Balancing Economic, Legal, and Social Macrosystems Based on Modelling Decision Processes

On the Concept of Equilibrium in Sanctions and Countersanctions in a Differential Game

“…Thus, under Conditions (11), the differential game Γ has no Nash equilibrium. At the same time, the strategy U T 1 ÷ αx∀α ≥ α (1) (U P , t 0 , x 0 ) implements the sanction of Player 1 to the Paretomaximal (efficient) strategy profile U P ; see (5).…”

“…In the lemmas below, the initial position (t 0 , x 0 ) is fixed and coincides with the one from Lemma 5; in addition, the sanction strategy U T 1 ÷ αx of Player 1 has a constant scalar α = α (1) . Recall that Conditions (11) are assumed to hold without special mention. Well, Lemma 5 establishes the following result.…”

“…In contrast to other research, this article offers an economical and legal substantiation of theoretical and game constructions modeling the process of application of sanctions and countersanctions, using in general terms the idea of systemic balance of three macrosystems: economic, legal, and social [11]. In a particular case, the economic-legal substantiation of the proposed theoretical-game model of the balance of sanctions and countersanctions is based on the use of the legal concept of sanctions as a component of the definition of legal responsibility of subjects, i.e., in practical terms, the regulator implements the principle of inevitability of legal responsibility, in particular, sanctions for offense and crime (can be a departure from the established in the legal prescriptions rules, rules of social and economic relations), which is manifested in the application in the description of real socio-economic processes, various concepts of static and active equilibriums.…”

“…These items (1) and ( 2) involve the definiteness of quadratic forms figuring in the integral terms of the payoff functions (3). From this point onwards, assume that Conditions (11)…”

“…By analogy with Lemmas 5 and 6, we will establish another important result below. Recall that an initial position (t 0 , x 0 ), a continuous matrix Θ P (t) of dimensions n × n, and an incomplete countersanction strategy U C 2 ÷ α (2) x figuring in Lemmas 5 and 6 are assumed to be fixed, and Conditions (11) are assumed to hold. Lemma 7.…”