Softarisons: theory and practice

This paper introduces the concept of softarison. Softarisons merge soft set theory with the theory of binary relations. Their purpose is the comparison of alternatives in a parameterized environment. We develop the basic theory and interpretations of softarisons. Then, the normative idea of ‘optimal’ alternatives is discussed in this context. We argue that the meaning of ‘optimality’ can be adjusted to fit in with the structure of each problem. A sufficient condition for the existence of an optimal alternative for unrestricted sets of alternatives is proven. This result means a counterpart of Weierstrass extreme value theorem for softarisons; thus, it links soft topology with the act of choice. We also provide a decision-making procedure—the minimax algorithm—when the alternatives are compared through a softarison. A case-study in the context of group interviews illustrates both the application of softarisons as an evaluation tool, and the computation of minimax solutions.

Mechanism for shifting Nash equilibrium trajectories to cooperative Pareto solutions in dynamic bimatrix games

In the paper, constructions of the generalized method of characteristics are applied for calculating the generalized minimax (viscosity) solutions of Hamilton-Jacobi equations in dynamic bimatrix games. The structure of the game presumes interactions of two players in the framework of the evolutionary game model. Stochastic contacts between players occur according to the dynamic process, which can be interpreted as a system of Kolmogorov's differential equations with controls instead of probability parameters. It is assumed that control parameters are not fixed and can be constructed by the feedback principle. Two types of payoff functions are considered: short-term payoffs are determined in the current moments of time, and long-term payoffs are determined as limit functionals on the infinite time horizon. The notion of dynamic Nash equilibrium in the class of controlled feedbacks is considered for the long-term payoffs. In the framework of constructions of dynamic equilibrium, the solutions are designed on the basis of maximization of guaranteed payoffs. Such guaranteeing strategies are built in the framework of the theory of minimax (viscosity) solutions of Hamilton-Jacobi equations. The analytical formulas are obtained for the value functions in the cases of different orientations for the “zigzags” (broken lines) of acceptable situations in the static game. The equilibrium trajectories generated by the minimax solutions shift the system in the direction of cooperative Pareto points. The proposed approach provides new qualitative properties of the equilibrium trajectories in the dynamic bimatrix games which guarantee better results of payoffs for both players than static Nash equilibria. As an example, interactions of two firms on the market of innovative electronic devices are examined within the proposed approach for treating dynamic bimatrix games.