  The Roamn domination problem is one important combinatorial optimization
problem that is derived from an old story of defending the Roman Empire and now
regains new significance in cyber space security, considering backups in the
face of a dynamic network security requirement. In this paper, firstly, we
propose a Roman domination game (RDG) and prove that every Nash equilibrium
(NE) of the game corresponds to a strong minimal Roman dominating function
(S-RDF), as well as a Pareto-optimal solution. Secondly, we show that RDG is an
exact potential game, which guarantees the existence of an NE. Thirdly, we
design a game-based synchronous algorithm (GSA), which can be implemented
distributively and converge to an NE in $O(n)$ rounds, where $n$ is the number
of vertices. In GSA, all players make decisions depending on the local
information. Furthermore, we enhance GSA to be enhanced GSA (EGSA), which
converges to a better NE in $O(n^2)$ rounds. Finally, we present numerical
simulations to demonstrate that EGSA can obtain a better approximate solution
in promising computation time compared with state-of-the-art algorithms.
