The Hirsch Conjecture for Dual Transportation Polyhedra

An algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most (m-1) (n-1) extreme edges

The Theory of Apportionment

A key problem area at IIASA is the study of how goods and resources -- as well as "bads" such as costs, pollution, and risks -- can or should be shared among different nations, groups, or individuals. This raises the question of what is meant by a fair division -- and, if this question can be answered at all, how fair divisions can be achieved. One of the situations studied in the System and Decision Sciences Area was how to allocate or "apportion" discrete entities in proportion to predetermined claims, a problem which encompasses many situations including for example the apportionment of political representation among different regions and constituencies. The result of this study was the development of a general theory to deal with such problems, together with concrete criteria of fairness which will hopefully prove useful to analyzing larger classes of problems

The Webster Method of Apportionment

Several results concerning the problem of U.S. Congressional apportionment are given which together indicate that a method first proposed by Daniel Webster (also known as "Major Fractions") is fairest judged on the basis of common sense, Constitutional requirement, and precedent

A Problem of Fair Division: Apportioning the European Parliament

In this paper certain natural and desirable properties for apportionment are formulated, and it is shown that they lead to a unique procedure for determining solutions. This procedure is then applied to the case of the European Parliament

Criteria for Proportional Representation

Methods to allocate seats in proportional representation systems are investigated in terms of several underlying common-sense properties. In particular, the idea of stability is introduced, and the method of Jefferson (or d'Hondt) is characterized

On Huntington Methods of Apportionment

A (generalized) Huntington method for apportioning representatives among states, or seats among parties, is one which distributes seats one by one by using a rank index that determines how deserving a state, or party, is to receive the next available seat. A characterization of these methods is given by two basic properties: consistency and house monotonicity. The arguments used to establish this result are combinatorial in nature and use classical theorems concerning partial orders and their representation by a real-valued function

Fair Representation in the European Parliament

The number of seats currently apportioned to each nation in the European Parliament is far from the ideal of one-man, one-vote. It is likely to become even more so as populations shift and new nations join. As European rather than national interests become more dominant pressure is likely to build for representation more closely tied to populations. The merits of different approaches to meeting fair representation are evaluated and compared in terms of basic principles that underlie one-man, one-vote

Apportionment Schemes and the Quota Method

This paper answers criticisms recently leveled at the Quota Method for congressional apportionment, and reconsiders the relative merits of various axioms and methods

Stability, Coalitions, and Schisms in Proportional Representation Systems

Methods to allocate seats in proportional representation systems are investigated in terms of underlying common-sense properties. Important among these are concepts of stability, coalition encouragement, and schism encouragement. In addition, a new concept of uniformity is introduced which seems inherent in the very idea of the word "method", and it is shown that this concept is essentially equivalent to a previously investigated property called consistency. These and other criteria are shown to uniquely determine certain methods. In particular, the Jefferson method (incorrectly credited to d'Hondt) and the Quota method are given characterizations which commend them as the principal candidates for use in proportional representation systems

Quotatone Apportionment Methods

It has recently been pointed out that there exists more than one house monotone apportionment method satisfying quota. This paper gives a simple characterization of all such methods as an immediate consequence of the Quota method's existence. Further, a manner of exposition is formulated which unites several key house monotone apportionment methods, thus clearly showing their connections