The paper constructs nine novel seat apportionment models based on the \(\chi ^2\) statistic as a descriptive measure of apportionment bias, systematically addressing the shortcomings of traditional apportionment methods in terms of fairness and accuracy. For each model, not only is a rigorous theoretical derivation presented, but an innovative, unified algorithmic framework for solving them, along with its practical implementation steps, is also proposed. Through the creation of multiple sets of representative both theoretical analysis and empirical verification perspectives, this study conducts an exhaustive comparison between the proposed models and currently prevalent mainstream apportionment approaches (including the Hamilton method, Webster method, Huntington-Hill method, etc.). Further systematic simulation analysis verifies the practicality and robustness of these models in stability tests of fairness indicators. This research not only provides new theoretical tools for the seat apportionment problem but also opens up new avenues for studying related resource apportionment problems through its proposed unified algorithmic framework. We use the data of the Legislative Council (LegCo) of Hong Kong SAR and the election of German as testing ground and basis for the comparison. Our findings may be useful in assessing electoral designs in other contexts as well.
