A joint analysis of the common ratio and common difference effects

There are three elementary premises assumed by those involved in the decision-making process. These are: a preference for sure rewards over risky ones, rewards sooner rather than later, and larger rewards to smaller ones (Luckman et al., 2020). However, this process becomes more difficult when risk, time and amount dimensions are combined.

In this way, risk and time preferences reveal different aspects of the decision-maker’s behaviour. In effect, whereas risk preferences are associated with uncertainty tolerance, time preferences are related to the degree of impatience. Risk and time preferences have been typically assessed by using two well-known classical models, namely the expected utility (EU) model by Von Neumann and Morgenstern (1944) and the discounted utility (DU) model by Samuelson (1937), respectively. However, few models have intended to establish a joint theoretical framework able to explain the assessment of risky dated rewards and to identify certain paradoxes present when dealing with risk and time preferences, separately. In effect, the two aforementioned models estimate the sum of the utilities of potential rewards based on the hypotheses of perfect rationality and preference consistency, without taking into account certain violations of these assumptions.

Therefore, the objective of this paper is to fill the gap in the existing literature on this topic by considering the assessment of risky dated preferences. The significance of this topic lies in the absence of consensus in the existing literature about how people make decisions on delayed and risky prospects. This could be done either by comparing the attribute values of their risky/intertemporal options (i.e. by comparing risks to risks, amounts to amounts and delays to delays) or by comparing both options by determining the utility of the reward through the combination of both models (Luckman et al., 2020). Chakraborty (2021) link time delays and perceived risk by indicating that people prefer immediacy and certainty, whilst Johnson et al. (2020) found no evidence of a singular process behind these tendencies. Sozou (1998), Halevy (2008) and Somasundaram and Eli (2022) argue that as delays increase, the perceived risk rises, making present consumption certain and future consumption uncertain, thereby relating delayed rewards with risk perception. Conversely, Rachlin and Siegel (1994), Rachlin et al. (2000) argue that risk preferences may be translated into temporal patterns.

Despite applying to different domains, a few models have been developed to combine risk and delay information into a single attribute dimension, by transforming either delays into perceived risks or probabilities into inferred delays until receipt (Baucells and Heukamp, 2010; Yi et al., 2006). Indeed, the development of theories which explain and relate risky delayed choices in a unified model has gained increasing interest. An example of this is the so-called attribute-based RITCH model (Luckman et al., 2020). In the same way, Baucells et al. (2022) have presented a comprehensive modelling attempt to combine risk and time preferences, the so-called discounted incremental utility model. However, in this paper, we will follow the methodology introduced by Baucells and Heukamp (2010, 2012) by focusing on the trade-off between the three components of any risk-time model: amount, time and probability. Therefore, this paper builds upon classical and contemporary decision-making models, contributing to the growing body of work which explores the complex interaction between temporal and risk preferences.

In addition, this methodology will allow us to demonstrate that some anomalies present in the field of risk and time preferences, considered as watertight compartments, are essentially the same when considering the trade-off between the aforementioned components of the model. Specifically, we will focus on the parallelism between the so-called common ratio and common difference effects. To study this interaction, we start with some common points between the risk and time effect in the decision-making process. In 1991, Prelec and Loewenstein were the first in observing that there are some fundamental psychological properties in prospect evaluation when they are risky or delayed. Specifically, these properties are focused on pointing out a fundamental link: the parallelism and correlation between anomalies in preferences under risk and under time (Leland and Schneider, 2017; Cruz Rambaud et al., 2023). Some violations of expected utility and discounted utilities respectively are listed below together with their mathematical definitions. It must be taken into account that the delayed-risky rewards are represented by a triple (x, p, t), where (for a summary of symbol and abbreviation meanings, see  Appendix):

• x ∈ X is the amount;

• p ∈ [0,1] is the probability of reaching the amount x at time t: and

• t ∈ T is the moment of availability.

