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In this paper, a stochastic game model of self-organization of strategies of stochastic game of mobile agents in the form of cyclic behavioral patterns, which consist of coordinated strategies for moving agents in a limited discrete space, is developed. The behavioral pattern of a multi-agent system is a visualized form of orderly movement of agents that arises from their initial chaotic movement during the learning of a stochastic game. The mobility of multi-step stochastic game agents is ensured by the fact that in discrete moments of time they randomly, simultaneously and independently choose their own pure strategy of moving in one of the possible directions. Current player payments are defined as loss functions that depend on the strategies of neighboring players. These functions are formed from the penalty for irregular spacing of agents in a limited discrete space and the penalty for collisions when moving agents. Random selection of players’ strategies aims to minimize their average loss functions. The generation of sequences of pure strategies is performed by a discrete distribution based on the vectors of mixed strategies. The elements of the vectors of mixed strategies are the conditional probabilities of choosing the appropriate pure displacement strategies. Mixed strategies change over time, adaptively taking into account the value of current losses. This provides an increase in the probability of choosing those strategies that lead to a decrease in the functions of average losses. The dynamics of the vectors of mixed strategies is determined by the Markov recurrent method, for the construction of which a stochastic approximation of the modified condition of complementary non- rigidity, which is valid at Nash equilibrium points, is performed, and a projection operator for expandable unit epsilon simplex is applied. The convergence of the recurrent game method is ensured by compliance with the fundamental conditions and limitations of stochastic approximation. The stochastic game begins with untrained mixed strategies that set a chaotic picture of moving agents. During the learning of the stochastic game, the vectors of mixed strategies are purposefully changed so as to ensure an orderly, conflict-free movement of agents. As a result of computer simulation of stochastic game, cyclic patterns of self-organization of mobile agents on the surface of a discrete torus and within a rectangular region on a plane are obtained. The reliability of the experimental studies was confirmed by the similarity of the obtained patterns of moving agents for different sequences of random variables. The results of the study are proposed to be used in practice for the construction of distributed systems with elements of self-organization, solving various flow and transport problems and collective decision-making in conditions of uncertainty.
In this paper, a stochastic game model of self-organization of strategies of stochastic game of mobile agents in the form of cyclic behavioral patterns, which consist of coordinated strategies for moving agents in a limited discrete space, is developed. The behavioral pattern of a multi-agent system is a visualized form of orderly movement of agents that arises from their initial chaotic movement during the learning of a stochastic game. The mobility of multi-step stochastic game agents is ensured by the fact that in discrete moments of time they randomly, simultaneously and independently choose their own pure strategy of moving in one of the possible directions. Current player payments are defined as loss functions that depend on the strategies of neighboring players. These functions are formed from the penalty for irregular spacing of agents in a limited discrete space and the penalty for collisions when moving agents. Random selection of players’ strategies aims to minimize their average loss functions. The generation of sequences of pure strategies is performed by a discrete distribution based on the vectors of mixed strategies. The elements of the vectors of mixed strategies are the conditional probabilities of choosing the appropriate pure displacement strategies. Mixed strategies change over time, adaptively taking into account the value of current losses. This provides an increase in the probability of choosing those strategies that lead to a decrease in the functions of average losses. The dynamics of the vectors of mixed strategies is determined by the Markov recurrent method, for the construction of which a stochastic approximation of the modified condition of complementary non- rigidity, which is valid at Nash equilibrium points, is performed, and a projection operator for expandable unit epsilon simplex is applied. The convergence of the recurrent game method is ensured by compliance with the fundamental conditions and limitations of stochastic approximation. The stochastic game begins with untrained mixed strategies that set a chaotic picture of moving agents. During the learning of the stochastic game, the vectors of mixed strategies are purposefully changed so as to ensure an orderly, conflict-free movement of agents. As a result of computer simulation of stochastic game, cyclic patterns of self-organization of mobile agents on the surface of a discrete torus and within a rectangular region on a plane are obtained. The reliability of the experimental studies was confirmed by the similarity of the obtained patterns of moving agents for different sequences of random variables. The results of the study are proposed to be used in practice for the construction of distributed systems with elements of self-organization, solving various flow and transport problems and collective decision-making in conditions of uncertainty.
This paper proposes a new application of the stochastic game model to solve the problem of self- organization of the Hamiltonian cycle of a graph. To do this, at the vertices of the undirected graph are placed game agents, whose pure strategies are options for choosing one of the incident edges. A random selection of strategies by all agents forms a set of local paths that begin at each vertex of the graph. Current player payments are defined as loss functions that depend on the strategies of neighboring players that control adjacent vertices of the graph. These functions are formed from a penalty for the choice of opposing strategies by neighboring players and a penalty for strategies that have reduced the length of the local path. Random selection of players’ pure strategies is aimed at minimizing their average loss functions. The generation of sequences of pure strategies is performed by a discrete distribution built on the basis of dynamic vectors of mixed strategies. The elements of the vectors of mixed strategies are the probabilities of choosing the appropriate pure strategies that adaptively take into account the values of current losses. The formation of vectors of mixed strategies is determined by the Markov recurrent method, for the construction of which the gradient method of stochastic approximation is used. During the game, the method increases the value of the probabilities of choosing those pure strategies that lead to a decrease in the functions of average losses. For given methods of forming current payments, the result of the stochastic game is the formation of patterns of self-organization in the form of cyclically oriented strategies of game agents. The conditions of convergence of the recurrent method to collectively optimal solutions are ensured by observance of the fundamental conditions of stochastic approximation. The game task is extended to random graphs. To do this, the vertices are assigned the probabilities of recovery failures, which cause a change in the structure of the graph at each step of the game. Realizations of a random graph are adaptively taken into account when searching for Hamiltonian cycles. Increasing the probability of failure slows down the convergence of the stochastic game. Computer simulation of the stochastic game provided patterns of self-organization of agents’ strategies in the form of several local cycles or a global Hamiltonian cycle of the graph, depending on the ways of forming the current losses of players. The reliability of experimental studies is confirmed by the repetition of implementations of self-organization patterns for different sequences of random variables. The results of the study can be used in practice for game-solving NP-complex problems, transport and communication problems, for building authentication protocols in distributed information systems, for collective decision-making in conditions of uncertainty.
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