${\hat{\sigma}^{{\Gamma}}}$ is strongly rational with respect to deviation set ${\Phi}$ and the players are rational with respect to ${\Phi}$, then they do not prefer ${{\sigma^{{\Gamma}}}}$ over ${\hat{\sigma}^{{\Gamma}}}$.

This is immediate as ${w^*}\in{V}$.

Phrased another way, a strongly rational prediction is no worse than the true behavior.

If a prediction ${\hat{\sigma}^{{\Gamma}}}$ is strongly rational with respect to deviation set ${\Phi}$ and the true behavior is an ${\varepsilon}$-equilibrium with respect to ${\Phi}$ under utility function ${w^*}\in{V}$, then ${\hat{\sigma}^{{\Gamma}}}$ is also an ${\varepsilon}$-equilibrium.

Again, the proof is immediate as ${\operatorname{Regret}^{{\Gamma}}_{{\Phi}}({\hat{\sigma}^{{\Gamma}}}|{w^*})} \le
{\operatorname{Regret}^{{\Gamma}}_{{\Phi}}({{\sigma^{{\Gamma}}}}|{w^*})} \le {\varepsilon}$.

Conversely, if we are uncertain about the true utility function we [**must**]{} assume strong rationality or we risk predicting less desirable behavior.

If a prediction ${\hat{\sigma}^{{\Gamma}}}$ is not strongly rational with respect to deviation set ${\Phi}$ and the players are rational, then there exists a ${w^*}\in{V}$ such that ${{\sigma^{{\Gamma}}}}$ is preferred to ${\hat{\sigma}^{{\Gamma}}}$.

The proof follows from the negation of the definition of strong rationality.

By restricting our attention to strongly rational behavior, at worst agents acting according to their unknown true preferences will be indifferent between our predictive distribution and their true behavior. That is, strong rationality is necessary and sufficient under the assumption players are rational given 