A partial-state space model of unawareness
Wesley H. Holliday
TL;DR
The paper introduces a partial-state space model of unawareness that extends Aumann-style knowledge to an awareness operator defined on events via a possibility correspondence. By employing a forcing-like partial order and regular open sets, the authors build a Boolean algebra of events and establish a representation theorem showing any epistemic awareness algebra can be represented by an epistemic possibility frame with awareness, knowledge, and belief correspondences. This approach generalizes prior models (e.g., HMS) and escapes the DLR impossibility through a carefully weakened axiom, showing unawareness of events can be modeled without collapsing into definability from knowledge alone. The framework promises versatility for multi-agent scenarios and opens avenues for logical languages, probabilistic extensions, and sentence-awareness integration, broadening the toolkit for reasoning about uncertainty and unawareness in economics and related fields.
Abstract
We propose a model of unawareness that remains close to the paradigm of Aumann's model for knowledge [R. J. Aumann, International Journal of Game Theory 28 (1999) 263-300]: 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 [A. Heifetz, M. Meier, and B. Schipper, Journal of Economic Theory 130 (2006) 78-94], 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 may be 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 in set theory. Second, we depart from a tradition in the literature, initiated by [S. Modica and A. Rustichini, Theory and Decision 37 (1994) 107-124] and adopted by Heifetz et al. and [J. Li, Journal of Economic Theory 144 (2009) 977-993], 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 in [Dekel, Lipman, and Rustichini, Econometrica 66 (1998) 159-173] 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.