\frac{1}{2} (
      M_{\mu_*} - G_t(\mu_*, \pi, s_0) +
      M_{\mu_*^-} - G_t(\mu_*^{-}, \pi, s_0)
      )
    & \text{max dominates the mean} \\
        & = \frac{1}{2}(M_{\mu_*} + M_{\mu_*^-} - t)
    & \text{by \cref{eq:sumt}} \\
        &= \frac{1}{2}(M_{\mu_*} + t - m_{\mu_*} - t)
    & \text{by \cref{eq:M-to-m}} \\
        & = \frac{1}{2} \max_{\check\pi} \Reg(\mu_*, \check\pi, s_0, t)
    & \text{by \cref{eq:max-regret}}\\
        & = \frac{1}{2} \max_{\check\pi} \Reg(\M, \check\pi, s_0, t).
    & \text{ by definition of $\mu_*$ } 
  \end{aligned}$$ That is, the regret of any policy $\pi$ is at least half of the regret of a worst policy $\check\pi$.

For the robot in the shower from \[ex:db\], the result means that if it tries to optimise observed reward by standing in the shower, then it performs poorly according to the hypothesis that “shower-induced” reward is corrupt and bad. But if instead the robot tries to optimise reward in some other way, say baking cakes, then (from the robot’s perspective) there is also the possibility that “cake-reward” is corrupt and bad and the “shower-reward” is actually correct. Without additional information, the robot has no way of knowing what to do.

The result is not surprising, since if all corruption functions are allowed in the class $\D$, then there is effectively 