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Simple Games
Simple games are mathematical structures inspired by voting systems in which a single alternative, such as a bill, is pitted against the status quo. The first in-depth mathematical study of the subject as a coherent subfield of finite combinatorics--one with its own organized body of techniques and results--this book blends new theorems with some of the striking results from threshold logic, making all of it accessible to game theorists. Introductory material receives a fresh treatment, with an emphasis on Boolean subgames and the Rudin-Keisler order as unifying concepts. Advanced material focuses on the surprisingly wide variety of properties related to the weightedness of a game. A desirab...
The Future of Economic Design
This collection of essays represents responses by over eighty scholars to an unusual request: give your high level assessment of the field of economic design, as broadly construed. Where do we come from? Where do we go from here? The book editors invited short, informal reflections expressing deeply felt but hard to demonstrate opinions, unsupported speculation, and controversial views of a kind one might not normally risk submitting for review. The contributors – both senior researchers who have shaped the field and promising, younger researchers – responded with a diverse collection of provocative pieces, including: retrospective assessments or surveys of the field; opinion papers; reflections on critical points for the development of the discipline; proposals for the immediate future; "science fiction"; and many more. The readers should have fun reading these unusual pieces – as much as the contributors enjoyed writing them.
Mathematics and Democracy
Voters today often desert a preferred candidate for a more viable second choice to avoid wasting their vote. Likewise, parties to a dispute often find themselves unable to agree on a fair division of contested goods. In Mathematics and Democracy, Steven Brams, a leading authority in the use of mathematics to design decision-making processes, shows how social-choice and game theory could make political and social institutions more democratic. Using mathematical analysis, he develops rigorous new procedures that enable voters to better express themselves and that allow disputants to divide goods more fairly. One of the procedures that Brams proposes is "approval voting," which allows voters to vote for as many candidates as they like or consider acceptable. There is no ranking, and the candidate with the most votes wins. The voter no longer has to consider whether a vote for a preferred but less popular candidate might be wasted. In the same vein, Brams puts forward new, more equitable procedures for resolving disputes over divisible and indivisible goods.
The Mathematics of Decisions, Elections, and Games
This volume contains the proceedings of two AMS Special Sessions on The Mathematics of Decisions, Elections, and Games, held January 4, 2012, in Boston, MA, and January 11-12, 2013, in San Diego, CA. Decision theory, voting theory, and game theory are three intertwined areas of mathematics that involve making optimal decisions under different contexts. Although these areas include their own mathematical results, much of the recent research in these areas involves developing and applying new perspectives from their intersection with other branches of mathematics, such as algebra, representation theory, combinatorics, convex geometry, dynamical systems, etc. The papers in this volume highlight and exploit the mathematical structure of decisions, elections, and games to model and to analyze problems from the social sciences.
How to Cut a Cake
Welcome back to Ian Stewart's magical world of mathematics! This is a strange world of never-ending chess games, empires on the moon, furious fireflies, and, of course, disputes over how best to cut a cake. Each quirky tale presents a fascinating mathematical puzzle — challenging, fun, and also introducing the reader to a significant mathematical problem in an engaging and witty way.
The Mathematics of Preference, Choice and Order
Peter Fishburn has had a splendidly productive career that led to path-breaking c- tributions in a remarkable variety of areas of research. His contributions have been published in a vast literature, ranging through journals of social choice and welfare, decision theory, operations research, economic theory, political science, mathema- cal psychology, and discrete mathematics. This work was done both on an individual basis and with a very long list of coauthors. The contributions that Fishburn made can roughly be divided into three major topical areas, and contributions to each of these areas are identi?ed by sections of this monograph. Section 1 deals with topics that are included in the general areas of utility, preference, individual choice, subjective probability, and measurement t- ory. Section 2 covers social choice theory, voting models, and social welfare. S- tion 3 deals with more purely mathematical topics that are related to combinatorics, graph theory, and ordered sets. The common theme of Fishburn’s contributions to all of these areas is his ability to bring rigorous mathematical analysis to bear on a wide range of dif?cult problems.
The Mathematics of Voting and Elections: A Hands-On Approach
The Mathematics of Voting and Elections: A Hands-On Approach, Second Edition, is an inquiry-based approach to the mathematics of politics and social choice. The aim of the book is to give readers who might not normally choose to engage with mathematics recreationally the chance to discover some interesting mathematical ideas from within a familiar context, and to see the applicability of mathematics to real-world situations. Through this process, readers should improve their critical thinking and problem solving skills, as well as broaden their views of what mathematics really is and how it can be used in unexpected ways. The book was written specifically for non-mathematical audiences and r...
Communicating Mathematics
This volume contains the proceedings of a conference held in July, 2007 at the University of Minnesota, Duluth, in honor of Joseph A. Gallian's 65th birthday and the 30th anniversary of the Duluth Research Experience for Undergraduates. In keeping with Gallian's extraordinary expository ability and broad mathematical interests, the articles in this volume span a wide variety of mathematical topics, including algebraic topology, combinatorics, design theory, forcing, game theory, geometry, graph theory, group theory, optimization, and probability. Some of the papers are purely expository while others are research articles. The papers are intended to be accessible to a general mathematics audience, including first-year or second-year graduate students. This volume should be especially useful for mathematicians seeking a new research area, as well as those looking to enrich themselves and their research programs by learning about problems and techniques used in other areas of mathematics.
Handbook of Computational Social Choice
The rapidly growing field of computational social choice, at the intersection of computer science and economics, deals with the computational aspects of collective decision making. This handbook, written by thirty-six prominent members of the computational social choice community, covers the field comprehensively. Chapters devoted to each of the field's major themes offer detailed introductions. Topics include voting theory (such as the computational complexity of winner determination and manipulation in elections), fair allocation (such as algorithms for dividing divisible and indivisible goods), coalition formation (such as matching and hedonic games), and many more. Graduate students, researchers, and professionals in computer science, economics, mathematics, political science, and philosophy will benefit from this accessible and self-contained book.
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