New concept for the value function of prospect theory

Abstract

In prospect theory, the value function is typically concave for gains and convex for losses, with losses usually having a steeper slope than gains. The neural system responds differently to losses and gains. Five new studies on neurons related to this issue have examined neuronal responses to losses, gains, and reference points. This study investigated a new concept of the value function. A value function with a neuronal cusp may exhibit variations and behavioral cusps associated with catastrophic events, potentially influencing a trader's decision to close a position. Additionally, we have conducted empirical studies on algorithmic trading strategies that employ different value function specifications.

Full text

https://www.aimspress.com/journal/QFE
QFE, 8(4): 733–756.
DOI: 10.3934/QFE.2024028
Received: 15 August 2024
Revised: 24 October 2024
Accepted: 12 November 2024
Published: 14 November 2024
Research article
New concept for the value function of prospect theory
Kazuo Sano*
SOBIN Institute LLC, 3-38-7 Keyakizaka, Kawanishi, Hyogo, 666-0145, Japan
*Correspondence: Email: sano@sobin.org.
Abstract: In prospect theory, the value function is typically concave for gains and convex for losses,
with losses usually having a steeper slope than gains. The neural system responds differently to losses
and gains. Five new studies on neurons related to this issue have examined neuronal responses to
losses, gains, and reference points. This study investigated a new concept of the value function. A value
function with a neuronal cusp may exhibit variations and behavioral cusps associated with catastrophic
events, potentially influencing a trader’s decision to close a position. Additionally, we have conducted
empirical studies on algorithmic trading strategies that employ different value function specifications.
Keywords: anterior cingulate cortex (ACC); lateral habenula (LHb); orbitofrontal cortex (OFC);
dorsal striatum (DS); ventral striatum (VS); anterior insular cortex (AIC)
JEL Codes: G10, G17, G40, G41
1. Introduction
Prospect theory (Tversky and Kahneman (1974); Kahneman and Tversky (1979); Tversky and
Kahneman (1992)) is one of the most influential theories of human decision-making (Thaler (1989);
Camerer (2004); Ariely et al. (2005); Liberman et al. (2005); Lejarraga and Hertwig (2017); Harinck
et al. (2007); Mogiliansky et al. (2009); MacGraw et al. (2010); Goyal et al. (2021); Walther and
M¨
unster (2021); Linzmajer et al. (2021); Mengov et al. (1982); Blavatskyy (2021)). According to
this theory, individuals assess their losses and gains asymmetrically. Experimental studies have shown
that the key behavioral assumption of expected utility theory, the so-called “independence axiom,” is
systematically violated in practice (Neumann and Morgenstern (1947)). The value function is
typically concave for gains and convex for losses, with losses typically having a steeper slope than
gains. Decision weights are often lower than the corresponding probabilities, except in the
low-probability range. Given these findings, the empirical relevance of numerous studies on the
behavior of economic agents under uncertainty that utilize expected utility analysis has been debated.

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In economics, however, it is assumed that economic agents are typically rational (Stigler and
Becker (1977)). Economists have long debated the issue of time preference (Strotz (1955); Thaler
and Shefrin (1981); Thaler (1981); Weitzman (2001); Nowaihi and Dhami (2006); Haigh and List
(2005); Graham and Snower (2013); Andreoni et al. (2015); Larson et al. (2016); Cohen et al.
(2020)), as well as the validity of prospect theory. Machina (1982) showed that the basic concepts,
tools, and results of his generalized expected utility analysis are independent of the independence
axiom but may be derived from a much weaker assumption of smoothness of preferences over
alternative probability distributions. He also demonstrated that a simple model of preferences could
be constructed to associate a wide body of observed behavior with risk, including the Allais paradox,
may be constructed. Thus, the hypothesis that an individual maximizes a fixed preference functional
defined over probability distributions may hold true for many apparently independent aspects of risk
behavior. However, this may reflect only a part of human innate nature and may represent a
mathematical abstraction of human rationality in economic phenomena.
