Journal of Quantum Information Science, 2012, 2, 119-122
http://dx.doi.org/10.4236/jqis.2012.24018 Published Online December 2012 (http://www.SciRP.org/journal/jqis)
An Experimental Comparison of Quantum Decision
Theoretical Models of Intertemporal
Choice for Gain and Loss
Taiki Takahashi1, Hiroshi Nishinaka2, Takaki Makino2, Ruokang Han1, Hiroki Fukui2
1Department of Behavioral Science, Center for Experimental Research in Social Science, Hokkaido University, Sapporo, Japan
2Department of Forensic Psychiatry, National Institute of Mental Health National Center of Neurology and Psychiatry, Tokyo, Japan
Received September 10, 2012; revised October 13, 2012; accepted October 24, 2012
ABSTRACT
In mathematical physics and psychology, “quantum decision theory” has been proposed to explain anomalies in human
decision-making. One of such quantum models has been proposed to explain time inconsistency in human decision over
time. In this study, we conducted a behavioral experiment to examine which quantum decision models best account for
human intertemporal choice. We observed that a q-exponential model developed in Tsallis’ thermodynamics (based on
Takahashi’s (2005) nonlinear time perception theory) best fit human behavioral data for both gain and loss, among
other quantum decision models.
Keywords: Discounting; Neuroeconomics; Econophysics; Quantum Decision Theory; Tsallis’ Statistics
1. Introduction
Intertemporal choice and temporal discounting (devalue-
tion of delayed reward as delay until its receipt increases)
have been attracting attention in quantum decision theory
[1], econophysics [2-4], and neuroeconomics [5]. It is to
be noted that intertemporal choice has originally been
investigated in economics [6]. In these disciplines, sev-
eral mathematical models have been proposed to account
for human intertemporal choice. Notably, [7] proposed
intertemporal choice models based on quantum theoretic-
cal foundations (“quantum decision theory”). In mathe-
matical physics and psychology, recent studies developed
this type of quantum decision models in order to explain
various anomalies in human decision and cognition [7-9].
However, to date, no study has experimentally examined
the explanatory powers of the quantum decision theo-
retical models for human intertemporal choice. This issue
is important for future studies in quantum information
and decision theory in mathematical physics, econo-
physics, and neuroeconomics. In this study, we experi-
mentally examined explanatory powers of intertemporal
choice models based on quantum decision theory, in re-
lation to psychophysical theory of intertemporal choice
[10] based on Tsallis’ thermostatistics [2].
Intertemporal and Probabilistic Choices
In standard economics, the following exponential time-
discount model was proposed [11]:



 
0exp e VD VkD

(1)
where V(D) is a subjective value of monetary outcome
subject receives or pays at delay D and impulsivity pa-
rameter eis an exponential time-discount rate. In be-
havioral neuroeconomics, a hyperbolic discounting mo-
del has been widely employed [5,6]:
k
 
0
1h
V
VD kD  (2)
where V(D) is a subjective value of outcome which sub-
ject receives/pays at delay D and impulsivity parameter
is a hyperbolic time-discount rate.
h
k
In econophysics, Cajueiro [2] proposed a q-exponent-
tial model based on Tsallis’ statistics [12] to generalize
exponential and hyperbolic discounting:
 
 

11
0
11 q
q
V
VD
kqD



(3)
where V(D) is a subjective value of outcome which sub-
ject receives/pays at delay D and impulsivity parameter
q is a q-exponential time-discount rate at delay D = 0.
Parameter q generalizes the exponential function. Equa-
tion (3) is equivalent to an exponential model Equation
(1) when q is 1 and equivalent to a hyperbolic model
Equation (2) when q is 0. When q is less than 1, the
k
C
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T. TAKAHASHI ET AL. 120
agent’s temporal discounting is referred to as “decreasing
impatience”, which may result in preference reversal
over time.
Recently, on the other hand, Yukalov and Sornett [1]
proposed several types of intertemporal choice models
based on quantum decision theory (QDT). In their mod-
els, it is basically assumed that
 
00
dπ,,π,
dttk tttt t 0
(4)
where

0
π,tt is a “prospect state” and is a
time-dependent time-decay (discount) rate. It is to be
noted that in Yukalov and Sornett’s temporal discounting
model, temporal discounting is assumed to occur due to a
decrease in probability as delay increases, in line with
Sozou’s evolutionary theory of hyperbolic discounting
[13,14]. Based on this assumption, Yukalov and Sornett
[1] derived several intertemporal choice models (i.e.,
discount functions, in terms of economic theory). The
general form for the Yukalov-Sornett time-discount func-
tions is

