Michel Balinski

A Theory of Measuring, Electing and Ranking

The impossibility theorems that abound in the theory of social choice show that there can be no satisfactory method for electing and ranking in the context of the traditional, 700-year-old model. A more realistic model, whose antecedents may be traced to Laplace and Galton, leads to a new theory that avoids all impossibilities with a simple and eminently practical method, “the majority judgement.” It has already been tested.

On the Assignment Polytope

An expository, completely elementary and self-contained account is given describing several properties of the constraint polytope of the assignment problem. In particular, it is shown that the “Hirsch conjecture” holds, and that to go from any one extreme point to any other, at most 2 extreme edges need to be traversed.

A competitive (dual) simplex method for the assignment problem

Abstract“Where there is abundance of mystery and confusion in every direction, the truth seldom remains hidden for long. Its a matter of having plenty of angles to go at it from. Only the utterly simple crimes - the simplex crimes, you may say - have the trick of remaining baffling.” - Sir John (from Michael Innes,The Open House (A Sir John Appleby Mystery), Penguin Books, 1974).A dual simplex method for the assignment problem leaves open to choice the activity (i,j) of rowi and columnj that is to be dropped in pivoting so long asxij < 0. A choice (i,j) over columnsj having at least 3 basic activities that minimizesxij is shown to converge in at most (2n-1) pivots, and at most O(n3) time, and it is argued that on average the number of pivots is at mostn logn.

The Quota Method of Apportionment

The problem of apportionment is explained together with an account of the methods used by the United States Congress beginning with the first decennial apportionment of 1792. Fairness and historica...

Many-to-many matching: stable polyandrous polygamy (or polygamous polyandry)

Abstract The major results known for the marriage and university admissions problems — the one-to-one and many-to-one stable matching problems — are shown to have equivalents in the general many-to-many setting. Some of these results depend upon a particular, natural definition of individual preferences over sets of mates: notably, characterizations of “optimal” stable assignments in terms of “efficiency”, “monotonicity”, and “strategy-proofness”.

Algorithms for proportional matrices in reals and integers

LetR be the set of nonnegative matrices whose row and column sums fall between specific limits and whose entries sum to some fixedh > 0. Closely related axiomatic approaches have been developed to ascribe meanings to the statements: the real matrixf ∈ R and the integer matrixa ∈ R are “proportional to” a given matrixp ≥ 0.These approaches are described, conditions under which proportional solutions exist are characterized, and algorithms are given for finding proportional solutions in each case.

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 Stable Allocation (or Ordinal Transportation) Problem

The stable allocation problem generalizes the 0,1 stable matching problems one-to-one, one-to-many, and many-to-many to the allocation of real valued hours or quantities. A strongly polynomial algorithm proves the existence of “stable allocations.” The set of stable allocations is shown to be a distributive lattice in general, but in the “nondegenerate” case it is a complete linear order. Indeed, in the generic case, when a problem is “strongly nondegenerate,” there exists a single stable allocation. A simple algorithm finds “row-optimal” and “column-optimal” stable allocations, given any stable allocation. When a problem is nondegenerate it finds all stable allocations.

Parametric methods of apportionment, rounding and production

Abstract The class of “parametric” methods of apportionment, of rounding, or for minimizing the variation of production rates in just-in-time production systems is characterized in several different ways that depend on the underlying qualitative behavior of its solutions.

Election by Majority Judgment: Experimental Evidence

The majority judgement is a method of election. It is the consequence of a new theory of social choice where voters judge candidates instead of ranking them. The theory is explained elsewhere [2, 4]. This article describes and analyzes electoral experiments conducted in parallel with the last two French presidential elections to: (1) show that the majority judgement is a practical method, (2) describe it and its salient properties, (3) establish that it escapes the classical paradoxes, (4) illustrate how in practice the well known electoral mechanisms all fail to meet important criteria. The demonstrations introduce new concepts and methods.