# Discrete versus Continuous Algorithms in Dynamics of Affective Decision Making
## 2 Affective Decision Making by Individuals  
The usual approach to decision making is based on constructing a utility functional for each of the alternatives from the considered set [21; 22]. In order to include the role of emotions, the expected utility is modified by adding the terms characterizing the influence of emotions [23; 24; 25; 26]. Thus, one tries to incorporate into utility at once both sides of decision making: rational reasoning, based on logical normative rules, and irrational unconscious emotions, such as joy, sadness, anger, fear, happiness, disgust, and surprise. The alternative that corresponds to the largest expected utility is treated as optimal and is certainly to be preferred. The approach we are using is principally different in several aspects: (i) This is a probabilistic theory, where the main characteristics are the probabilities of choosing each of the given alternatives. (ii) The probability of a choice is the sum of a utility factor, describing the probability of a choice based on rational reasoning, and an attraction factor, characterizing the influence of emotions. (iii) The optimal, or more correctly, a stochastically optimal alternative, is that which is associated with the largest probability. The mathematically rigorous axiomatic formulation of the theory has been carried out in Refs. [9; 10; 11]. The theory starts with the process of making decisions by separate individuals. Here, we state the main points of the approach in order that the reader could better understand the extension to decision making by a society, as is presented in this paper. First of all, decision making is understood as a probabilistic process. Let us consider decision makers choosing between the alternatives from a set  
\[\mathbb{A}\ =\ \{A_{n}:\ n=1,2,\ldots,N_{A}\}. \tag{1}\]  
The decision makers are considered as separate agents making decisions independently from each other. Equivalently, it is possible to keep in mind a single decision maker deciding on the given alternatives. The aim is to define the probability \(p(A_{n})\) of choosing an alternative \(A_{n}\). This probability can be understood as either the fraction of agents choosing this alternative or the frequency of choices of the alternative \(A_{n}\) by a separate decision maker. Of course, the probability is normalized:  
\[\sum_{n=1}^{N_{A}}p(A_{n})\ =\ 1\,\qquad 0\ \leq\ p(A_{n})\ \leq\ 1. \tag{2}\]  
The process of taking decisions consists of two sides. One evaluates the utility of alternatives as well as the attractiveness of alternatives that is influenced by emotions with respect to the choice of the alternatives. Therefore, the probability \(p(A_{n})\) of choosing an alternative \(A_{n}\) is a behavioral probability consisting of two terms: a utility factor \(f(A_{n})\) and an attraction factor \(q(A_{n})\):  
\[p(A_{n})\ =\ f(A_{n})+q(A_{n}). \tag{3}\]  
The utility factor \(f(A_{n})\) shows the rational probability of choosing an alternative \(A_{n}\) being based on the rational evaluation of the alternative utility, with the normalization  
\[\sum_{n=1}^{N_{A}}f(A_{n})\ =\ 1\,\qquad 0\ \leq\ f(A_{n})\ \leq\ 1. \tag{4}\]  
The attraction factor characterizes the influence of emotions in the process of choice of the alternative \(A_{n}\). Emotions can be positive or negative. For instance, the positive emotions are joy, happiness, pride, calm, serenity, love, gratitude, cheerfulness, euphoria, satisfaction (moral or physical), inspiration, amusement, pleasure, etc. Examples of negative emotions are sadness, anger, fear, disgust, guilt, shame, anxiety, loneliness, disappointment, etc.