 If furthermore \(\rho_{i+1} \in \sigma_0(\rho_i)\) whenever \(\rho_i\) corresponds to a disjunction or an existential quantifier, we say that \(\overline \rho\) respects the strategy \(\sigma_0\); and as mentioned, we write \(\Pi(G_M(\phi), \sigma_0)\) for the set of all plays over \(G_M(\phi)\) which respect \(\sigma_0\).