  We study a problem of designing replication-proof bandit mechanisms when
agents strategically register or replicate their own arms to maximize their
payoff. We consider Bayesian agents who are unaware of ex-post realization of
their own arms' mean rewards, which is the first to study Bayesian extension of
Shin et al. (2022). This extension presents significant challenges in analyzing
equilibrium, in contrast to the fully-informed setting by Shin et al. (2022)
under which the problem simply reduces to a case where each agent only has a
single arm. With Bayesian agents, even in a single-agent setting, analyzing the
replication-proofness of an algorithm becomes complicated. Remarkably, we first
show that the algorithm proposed by Shin et al. (2022), defined H-UCB, is no
longer replication-proof for any exploration parameters. Then, we provide
sufficient and necessary conditions for an algorithm to be replication-proof in
the single-agent setting. These results centers around several analytical
results in comparing the expected regret of multiple bandit instances, which
might be of independent interest. We further prove that exploration-then-commit
(ETC) algorithm satisfies these properties, whereas UCB does not, which in fact
leads to the failure of being replication-proof. We expand this result to
multi-agent setting, and provide a replication-proof algorithm for any problem
instance. The proof mainly relies on the single-agent result, as well as some
structural properties of ETC and the novel introduction of a restarting round,
which largely simplifies the analysis while maintaining the regret unchanged
(up to polylogarithmic factor). We finalize our result by proving its sublinear
regret upper bound, which matches that of H-UCB.
