adversarially but subject to strong rationality.

The proof of Lemma \[lem:maxent\] follows immediately from the result of @grunwald2003.

Dual Optimization
-----------------

In this section, we will derive and describe a procedure for optimizing the dual program for solving the MaxEnt ICE optimization problem. We will see that the dual multipliers can be interpreted as utility vectors and that optimization in the dual has computational advantages. We begin by presenting the dual program.

The dual maximum entropy ICE optimization problem is the following non-smooth, but convex program: $$\begin{aligned}
{\operatornamewithlimits{minimize}}_{{\theta_{{f}}}\in{{\mathcal{K}}^{**}}} & \;\; \sum_{{f}\in{\Phi}}{\operatorname{Regret}^{{\Gamma}}_{{\Phi_{{f}}}}({{\sigma^{{\Gamma}}}}|{\theta_{{f}}})} + \log{Z^{{\Gamma}}({\theta})}, \mbox{~where} \\ 
{Z^{{\Gamma}}({\theta})} & = \sum_{{a}\in{\mathcal A}}\exp\left(-\sum_{{f}\in{\Phi}}{r^{{\Gamma}}_{{f}}({a}|{\theta_{{f}}})} \right).  \end{aligned}$$ \[thm:dual\]

We derive the dual in the Appendix.

As the dual’s feasible set has non-empty relative interior, strong duality holds by Slater’s condition—there is no duality gap. We can also use a dual solution to recover ${\hat{\sigma}^{{\Gamma}}}$.

Strong duality holds for the maximum entropy ICE optimization problem and given optimal dual weights ${\theta^*}$, the maximum entropy ICE joint-strategy ${\hat{\sigma}^{{\Gamma}}}$ is $$\begin{aligned}
{\hat{\sigma}^{{\Gamma}}({a})} & \propto \exp\left(-\sum_{{f}\in{\Phi}}{r^{{\Gamma}}_{{f}}({a}|{\theta^*_{{f}}})} \right).
\label{eqn:dualtoprimal} \end{aligned}$$

Let ${\hat{\sigma}^{{\Gamma}}}$ be the prediction given the current dual weights, ${\theta}$, as from Equation . ${f^*}\leftarrow {\operatornamewithlimits{argmax}}_{{f'}\in{\Phi_{{f}}}}{r^{{\Gamma}}_{{f'}}({{\sigma^{{\Gamma}}}}|{\theta_{{f}}})}$ ${g^f}\leftarrow {r^{{\Gamma}}_{{f^*}}({{\sigma^{{\Gamma}}}})} - {r^{{\Gamma}}_{{f}}({\hat{\sigma}^{{\Gamma}}})}$ ${g}$

\[alg:dual\]

The dual formulation of our program has 