The paper addresses the problem of emulating a regular register in a synchronous distributed system where clients invoking ${\sf read}()$ and ${\sf write}()$ operations are anonymous while server processes maintaining the state of the register may be compromised by rational adversaries (i.e., a server might behave as \emph{rational malicious Byzantine} process). We first model our problem as a Bayesian game between a client and a rational malicious server where the equilibrium depends on the decisions of the malicious server (behave correctly and not be detected by clients vs returning a wrong register value to clients with the risk of being detected and then excluded by the computation). We prove such equilibrium exists and finally we design a protocol implementing the regular register that forces the rational malicious server to behave correctly.