the demonstrations by additionally preserving the players’ utilities along side the constraints prescribed by the ICE polytope. A joint strategy is utility-preserving under all utility functions if $$\begin{aligned}
\forall {w}\in{V},{i}\in{\mathcal N},\;\;{{u^{{\Gamma}}_{{i}}}({\hat{\sigma}^{{\Gamma}}}|{w})} & = {{u^{{\Gamma}}_{{i}}}({{\sigma^{{\Gamma}}}}|{w})}.\end{aligned}$$ As with the correspondence between strong rationality and the ICE polytope, utility preservation can be represented as a set of linear equality constraints. These utility feature matching constraints are exactly the basis of many methods of inverse optimal control [@abbeel2004; @ziebart2008].

A joint strategy is utility-preserving under all utility functions if and only if $$\begin{aligned}
{i}\in{\mathcal N},\;\;{{u^{{\Gamma}}_{{i}}}({\hat{\sigma}^{{\Gamma}}})} & = {{u^{{\Gamma}}_{{i}}}({{\sigma^{{\Gamma}}}})}.\end{aligned}$$

The proof is due to @abbeel2004.

A notable choice for ${\Phi_{{f}}}$ is we compare each deviation only to itself. As a consequence this enforces a stronger constraint that the regret under each deviation, and in turn the overall regret, is the same under our prediction and the demonstrations. That is, ${\hat{\sigma}^{{\Gamma}}}$ is [****]{} as for all ${w}\in{V}$, $$\begin{aligned}
{\operatorname{Regret}^{{\Gamma}}_{{\Phi}}({\hat{\sigma}^{{\Gamma}}}|{w})} & = {\operatorname{Regret}^{{\Gamma}}_{{\Phi}}({{\sigma^{{\Gamma}}}}|{w})}.\end{aligned}$$ Thus, regret-matching preserves the equilibrium qualities of the demonstrations.

Unlike the correspondence between the ICE polytope and strong rationality, matching the regret features for each deviation is **not** required for a strategy to match the regrets of the demonstrations. That is, the converse does 