''Beliefs about beliefs'' without probabilities

This paper constructs a space of states of the world representing the exhaustive uncertainty facing each player in a strategic situation. The innovation is that preferences are restricted primarily by ''regularity'' conditions and need not conform with subjective expected utility theory. The construction employs a hierarchy of preferences, rather than of beliefs as in the standard Bayesian model. The framework is sufficiently general to accommodate uncertainty averse preferences, such as exhibited in the Ellsberg paradox, and to allow common knowledge of expected utility (or Choquet expected utility) to be well-defined formally. Applications include the provision of (i) foundations for a Harsanyi-style game of incomplete information, and (ii) a rich framework for the axiomatization of solution concepts for complete information normal form games.