Meyer and Pagel (2022) explored the impact of stock price fluctuations on individual investor
behavior, drawing on past literature and debates. Standard economic theory suggests that individuals
should respond rationally to changes in stock value, but behavioral economics highlights how
emotional reactions can influence investment decisions. They analyzed reinvestment behavior
following forced liquidations, demonstrating that losses significantly decrease the likelihood of
reinvestment. This finding provides empirical support for these theories and contributes to
understanding how past investment experiences shape future financial behaviors. The study employs
mutual fund liquidations as exogenous shocks to analyze responses to realized gains and losses. They
demonstrate that individuals are less likely to reinvest after experiencing losses, indicating a shift in
preferences and increased risk aversion. This research enhances our understanding of how past
experiences with investments shape future financial behaviors and risk tolerance, ultimately providing
insights into broader household finance phenomena, such as stock market non-participation.
If the expected utility theory does not hold for investor’s decisions, does the neural system affect
the investor trading decisions? Does the neural system respond differently to gains and losses? This
study presents hypotheses to address these issues. In general, we examine the outcome of our choice
and adjust subsequent choice behavior using the outcome information to choose an appropriate action.
Five significant studies on neurons (Kawai et al. (2015); Yamada et al. (2021); Imaizumi et al. (2022);
Yang et al. (2022); Ferrari-Toniolo and Schultz (2023)) have examined neuronal responses to loss and
gain. These studies suggest that two different neural systems may respond to loss and gain, resulting
in a value function with a cusp as a reference point.
This study addresses the reference point and cusps of value functions, and the following questions.
When does a trader exit one’s position? Why would a trader ever close a position when losses and gains
are continuously changing? The reference point is a neuronal cusp, and also, the value function could
have other behavior cusps (Rosales-Ruiz and Baer (1997); Bosch and Fuqua (2001); Bosch and
Hixson (2004); Smith et al. (2006)) with catastrophe (Whitney (1955); Zeeman (1976)).
2. Materials and methods
This article provides a comprehensive review and synthesis of existing literature on the topic of
the neural basis of the value function and decision-making. The article focuses on prospect theory and
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the neural systems involved in evaluating gains and losses relative to a reference point. Five recent
studies that have examined neuronal responses to gains, losses, and reference points are reviewed, and
their findings are integrated into a broader framework of decision-making.
The article provides a detailed overview of the neural systems involved in decision-making,
including the anterior cingulate cortex (ACC), lateral habenula (LHb), orbitofrontal cortex (OFC),
dorsal striatum (DS), ventral striatum (VS), and anterior insular cortex (AIC). It also discusses the
role of these regions in the valuation of gains and losses, as well as the effects of reference points and
the shape of the value function.
In addition, the article introduces a novel concept of the value function, marked by a neuronal
cusp. This concept suggests a potential link between catastrophic behavior and position closure in
volatile market conditions. The article includes original data analysis using TOPIX tick data and
highlights the need for further research to fully elucidate the implications of this concept for decision-
making and financial risk management. We will conduct empirical studies on algorithmic trading
strategies that employ different value function specifications.
3. Response of neural systems to loss and gain
3.1. Loss side
Recent attention has focused on the critical roles of both the lateral habenula (LHb) and the
anterior cingulate cortex (ACC) in monitoring negative outcomes. Kawai et al. (2015) investigated
how LHb and ACC contribute to behavioral adjustments using single-unit activity recordings from
these areas in monkeys performing a reversal learning task. In this task, monkeys had to switch their
choice if their previous choice consistently led to no reward in prior attempts. The researchers found
that ACC neurons stored outcome information accumulated across multiple trials, whereas LHb
neurons quickly detected ongoing negative outcomes. Only ACC neurons, not LHb neurons, signaled
a change in behavior in the subsequent trial. Despite both LHb and ACC signaling negative outcomes,
the study indicated that these brain regions influence behavioral adjustments in distinct manners.
Additionally, the researchers observed that ACC neurons associated with negative outcomes
consistently integrated the impact of these experiences on their responses to no-reward situations.
This integration corresponded to the monkeys’ shift from their initial choice to an alternative. They
further tested whether neuronal activity reflecting negative outcome experiences predicted behavioral
shifts, finding consistency with the loss side of the value function (refer to Figure 4 of Kawai et al.
(2015)). From a loss perspective, rapid detection of harm or danger is deemed more adaptive for
survival than heightened sensitivity to stimuli.
