Weighted Bankruptcy with Discrete Estates: Apportionments and Lotteries

In this chapter we study proportional allocation methods for discrete weighted bankruptcy problems, i.e., weighted bankruptcy problems with a discrete (‘indivisible’) estate, such as horses or seats in a parliament. The major part of this chapter is devoted to an overview of apportionment theory which, in its canonical formulation, is the systematic study of methods to allocate, proportionally, a number of seats to states on the basis of their populations. However, as Balinski and Young (Fair Representation. Brookings Institution Press, Washington D.C., 2001) explain ‘any problem in which objects are to be allocated in non-negative integers proportionally to some numerical criterion belongs to this class, and [apportionment] theory applies to it’. In particular, apportionment theory applies to discrete weighted bankruptcy problems, as we explain in Sect. 10.1. In Sect. 10.2, we discuss various apportionment methods, which output an allocation—or a set of allocations in case there are ties—of the discrete estate. Another way to allocate a discrete estate is via a lottery. In Sect. 10.3, we discuss a lottery method to allocate the estate of a discrete weighted bankruptcy problem in a fair, i.e., ‘proportional’ way.