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Apportionment of representatives is a basic rule of everyday politics. By definition, this basic rule is a constitutional stage problem and should be decided behind the veil of uncertainty. To bring apportionment closer to quotas, we introduce f-divergence for utilitarianism and Bregman divergence for consistent optimization. Even in our less restricted condition, we find that we must use α-divergence for optimization and show that the minimization of α-divergence induces the same divisor methods that correspond to the maximization of the Kolm–Atkinson social welfare function (or the expected utility function), which is bounded by constant relative risk aversion.

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... In this study, we relied on a measure of malapportionment that does not distinguish between apportionment (how many seats to distribute to each administrative units) and boundary drawing (how to draw boundaries within an administrative unit). Such a measure had been an industry standard until recently but scholars are beginning to distinguish between the two (Wada, 2016). It is possible that the political motivations behind apportionment and boundary drawing are different. ...

This article examines electoral malapportionment by illuminating the relationship between malapportionment level and democracy. Although a seminal study rejects this relationship, we argue that a logical and empirically significant relationship exists, which is curvilinear and is based on a framework focusing on incumbent politicians' incentives and the constraints they face regarding malapportionment. Malapportionment is lowest in established democracies and electoral authoritarian regimes with an overwhelmingly strong incumbent; it is relatively high in new democracies and authoritarian regimes with robust opposition forces. The seminal study's null finding is due to the mismatch between theoretical mechanisms and choice of democracy indices. Employing an original cross-national dataset, we conduct regression analyses; the results support our claims. Furthermore, on controlling the degree of democracy, the single-member district system's effects become insignificant. Australia, Belarus, the Gambia, Japan, Malaysia, Tunisia, and the United States illustrate the political logic underlying curvilinear relations at democracy's various levels.

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.

We investigate the notion o0f the Nash social welfare function and make a fundamental assumption that there exists a distinguished alternative called an origin, which represents one of the worst states for all individuals in the society. Under this assumption, in Sections 1 and 2, we formulate several rationality criteria that a reasonable social welfare function should satisfy. Then we introduce the Nash social welfare function and the Nash social welfare indices which are the images of the welfare function. The function is proved to satisfy the criteria. In Section 3 it is shown that the Nash social welfare function is the unique social welfare function that satisfies the criteria. Then, in Section 4, we examine two examples which display plausibility of the welfare function.

Over two centuries of theory and practical experience have taught us that election and decision procedures do not behave as expected. Instead, we now know that when different tallying methods are applied to the same ballots, radically different outcomes can emerge, that most procedures can select the candidate, the voters view as being inferior, and that some commonly used methods have the disturbing anomaly that a winning candidate can lose after receiving added support. A geometric theory is developed to remove much of the mystery of three-candidate voting procedures. In this manner, the spectrum of election outcomes from all positional methods can be compared, new flaws with widely accepted concepts (such as the "Condorcet winner") are identified, and extensions to standard results (e.g. Black's single-peakedness) are obtained. Many of these results are based on the "profile coordinates" introduced here, which makes it possible to "see" the set of all possible voters' preferences leading to specified election outcomes. Thus, it now is possible to visually compare the likelihood of various conclusions. Also, geometry is applied to apportionment methods to uncover new explanations why such methods can create troubling problems.

This paper links Stolarsky mean apportionment methods, which include the USÂ House of Representatives, the Saint-Lague, and the dâ€™Hondt methods, to Kolmâ€“Atkinson social welfare maximization and to generalized entropy minimization. Within this class, the logarithmic mean apportionment method is the most unbiased one that assigns at least one seat to each state.

A divergence measure between two probability distributions or positive arrays (positive measures) is a useful tool for solving optimization problems in optimization, signal processing, machine learning, and statistical inference. The Csiszar f -divergence is a unique class of divergences having information monotonicity, from which the dual alpha geometrical structure with the Fisher metric is derived. The Bregman divergence is another class of divergences that gives a dually flat geometrical structure different from the alpha -structure in general. Csiszar gave an axiomatic characterization of divergences related to inference problems. The Kullback-Leibler divergence is proved to belong to both classes, and this is the only such one in the space of probability distributions. This paper proves that the alpha -divergences constitute a unique class belonging to both classes when the space of positive measures or positive arrays is considered. They are the canonical divergences derived from the dually flat geometrical structure of the space of positive measures.

Thesis (A.B., Honors)--Harvard University, 1942.

AN EMPIRICAL COMPARISON IS GIVEN OF TWO SCHEMES OF CONGRESSIONAL APPORTIONMENT: Huntington's generally accepted Equal Proportion scheme, and a new "house-monotone" Quota scheme. Evidence is given to show that Quota tends to underrepresent small states.

The Sainte-Laguë Method for Solving an Apportionment Problem (in Japanese)

- Wada

- Saari