A theory of desirable things
This work provides a foundational framework for modeling beliefs in imprecise probabilities, but it is incremental as it extends existing theories without addressing a specific practical problem.
The paper introduces a theory of desirable things to model a subject's beliefs about desirability, applicable to various entities like gambles or pizzas, and establishes that sets of desirable sets can be represented by sets of desirable things.
Inspired by the theory of desirable gambles that is used to model uncertainty in the field of imprecise probabilities, I present a theory of desirable things. Its aim is to model a subject's beliefs about which things are desirable. What the things are is not important, nor is what it means for them to be desirable. It can be applied to gambles, calling them desirable if a subject accepts them, but it can just as well be applied to pizzas, calling them desirable if my friend Arthur likes to eat them. Other useful examples of things one might apply this theory to are propositions, horse lotteries, or preferences between any of the above. Regardless of the particular things that are considered, inference rules are imposed by means of an abstract closure operator, and models that adhere to these rules are called coherent. I consider two types of models, each of which can capture a subject's beliefs about which things are desirable: sets of desirable things and sets of desirable sets of things. A crucial result is that the latter type can be represented by a set of the former.