that ${\hat{\sigma}^{{\Gamma}}}$ and ${\eta}$ satisfy the standard ICE polytope. \[c:standardice\]

We now show a more general result that implies Corollary \[c:standardice\]. We start by generalizing the notion of strong rationality by restricting ${w^*}$ to be in a known set ${\mathcal{K}}\subseteq{V}$. We say a prediction ${\hat{\sigma}^{{\Gamma}}}$ is ${\mathcal{K}}$-strongly rational with respect to deviation set ${\Phi}$ if $$\begin{aligned}
 \forall
{w}\in{\mathcal{K}},\;\;{\operatorname{Regret}^{{\Gamma}}_{{\Phi}}({\hat{\sigma}^{{\Gamma}}}|{w})}& \le
{\operatorname{Regret}^{{\Gamma}}_{{\Phi}}({{\sigma^{{\Gamma}}}}|{w})}.\end{aligned}$$ If ${\mathcal{K}}$ is convex with non-empty relative interior and $0\in{\mathcal{K}}$, we derive the ${\mathcal{K}}$-ICE polytope.

$$\begin{aligned}
{r^{{\Gamma}}_{{f}}({\hat{\sigma}^{{\Gamma}}})} - {\mathbb{E}_{g{\sim}{\eta_{{f}}}}\left[{r^{{\Gamma}}_{g}({{\sigma^{{\Gamma}}}})}\right]}& \in -{\mathcal{K}^*}, &\forall {f}\in{\Phi}\\
{\eta_{{f}}}&\in{\Delta_{{\Phi_{{f}}}}}, &\forall {f}\in{\Phi}\\
{\hat{\sigma}^{{\Gamma}}}&\in{\Delta_{{\mathcal A}}}. \end{aligned}$$

Note that the above constraints are linear in ${\hat{\sigma}^{{\Gamma}}}$ and ${\eta_{{f}}}$, and ${\mathcal{K}^*}$, the dual cone, is convex. The following theorem shows the equivalence of the ${\mathcal{K}}$-ICE polytope and ${\mathcal{K}}$-strong rationality.

A prediction ${\hat{\sigma}^{{\Gamma}}}$ is ${\mathcal{K}}$-strongly rational with respect to deviation set ${\Phi}$ if and only if for all ${f}\in{\Phi}$ there exists ${\eta_{{f}}}\in{\Delta_{{\Phi}}}$ such that ${\hat{\sigma}^{{\Gamma}}}$ and ${\eta}$ satisfy the ${\mathcal{K}}$-ICE polytope. \[thm:kice\]

The proof is provided in the Appendix.

By choosing ${\mathcal{K}}={V}$, then ${\mathcal{K}^*}=\{0\}$ and the polytope reduces to the standard ICE polytope. Thus, Corollary \[c:standardice\] follows directly from Theorem \[thm:kice\]. By choosing ${\mathcal{K}}$ to be the positive orthant, ${\mathcal{K}}={\mathcal{K}^*}={{\mathbb R}^{K}_{+}}$, the polytope reduces to the following inequalities. Here, we 