Although they studied monkeys and the same experiments cannot be repeated on humans, earlier
studies have shown strong similarities. Such similarities are seen in human event-related potentials
(ERP) and functional magnetic resonance imaging (fMRI) studies (see the comprehensive review by
Pammi and Miyapuram (2012)). As Kawai et al. (2015) stated, the ACC is crucial for monitoring and
adjustment. Studies using human event-related potentials and functional magnetic resonance imaging
(fMRI) have shown that the ACC is activated when subjects receive negative feedback after an
inappropriate behavioral response. However, the crucial role played by the LHb in monitoring
negative outcomes has evoked interest. As the ACC sends projections directly to the LHb, both
structures can communicate through their reciprocal relationship. Therefore, the LHb and ACC may
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work together to monitor the negative outcome and alter subsequent choice behavior.
Prior to these recent studies, it was already known that the ventromedial prefrontal cortex
(vmPFC) and amygdala are involved in losses. The vmPFC plays a crucial role in encoding value
representations and decision-making processes. It integrates information about potential losses,
influencing how individuals evaluate different options. Its involvement in assessing the emotional
significance of outcomes makes it essential for understanding loss-related behavior (Bechara et al.
(1994); Padoa-Schioppa and Assad (2006)). The amygdala, on the other hand, is key in processing
emotional responses, particularly fear and anxiety associated with losses. It contributes to the urgency
and salience of loss experiences, which can significantly impact decision-making and risk assessment
(LeDoux (2000); Phelps and LeDoux (2005)).
This can be understood by imagining how humans ceased living in trees and transitioned to
bipedal walking. We have chosen survival strategies that enhance the brain’s development. Frogs,
unlike monkeys and humans, adopted a different evolutionary strategy and aimed to improve by
increasing the size of their brains. However, this approach proved unsuccessful, leading them to
change their survival strategy by camouflaging, thus saving energy by reducing the size of their brains
(Liao et al. (2022)).
3.2. Gain side
In their studies on brain reward circuitry, Yamada et al. (2021) and Imaizumi et al. (2022)
proposed a neuronal prospect theory model. Using theoretical accuracy equivalent to that of human
neuroimaging studies from a gain perspective, they showed that single-neuron activity in four core
reward-related cortical and subcortical regions represents a subjective assessment of risky gambles in
monkeys. Their studies focused on the central part of the orbitofrontal cortex (cOFC, area 13 M), the
medial part of the OFC (mOFC, area 14 O), dorsal striatum (DS, the caudate nucleus), and ventral
striatum (VS). Similar to the prospect theory framework, the attractiveness of probabilistic rewards
parameterized as a product of utility and probability weighting functions is represented by the activity
pattern of single neurons in monkeys passively viewing a lottery. Their study showed that the varied
patterns of valuation signals were distributed in most regions of the reward circuitry and not localized.
By combining these data, a network model was built to reconstruct the animals’ risk preferences and
subjective probability weighting as demonstrated by their choices. Distributed neural coding thus
explains how subjective valuations under risk are computed. A logistic function was used as the value
function. However, the logistic function lacks a reference point, which is obvious because the
researchers only focused on neuronal responses on the gain side.
Their findings about the characteristics of the gain side of the value function may be reliable. In
contrast to the logistic function of the loss side, that of the gain side may seem appropriate, considering
the threshold of response in the region that evokes a response to a stimulus. It is well known that
fish can reduce predation risk through learning. Wallerius et al. (2020) showed that fish can acquire
social information about a threat not only through privately acquired learned experiences but also by
observing the conspecifics’ response to threat and using such open information to alter future behavior
through learning. This learning from the school of fish also applies to investor groups in the securities
market. If you are a smart investor like Buffett, you will wait before buying a stock even if you find it
very attractive. It could be a trap. It may be better for survival to avoid unpleasant situations promptly
and react slowly to favorable situations.
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3.3. Reference point
Yang et al. (2022) used a token gambling task to show that monkeys altered their risk preference
across wealth levels and gain/loss contexts. Consistent with the concept suggested by prospect theory,
they proposed that the AIC neurons may encode behaviorally relevant value information. The AIC
would represent both the subject’s current state (the reference point) and reference-dependent value
signals that vary in the loss or gain context (asymmetrical value functions in loss and gain), which
would together influence a subject’s risk attitude in decision-making. To better understand the effect of
tokens on different components, they modeled the behavior for each start token number separately. In
the gain context, both monkeys were risk-seekers when the starting token number was low; however,
both demonstrated risk-neutral or risk-averse behavior when the start token number increased. This
result is consistent with Yamada et al. (2021) and Imaizumi et al. (2022). Moreover, Yang et al. (2022)
showed, in monkey G, the utility functions (value functions) in monkey G were consistently steeper for
losses than for gains. Thus, monkey G showed loss aversion behavior. However, they showed that in
monkey O, the utility functions were not consistently steeper for losses than for gains, indicating that
monkey O was equally sensitive to gains and losses and thus was not averse to loss. This is similar to
the context where there are different types of investors.
