We start with the models that are familiar to philosophical logicians (van Benthem 2010) and computer scientists (Fagin et al. 1995). These models were introduced to game theory by Robert Aumann (1976) in his seminal paper Agreeing to Disagree (see Vanderschraaf &amp; Sillari 2009, Section 2.3, for a discussion of this result). First, some terminology: Given a set \(W\) of states, or possible worlds, let us call any subset \(E\subseteq W\) an event or proposition. Given events \(E\subseteq W\) and \(F\subseteq W\), we use standard set-theoretic notation for intersection (\(E\cap F\), read “\(E\) and \(F\)”), union (\(E\cup F\), read “\(E\) or \(F\)”) and (relative) complement (\(-{E}\), read “not \(E\)”). We say that an event \(E\subseteq W\) occurs at state \(w\) if \(w\in E\). This terminology will be crucial for studying the following models.