the time-based accounts, we specify the functional

forms of psychological time and temporal discounting, in order

to future parametric applications in behavioral economics and

neuroeconomics, although the main conclusions of the present

study do not depend on the precise functional forms.

Time Discounting with Psychological Time

Takahashi (2005) proposed that subjects utilize the following

psychological time in their intertemporal choice:

  

ln 1.DD (1)

where τ(D) is subjective time (or time-weighting) of delay D

when the delayed outcome is obtained. If we further assume

that subject discount the delayed outcome x exponentially with

the psychological time, we obtain

















,,0expVxDVxk D. (2)

where V(x, D) is the time-discounted value of the delayed out-

come x obtained at delay D. Equation (2) is hyperbolic in

physical delay D (Takahashi, 2005). It is to be noted that Equa-

tion (2) is equivalent to Loewenstein and Prelec’s generalized

hyperbola (Loewenstein & Prelec, 1992) and the “q-exponential”

time-discount model developed in econophysics (Cajueiro,

2006; Takahashi et al., 2007; Takahashi, 2007; Takahashi et al.,

2008; Destefano & Martinez, 2011). Takahashi (2005) claimed

that Equation (2) can account for decreasing impatience (hy-

perbolic discounting) over physical time, although the discount

rate with respect to psychological time (k > 0) is independent of

psychological time, and Zauberman et al. (2009) confirmed this

claim in their behavioral economic experiment.

Tempospect Theory of Intertemporal Choice

Scholten and Read (2010) and Takahashi (2006) stated that

intertemporal choice occurs with respect to time-intervals rather

than delay (time-point) per se at which the delayed outcome is

obtained. Therefore, it is natural to generalize Equation (1) to

the following equations:









ln 1



  (for ΔD > 0). (3)













ln 1

DD  (for ΔD < 0). (4)

where ∆τ(∆D) is subjective time-interval (what we call “tem-

pospect”) between options which are obtained at the physical

time-interval of ∆D (∆D > 0 and ∆D < 0 correspond to “delay”

and “speedup” of the delayed option, respectively). Here we

make a natural assumption, as is the case with the gain-loss

asymmetrical value function in Kahneman-Tversky’s prospect

thery (Kahneman & Tversky, 1979; Tversky & Kahneman,

1992) in which an increase in outcomes less dramatically im-

pacts subjective valuation than a decrease in outcomes (referred

to as “loss aversion”), that ∆τ is larger for “delay” (an increase

in delay until receipt, which may be perceived as “loss”) than

“speedup” (a decrease in delay until receipt, which may be

perceived as “gain”), when the outcome is positive (gain). Op-

positely, ∆τ may be smaller for “dalay” than for “speedup”

when the outcome is negative (loss), because an increase in

delay in waiting for loss may be perceived as gain. Under these

assumptions, the time-interval discount function may be:

















,,expVxtD VxtkD  (for ∆D > 0). (5)

















,,expVxtD VxtkD  (for ∆D < 0). (6)

where V(x, t + ∆D) is the subjective value of the delayed out-

come obtained at delay t + ∆D. As can be seen from Equation

(5), when ∆τ is large, the subjective value of the outcome is

small. Considering the assumption that ∆τ may be larger for

“delay” than “speedup” in waiting for delayed gain, we can

suppose that subjective value of the delayed reward changes

more dramatically when the option is delayed than sped up.

Concerning the characteristics of the outcome x, when x is

negative (loss), ∆D may have smaller impact on decision over

time, than when x is positive (gain), because subjective valua-

tion of the outcome, rather than ∆D, may have stronger impact

when x is negative, due to loss aversion in the value function in

prospect theory (Kahneman & Tversky, 1979; Tversky & Kah-

neman, 1992). This can explain the sign effect in intertemporal

556

T. TAKAHASHI, R. HAN

choice. Also, when the outcome is large, ∆D may have smaller

impact on decision over time, again the subjective value of the

outcome, rather than ∆D, may have stronger impact. This may

account for the magnitude effect in intertemporal choice. Taken

together, our present “tempospect” theory can account for the

important anomalies in intertemporal choice.

Implications for Behavioral Economics and

Neuroeconomics

Since Loewenstein and Prelec (1992)’s proposal, studies in

intertemporal choice have mainly focused on the value-based

account of the anomalies in intertemporal choice. Subsequent

studies examined the roles of the shape of value functions

which can explain the anomalies in intertemporal choice (Al-

Nowaihi & Dhami, 2006; Al-Nowaihi & Dhami, 2009). Our

present study emphasizes, in line with perspectives proposed by

Scholten and Read’s group (Scholten & Read, 2006, 2010), the

roles of psychological time regarding time-intervals between

options. By estimating the functional forms of the psychologi-

cal time for time-intervals, we will be able to establish more

precise functional forms of temporal discounting. Also, inves-

tigations into neural processing underlying psychological time

in decision over time may help establish more effective medical

treatments for addiction and other impulsive and problematic

behaviors observed in psychiatric illnesses (Takahashi, 2009,

for a review).

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