there is a fixed amount of utility divided among the players. That is, if there is a constant $C$ such that $\forall{a}\in{\mathcal A}$, $$\begin{aligned}
\sum_{{i}\in{\mathcal N}}{{u^{{\Gamma}}_{{i}}}({a}|{w^*})} & = C.\end{aligned}$$

In settings where the players act independently, we use external regret to measure a profile’s stability, which corresponds with the famous [****]{} solution concept [@osborne1994course]. By using the ICE polytope with external regret, we can recover a Nash equilibrium if one is demonstrated in a constant-sum game.

If the true behavior is an ${\varepsilon}$-Nash equilibrium in a two-player constant-sum game ${\Gamma}$, then the marginal strategies formed from a prediction ${\hat{\sigma}^{{\Gamma}}}$ that satisfies the standard ICE polytope where ${\Phi}= {\Phi^{\mbox{ext}}}$ and $\forall {f}\in{\Phi}, {\Phi_{{f}}}={\Phi}$ is a $2{\varepsilon}$-Nash equilibrium. \[thm:nash\]

The proof is provided in the Appendix.

In general, there can be infinitely many correlated equilibrium with vastly different properties. One such property that has received much attention is the social welfare of a joint strategy, which refers to the total utility over all players. Our strong rationality assumption states that the players have no preference on which correlated equilibrium is selected, and thus without modification cannot capture such a concept should it be demonstrated. We can easily maintain the social welfare of 