• (1) Common ratio and common difference effects. The common ratio effect (for risk) holds if, for every x < y and θ ∈ (0,1), one has:

The common difference effect (for time) holds if, for every x < y and Δ ∈ (0, ∞), one has:

• (2) Common consequence and cancellation effects. The common consequence effect (for risk) holds if, for every x < y, p < q and p + q = 1, one has:

which is the same as:

but, by removing the reward (x,  q, t) from both sides of the equivalence, one has:

The cancellation effect (for time) holds if, for every x < y < z and t < s, one has:

but, by removing the reward (y, p, t) from both sides of the equivalence, one has:

• (3) Peanuts and magnitude effects. The peanuts effect (for risk) holds if, for every x < y and α ∈ (1, ∞), one has:

The magnitude effect (for time) holds if, for every x < y and α ∈ (1,∞), one has:

Other effects which may be related are:

• the fourfold pattern of risk attitudes (for risk) and the bias toward concentration (for time);

• the downside risk framing effect (for risk) and the opportunity cost framing effect (for time); and

• the gain-loss labelling effect (for risk) and the date-delay labelling effect (for time).

Leland and Schneider (2017) have observed that subjects showing one of these biases tend also to exhibit the other. As a consequence, they introduce a model of temptation-biased preferences which generalises quasi-hyperbolic discounting and quasi-rank-dependent probability weighting. Specifically, the model becomes quasi-hyperbolic discounting for choices involving only time and quasi-rank-dependent probability for choices involving only risk. The model explains empirically-observed interactions between risk and time preferences, and empirically-observed correlations between violations referred to expected and discounted utilities.

The primary purpose of this paper is to understand the relationship and interaction between risk and time in intertemporal decision-making, where decisions about future rewards involve both delays and probabilities. Although the existing model suggests that these two domains are correlated, this relationship has remained unproven. This study aims to analyse how risk and time preferences may be related to assess possible theoretical implications.

Recent research shows an increasing interest in linking delayed and risky prospects, but there is still no comprehensive evidence of this relationship. The research gap addressed by this paper is the lack of a theoretical framework which clearly evidences how risk and time influence each other and interact within the decision-making process. Although some models have been developed to combine risk and time information into a single attribute dimension (such as the RITCH model), there is still a need for a more detailed mathematical and theoretical approach which demonstrates how specific effects and biases in DU and EU models are essentially the same.

This study aims to fill this gap by providing a theoretical framework which clearly evidences the correlation between non-standard behaviours of risk and time in intertemporal decisions. The main contribution of this paper is to demonstrate that one component of rewards (amount, time, or probability) can be transformed and incorporated into another component. To do this, it illustrates the trade-off between amount and time, amount and probability, and time and probability. This transformation is formalised through theorems which link anomalies between the EU and DU models. Specifically, Theorem 1 derives the common difference effect starting from the common delay effect, and Theorem 2 deduces the common ratio effect starting from the common difference effect. This insight provides a novel theoretical framework for understanding how DU and EU models are intrinsically related through the interaction of their respective behavioural anomalies, bridging certain concepts involving these traditionally-considered separate domains, based on risk and time.

The value of this paper lies in the explicit identification of these transformations, and deriving implications which connect well-known anomalies in EU and DU models, such as the common ratio, common delay and common difference effects, within a unified framework. In addition, this paper provides a foundation for building an integrative DU and EU decision-making model which considers the risk-time trade-offs based on the intersections between temporal and probabilistic decision biases.

After the Introduction, this paper is organised as follows (see Figure 1). Section 2 presents a review of the existing literature which deals with the interrelation between risk and time in contexts of intertemporal choice where the involved dated rewards are probabilistic. This review provides the necessary theoretical foundation for the analysis which follows. Section 3 analyses the framework and general results. Starting with some preliminary concepts, it examines the common difference effect and common ratio effect based on the probability-time trade-off axiom introduced by Baucells and Heukamp (2010). Specifically, it presents the theoretical development of the trade-offs between amount and time (ATTO), amount and probability (APTO) and probability and time (PTTO). Section 4 offers the main findings of this study. First, the derivation of the common difference effect starting from the delay effect is explored, followed by the derivation of the common ratio effect starting from the common difference effect. These implications are formalised in theorems 1 and 2 respectively, by offering a novel theoretical relationship which reinforces the existing framework by connecting these effects logically and mathematically. Section 5 discusses the theoretical and practical implications of this study and presents the main conclusions which summarise the core contributions and suggest directions for future research.