4. Results
4.1. Value function
A new concept of the value function can be readily derived by integrating these studies on the
neural systems’ responses to losses and gains. Kahneman and Tversky (1979) proposed that the value
function is: (1) defined on deviations from the reference point; (2) concave for gains and convex for
losses; and (3) steeper for losses than for gains. However, they have also noted that “most utility
functions for gains were concave, most functions for losses were convex, and only three individuals
exhibited risk aversion for both gains and losses. With a single exception, utility functions were
considerably steeper for losses than for gains” (Kahneman and Tversky (1979), 280). The most
important point is that the reference point of the value function is non-differentiable. In fact, it is
illustrated that way in Figure 3—a hypothetical value function of their article (279). Although it is not
heavily emphasized in economics textbooks, the discovery of a non-differentiable reference point in a
value function is remarkable. A Reference point is precisely the neuronal cusp where different
nervous systems intersect.
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VALUE
GAINS
LOSSES
A
B
C
D
E
F
Figure 1. Value functions with a reference point as a neuronal cusp. A: Risk loving type,
B: Standard type, C: Kahneman and Tversky’s original type, D: Standard type, E: Steeper
type than standard, F: Reverse type of the original. We can graph any other type in the
area (neural system) and how it responds to loss and gain. Note that the reference point of
the value function is non-differentiable. A type that exhibits risk aversion on both the gain
and loss sides near the reference point (C and F in Figure 1) would, in reality, refrain from
purchasing risky assets.
Yang et al. (2022) also observed variations in the value function using two examples. Ferrari-
Toniolo and Schultz (2023) explored how signals from individual brain cells involved in behavior can
vary. This variation could be attributed to the unreliability of single neurons or to a more complex
system where different neurons contribute distinct pieces of information. This concept might also hold
true for brain activity related to economic decisions and the associated rewards. The value of a reward
is subjective, depending on both its size and the probability of obtaining it. To understand how the brain
represents these subjective values, researchers rely on the continuity axiom. This principle is tested by
presenting animals with various rewards of different sizes and probabilities. If animals comprehend
the choices and make meaningful decisions, their behavior should adhere to the continuity axiom.
Ferrari-Toniolo and Schultz (2023) explored how the brain represents subjective value in
economic decision-making. Their work suggests that signals from individual neurons might be
unreliable, but the brain seems to overcome this by using a “population code.” This code combines
information from many neurons to accurately represent subjective value, even if the individual
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neurons themselves have noisy or inconsistent signals.
They tested this idea in the orbitofrontal cortex (OFC), a brain region crucial for economic
decisions in monkeys. They found that while individual neurons in the OFC displayed a variety of
value signals, these signals often didn’t directly reflect the monkey’s choices. However, when they
considered the activity of entire populations of neurons, a clear pattern emerged. The combined signal
from the population accurately matched the monkey’s choices, suggesting that the OFC uses a
complex code to represent subjective value despite the limitations of individual neurons.
4.2. Behavior cusps
In this section, we present one theoretical hypothesis regarding traders’ decision-making. When
do traders exit their position? Why would a trader close a position at some point when losses and gains
are continuously changing?
This problem can be addressed using cusp catastrophe theory. In this theory, the concepts of
control space and behavior space are crucial for understanding how systems respond to changes. In our
example, the variables in the control space are gains and losses, while the behavior space is represented
by the surface of a function, illustrating how behavior changes in response to variations in gains and
losses. A control space refers to the range of variables or parameters that can be manipulated or
controlled within the system. These variables can influence the system’s behavior, often leading to
different outcomes based on their values. Behavior space, on the other hand, represents the observable
outcomes or behaviors of the system as a result of changes in the control variables. It reflects how the
system behaves under various conditions defined by the control space. The relationship between these
two spaces is significant: changes in the control space can lead to sudden shifts in the behavior space,
particularly when critical thresholds are crossed. This dynamic interaction helps to explain why small
adjustments in control parameters can sometimes result in abrupt and significant changes in behavior,
highlighting the nonlinear nature of the system.
