An Experimental Comparison of Quantum Decision Theoretical Models of Intertemporal Choice for Gain and Loss

doi:10.4236/jqis.2012.24018

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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

Email: [email protected]

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]:

 

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.

In econophysics, Cajueiro [2] proposed a q-exponent-

tial model based on Tsallis’ statistics [12] to generalize

exponential and hyperbolic discounting:

 



11 q

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

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

 

dπ,,π,

dttk tttt

t 0

(4)

where



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



,ktt



 



,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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