# Discrete versus Continuous Algorithms in Dynamics of Affective Decision Making
## 6 Comparison of Discrete versus Continuous Algorithms
Solution \(p_{1}^{dis}(t)\), and solution \(p_{1}^{com}(t)\) monotonically tend to the same limit \(p_{1 dis}^{*}=p_{1 com}^{*}=0.209\); **(d)** Solutions \(p_{2}^{dis}(t)\) (solid line) and \(p_{2}^{com}(t)\) (dashed-dotted line) for the herding parameters \(\varepsilon_{1}=0.3\) and \(\varepsilon_{2}=0.1\). Solution \(p_{2}^{dis}(t)\), and solution \(p_{2}^{com}(t)\) monotonically tend to the same limit \(p_{2 dis}^{*}=p_{2 com}^{*}=0.628\). Discrete and continuous probabilities tend to common fixed points, but in a different way. Figure 9 gives an example where continuous solutions for both groups, with long-term and short-term memory, can 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. Finally, Figures 10 and 11 demonstrate the possibility of chaotic behavior for discrete solutions, when, for the same parameters, continuous solutions smoothly converge to stable nodes. \[p_{2}^{*}\ =\ (1-\varepsilon_{2})(f_{2}+q_{2}^{*})+\varepsilon_{2}(f_{1}+q_{ 1}^{*}), \tag{46}\]0 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 \(\varepsilon_{1}=1\) and \(\varepsilon_{2}=0.7\): (**a**) Solutions \(p_{1}^{dis}(t)\) (solid line) and \(p_{1}^{con}(t)\) (dashed-dotted line). Discrete solution \(p_{1}^{dis}(t)\) chaotically oscillates and continuous solution \(p_{1}^{con}(t)\) tends to the limit \(p_{1}^{*}{}_{con}=0.735\); (**b**) Discrete solution \(p_{2}^{dis}(t)\) (solid line) and continuous solution \(p_{2}^{con}(t)\) (dashed-dotted line). Discrete solution \(p_{2}^{dis}(t)\) chaotically oscillates, while continuous solution \(p_{2}^{con}(t)\) tends to the limit \(p_{2}^{*}{}_{con}=0.360\). Discrete solutions are chaotic, while, for the same parameters, continuous solutions smoothly converge to stable nodes. \[p_{2}^{*}\ =\ (1-\varepsilon_{2})(f_{2}+q_{2}^{*})+\varepsilon_{2}(f_{1}+q_{ 1}^{*}), \tag{46}\]1 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 \(\varepsilon_{1}=\varepsilon_{2}=1\): (**a**) Solution to discrete Equation (37) \(p_{1}^{dis}(t)\) (solid line) oscillates, but solution \(p_{2}^{dis}(t)\) (dashed-dotted line) tends to the fixed point \(p_{2}^{*}=f_{1}=0.1\); (**b**) Solutions to continuous Equation (43) \(p_{1}^{con}(t)\) (solid line) and \(p_{2}^{con}(t)\) (dashed-dotted line) tend to the same fixed point \(p_{1}^{*}=p_{2}^{*}=f_{2}+q_{2}=0.93\). 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. Summarizing the possible types of behavior, we see that continuous decision making always displays smooth behavior of probabilities for both groups, with either long-term or short-term memory always converging to a stable node. However, discrete decision making can exhibit, for the same probabilities, a larger variety of behavior types, which can be smooth, tending to a stable node, or oscillating, hence tending to a stable focus, or even chaotic. As far as the temporal behavior of the probabilities of choosing the related alternatives for discrete and continuous decision making can be essentially different, the natural question arises: Which of the algorithms, discrete or continuous, better corresponds to the real decision making of social groups? It seems there are activities, such as car driving, where decisions can be well approximated by a continuous process. At the same time, it looks like such processes can be described by a series of decisions occurring discretely, although with rather small time intervals between the subsequent steps.