Recently, cusp catastrophe theory has been applied in various fields. Barnik and Vosvrda (2009)
made the first attempt to fit a stochastic cusp catastrophe model to stock market data. The authors
took a macroeconomic perspective and applied stochastic cusp catastrophe theory to understand the
dynamics of the entire stock market and the mechanisms behind market crashes. Rather than focusing
on the behavior or decision-making of individual traders, the research emphasizes the nonlinear
dynamics of the market as a whole. Therefore, it is concerned with macroeconomic factors and
market movements. Wei et al. (2024) investigated the factors that lead to sudden shifts in online
impulsive buying behavior. The authors developed a qualitative cusp catastrophe model integrated
with multi-agent simulations to analyze how various elements—such as consumer emotions, social
influence, and external triggers—interact to create abrupt changes in purchasing decisions. By
simulating different scenarios, the study highlights the conditions under which impulsive buying can
escalate dramatically. Key findings indicate that certain thresholds can prompt significant changes in
consumer behavior, suggesting that understanding these cusp points can help businesses better predict
and respond to shifts in online shopping patterns. The research contributes valuable insights into the
psychology of online consumers and the dynamics of e-commerce, offering practical implications for
marketers and platform designers. Chen et al. (2014) presented a novel application of the cusp
catastrophe model to explore nonlinear health outcomes in nursing contexts. The authors argue that
traditional linear models may not adequately capture the complexity of health outcomes influenced by
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various factors, such as patient characteristics, environmental conditions, and nursing interventions.
By employing the cusp catastrophe model, they aim to identify critical thresholds that can lead to
sudden changes in health outcomes, providing a more nuanced understanding of patient responses to
care. Through simulations and case studies, the research demonstrates how this nonlinear approach
can reveal insights into the dynamics of health outcomes, highlighting the importance of recognizing
tipping points in patient care. The findings suggest that using the cusp catastrophe model could
enhance nursing research and practice by improving the prediction of health outcomes and guiding
interventions more effectively. Overall, the paper contributes to the growing body of literature on
innovative modeling techniques in healthcare.
The logical connection between psychology and the behavioral cusp model lies in the
understanding of how abrupt changes in behavior can be influenced by psychological factors and
neural mechanisms. The behavioral cusp model helps explain how small changes in control variables
(like emotions or external stimuli) can lead to sudden shifts in behavior, such as impulsive buying or
risk-taking. The model posits that there are critical thresholds that, when crossed, result in significant
changes in behavior. This concept aligns with psychological theories that emphasize the role of
cognitive and emotional thresholds in decision-making.
Neuroscientific research has identified specific brain regions involved in processing rewards and
risks. These areas are critical for evaluating control variables and can show changes in activity when
individuals approach behavioral thresholds. The role of neurotransmitters like dopamine can illustrate
how psychological states (like motivation or pleasure) interact with control parameters in the cusp
model. For instance, increased dopamine levels may lower the threshold for engaging in impulsive
behaviors.
By integrating insights from psychology and neuroscience, we can better understand how mental
states and brain activity influence behavioral responses in the context of the cusp model. This
interdisciplinary approach can help in developing interventions or predicting behavior in various
settings, such as marketing or clinical psychology.
The connection between psychology and the behavioral cusp model is grounded in the
understanding that psychological factors and neural mechanisms interplay to shape behavior. By
studying these relationships, we can gain deeper insights into the dynamics of decision-making and
the conditions under which sudden behavioral changes occur.
Now, we obtain the 3D image of the value function with cusps near the decision point (Figure
2). The words “anger” and “fear” on the losing side are used in the famous dog example by Zeeman
(1976). For example, when a stock trader exits the position, she jumps and fixates on loss or gain. The
value function, therefore, should have cusps corresponding to unstable psychology. It is well known
that investor psychology is easily affected. We can assume that these cusps are also moving. They
could be a type of noise that can be synchronized and influence investors’ decisions. Indeed, when
viewed individually, there are no such cusps in trading driven by mechanical algorithms. However,
when viewed collectively, even trading algorithms driven by AI may involve interactions between
individual trades, leading to the potential for instantaneous synchronization that could be modeled as
a mathematical catastrophe.
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Behavior Space
Anger
Fear
Value
Hold
Sell
Value
Gains
Smile
Losses
Hold
Value
Control Space
Behavior Space
Peace
Sell
+
-
Figure 2. Value function with cusps near the decision point: control space and behavior
space.