This section presents a comprehensive view of the literature on risk and time preferences in decision-making. The references are systematically organised to start with fundamental decision-making models and progress to advanced approaches which examine the relationship between risk and time preferences. Finally, attention is drawn to the existing gaps in this field which need to be addressed by further research.

Neoclassical descriptive theories are essential when considering the subject of reward valuation. They serve as points of reference for building descriptive theories which reflect people’s behaviour in real-life scenarios. Samuelson (1937) introduced the exponential discount model, represented by the function δ^t, where δ is the discount factor and t is the time horizon. This model is widely used to assess reward utility flows which involve an intertemporal dimension. However, it assumes a linear perception of time, which limits its ability to appreciate any bias in human behaviour. Similarly, Von Neumann and Morgenstern’s (1944) expected utility model is used to assess decisions under risk by evaluating the potential outcomes based on their probabilities and associated utilities. This model assumes that people behave rationally and maintain consistent preferences. However, this does not reflect how decisions are often made in real life.

Kahneman and Tversky (1979) highlighted the importance of distinguishing between normative and descriptive theories in economics. Whereas normative theories such as expected utility are designed to guide rational decision-making, descriptive theories reflect real-world behaviours, including preference reversals which deviate from rational assumptions. These ideas laid the foundation for behavioural economics, which incorporates psychological insights into economic theory. Thaler (2018) further argued for the relevance of behavioural theory in finance, a field traditionally resistant to such perspectives, emphasising the need for models which address biases and irrational tendencies in financial decisions.

More recently, Abe and Kaneko (2022) have defended the value of predicting potential deviations from normative theories, such as preference reversals. These frameworks gain explanatory power by integrating psychological realism into models like the expected or discount utility models, bridging the gap between theoretical assumptions and observed human behaviour.

Research on decision-making models is essential for the accurate prediction of behaviours and minimising biases (Green and Armstrong, 2015). Developing refined models is crucial for effective decision-making in uncertain environments. To ensure maximum effectiveness, these models should be rooted in a simplified framework as excessively complex models can hinder their application by some decision-makers (Jin and Xu, 2024c, 2024d). Loewenstein and Prelec (1992), together with Phelps and Pollak (1968), introduce the hyperbolic and the quasihyperbolic discounting models, respectively reflecting more accurately present-biased preferences compared to the exponential model. These models also address phenomena such as dynamic inconsistency and preference reversals over time in decision-making. Takahashi et al. (2008) supported the use of psychophysical laws, especially the Weber-Fechner (Fechner, 1860) and Stevens power laws (Stevens and Galanter, 1957), both well documented for neurophysiological stimuli such as heat, sound, and light. These principles may also be used to model how people perceive time, which is relevant for understanding time discounting. Reinforcing this approach, Kim and Zauberman (2009) suggested that the Weber-Fechner and Stevens power laws, fundamental principles in the field of psychophysics, can help explain distorted perceptions of time. In this sense, Cruz Rambaud and Sánchez Pérez (2013) adapted these laws to improve discounting models, integrating subjective time perception in decision-making.

Jin and Xu (2024a, 2024b) underscored the transformative potential of machine learning and Gaussian process regression to improve the predictive accuracy of decision models. Integrating these advanced methods into decision-making processes offers the potential for significant advancements in predictive capabilities, reshaping the field through increased accuracy and adaptability.

In examining the relationship between temporal and risk preferences, it is essential to contextualise current perspectives on how time and risk interact in decision-making. Sozou (1998) propose that risk perception increases with delays; individuals perceive immediate consumption as secure, whilst future consumption is inherently uncertain. This idea is echoed by Halevy (2008), who relates delayed rewards to an inherent perception of risk, given that events between the present date and the promised date could interfere with the process of acquiring the reward. In contrast, Rachlin and Siegel (1994), Rachlin et al. (2000) suggest that risk preferences may be manifested in temporal patterns, or in other words, that time preferences could reflect preferences for probabilistic rewards. According to this view, the utility of uncertain rewards could be estimated by the average waiting time required for a successful outcome.

