# Discrete versus Continuous Algorithms in Dynamics of Affective Decision Making

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## Abstract

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## 1. Introduction

## 2. Affective Decision Making by Individuals

## 3. Discrete Dynamics in Affective Decision Making

## 4. Two Groups with Binary Choice

## 5. Continuous Dynamics of Affective Decision Making

## 6. Comparison of Discrete versus Continuous Algorithms

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Solutions to discrete Equation (37) and to continuous Equation (43) for the initial conditions ${f}_{1}=0.4$, ${f}_{2}=0.1$, ${q}_{1}=0.59$, and ${q}_{2}=0.6$, in the absence of herding effect, when ${\epsilon}_{1}={\epsilon}_{2}=0$: (

**a**) Discrete solution ${p}_{1}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line). Both solutions tend to the same fixed point ${p}_{1}^{\ast}=0.4$; (

**b**) Discrete solution ${p}_{2}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line). Both solutions tend to the same fixed point ${p}_{2}^{\ast}=0.636$, which is a stable node.

**Figure 2.**Solutions to discrete Equation (37) and to continuous Equation (43) for the initial conditions ${f}_{1}=0.8$, ${f}_{2}=0.9$, ${q}_{1}=0.19$, and ${q}_{2}=-0.8$, when there is no herding effect, hence ${\epsilon}_{1}={\epsilon}_{2}=0$: (

**a**) Discrete solution ${p}_{1}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line). Both solutions tend to the same fixed point ${p}_{1}^{\ast}=0.8$; (

**b**) Discrete solution ${p}_{2}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line). Probability ${p}_{2}^{con}\left(t\right)$ tends monotonically, while ${p}_{2}^{dis}\left(t\right)$ tends with oscillations to the same fixed point ${p}_{2}^{\ast}=0.377$. Discrete and continuous solutions tend to the same fixed point, but for the agents with long-term memory the fixed point is a stable node, however for the agents with short-term memory, the continuous solution tends to a stable node, while for the discrete solution, to a stable focus.

**Figure 3.**Solutions to discrete Equation (37) and to continuous Equation (43) for the initial conditions ${f}_{1}=0.8$, ${f}_{2}=1$, ${q}_{1}=0.1$, and ${q}_{2}=-0.99$, in the absence of herding effect, when ${\epsilon}_{1}={\epsilon}_{2}=0$: (

**a**) Discrete solution ${p}_{1}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line). Solutions ${p}_{1}^{con}\left(t\right)$ and ${p}_{1}^{dis}\left(t\right)$ tend to the same fixed point ${p}_{1}^{\ast}=0.8$; (

**b**) Discrete solution ${p}_{2}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line). Solution ${p}_{2}^{con}\left(t\right)$ tends to the fixed point ${p}_{2}^{\ast}=0.366$, whereas ${p}_{2}^{dis}\left(t\right)$ oscillates around ${p}_{2}^{\ast}$ with the constant amplitude. For the agents with long-term memory, both probabilities, discrete and continuous, tend to the same stable node, but for the agents with short-term memory, the fixed point for discrete probability is a stable limit cycle, while the continuous probability tends to a stable node.

**Figure 4.**Solutions to discrete Equation (37) and to continuous Equation (43) for the initial conditions ${f}_{1}=0.3$, ${f}_{2}=0$, ${q}_{1}=0.699$, and ${q}_{2}=0.98$, without the herding effect, when ${\epsilon}_{1}={\epsilon}_{2}=0$: (

**a**) Discrete solution ${p}_{1}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line). Solutions ${p}_{1}^{con}\left(t\right)$ and ${p}_{1}^{dis}\left(t\right)$ tend to the same fixed point ${p}_{1}^{\ast}=0.3$; (

**b**) Discrete solution ${p}_{2}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line). Solution ${p}_{2}^{con}\left(t\right)$ tends to ${p}_{2}^{\ast}=0.699$, whereas ${p}_{2}^{dis}\left(t\right)$ oscillates around ${p}_{2}^{\ast}$, starting at a finite time and continues oscillating for $t\to \infty $ with a constant amplitude. The fixed points of agents with long-term memory coincide for discrete and continuous solutions, both being stable nodes, while for agents with short-term memory, the continuous solution tends to a stable node, whereas the discrete one oscillates.

