# Discrete versus Continuous Algorithms in Dynamics of Affective Decision Making
## 6 Comparison of Discrete versus Continuous Algorithms
It may happen that, despite the small time intervals, the discrete and continuous decision algorithms lead to different conclusions. From our point of view, the discrete algorithm is preferable, since decisions, anyway, are complex, discrete actions composed of several subactions: receiving information, processing this information, and making a decision, so that always there is a delay time from the start of receiving information to the moment of making a decision. The continuous algorithm can provide a reasonable approximation in some cases, although sometimes can result in wrong conclusions. \[p_{2}^{*}\ =\ (1-\varepsilon_{2})(f_{2}+q_{2}^{*})+\varepsilon_{2}(f_{1}+q_{ 1}^{*}), \tag{46}\]2 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 \(p_{1}^{dis}(t)\) (solid line) and continuous solution \(p_{1}^{con}(t)\) (dashed-dotted line) for the herding parameters \(\varepsilon_{1}=\varepsilon_{2}=1\). Solution \(p_{1}^{con}(t)\) tends to the limit \(p_{1}^{*}\)com = 0.246, while \(p_{1}^{dis}(t)\) chaotically oscillates around \(p_{1}^{*}\)dis; **(b)** Discrete solution \(p_{2}^{dis}(t)\) (solid line) and \(p_{2}^{con}(t)\) (dashed-dotted line) for the herding parameters \(\varepsilon_{1}=\varepsilon_{2}=1\). Solutions \(p_{2}^{dis}(t)\) and \(p_{2}^{con}(t)\) tend to the same limit \(p_{2}^{*}\)dis = \(p_{2}^{*}\)com = \(f_{1}=0.6\); **(c)** Solutions \(p_{1}^{dis}(t)\) (solid line) and \(p_{1}^{con}(t)\) (dashed-dotted line) for the herding parameters \(\varepsilon_{1}=1\) and \(\varepsilon_{2}=0.8\). Solution \(p_{1}^{con}(t)\) tends to the limit \(p_{1}^{*}\)com = 0.210, but solution \(p_{1}^{dis}(t)\) chaotically oscillates for all times \(t\to\infty\); **(d)** Solutions \(p_{2}^{dis}(t)\) (solid line) and \(p_{2}^{con}(t)\) (dashed-dotted line) for the herding parameters \(\varepsilon_{1}=1\) and \(\varepsilon_{2}=0.8\). Solution \(p_{2}^{con}(t)\) tends to the limit \(p_{2}^{*}\)com = 0.522, while solution \(p_{2}^{dis}(t)\) chaotically oscillates around \(p_{2}^{*}\)dis. Examples of chaotic behavior of discrete solutions. When a probability \(p_{j}(t)\) converges to a stable node, the corresponding stationary limit \(p_{j}^{*}\) plays the role of the optimal decision taken after multiple steps of decision making, including the exchange of information with other agents of all groups, taking account of agents' emotions, and the tendency of the agents to herding. When a probability oscillates either periodically or chaotically, this implies that the agents are not able to come to a decision, but cannot stop hesitating. There exist numerous examples of chaotic behavior of decision making in medicine, economics, and different types of management [76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87]. The mathematical reason why the considered continuous solutions for the probabilities cannot display chaos is as follows. The probabilities, by definition, are bounded, hence Lagrange stable. Then, for a plane motion, the Poincare-Bendixson theorem tells us that if a trajectory of a continuous two-dimensional dynamical system is Lagrange stable, then it approaches either a stable node or a limit cycle [75]. However, for discrete equations, there is no such theorem, and a discrete dynamical system can exhibit chaos.