# Discrete versus Continuous Algorithms in Dynamics of Affective Decision Making
## 6 Comparison of Discrete versus Continuous Algorithms
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 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 \(\varepsilon_{1}=\varepsilon_{2}=0\): (**a**) Discrete solution \(p_{1}^{dis}(t)\) (solid line) and continuous solution \(p_{1}^{com}(t)\) (dashed-dotted line). Both solutions tend to the same fixed point \(p_{1}^{*}=0.8\); (**b**) Discrete solution \(p_{2}^{dis}(t)\) (solid line) and continuous solution \(p_{2}^{com}(t)\) (dashed-dotted line). Probability \(p_{2}^{com}(t)\) tends monotonically, while \(p_{2}^{dis}(t)\) tends with oscillations to the same fixed point \(p_{2}^{*}=0.377\). 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 4 shows that the fixed points of agents with long-term memory can 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 solution at the beginning almost coincides with the continuous one, but starts oscillating from a finite time and after this continues oscillating for all times. Figure 5 explains that discrete and continuous probabilities, though both being stable nodes, tend to different fixed points, which do not coincide. This happens in the presence of a strong herding effect. 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 \(\varepsilon_{1}=\varepsilon_{2}=0\): **(a)** Discrete solution \(p_{1}^{dis}(t)\) (solid line) and continuous solution \(p_{1}^{con}(t)\) (dashed-dotted line). Solutions \(p_{1}^{con}(t)\) and \(p_{1}^{dis}(t)\) tend to the same fixed point \(p_{1}^{*}=0.3\); **(b)** Discrete solution \(p_{2}^{dis}(t)\) (solid line) and continuous solution \(p_{2}^{con}(t)\) (dashed-dotted line). Solution \(p_{2}^{con}(t)\) tends to \(p_{2}^{*}=0.699\), whereas \(p_{2}^{dis}(t)\) oscillates around \(p_{2}^{*}\), 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 \(\varepsilon_{1}=\varepsilon_{2}=1\):  
(a) Discrete solution \(p_{1}^{dis}(t)\) (solid line) tends to the fixed point \(p_{1 dis}^{*}=0.5\) and continuous solution \(p_{1}^{con}(t)\) (dashed-dotted line) tends to the fixed point \(p_{1 com}^{*}=f_{2}+q_{2}=0.8=p_{2 com}^{*}\); (b) Discrete solution \(p_{2}^{dis}(t)\) (solid line) tends to \(p_{2 dis}^{*}=1\), while continuous solution \(p_{2}^{con}(t)\) (dashed-dotted line) tends to \(p_{2 com}^{*}=p_{1 com}^{*}=0.8\); (c) Solutions \(p_{1}^{dis}(t)\) and \(p_{2}^{dis}(t)\); (d) Solutions \(p_{1}^{con}(t)\) and \(p_{2}^{con}(t)\). For \(t\to\infty\), solutions \(p_{1}^{con}(t)\) and \(p_{2}^{con}(t)\) tend to the same fixed point \(p_{1 com}^{*}=p_{2 com}^{*}=f_{2}+q_{2}=0.8\), however solution \(p_{1}^{dis}(t)\) tends to \(p_{1 dis}^{*}=0.5\), whereas solution \(p_{2}^{dis}(t)\) tends to \(p_{2 dis}^{*}=1\).