**Figure 5.**Solutions to discrete Equation (37) and to continuous Equation (43) for the initial conditions ${f}_{1}=1$, ${f}_{2}=0.2$, ${q}_{1}=-0.9$, and ${q}_{2}=0.6$, in the presence of strong herding effect, when ${\epsilon}_{1}={\epsilon}_{2}=1$: (

**a**) Discrete solution ${p}_{1}^{dis}\left(t\right)$ (solid line) tends to the fixed point ${p}_{1dis}^{\ast}=0.5$ and continuous solution ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line) tends to the fixed point ${p}_{1con}^{\ast}={f}_{2}+{q}_{2}=0.8={p}_{2con}^{\ast}$; (

**b**) Discrete solution ${p}_{2}^{dis}\left(t\right)$ (solid line) tends to ${p}_{2dis}^{\ast}=1$, while continuous solution ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line) tends to ${p}_{2con}^{\ast}={p}_{1con}^{\ast}=0.8$; (

**c**) Solutions ${p}_{1}^{dis}\left(t\right)$ and ${p}_{2}^{dis}\left(t\right)$; (

**d**) Solutions ${p}_{1}^{con}\left(t\right)$ and ${p}_{2}^{con}\left(t\right)$. For $t\to \infty $, solutions ${p}_{1}^{con}\left(t\right)$ and ${p}_{2}^{con}\left(t\right)$ tend to the same fixed point ${p}_{1con}^{\ast}={p}_{2con}^{\ast}={f}_{2}+{q}_{2}=0.8$, however solution ${p}_{1}^{dis}\left(t\right)$ tends to ${p}_{1dis}^{\ast}=0.5$, whereas solution ${p}_{2}^{dis}\left(t\right)$ tends to ${p}_{2dis}^{\ast}=1$. Discrete and continuous probabilities, though both being stable nodes, but tend to different fixed points.

**Figure 6.**Solutions to discrete Equation (37) and to continuous Equation (43) for the initial conditions ${f}_{1}=0.6$, ${f}_{2}=1$, ${q}_{1}=0.39$, and ${q}_{2}=-0.9$: (

**a**) Discrete solution ${p}_{1}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}={\epsilon}_{2}=1$. Solution ${p}_{1}^{con}$(t) tends to the fixed point ${p}_{1con}^{\ast}=0.280$, whereas solution ${p}_{1}^{dis}\left(t\right)$ oscillates with a constant amplitude around ${p}_{1con}^{\ast}$ for $t\to \infty $; (

**b**) Discrete solution ${p}_{2}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}={\epsilon}_{2}=1$. Solutions ${p}_{2}^{dis}\left(t\right)$ and ${p}_{2}^{con}\left(t\right)$ tend to the same fixed point ${p}_{2dis}^{\ast}={p}_{2con}^{\ast}={f}_{1}=0.6$; (

**c**) Discrete solution ${p}_{1}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}=0.9$ and ${\epsilon}_{2}=0.8$. Solution ${p}_{1}^{dis}\left(t\right)$ oscillates, and solution ${p}_{1}^{con}\left(t\right)$ monotonically tends to the fixed point ${p}_{1con}^{\ast}=0.265$; (

**d**) Discrete solution ${p}_{2}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}=0.9$ and ${\epsilon}_{2}=0.8$. Solution ${p}_{2}^{dis}\left(t\right)$ oscillates, and solution ${p}_{2}^{con}\left(t\right)$ monotonically tends to the limit ${p}_{2con}^{\ast}=0.525$. The behavior of discrete and continuous solutions is qualitatively different.

