## Multiple Weighted Expected Utility Theory

##### Abstract

Multiple Weighted Expected Utility Theory
Chapter 1: Expected Utility without the Independence and Completeness Axioms
This paper establishes an intuitive foundation for modeling preferences which do not obey either completeness or independence. From the existing literature, we know that preferences which violate the independence axiom, as demonstrated by the famous Allais paradox, are represented by a utility function that is not linear in the probabilities, and hence individual components cannot be freely substituted for equivalent ones. On the other hand, incomplete preferences can only be represented by a set of utility functions, and whenever they disagree on how to rank a pair of lotteries, the decision maker can exhibit neither preference nor indifference, and must judge the alternatives to be incomparable. Therefore, preferences that violate both of these axioms simultaneously must have a utility representation that exhibits both multiplicity and non-linearity, reflecting an internal consensus among a set of decision criteria, none of which admit the strong substitution property implied by the independence axiom.
Chapter 2: Weighted Expected Utility without the Completeness Axiom
This paper axiomatizes a utility representation for incomplete preferences that violate independence and satisfy only a weaker ratio substitution property. I show that such a representation is given by a set of weighted linear utility functions, each of which generates an indifference map consisting of a set of projectively parallel indifference curves originating from a source point outside of the simplex of lotteries. The overall indifference map is therefore constructed by superimposing the maps corresponding to the individual decision criteria, with the locations of the source points determining the decision maker’s sets of utility and weight functions. These respectively represent a range of conflicting tastes and a set of disparate perceptions of mixture distorting the evaluation of subjective probabilities, and incompleteness may arise from any disagreement in either or both.
Chapter 3: Incomplete Preferences with Conflicting Tastes and Perceptions
This paper considers variations of the general model where the multiplicity of decision criteria is restricted to either the utility or weight functions alone. I find that these cases differ fundamentally in how the level of indecision exhibited by the decision maker varies with her point of reference, standing in contrast to standard multi-utility models where the degree of indecision must remain constant throughout the space of lotteries. In the multiple utility, single weight case the decision maker is unsure only of her tastes, so that her inability to rank alternatives derives purely from her incapacity to evaluate the component outcomes and is hence mitigated by mixture of prospects. On the other hand, in the single utility, multiple weight case the decision criteria disagree only on perception of risk, so that incompleteness is instead exacerbated by mixture. This allows us to discern the composition of an individual’s internal decision making process from patterns of observed choice or lack thereof.