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Figure 3. Sample image of the cusp catastrophe: f(x,u,v)=x4+ux2+vx, by Eric W.
Weisstein (August, 2021); https://mathworld.wolfram.com/CuspCatastrophe.html.
4.3. Data analysis
The existence of diverse value functions at the neuronal level implies the existence of diverse
types of investors in the market. In this section, we investigate what types of algorithms are
advantageous in short-term algorithmic trading unrelated to behavioral cusps. According to the
efficient market hypothesis, the existence of consistently advantageous trading algorithms implies the
existence of arbitrage opportunities.
Shefrin and Statman (1985) explored the disposition effect, which is the tendency of investors to
sell winning stocks too early and hold onto losing stocks for too long. This behavior contradicts rational
expectations and tax-efficient strategies. The authors attribute this to prospect theory’s value function,
where investors are more sensitive to losses than gains. The fear of realizing losses (loss aversion)
leads to holding onto losers, while the diminishing sensitivity to gains prompts premature selling of
winners. This aligns with the investment wisdom of “cut losses short, let profits run,” emphasizing the
importance of swift loss realization and allowing gains to potentially grow further.
The diversity of value functions manifests itself as differences in risk preference near the reference
point. For instance, a type that exhibits risk aversion on both the gain and loss sides near the reference
point (C and F in Figure 1) would, in reality, refrain from purchasing risky assets. In this analysis, we
simplify the characteristics of the value function near the reference point and compare the performance
of three types: a type whose position liquidation timing is symmetrical with respect to gains and
losses, a type whose gain realization is faster than loss realization, and the reverse type. Specifically,
we examine scenarios where:
1. gains and losses are both 1%,
2. gains are 1.5% and losses are 0.5%, and
3. the reverse, where gains are 0.5% and losses are 1.5%.
We will conduct this analysis using TOPIX tick data from the JPX Data Cloud (Figure 4).
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(a) 2021/8(b) 2021/9
(c) 2021/10 (d) 2021/11
(e) 2021/12
Figure 4. TOPIX: August to December in 2021 (https://db-ec.jpx.co.jp).
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Let’s explain the analysis method. There are various programming codes to execute this
simulation, but considering processing speed, parallel processing is essential. We used R version 4.4.0
along with two libraries, foreach and pforeach (https://github.com/hoxo-m/pforeach). We will
randomly extract 1000 ticks from each month’s TOPIX tick data. The data period is from August to
December in 2021, with data sizes of 377,902 ticks in August, 359,907 ticks in September, 377,899 in
October, 359,908 ticks in November, and 395,895 ticks in December. From those, we will select the
tick that first reaches a 1% increase or decrease, and calculate the profit or loss if we were to settle at
that point (type 1). Similarly, we will calculate the profit and loss for cases where we settle at a 1.5%
increase and a 0.5% decrease (type 2), as well as a 0.5% increase and a 1.5% decrease (type 3). We
will repeat this process 1000 times and compare the performance of the trades. These parameters
represent a first-order approximation of the local shape of the value function near the reference point.
Considering that the global properties of the value function are believed to exhibit significant
variation, this paper aims to compare the performance of short-term trades based on differences in the
local characteristics of the value function near the reference point.
The results of this Monte Carlo simulation are shown in Figures 5 through 9. It is evident, even
without statistical analysis, that there is no consistently superior trading algorithm.
As one might easily surmise, in a rising market, delaying profit-taking could lead to greater gains,
while in a falling market, quicker stop-losses would curb losses. However, this is impossible without a
rational and theoretical method to determine market trends. Therefore, in trading an index like TOPIX,
it is not possible to obtain arbitrage profits with a simple mechanical algorithm like the one adopted
here. In this sense, the efficient market hypothesis can be considered valid.
Table 1. The mean and variance of gains obtained from Monte Carlo simulation.
Type (gain, loss) Type 1 (1.0, 1.0) Type 2 (1.5, 0.5) Type 3 (0.5, 1.5)
Month (ticks) Mean Variance Mean Variance Mean Variance
August (377,902) 4,266.31 365,252.1 3,083.55 342,724.7 1,387.24 288,850.7
September (359,907) 2,422.43 601,216.4 6,421.68 588,405.1 1,261.64 429,503.6
October (377,899) 3,649.76 408,103.9 −712.69 295,436.6 3,370.87 342,059.1
November (359,908) −9,959.35 348,855.9 −7,480.95 131,341.4 −8648.46 422,268.2
December (395,895) 7,431.03 341,639.2 3,452.62 322,042.6 4,864.27 304,590.1
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(a) 1.0% gain or loss (b) 1.5% gain and 0.5% loss
(c) 0.5% gain and 1.5% loss
Figure 5. Trading gains in August, 2021.
