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Ideally, a representative democracy awards a genuine vote to each adult. We study this issue in competitive democracies with an election model combining district apportionment and proportional representation. Four classic seat allocation rules, including d'Hondt, are reframed as Dutch auctions, allowing important properties to be easily derived. The pros and cons of these methods are stated in terms of economic inequality; Sainte Laguë's is shown to best carry the genuine vote ideal, both for elections and for apportionment. We next expound the interplay between these two components in generating an inequitable treatment of voters and develop the scale-free index of inequity best fitted to their concern. We apply it to 40 countries for the apportionment of electoral districts. Lastly, we compute the same inequity index for recent parliamentary elections in 80 countries, finding that the majority system mistreats electors, thus putting a 'price' on government stability. [ABSTRACT FROM AUTHOR]

We establish a link between multiwinner elections and apportionment problems by showing how approval-based multiwinner election rules can be interpreted as methods of apportionment. We consider several multiwinner rules and observe that some, but not all, of them induce apportionment methods that are well-established in the literature and in the actual practice of representation, be it proportional or non-proportional. For instance, we show that proportional approval voting induces the D’Hondt method and that Monroe’s rule induces the largest remainder method. Our approach also yields apportionment methods implementing degressive proportionality. Furthermore, we consider properties of apportionment methods and exhibit multiwinner rules that induce apportionment methods satisfying these properties. [ABSTRACT FROM AUTHOR]

Using the Nash product (Nash Social Welfare Function) as a micro foundation, we create a decomposable index to evaluate the unfairness of representation in electoral districts. Using this index, we decompose the factors of such unfairness into apportionment and districting. We explore the situations of New Zealand, the United States, Australia, the United Kingdom, Japan, and Canada. We then provide an integer solution that minimizes the index of the apportionment, which can be obtained by the divisor method (Balinski and Young, 1982) using a logarithmic mean. [ABSTRACT FROM PUBLISHER]

We prove that cumulative voting usually enables a minority population to achieve political representation corresponding to apportionment by Webster's method, which minimizes the absolute difference of per capita representation between the minority and the remaining majority population. The minority, of arbitrary size, can generally attain its 'Webster-fair' share of n seats with probability greater than 75 per cent and otherwise, with probability at most ¼ ⋅ n/n+1, the minority can attain just one seat less than its Webster-fair representation. Furthermore, for two subpopulations, the potential representation yielded by cumulative voting is identical to that obtained from apportionment by Jefferson's method, and for more than two subpopulations the potential representation by cumulative voting cannot be greater than that of Jefferson apportionment. These results confirm the potential of cumulative voting to yield representation proportional or nearly proportional to population, and the results counter claims or concerns that cumulative voting would be unfairly advantageous to minority populations. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of two-dimensional proportional representation most commonly encountered when seats are to be allocated in a region where voters are classified according to the double cleavage of the constituency in which they vote and the party of their choice. A priority is set here on marginal proportionality: seats are dealt out to constituencies and to parties at the global level first. It is proven that, under a very weak condition, a biproportional delegation always exists, i.e. a representation matrix matching imposed margins and which is a controlled rounding of the corresponding solution to the well-known biproportional problem. Two versions of a construction process for such a biproportional delegation are proposed and they are simulated on Belgian electoral data covering the last four general elections. Though no perfect system exists for this type of representation, comparisons with the current system plead in favor of the use of biproportional delegations. [ABSTRACT FROM AUTHOR]