Evaluation of thresholds for power mean-based and other divisor methods of apportionment

For divisor methods of apportionment with concave up or concave down rounding functions, we prove explicit formulas for the threshold values—the lower and upper bounds for the percentage of population that are necessary and sufficient for a state to receive a particular number of seats. Among the rounding functions with fixed concavity are those based on power means, which include the methods of Adams, Dean, Hill–Huntington, Webster, and Jefferson. The thresholds for Dean’s and Hill–Huntington’s methods had not been evaluated previously. We use the formulas to analyze the behavior of the thresholds for divisor methods with fixed concavity, and compute and compare threshold values for Hill–Huntington’s method (used to apportion the US House of Representatives).