Refining the Lorenz-ranking of rules for claims problems on restricted domains

The comparison of the central rules for claims problems, according to the Lorenz order, has been studied not only on the entire set of problems but also on some restricted domains. We provide new characterizations of the adjusted proportional rule as being Lorenz-maximal or Lorenz-minimal within a class of rules on the half-domains. Using this result, we rank the adjusted proportional, the minimal overlap, and the average-of-awards rules by analyzing whether or not these rules satisfy progressivity and regressivity on the half-domains. We also find that the adjusted proportional rule violates two well-known claim monotonicity properties.