Andreoni and Sprenger (2012) further explore the link between risk and time preferences, noting that whilst these are indeed connected, their experimental findings indicate a distinction between the two. Referencing the Allais (1953) paradox, they observe that discounted expected utility theory performs well in risk-related contexts, but when an element of uncertainty is included, the common ratio predictions fail dramatically. Johnson et al. (2020) contribute to this analysis by identifying a correlation between the tendency to tolerate delay and risk, though they do not find evidence of a single underlying process which encourages these tendencies. Instead, they suggest a complex interaction, highlighting the need for further research to better understand the delicate relationship between time and risk preferences.

Somasundaram and Eli (2022) provide empirical support for the connection between time delay and probability, showing that time delay affects preferences similarly to a reduced probability of receiving an outcome. In this study, participants were relatively insensitive to delays for rewards with low probabilities, but they became more sensitive to time delay as the probability of winning increased. This effect was particularly pronounced in scenarios such as public lotteries with minimal odds of winning, where the waiting time for rewards became less relevant, underscoring the complex dynamics between time delay and perceived risk.

As mentioned at the end of the Introduction, recent research has highlighted a strong connection between nonstandard behaviours in risk and time (Leland and Schneider, 2017; Cruz Rambaud et al., 2023). One of the most recognised biases in the risk domain is the certainty effect, also known as the common ratio effect, which illustrates how people tend to overvalue certain choices disproportionally. This preference for certainty challenges the rationality assumptions underpinning expected utility theory, as initially observed by Allais (1953). Saito (2011) notes that when faced with a choice between a low but certain reward and a higher, riskier one, decision-makers may initially prefer the certain option. However, when a slight risk is added to a certain choice, they often prefer the higher-risk alternative.

Analogously, in the domain of temporal decisions, the present bias (also referred to as the common difference effect or immediacy effect) illustrates how preferences may shift over time due to decision-makers placing a higher value on immediate rewards. This bias leads to dynamic inconsistency, where an individual’s preference for the same outcome changes depending on the timing. Keren and Roelofsma (1995) explored this immediacy effect, showing that individuals prefer a guaranteed immediate reward over a slightly larger but delayed one – a preference which reverses if a common delay is added to both options. This effect suggests that introducing minor delays to an immediate reward can make people more willing to choose the most delayed option.

Several studies further connect the certainty effect and present bias, suggesting that time delays and risk share similar behavioural effects. Noussair and Wu (2006) propose that delaying a reward can increase risk tolerance, thus reducing the preference for certain outcomes. Their findings align with the common ratio effect, indicating that individuals are disproportionately sensitive to certainty regarding risk and present bias in time. Baucells and Heukamp (2010) investigate this relationship by examining how introducing risk into different prospective choices affects preferences. They found that the immediacy effect diminishes significantly when the probability of receiving a reward is high (p  = 0.9) and almost disappears when the probability drops to moderate levels (p  = 0.5). Their results illustrate how the immediacy effect varies according to the level of certainty involved, reinforcing the connection between risk and temporal preferences.

Building on these insights, Baucells and Heukamp (2012) demonstrate that adding a common delay to two risky choices has an effect on preferences similar to that of applying a common probability (that is the common ratio effect). Their empirical studies show that proportional changes in probabilities or the introduction of equal time delays influence disproportionately preferences between alternatives. This suggests a complex interaction between risk and time which requires decision models capable of addressing both effects simultaneously.

Finally, Chakraborty (2021) discusses how individuals tend to favour both certainty and immediacy, linking time delays and perceived risk. They argue that only present consumption is perceived as sure, whilst future consumption is viewed as uncertain, thus associating delayed rewards with perceived risk. Furthermore, they suggest that a weak certain bias is behaviourally equivalent to a weak present bias, indicating a shared underlying mechanism in how time and risk preferences shape decision-making.

