this distribution is a convex optimization problem. The resulting log-linear family of distributions ([*e.g.*]{}, logistic regression, Markov random fields, conditional random fields) are widely used within statistical machine learning.

In the context of multi-agent behavior, the principle of maximum entropy has been employed to obtain correlated equilibria with predictive guarantees in normal-form games when the utilities are known [*a priori*]{} [@ortiz2007]. We will now leverage its power with our rationality assumption to select predictive distributions in games where the utilities are unknown, but the important features that define them are available.

For our problem, the constraints are precisely that the distribution is in the ICE polytope, ensuring that whatever we predict has no more regret than the demonstrated behavior.

The primal maximum entropy ICE optimization problem is $$\begin{aligned}
{\operatornamewithlimits{maximize}}_{{\hat{\sigma}^{{\Gamma}}},{\eta}} &\;\; {H^{{\Gamma}}({\hat{\sigma}^{{\Gamma}}})} \quad\mbox{subject to:}\\
{r^{{\Gamma}}_{{f}}({\hat{\sigma}^{{\Gamma}}})} - {\mathbb{E}_{g{\sim}{\eta_{{f}}}}\left[{r^{{\Gamma}}_{g}({{\sigma^{{\Gamma}}}})}\right]} & \in -{\mathcal{K}^*}, &\forall {f}\in{\Phi}\\
{\eta_{{f}}}&\in{\Delta_{{\Phi_{{f}}}}}, &\forall {f}\in{\Phi}\\
{\hat{\sigma}^{{\Gamma}}}&\in{\Delta_{{\mathcal A}}}.  \end{aligned}$$

This program is convex, feasible, and bounded. That is, it has a solution and is efficiently solvable using simple techniques in this form.

Importantly, the maximum entropy prediction enjoys the following guarantee:

\[lem:maxent\] The maximum entropy ICE distribution minimizes over all strongly rational distributions the worst-case log-loss, ${\mathbb{E}_{{a}{\sim}{\sigma^{{\Gamma}}}}\left[-\log_2
{\hat{\sigma}^{{\Gamma}}({a})}\right]}$, when ${{\sigma^{{\Gamma}}}}$ is chosen 