 The first generalization, which is most common in applications of modal probabilistic logic, is to allow the distributions to be indexed by two sets rather than one. The first set is the set \(W\) of worlds (the base set of the model), but the other is an index set \(A\) often to be taken as a set of actions, agents, or players of a game. Formally, \(\mathcal{P}\) associates a distribution \(\mathcal{P}_{a,w}\) over \(W\) for each \(w\in W\) and \(a\in A\). For the language, rather than involving formulas of the form \(P(\phi)\ge q\), we have \(P_a(\phi)\ge q\), and \((M,w)\models P_a(\phi)\ge q\) if and only if \(\mathcal{P}_{a,w}(\{w'\mid (M,w')\models \phi\}) \ge q\).