This subsection shows the evolution of models and theories, transitioning from classical to contemporary normative approaches. These approaches capture the complexity of human decision-making by emphasising the significant contributions of each author. From the literature review, it can be seen that models which incorporate both time and risk have been developed with empirical evidence showing that the perception of time and risk influence each other.

Furthermore, behavioural theories reveal biases such as a preference for certainty and present bias. Integrating psychophysical laws and advanced machine learning methods in decision-making has enabled more precise predictions for decisions under uncertainty, thus combining the theory and real-world applications. This interdisciplinary approach underscores the potential for refined models which accommodate human psychological factors within decision frameworks.

Looking ahead, several key areas need further study to develop a more unified and practical understanding of these complexities. There is a pressing need for comprehensive models which encapsulate the delicate interaction between risk and time preferences in different contexts. In addition, undertaking further empirical studies to investigate the impact of these preferences on specific domains such as health and finance could provide valuable insights. This research focus would help to build a more robust foundation for understanding how psychological biases shape decisions, ultimately contributing to more effective interventions and policy recommendations.

Definition 1.

A discount function is a continuous real-valued function:

such that:

where:

• x ∈ X is the amount,

• p ∈ [0,1] is the probability of reaching the amount x at time t, and

• t ∈ T is the availability instant of the reward (x, p, t).

Moreover, V satisfies the following conditions:

1. V is increasing with x and p.

2. V is decreasing with t.

3. For every p ∈ [0, 1] and t ∈ T , V(0, p, t) = 0.

4. For every x ∈ X and t ∈ T , V(x, 0, t) = 0.

If T = [t₀, +∞] denote by L(x, p) the limit $limt →+∞V(x,p,t)$⁠. The following definition is therefore provided.

Definition 2.The discount function V is said to be regular (resp. singular) if, for every x ∈ X and p ∈ [0, 1], L(x, p) = 0 (resp. L(x, p) ≠ 0).

In this area, the study of the probability-time trade-off axiom of Baucells and Heukamp (2012) is particularly significant as it reveals a constant trade-off between probability and time delay for a specific outcome size:

Axiom 1 (Probability–time trade-off).

For every x ∈ X, p, q, θ ∈ (0,1), t, s ∈ T and Δ > 0,

implies

Proposition 1. A necessary condition for V satisfying the probability-time trade-off axiom is that the following equation holds:

Proof. By the probability-time trade-off axiom,

where θ only depends on x and Δ. The former equation can be subsequently written as follows:

and:

Observe that, if Δ → 0, then θ → 1. Therefore:

which leads to:

as required. ^▪

Example 1. Consider the non-separable discount function V(x, p, t) = px exp{-kxt}, with k > 0 and θ(x, Δ) = exp{-kxΔ}. Obviously, V satisfies the probability-time trade-off axiom with respect to θ. Observe that Proposition 1 holds because:

• $∂V(x,p,t)∂p=xexp⁡{−kxt}$ .

• $∂θ∂Δ|Δ=0=−kx$ .

• $∂V(x,p,t)∂t=−kpx2exp⁡{−kxt}$ .

Therefore, equation (1) can be easily verified.

This axiom of Baucells and Heukamp (2012) demonstrates that the introduction of risk and delay have similar effects on behaviour, and there is a consistent way in which both may be traded in the behavioural domain (Bradford et al., 2019). Thus, this axiom implies calibration properties which Chakraborty (2021) uses to specify a single discount factor for any decision maker.

Let (y, p₀, r) be a reward and z an amount greater than y (z > y). Consider the real-valued function $gz,p0(t):=V(z,p0,t)$⁠. If, for every x ∈ X, $L(x,p0)=lim⁡t→+∞gx,p0(t)$ is independent of x, then the following double inequality holds:

So, as $gz,p0$ is continuous, by the so-called Intermediate Value Theorem there will be s > r such that $gz,p0(s)=V(y,p0,r)$⁠. Therefore:

which demonstrates the amount-time trade-off (ATTO). In effect, if V is regular, Figure 2 displays the former reasoning.

