By Donald G. Saari
A shock is how the complexities of vote casting thought might be defined and resolved with the cozy geometry of our 3-dimensional international. This e-book is directed towards scholars and others wishing to benefit approximately vote casting, specialists will realize formerly unpublished effects. as an instance, a brand new profile decomposition speedy resolves centuries previous controversies of Condorcet and Borda, demonstrates, that the scores of pairwise and different equipment fluctuate simply because they depend upon various details, casts sequence doubt at the reliability of a Condorcet winner as a customary for the sphere, makes the well-known Arrow`s Theorem predictable, and simplifies the development of examples. The geometry unifies likely disparate themes as manipulation, monotonicity, or even the apportionment problems with the USA ideally suited Court.
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Additional info for Basic Geometry of Voting
The reader familiar with the connection between convexity and linear mappings (or those willing to return later) should move on to Sect. 1. 1 Convexity and Linear Mappings An important geometric concept used repeatedly throughout this book is convexity. Convexity. A set is convex if for any two points in the set, the straight line connecting these points also is in the set. As examples, the first two sets in Fig. 1 are convex. Intuitively, convexity is ensured because all "bulges" are outwards.
2. 4 WHAT KIND OF GEOMETRY? 24 More general than a straight line is a linear mapping - it generalizes the single variable equation y = ax where a is a given constant, x is the input, or independent variable, and y is the output, or dependent variable. In voting, the inputs often correspond to how many voters have each of the possible rankings of the candidates, and the outputs represent the election tallies. The algebra, then, is the set of linear equations defined by specified constants ai,j + al,2 X 2 + ...
The Money Pump. The Condorcet cycle C1 )0- C2, C2 )0- C3, C3 )0- C1 created by = (~, 0, ~, 0, ~, 0, ~) is particularly bothersome because it violates intuition and expectations about rational behavior. Some of us, after all, claim to be "rational individuals" who affiliate only with rational behaving groups. A Turing test of this claim means it is impossible to decide whether a given set of outcomes are the decisions of a rational individual or the group. First, we need a description of rational behavior.