explicitly assume the utility to be a positive linear function of the features.

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

Predictive behavior within the ICE polytope will retain the quality of the demonstrations provided. The following corollaries formalize this guarantee.

If the true behavior is an ${\varepsilon}$-correlated equilibrium under ${w^*}$ in game ${\Gamma}$, then a prediction ${\hat{\sigma}^{{\Gamma}}}$ that satisfies the standard ICE polytope where ${\Phi}= {\Phi^{\mbox{swap}}}$ and $\forall
{f}\in{\Phi}, {\Phi_{{f}}}={\Phi}$ is also an ${\varepsilon}$-correlated equilibrium.

This follows immediately from the definition of an approximate correlated equilibrium.

If the true behavior is an ${\varepsilon}$-correlated equilibrium under ${w^*}$ in game ${\Gamma}$, then a prediction ${\hat{\sigma}^{{\Gamma}}}$ that satisfies the standard ICE polytope where ${\Phi}= {\Phi^{\mbox{int}}}$ and $\forall
{f}\in{\Phi}, {\Phi_{{f}}}={\Phi}$ is also an ${A}{\varepsilon}$-correlated equilibrium.

This follows immediately from Lemma \[lemma:internalapproxswap\].

In two-player constant-sum games, we can make stronger statements about our predictive behavior. In particular, when these requirements are satisfied we may reason about games without coordination. That is, each player chooses their action independently using their [****]{}, ${\sigma^{{\Gamma}}_{{i}}}$ a distribution over ${A_{{i}}}$. A [****]{} ${\sigma^{{\Gamma}}}$ consists of a strategy for each player. It defines a joint-strategy with no coordination between the players.

A game is constant-sum if 