not hold.[^1]

A prediction ${\hat{\sigma}^{{\Gamma}}}$ matches the regret of ${{\sigma^{{\Gamma}}}}$ for all ${w}\in{V}$ does not necessarily match the regret features of ${{\sigma^{{\Gamma}}}}$.

We use both utility and regret matching in our final set of experiments. The former for predictive reasons, the latter to allow for the use of smooth minimization techniques.

The Principle of Maximum Entropy
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As we are interested in the problem of statistical prediction of strategic behavior, we must find a mechanism to resolve the ambiguity remaining after accounting for the rationality constraints. The [****]{}, due to @jaynes1957, provides a well-justified method for choosing such a distribution. This choice leads to not only statistical guarantees on the resulting predictions, but to efficient optimization.

The [****]{} of a joint-strategy ${\sigma^{{\Gamma}}}$ is $$\begin{aligned}
{H^{{\Gamma}}({\sigma^{{\Gamma}}})} & = {\mathbb{E}_{{a}{\sim}{\sigma^{{\Gamma}}}}\left[-\log {{\sigma^{{\Gamma}}}({a})}\right]},\end{aligned}$$ and the principle of maximum entropy advocates choosing the distribution with maximum entropy subject to known constraints [@jaynes1957]. That is, $$\begin{aligned}
\sigma_{\text{MaxEnt}} & = {\operatornamewithlimits{argmax}}_{{\sigma^{{\Gamma}}}\in{\Delta_{{\mathcal A}}}} {H^{{\Gamma}}({\sigma^{{\Gamma}}})}, \quad\mbox{subject to:} \\ 
               & \; g({\sigma^{{\Gamma}}}) = 0 \text{ and } h({\sigma^{{\Gamma}}}) \leq 0.  \end{aligned}$$ The constraint functions, $g$ and $h$, are typically chosen to capture the important or most salient characteristics of the distribution. When those functions are affine and convex respectively, finding 