on the prescribed action. A [****]{}, denoted ${\Phi}$, is a set of deviation functions. We call the set of all switch deviations the internal deviation set, ${\Phi^{\mbox{int}}}$, and the set of all fixed deviations the external deviation set, ${\Phi^{\mbox{ext}}}$. The set ${\Phi^{\mbox{swap}}}$ is the set of all deterministic deviations. Given that the other players indeed play their recommended actions, there is no strategic advantage to considering randomized deviations.

The benefit of applying deviation ${f_{{i}}}$ when the players jointly play ${a}$ is known as [****]{}. We write the instantaneous regret features as $$\begin{aligned}
{r^{{\Gamma}}_{{f_{{i}}}}({a})} &= {{u^{{\Gamma}}_{{i}}}({f_{{i}}({a})})} - {{u^{{\Gamma}}_{{i}}}({a})},\end{aligned}$$ and the instantaneous regret under utility function ${w}$ as $$\begin{aligned}
{r^{{\Gamma}}_{{f_{{i}}}}({a}|{w})} &= {{u^{{\Gamma}}_{{i}}}({f_{{i}}({a})}|{w})} - {{u^{{\Gamma}}_{{i}}}({a}|{w})} =
{\langle {r^{{\Gamma}}_{{f_{{i}}}}({a})},{w}\rangle}.\end{aligned}$$

More generally, we can consider broader classes of deviations than the two we have mentioned. Conceptually, a deviation is a strategy modification and its regret is its benefit to a particular player. As we will ultimately only work with the regret features, we can now suppress the implementation details while bearing in mind that a deviation typically has these prescribed semantics. That is, a [****]{} ${f}\in{\Phi}$ has associated instantaneous regret features, ${r^{{\Gamma}}_{{f}}(\cdot)}$, and instantaneous regret, ${r^{{\Gamma}}_{{f}}(\cdot|{w})}$.

As a player is only privileged to its own 