**Figure 7.**Solutions to discrete Equation (37) and to continuous Equation (43) for the initial conditions ${f}_{1}=0$, ${f}_{2}=0.1$, ${q}_{1}=0.93$, and ${q}_{2}=0.899$: (

**a**) Discrete solution ${p}_{1}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}={\epsilon}_{2}=1$. Solutions ${p}_{1}^{con}\left(t\right)$, monotonically, and ${p}_{1}^{dis}\left(t\right)$, with oscillations, tend to the same limit ${p}_{1dis}^{\ast}={p}_{1con}^{\ast}=0.526$; (

**b**) Solutions ${p}_{2}^{dis}\left(t\right)$ (solid line) and ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}={\epsilon}_{2}=\phantom{\rule{3.33333pt}{0ex}}1$. Solutions ${p}_{2}^{dis}\left(t\right)$ and ${p}_{2}^{con}\left(t\right)$ tend to the same limit ${p}_{2dis}^{\ast}={p}_{2con}^{\ast}={f}_{1}=0$; (

**c**) Solutions ${p}_{1}^{dis}\left(t\right)$ (solid line) and ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}=0.3$ and ${\epsilon}_{2}=0.1$. Solution ${p}_{1}^{dis}\left(t\right)$, and solution ${p}_{1}^{con}\left(t\right)$ monotonically tend to the same limit ${p}_{1dis}^{\ast}={p}_{1con}^{\ast}=0.209$; (

**d**) Solutions ${p}_{2}^{dis}\left(t\right)$ (solid line) and ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}=0.3$ and ${\epsilon}_{2}=0.1$. Solution ${p}_{2}^{dis}\left(t\right)$, and solution ${p}_{2}^{con}\left(t\right)$ monotonically tend to the same limit ${p}_{2dis}^{\ast}={p}_{2con}^{\ast}=0.628$. Discrete and continuous probabilities tend to common fixed points, but in a different way.

**Figure 8.**Solutions to discrete Equation (37) and continuous Equation (43) for the initial conditions ${f}_{1}=0.3$, ${f}_{2}=0$, ${q}_{1}=0.699$, and ${q}_{2}=0.99$, with the herding parameters ${\epsilon}_{1}=0.9$ and ${\epsilon}_{2}=0.8$: (

**a**) Solutions ${p}_{1}^{dis}\left(t\right)$ (solid line) and ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line). Solutions ${p}_{1}^{dis}\left(t\right)$ and ${p}_{1}^{con}\left(t\right)$ tend to the same limit ${p}_{1}^{\ast}={f}_{2}+{q}_{2}=0.99$; (

**b**) Solutions ${p}_{2}^{dis}\left(t\right)$ (solid line) and ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line). Solutions ${p}_{2}^{dis}\left(t\right)$ and ${p}_{2}^{con}\left(t\right)$ tend to the same limit ${p}_{2}^{\ast}={f}_{2}+{q}_{2}=0.99$. Note that here ${p}_{1}^{\ast}={p}_{2}^{\ast}$. All probabilities for the groups with long-term memory as well as short-term memory, for discrete as well as continuous solutions, tend to the common fixed point.

**Figure 9.**Solutions to discrete Equation (37) and to continuous Equation (43) for the initial conditions ${f}_{1}=0.1$, ${f}_{2}=0$, ${q}_{1}=0.899$, and ${q}_{2}=0.93$, with the herding parameters ${\epsilon}_{1}={\epsilon}_{2}=1$: (

**a**) Solution to discrete Equation (37) ${p}_{1}^{dis}\left(t\right)$ (solid line) oscillates, but solution ${p}_{2}^{dis}\left(t\right)$ (dashed-dotted line) tends to the fixed point ${p}_{2}^{\ast}={f}_{1}=0.1$; (

**b**) Solutions to continuous Equation (43) ${p}_{1}^{con}\left(t\right)$ (solid line) and ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line) tend to the same fixed point ${p}_{1}^{\ast}={p}_{2}^{\ast}={f}_{2}+{q}_{2}=0.93$. Continuous solutions for both groups, with long-term and short-term memory, tend to coinciding limits, while the related discrete solutions for these groups are very different: One solution permanently oscillates, and the other tends to a stable node.

