Abstract

We study the dynamics of the so-called Game of Cards by using tools developed in the context of discrete dynamical systems. We extend a result of \textit{J. Desel, E. Kindler, T. Vesper}, and \textit{R. Walter} [“A simplified proof for the self-stabilizing protocol: a game of cards”, Inf. Proc. Lett. 54, 327–328 (1995; Zbl 1004.68506)] and \textit{S.-T. Huang} [“Leader election in uniform rings”, ACM Trans. Program. Lang. Syst. 15, 563–573 (1993)] (the last one in the context of distributed systems) that established a necessary and sufficient condition for the game to converge. We precisely describe the lattice structure of the set of configurations and we state bounds for the convergence time.