 Since in words, \(\mathcal{H}_i (\omega)\) is the intersection of all propositions which \(i\) knows at \(\omega , \mathcal{H}_i (\omega)\) is the smallest proposition in \(\Omega\) that \(i\) knows at \(\omega\). Put another way, \(\mathcal{H}_i (\omega)\) is the most specific information that \(i\) has about the possible world \(\omega\). The intuition behind assigning agents private information systems is that while an agent \(i\) may not be able to perceive or comprehend every last detail of the world in which \(i\) lives, \(i\) does know certain facts about that world. The elements of \(i\)’s information system represent what \(i\) knows immediately at a possible world. We also have the following: