Regular choice systems: A general technique to represent them by random variables

- The possibility of random utility representations for incomplete regular choice systems is explored.
- The theory of convex polytopes is utilized.
- Conditions similar to the Block/Marschak conditions for a complete choice system are derived.
- The proposed technique depends on the Möbius function and Möbius inversion.

Regular choice systems and their random utility representations are investigated. A generalization of the derivation of the Block-Marschak conditions, based on the Möbius function of a partial order is presented. The technique is demonstrated in connection with two examples. The first is similar to complete choice data. In the second example a complete characterization of the ensuing polytope is obtained including a procedure to explicitly derive a convex representation of a data matrix if it is in the polytope.