A new application of a stochastic game model for solving the problem of self-organization of the Hamiltonian graph cycle is proposed. To do this, game agents are placed at the vertices of an undirected graph, the pure strategies of which are options for choosing one of the incident edges. All agents’ random choice of strategies forms a set of local paths that begin at each vertex of the graph. Current player payouts are defined as loss-making functions that depend on the strategies of neighbouring players who control adjacent vertices of the graph. These functions are formed from a penalty for choosing opposing strategies by neighbouring players and a penalty for strategies that have shortened the length of the local path. Computer simulation of a stochastic game provided a pattern of self-organization of agents' strategies in the form of several local cycles or a global Hamiltonian cycle into the graph, depending on the ways in which the current losses of players are formed. The results of the study can be used in practice for game solutions of NP-complete problems, transport, and communication problems, for building authentication protocols in distributed information systems, and for collective decision-making under uncertainty.