Don’t let the lawyers do the math: Some problems of legislative districting in the UK and the USA

A fair allocation of electoral districts would give each citizen as near as possible an equal share of power, as measured by standard power indices which operationalize ‘power’ as ‘pivotality’. The practical difficulties of implementing this principle are discussed, with reference to the UK and the USA. The history of attempts to achieve equal districts in each country is reviewed. It is shown that equal-population (or equal electorate) districts are a necessary condition for each citizen to have an identical power index, but not a sufficient one; and that it is impossible to achieve descriptive representation (where each gender, ethnic group, and/or political party is represented in the legislature in the same proportion as in the population) while retaining single-member districts. Some open issues in the mathematics of redistricting are listed.