We provide conditions under which an incomplete strongly independent preorder on a convex set X can be represented by a set of mixture preserving real-valued functions. We allow X to be infi nite dimensional. The main continuity condition we focus on is mixture continuity. This is sufficient for such a representation provided X has countable dimension or satisfi es a condition that we call Polarization.

We study preferences over lotteries which do not necessarily satisfy completeness. We provide a characterization which generalizes Expected Utility theory. We show in particular that various sure-thing axioms are needed to guaranteee the representability in terms of utility intervals rather than numbers, and to provide a linear interval order representation which is very much in the spirit of Expected Utility theory.

In addition to addressing the usual topics of real analysis, this book discusses the elements of order theory, convex analysis, optimization, correspondences, linear and nonlinear functional analysis, fixed-point theory, dynamic programming ...

We present an axiomatic model of preferences over menus that is motivated by three assumptions. First, the decision maker is uncertain ex ante (i.e., at the time of choosing a menu) about her ex post (i.e., at the time of choosing an option within her chosen menu) preferences over options, and she anticipates that this subjective uncertainty will not resolve before the ex post stage. Second, she is averse to ex post indecisiveness (i.e., to having to choose between options that (...) she cannot rank with certainty). Third, when evaluating a menu she discards options that are dominated (i.e., inferior to another option whatever her ex post preferences may be) and restricts attention to the undominated ones. Under these assumptions, the decision maker has a preference for commitment in the sense of preferring menus with fewer undominated alternatives. We derive a representation in which the decision maker’s uncertainty about her ex post preferences is captured by means of a subjective state space, which in turn determines which options are undominated in a given menu, and in which the decision maker fears, whenever indecisive, to choose an option that will turn out to be the worst (undominated) one according to the realization of her ex post preferences. (shrink)