A partial-state space model of unawareness

We propose a model of unawareness that remains close to the paradigm of Aumann's (1999a) model for knowledge: just as Aumann uses a correspondence on a state space to define an agent's knowledge operator on events, we use a correspondence on a state space to define an agent's awareness operator on events. This is made possible by three ideas. First, like the model of Heifetz, Meier, and Schipper (2006), ours is based on a space of partial specifications of the world, partially ordered by a relation of further specification or refinement, and the idea that agents maybe aware of some coarser-grained specifications while unaware of some finer-grained specifications; however, our model is based on a different implementation of this idea, related to forcing inset theory. Second, we depart from a tradition in the literature, initiated by Modica and Rustichini (1994) and adopted by Heifetz et al. and Li (2009), of taking awareness to be definable in terms of knowledge. Third, we show that the negative conclusion of a well-known impossibility theorem concerning unawareness due to Dekel, Lipman, and Rustichini (1998) can be escaped by a slight weakening of a key axiom. Together these points demonstrate that a correspondence on a partial-state space is sufficient to model unawareness of events. Indeed, we prove a representation theorem showing that any abstract Boolean algebra equipped with awareness, knowledge, and belief operators satisfying some plausible axioms is representable as the algebra of events arising from a partial-state space with awareness, knowledge, and belief correspondences.