In the case of a singular discount function, there is a limit for the values of y and z, given by (see Figure A1):

Another way of analysing the relationship between x and t, given p₀, is by deriving the set of points (x, t) such that V(x, p₀, t) = k. By the former reasoning, V(x, p₀, t) = k describes a decreasing curve.

Let (y,q,t₀) be a reward and z an amount greater than y (z > y). Consider the real-valued function $gz,t0(p):=V(z,p,t0)$⁠. The following double inequality holds:

So, as $gz,p0$ is continuous, by the so-called Intermediate Value Theorem there will be $q′< q$ such that $gz,t0(q′)=V(y, q,t0)$⁠. Therefore:

which demonstrates the amount-probability trade-off (APTO). In effect, Figure A2 displays the former reasoning.

Another way of analysing the relationship between x and p, given t₀, is by deriving the set of points (x, p) such that V(x, p, t₀) = k. By the former reasoning, V(x, p, t₀) = k describes a decreasing curve.

Let (x₀, q, r) be a reward and q′ a probability greater than q (q′ > q). Consider the real-valued function $gx0,q′(t):=V(x0,q′,t)$⁠. The following double inequality holds:

So, as $gx0,q′$ is continuous, by the so-called Intermediate Value Theorem there will be s > r such that $gx0,q′(s)=V(x0,q,r)$⁠. Therefore:

which demonstrates the probability-time trade-off (PTTO). In effect, Figure A3 displays the former reasoning.

Another way of analysing the relationship between p and t, given x₀, is by deriving the set of points (p,t) such that V(x₀, p,t) = k.

Following Leland and Schneider (2017), the common ratio and the common difference effects can be defined as follows:

Axiom 2 (Common ratio effect).

For every (x, p, t), t ∈ (0, x), θ ∈ (0, 1) and q <  p, (y, p, t)∼(x, pθ, t) implies (y, q, t) ≤ (x,qθ, t).

Axiom 3 (Common difference effect).

For every (x, p, t), y ∈ (0, x), Δ ∈ (0, ∞) and s > t, (y, p, t) ∼ (x, p, t + Δ) implies (y, p, s) ≤ (x, p, s + Δ).

The next step is to propose a “trade-off” between p and t which keeps the indifference between the rewards involved.

Theorem 1.

If the partial function V(·,p,t) is convex, then the delay effect implies the common difference effect.

Proof. In effect, assume that (y,p,t) ∼ (x,p,t + Δ) (which obviously involves y < x). From the amount-probability trade-off, there exists θ ∈ (0,1) such that:

By the delay effect, for every s > t, one has:

By the probability-time trade-off, one has:

As V(·,p,t) is convex, then Δ’ > Δ (see Figure A4) and so (x,p,s + Δ’) ≺ (x,p,s + Δ). Finally, by transitivity:

which demonstrates the common difference effect. ▪

It must now be considered whether there is an analogous result for the common ratio effect. To do this, Schneider (2016) introduces the following axiom:

Axiom 4 (Interaction of risk with time preference).

For every (x,p,t), y ∈ (0,x), Δ ∈ (0,∞) and q < p, (y,p,t) ∼ (x,p,t + Δ) implies (y,q,t) ≤ (x,q,t + Δ).

Theorem 2. If the partial function V(·, p,t) is convex, then the interaction of risk with time preference implies the common ratio effect.

Proof. In effect, assume that (y,p,t) ∼ (x,pθ,t) (which obviously involves y < x). By the amount-time trade-off, there exists Δ ∈ (0, +∞) such that:

By the interaction of risk with time preference, for every q < p, one has:

By the probability-time trade-off, one has:

As V(·,p,t) is convex, then θ′ < θ and so (x,qθ′,t) ≺ (x,qθ,t). Finally, by transitivity:

which demonstrates the common ratio effect.

In the decision-making process, two different models, namely the DU and EU models, traditionally assess intertemporal and risky rewards respectively. However, recent research highlights an increasing interest in examining the relationship between the assessment of delayed and risky prospects. This paper introduces a novel mathematical framework which relates the key anomalies present in the DU and EU models, demonstrating how the effects of time and risk are intertwined in a manner previously unexplored. Specifically, it demonstrates how different reward components (amount, maturity and probability) can be mathematically transformed and integrated into another component.

