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

•The number of system states decreases effectively, which makes the theoretical analysis feasible when NN is larger.•The result of the theoretical analysis method in this paper is more accurate than the one based on the mean-field approach.•When NN is larger, the results of this study and the simulation results are in good agreement.

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 NN players were represented to be the system states. The number of system states was 2N2N types. However, the theoretical calculations could not be carried out when NN 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 NN individuals. Thus, the number of system states decreases from 2N2N types to N+1N+1 types. In this study, game AA and game BB based on the one-dimensional line and the randomized game A+BA+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 NN is larger.