**Figure 10.**Solutions to discrete Equation (37) and continuous Equation (43) for the initial conditions ${f}_{1}=0.2$, ${f}_{2}=0$, ${q}_{1}=-0.1$, and ${q}_{2}=0.999$, with the herding parameters ${\epsilon}_{1}=1$ and ${\epsilon}_{2}=0.7$: (

**a**) Solutions ${p}_{1}^{dis}\left(t\right)$ (solid line) and ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line). Discrete solution ${p}_{1}^{dis}\left(t\right)$ chaotically oscillates and continuous solution ${p}_{1}^{con}\left(t\right)$ tends to the limit ${p}_{1con}^{\ast}=0.735$; (

**b**) Discrete solution ${p}_{2}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line). Discrete solution ${p}_{2}^{dis}\left(t\right)$ chaotically oscillates, while continuous solution ${p}_{2}^{con}\left(t\right)$ tends to the limit ${p}_{2con}^{\ast}=0.360$. Discrete solutions are chaotic, while, for the same parameters, continuous solutions smoothly converge to stable nodes.

**Figure 11.**Solutions to discrete Equation (37) and to continuous Equation (43) for the initial conditions ${f}_{1}=0.6$, ${f}_{2}=1$, ${q}_{1}=0.3$, and ${q}_{2}=-0.999$: (

**a**) Discrete solution ${p}_{1}^{dis}\left(t\right)$ (solid line) and continuous solution ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}={\epsilon}_{2}=1$. Solution ${p}_{1}^{con}\left(t\right)$ tends to the limit ${p}_{1con}^{\ast}=0.246$, while ${p}_{1}^{dis}\left(t\right)$ chaotically oscillates around ${p}_{1dis}^{\ast}$; (

**b**) Discrete solution ${p}_{2}^{dis}\left(t\right)$ (solid line) and ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}={\epsilon}_{2}=1$. Solutions ${p}_{2}^{dis}\left(t\right)$ and ${p}_{2}^{con}\left(t\right)$ tend to the same limit ${p}_{2dis}^{\ast}={p}_{2con}^{\ast}={f}_{1}=0.6$; (

**c**) Solutions ${p}_{1}^{dis}\left(t\right)$ (solid line) and ${p}_{1}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}=1$ and ${\epsilon}_{2}=0.8$. Solution ${p}_{1}^{con}\left(t\right)$ tends to the limit ${p}_{1con}^{\ast}=0.210$, but solution ${p}_{1}^{dis}\left(t\right)$ chaotically oscillates for all times $t\to \infty $; (

**d**) Solutions ${p}_{2}^{dis}\left(t\right)$ (solid line) and ${p}_{2}^{con}\left(t\right)$ (dashed-dotted line) for the herding parameters ${\epsilon}_{1}=1$ and ${\epsilon}_{2}=0.8$. Solution ${p}_{2}^{con}\left(t\right)$ tends to the limit ${p}_{2con}^{\ast}=0.522$, while solution ${p}_{2}^{dis}\left(t\right)$ chaotically oscillates around ${p}_{2dis}^{\ast}$. Examples of chaotic behavior of discrete solutions.

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**MDPI and ACS Style**

Yukalov, V.I.; Yukalova, E.P.
Discrete versus Continuous Algorithms in Dynamics of Affective Decision Making. *Algorithms* **2023**, *16*, 416.
https://doi.org/10.3390/a16090416

**AMA Style**

Yukalov VI, Yukalova EP.
Discrete versus Continuous Algorithms in Dynamics of Affective Decision Making. *Algorithms*. 2023; 16(9):416.
https://doi.org/10.3390/a16090416

**Chicago/Turabian Style**

Yukalov, Vyacheslav I., and Elizaveta P. Yukalova.
2023. "Discrete versus Continuous Algorithms in Dynamics of Affective Decision Making" *Algorithms* 16, no. 9: 416.
https://doi.org/10.3390/a16090416