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(a) 1.0% gain or loss (b) 1.5% gain and 0.5% loss
(c) 0.5% gain and 1.5% loss
Figure 6. Trading gains in September, 2021.
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(a) 1.0% gain or loss (b) 1.5% gain and 0.5% loss
(c) 0.5% gain and 1.5% loss
Figure 7. Trading gains in October, 2021.
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(a) 1.0% gain or loss (b) 1.5% gain and 0.5% loss
(c) 0.5% gain and 1.5% loss
Figure 8. Trading gains in November, 2021.
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(a) 1.0% gain or loss (b) 1.5% gain and 0.5% loss
(c) 0.5% gain and 1.5% loss
Figure 9. Trading gains in December, 2021.
5. Discussion
Mechanical algorithms, such as those tested in this paper, lack psychological cusps. However,
AI-based algorithmic trading might incorporate unforeseen cusps. The synchronized trading of diverse
AI algorithms, learning from vast amounts of information concurrently, could significantly impact the
market. Therefore, a theoretical examination of the issue of human psychological cusps is deemed
crucial.
The concept of behavioral cusps remains a theoretical hypothesis. To fully embrace this new
perspective of the value function with cusps, another challenge must be addressed: the necessity of
employing terms like “anger” or “fear” to represent these cusps. Since the 19th century, there has
been ongoing debate regarding whether “emotions” are the cause or consequence of their associated
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behaviors. Anderson and Adolphs (2014) contended that emotional behaviors are a class of behaviors
that express internal emotional states. These states exhibit general functional and adaptive properties
that are consistent across all human emotions, such as fear or anger, and throughout phylogeny. Paul
and Mendl (2018) highlighted the complexity of defining emotions in the human context, a difficulty
that is magnified when dealing with animals incapable of self-expression. In such cases, notions of
internal or consciously experienced states remain inaccessible. Hence, it would be premature to infer
the influence of emotions solely from monkeys’ neuronal responses that appear evident to us.
The reference point within the value function could potentially be a cusp, representing a
behavioral change (Rosales-Ruiz and Baer (1997); Bosch and Fuqua (2001); Bosch and Hixson
(2004); Smith et al. (2006)) associated with a mathematical catastrophe (Whitney (1955); Zeeman
(1976)). This possibility arises from the substantial differences observed in the neural system’s
responses to losses and gains. We monitor the outcomes of our choices and adjust subsequent choices
using this outcome information to select appropriate actions. Studies by Kawai et al. (2015), Yamada
et al. (2021), Imaizumi et al. (2022), Yang et al. (2022), and Ferrari-Toniolo and Schultz (2023) have
examined neuronal responses to losses, gains, and reference points. These studies suggest that two
distinct neural systems might respond differently to losses and gains, resulting in a value function
characterized by a reference point acting as a neuronal cusp.
The introduction of cusps in the value function of prospect theory indeed raises important
questions regarding the axiom of continuity. In traditional utility theory, the axiom of continuity states
that small changes in outcomes should not lead to abrupt changes in preferences. However, the
presence of cusps implies that there are points where the value function has sharp turns, potentially
leading to discontinuities in how individuals evaluate changes in outcomes.
Value function characteristics
The value function in prospect theory is typically concave for gains and convex for losses,
exhibiting risk aversion in gains and risk-seeking behavior in losses. When cusps are introduced, the
function becomes non-differentiable at certain points, which can create scenarios where minor
changes in wealth lead to significant shifts in perceived value.
Decision-making under uncertainty
According to prospect theory, individuals evaluate potential outcomes based on perceived gains
and losses rather than final wealth. The introduction of cusps means that for certain ranges of outcomes,
a small change can drastically affect the decision-maker’s valuation. This can lead to inconsistent
choices when evaluating similar prospects, as the evaluation process may jump from one valuation to
another due to the cusp.
