 Slightly abusing notation, we say that a valuation \(V\) satisfies a formula \(\varphi\), denoted \(V \models \varphi\), whenever it makes \(\varphi\) true under the current assignment of propositions. In other words, propositional control games are played in one single world, and the individual assignments determine what propositions are true are that world. Denoting \(\mathcal{V}\) the set of all valuations and \(\mathcal{V}_i\) to the partial ones under the control of \(i\) we have the following.