0
,ktt


0
 

0
,1 n ttt t 

 (5)
where

and n are free parameters and the time-scale is
set at 1 day. [1] further derived several simplified tem-
poral discounting models (Models 1, 2, 2’, 3, 4, see [12],
for explicit functional forms of the QDT-based models)
based on reasonable approximations. However, little is
known regarding which model is the best for explaining
human intertemporal decision-making. Therefore, we ex-
perimentally investigated the explanatory powers of the
proposed models based on QDT and the q-exponential
time-discount model based on Tsallis’ thermostatistics
(and Takahashi’s logarithmic time perception theory of
“hyperbolic” discounting [10,15,16].
2. Method
2.1. Participants
Forty-one (22 male and 19 female) students were re-
cruited from Chuo University in Japan to take part in the
experiment. The mean age was 23.31 years old (standard
deviation = 3.20).
2.2. Procedure
The participants were asked to perform time discounting
tasks of both gain and loss. They were seated individu-
ally in a quiet room, facing the experimenter across a
table. Then they received a simple instruction that they
were asked to choose from a series of alternatives of
monetary amounts either gain or loss with certain prob-
abilities and imagine them, though hypothetical, as real
money in this experiment. The employed time discount-
ing tasks were the same as our previous studies [15,16].
2.3. Data Analysis
The procedures of data analysis in the present study is
similar to our previous studies [4,14]. Switching points
of the time discounting tasks were defined as the means
of the largest adjusting amount in which the standard
alternative choice and the smallest adjusting amount in
which the adjusting alternative choice. Indifference
points of individuals were calculated by averaging the
switching point in the ascending and descending adjust-
ing conditions. The indifference points of the group data
were obtained by calculating the medians of individuals’
indifference points in order to compare the goodness-
of-fit among the models based on QDT and Tsallis’ sta-
tistics at the group level. The fitness of each equation
was estimated with AIC (Akaike Information Criterion)
values, which is the most standard criterion for the fit-
ness of mathematical model to observed data. All statis-
tical procedures were conducted with R statistical lan-
guage.
3. Result
We present fitted parameters and goodness-of-fit (AIC)
in Tables 1 (gain) and 2 (loss). From these analyses, we
can see that the q-exponential time-discount model fit the
behavioral data best for both gain and loss, consistent
with logarithmic time-perception theory of intertemporal
choice proposed by [10]. It is also to be noted that q vales
in the q-exponential discounting were smaller than 1 for
both gain and loss, indicating decreasing impatience
(“hyperbolic” discounting) for both signed outcomes.
Furthermore, we can see that Model 2 and 4 are mathe-
matically equivalent (Tables 1 and 2).
4. Discussion
To our knowledge, this study is the first to experiment-
tally examine time discount models for gain and loss
proposed in the Quantum Decision Theory and the
q-exponential discounting model based on deformed al-
gebra developed in Tsallis’ thermostatistics. Our results
demonstrated that 1) Model 2 and 4 (temporal discount-
ing with non-zero limit at large delays) best fit among the
quantum decision theoretic intertemporal choice models,
and 2) the q-exponential model best fitted human tem-
poral discounting behavior for both gain and loss. It is to
be emphasized that, even when novel temporal discount-
ing models proposed in QDT, the q-exponential time-
discount model best accounted for human intertemporal
choice for both gain and loss, consistent with our previ-
ous studies [10,15,16]. Therefore, future studies in
econophysics, QDT, and neuroeconomics should model

T. TAKAHASHI ET AL. 121
Table 1. Parameters and AICs of temporal discounting for gain.
Model 1 Model 2 Model 2’ Model 3 Model 4 q-exponential
discounting
AIC 125.4823 103.1162 106.0987 122.3969 103.1162 99.93185
Parameter γ γ n γ n γ γ n k q
1.916 × (10)−4 1.418 × (10)−2 0.6.3458 1.024 × (10)−20.585476.395 × (10)−21.418 × (10)−20.63458 2.0941 × (10)−3−3.6593555
Model 1 is an exponential discounting model, Model 2 reduces to an exponential discounting at short delays, Model 2’ is a stretched exponential discounting
model, Model 3 is generalized hyperbolic function, Model 4 is close to an exponential model at short delays and tends to non-zero limit at large delays. See
Yukalov and Sornette (2009) for details. The q-exponential discounting based on Tsallis’ statics is derived from logarithmic time-perception theory of hyper-
bolic discounting (Takahashi, 2005). Note that smaller AIC indicates better fitting.
Table 2. Parameters and AICs of temporal discounting for loss.
Model 1 Model 2 Model 2’ Model 3 Model 4 q-exponential
discounting
AIC 105.9509 99.47295 99.35718 119.116899.47295 97.9409
Parameter γ γ n γ n γ γ n k q
5.667 × (10)−5 8.374 × (10)−4 0.3502031 8.169 × (10)−40.34733630.0245378.374 × (10)−40.350203 1.445 × (10)−4−2.74
Model 1 is an exponential discounting model, Model 2 reduces to an exponential discounting at short delays, Model 2’ is a stretched exponential discounting
model, Model 3 is generalized hyperbolic function, Model 4 is close to an exponential model at short delays and tends to non-zero limit at large delays. See
Yukalov and Sornette (2009) for details. The q-exponential discounting based on Tsallis’ statics is derived from logarithmic time-perception theory of hyper-
bolic discounting (Takahashi, 2005). Note that smaller AIC indicates better fitting.
human intertemporal choice with the q-exponential dis-
count model, which can be derived from Takahashi’s
nonlinear time perception theory of hyperbolic discount-
ing [10].
5. Acknowledgements
The research reported in this paper was supported by a
grant from the Grant-in-Aid for Scientific Research
(Global COE and on Innovative Areas, 23118001; Ado-
lescent Mind & Self-Regulation) from the Ministry of
Education, Culture, Sports, Science and Technology of
Japan.
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