In particular, this analysis reveals the trade-off between amount and time, amount and probability, and time and probability. In this way, theorems 1 and 2 formalise the interrelation between certain anomalies present in the DU and EU models by considering the interaction between risk and time in the preference relation. Specifically, Theorem 1 derives the common difference effect starting from the delay effect (both anomalies of the DU model) whilst Theorem 2 derives the common ratio effect (an anomaly of the EU model) starting from the common difference effect (an anomaly of the DU model). The theorems presented herein formalise and consider the interaction between temporal and probabilistic magnitudes and set a cohesive perspective which deepens theoretical insights into decision-making biases. From a graphical point of view, Figure 2 shows the two-dimensional representation of a regular discount function V(x,p,t) showing the trade-off between amount and time on the x-axis. Analogously, Figure A1 shows the corresponding representation of a singular discount function. Figure A2 shows the two-dimensional representation of a discount function displaying the trade-off between amount and probability on the x-axis. Figure A3 shows the effect of convexity on the trade-off between probability and time to explain the common ratio effect, and finally Figure A4 shows the two-dimensional representation of a discount function representing the trade-off between probability and time on the x-axis.

The implications of this framework in management are significant also in other applied fields, particularly in contexts where both time and risk are crucial variables, such as financial risk assessment and healthcare policy modelling. Intertemporal choice is a central element in management, economics and finance, influencing key decision-making processes such as investment, capital budgeting, financial planning and determining asset prices (Singh and Nandan, 2024).

This paper refines the decision-making process under conditions of delay and risk by exploring potential biases and their interactions. As a result, managers and investors need to consider applying risk-adjusted discounting models in project assessment and financial planning.

Another managerial implication is the provision of an improved understanding of consumer preferences which may help to establish loyalty and pricing strategies. These strategies need to balance immediate incentives with long-term commitment to improve customer retention and promote sustainable purchasing behaviour. These insights provide a framework for understanding how individuals and organisations manage the trade-offs between short-term and long-term, as well as between risky and certain outcomes in decision-making.

Future research could consider the practical applications of this integrated framework, investigating how time and risk biases manifest themselves in real-world decision environments. Furthermore, a study of the interaction between other anomalies of the DU and EU models, such as the common consequence effect (an anomaly of the EU model) and the cancellation effect (an anomaly of the DU model), could reveal further complexities in non-standard decision-making behaviour.

In conclusion, our findings offer the foundation for a comprehensive decision-making model which considers probability-time-amount trade-offs based on the intersections of temporal and probabilistic decision biases, offering deeper insights into real-world decision-making processes.

The authors are very grateful for the comments and suggestions offered by two anonymous referees. Moreover, they acknowledge the comments and suggestions offered by those who attended the DySES (Dynamics of Socio Economic Systems) conference held in the University of Almería (Spain), 17–20 October, 2023.

Competing interests statement: The authors declare no competing interests.

Financial support statement: The authors acknowledge the financial support from the Mediterranean Research Centre for Economics and Sustainable Development (CIMEDES) at the University of Almería (Spain).

DU

= Discounted utility;

EU

= Expected utility;

ATTO

= Amount-time trade-off;

APTO

= Amount-probability trade-off;

PTTO

= Probability-time trade-off;

V

= Discount function;

L

= Limit of a discount function at infinity;

X

= Set of amounts;

T

= Set of times;

x

= Name of the variable “amount”;

p

= Name of the variable “probability”;

t

= Name of the variable “time”;

x₀

= Fixed amount;

p₀

= Fixed probability;

t₀

= Fixed time;

y and z

= Generic amounts;

q and q’

= Generic probabilities;

r and s

= Generic times;

α

= Factor belonging to the interval (0, +∞);

θ

= Factor belonging to the interval (0,1);

Δ

= Summand belonging to the interval (0, +∞).

Source(s): Own elaboration

Figure A1 

Figure A2 

Figure A3 

Figure A4