Types and their associated image under \(\lambda_i\) encode the players’ (probabilistic) information about the others’ information. Indeed, each type is associated with a hierarchy of belief. More formally, recall that an event \(E\) for a type \(t_i\) is a set of pairs \((\sigma_{-j}, t_{-j})\), i.e., a set of strategy choices and types for all the other players. Given an event \(E\) for player \(i\), let \(\lambda_i(t_i)(E)\) denote the sum of the probabilities that \(\lambda_i(t_i)\) assigns to the elements of \(E\). The type \(t_i\) of player \(i\) is said to (all-out) believe the event \(E\) whenever \(\lambda_i(t_i)(E) = 1\). Conditional beliefs are computed in the standard way: type \(t_i\) believes that \(E\) given \(F\) whenever: