A new theoretical analysis approach for a multi-agent spatial Parrondo’s game

For the multi-agent spatial Parrondo’s games, the available theoretical analysis methods based on the discrete-time Markov chain were assumed that the losing and winning states of an ensemble of N players were represented to be the system states. The number of system states was 2 N types. However, the theoretical calculations could not be carried out when N became much larger. In this paper, a new theoretical analysis method based on the discrete-time Markov chain is proposed. The characteristic of this approach is that the system states are described by the number of winning individuals of all the N individuals. Thus, the number of system states decreases from 2 N types to N + 1 types. In this study, game A and game B based on the one-dimensional line and the randomized game A + B are theoretically analyzed. Then, the corresponding transition probability matrixes, the stationary distribution probabilities and the mathematical expectations are derived. Moreover, the conditions and the parameter spaces where the strong or weak Parrondo’s paradox occurs are given. The calculation results demonstrate the feasibility of the theoretical analysis when N is larger.