A unified approach to measuring unequal representation

The concept of unequal representation is commonly understood through the lenses of disproportionality and malapportionment, pertaining to inter-party and inter-district aspects, respectively. Popular indices used to measure such features are analyzed separately despite being mathematically identical. District-level wasted votes are not measured in terms of unequal representation, even though they can be conceptualized as intra-district unequal representation. A new component, intra-party unequal representation, which measures unequal representation across districts for voters who support each party, has not been considered to contribute to unequal representation. We propose a unified approach for measuring these components—disproportionality, malapportionment, wasted votes, and intra-party unequal representation—by using \({\alpha }\)-divergence. We show mathematically that the total of disproportionality and intra-party unequal representation equals that of malapportionment and wasted votes. We apply this approach to the Japanese political system and demonstrate the role of intra-party unequal representation in sustaining disproportionality in favor of the Liberal Democratic Party.