September 2010 Contents
FAIRNESS ISSUEAn Interview with Steven J. Brams
Michael A. Jones
Steven J. Brams, game theorist and political scientist, describes his role helping approval voting gain political traction. Also how to apply fair division procedures to resolve eBay buyer-seller disputes and how interdisciplinary seminars in the 1980s led not only to advances in research, but to significant pedagogical changes.
A Geometric Approach to Fair Division
We wish to divide a cake among some collection of people (who may have very different notions of the comparative value of pieces of cake) in a way that is both “fair” and “efficient”. We explore the meaning of these terms, introduce two geometric tools to aid our analysis, and present a proof (due to Dietrich Weller) that establishes the existence of cake divisions that are both fair and efficient.
Cutting Cakes Carefully
Ted Hill, Kent E. Morrison
This paper surveys the fascinating mathematics of fair division, and provides a suite of examples using basic ideas from algebra, calculus, and probability which can be used to examine and test new and sometimes complex mathematical theories and claims involving fair division. Conversely, the classical cut-and-choose and moving-knife algorithms show it is often possible to express practical, yet clean, clear, and beautiful logical conclusions without using highly technical language.
Two people take turns selecting from an even number of items. Their relative preferences over the items can be described as a permutation, then tools from algebraic combinatorics can be used to answer various questions. We describe each person's optimal selection strategies including how each could make use of knowing the other's preferences. We determine those permutations for which both players, under any strategy, get their best possible outcomes, and the same for worst possible outcomes. We also count the number of possible outcomes under optimal play, finding the ubiquitous Catalan numbers.
Who Does the Housework?
Ever had a roommate? Then you know that it can be difficult to share housekeeping duties. This article uses game theory to analyze situations in which there is a task to be done and two people who might do it. We then use our analysis to consider what game theory tells us about the gendered division of household labor.
Lewis Carroll, Voting, and the Taxicab Metric
The Dodgson winner seems very intuitive and reasonable: when a Condorcet winner doesn’t exist, pick the candidate that is closest, under some measure, to being a Condorcet winner. However, Dodgson’s method is computationally intensive. Approximate methods are more tractable. By placing these method in a geometric framework, we can understand how different approximations to the Dodgson winner might identify different candidates as the winner. Furthermore, our framework provides intuition concerning some unexpected properties of Dodgson’s method.
Gerrymandering and Convexity
Jonathan Hodge, Geoff Patterson, Emily Marshall
Convexity-based measures of shape compactness provide an effective way to identify irregularities in congressional district boundaries. A low convexity coefficient may suggest that a district has been gerrymandered, or it may simply reflect irregularities in the corresponding state boundary. Furthermore, the distribution of population within a district can either amplify or lessen the effects of boundary irregularities. As such, it is essential to take both population data and state boundaries into account when using convexity coefficients to detect gerrymandering. Our boundary-adjusted convexity coefficient provides an efficient way to do so, and it yields results similar to the shortest-path approach taken by Chambers and Miller.
Visualizing Elections Using Saari Triangles
Students sometimes have difficulty calculating the result of a voting system applied to a particular set of voter preference lists. Saari triangles offer a way to visualize the result of an election and make this calculation easier in the case of several important voting systems.
PROBLEMS AND SOLUTIONS
Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It) by William Poundstone
Reviewed by Samuel Goldberg