no knowledge of their true utility function.

Unfortunately, a direct translation of the strong rationality requirement into constraints on the distribution ${\hat{\sigma}^{{\Gamma}}}$ leads to a non-convex optimization problem as it involves products of varying utility vectors with the behavior to be estimated. Fortunately, we can provide an equivalent concise convex description of the constraints on ${\hat{\sigma}^{{\Gamma}}}$ that ensures any feasible distribution satisfies strong rationality. We denote this set of equivalent constraints as the [*Inverse Correlated Equilibria*]{} (ICE) polytope.

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

Here, we have introduced ${\Phi_{{f}}}$, the set of deviations that ${f}$ will be compared against. Our rationality assumption corresponds to when ${\Phi_{{f}}}={\Phi}$, but there are different choices that have reasonable interpretations as alternative rationality assumptions. For example, if each switch deviation is compared only against switches for the same player—a more restrictive condition—then the quality of the equilibrium is measured by the [*sum*]{} of all players’ regrets, as opposed to only the one with the most regret.

The following corollary equates strong rationality and the standard ICE polytope.

A prediction ${\hat{\sigma}^{{\Gamma}}}$ is 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 