Behavioral implications
The discontinuity introduced by cusps can result in framing effects, where the way a choice is
presented influences decision-making. For example, individuals may become overly sensitive to
specific outcomes associated with the cusps, leading them to prefer options that avoid loss even if they
are not optimal from a utility perspective.
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Impact on risk assessment
The presence of cusps could also distort risk perception, causing individuals to overreact to losses
near a cusp and underreact to losses further away. This could skew their decisions, making them less
stable and predictable.
In summary, while the introduction of cusps in the value function offers a nuanced understanding
of decision-making under risk, it challenges the axiom of continuity and suggests that individuals
may make choices based on abrupt changes in perceived value rather than smooth transitions. This
has significant implications for understanding behavior in economic contexts and the psychological
mechanisms behind risk assessment and decision-making.
While the study provides valuable insights into investor behavior, it is crucial to recognize that the
findings may not be universally applicable to all contexts or populations. Human behavior is influenced
by numerous factors, including psychological, social, and emotional components. The study may not
fully account for all these complexities, leading to oversimplifications in the interpretation of results.
To enhance the robustness of the findings, future research should aim to include diverse samples and
consider longitudinal studies that track changes over time. This would help in better understanding the
nuances of investor behavior and the factors influencing it.
6. Conclusions
Variations in value functions arising from neuron-level diversity can be considered to contribute
to the diversity of investors in the market. The presence of diverse investors is likely to lead to the
revitalization and stabilization of market transactions. On the other hand, if the psychological cusps of
investors synchronize, it will bring extreme price movements to the market, and the same possibility
exists for algorithmic trading, including AI. A pertinent issue is the need to clarify how our findings
have implications for investors in future markets. Fortunately, an R package for the cusp model has
been developed, providing concrete examples of analysis (Grasman et al. (2009)). Therefore, with
appropriate data, it is possible to conduct analyses on behavioral cusps. However, such analyses require
data on the decision-making processes of multiple traders, and obtaining and utilizing actual trading
records may be challenging due to privacy concerns. We encourage future researchers to explore these
avenues, as they could open new pathways for understanding market dynamics.
In economics, we assume the rationality of economic agents. For instance, theoretical economics
loses relevance without assumptions like utility maximization and profit maximization. Economists
have long debated the issue of time preference (Strotz (1955); Thaler and Shefrin (1981); Haigh
and List (2005)) and the validity of prospect theory. Machina (1982)’s generalized expected utility
analysis can be employed to construct a simple model of preferences that connects various observed
risk-related behaviors. However, it remains a mathematical and abstract theory of human rationality
within economic phenomena, representing only one facet of our innate nature. The existence of diverse
types of investors leads to variations in the valuation of risky assets in the market, which is thought
to stimulate trading activity. However, the synchronization of psychological cusps has the potential to
generate extreme price movements.
It is clear that humans are both rational and emotional. While these two aspects often conflict,
both are essential for maintaining human society. This insight was recognized by Adam Smith in the
18th century. Smith (1759, 1776), considered the father of economics, believed that the pursuit of
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752
self-interest should be moderated by a “fellow feeling.” However, contemporary economics focuses
solely on the rational behavior of self-interested economic agents, neglecting the existence and impact
of empathy.
Furthermore, Smith (1759) observation holds significant importance when considering the value
function. Human empathy—the ability to share and understand the emotions of others—is another
cornerstone of social interaction. Recent research has delved into the mechanisms underlying this
remarkable phenomenon, uncovering the intricate interplay of shared emotions, imitation, and learned
perceptual-motor links.
Imitation, another pillar of empathy, manifests as the automatic mirroring of observed behaviors.
Witnessing someone’s actions triggers the activation of corresponding motor representations in our
own brains, as if we were preparing to perform those actions ourselves. These internally generated
motor representations then guide our own actions, leading to the imitation of observed behaviors.
Recent studies on empathetic systems support Smith’s idea (Preston and de Waal FBM (2002);
de Wall (2009); Yamamoto (2016)), which include noise synchronization (Sano (2022)). Perhaps
it is time to revisit Smith’s concepts and rethink economics in terms of justice and empathy. This
perspective is lacking in modern microeconomic theory and could introduce a new field of economics.
Data availability
All data used in this paper can be obtained from JPX Data Cloud (https://db-ec.jpx.co.jp/).
Use of AI tools declaration
The author declares they have not used Artificial Intelligence (AI) tools in the creation of this
article.
Conflict of interest
The author declares no conflict of interest.
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