Two prisoners are told that they will be brought to a room and seated so that each can see the other. Hats will be placed on their heads; each hat is either red or green. The two prisoners must simultaneously submit a guess of their own hat color, and they both go free if at least one of them guesses correctly. While no communication is allowed once the hats have been placed, they will, however, be allowed to have a strategy session before being brought to the room. Is there a strategy ensuring their release? The answer turns out to be yes, and this is the simplest non-trivial example of a "hat problem." This book deals with the question of how successfully one can predict the value of an arbitrary function at one or more points of its domain based on some knowledge of its values at other points. Topics range from hat problems that are accessible to everyone willing to think hard, to some advanced topics in set theory and infinitary combinatorics. For example, there is a method of predicting the value f(a) of a function f mapping the reals to the reals, based only on knowledge of f's values on the open interval (a - 1, a), and for every such function the prediction is incorrect only on a countable set that is nowhere dense. The monograph progresses from topics requiring fewer prerequisites to those requiring more, with most of the text being accessible to any graduate student in mathematics. The broad range of readership includes researchers, postdocs, and graduate students in the fields of set theory, mathematical logic, and combinatorics. The hope is that this book will bring together mathematicians from different areas to think about set theory via a very broad array of coordinated inference problems.

Two prisoners are told that they will be brought to a room and seated so that each can see the other. Hats will be placed on their heads; each hat is either red or green. The two prisoners must simultaneously submit a guess of their own hat color, and they both go free if at least one of them guesses correctly. While no communication is allowed once the hats have been placed, they will, however, be allowed to have a strategy session before being brought to the room. Is there a strategy ensuring their release? The answer turns out to be yes, and this is the simplest non-trivial example of a hat problem.

This book deals with the question of how successfully one can predict the value of an arbitrary function at one or more points of its domain based on some knowledge of its values at other points. Topics range from hat problems that are accessible to everyone willing to think hard, to some advanced topics in set theory and infinitary combinatorics. For example, there is a method of predicting the value "f"("a") of a function f mapping the reals to the reals, based only on knowledge of "f"'s values on the open interval ("a" 1, "a"), and for every such function the prediction is incorrect only on a countable set that is nowhere dense.

The monograph progresses from topics requiring fewer prerequisites to those requiring more, with most of the text being accessible to any graduate student in mathematics. The broad range of readership includes researchers, postdocs, and graduate students in the fields of set theory, mathematical logic, and combinatorics. The hope is that this book will bring together mathematicians from different areas to think about set theory via a very broad array of coordinated inference problems. "

As a text for an undergraduate mathematics course for nonmajors, Mathematics and Politics requires no prerequisites in either area while the underlying philosophy involves minimizing algebraic computations and focusing instead on some conceptual aspects of mathematics in the context of important real-world questions in political science.

Five major topics are covered including a model of escalation, game theoretic models of international conflict, yes-no voting systems, political power, and social choice. Each topic is discussed in an introductory chapter and revisited in more depth in a later chapter. This new edition has added co-author, Allison Pacelli, and two new chapters on "Fairness" and "More Fairness." The examples and the exercises have been updated and enhanced throughout.

Reviews from first edition:

This book is well written and has much math of interest. While it is pitched at a non-math audience there is material here that will be new and interesting to the readers...

-Sigact News

For mathematicians, Taylor's book shows how the social sciences make use of mathematical thinking, in the form of axiomatic systems, and offers a chance to teach this kind of thinking to our students.

- The College Mathematics Journal

The writing is crisp and the sense of excitement about learning mathematics is seductive. The political conflict examples are well thought out and clear.

-Michael C. Munger

As a text for an undergraduate mathematics course for nonmajors, Mathematics and Politics requires no prerequisites in either area while the underlying philosophy involves minimizing algebraic computations and focusing instead on some conceptual aspects of mathematics in the context of important real-world questions in political science.

Five major topics are covered including a model of escalation, game theoretic models of international conflict, yes-no voting systems, political power, and social choice. Each topic is discussed in an introductory chapter and revisited in more depth in a later chapter. This new edition has added co-author, Allison Pacelli, and two new chapters on "Fairness" and "More Fairness." The examples and the exercises have been updated and enhanced throughout.

Reviews from first edition:

This book is well written and has much math of interest. While it is pitched at a non-math audience there is material here that will be new and interesting to the readers...

-Sigact News

For mathematicians, Taylor's book shows how the social sciences make use of mathematical thinking, in the form of axiomatic systems, and offers a chance to teach this kind of thinking to our students.

- The College Mathematics Journal

The writing is crisp and the sense of excitement about learning mathematics is seductive. The political conflict examples are well thought out and clear.

-Michael C. Munger

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 desirability relation orders the individuals or coalitions of a game according to their influence in the corresponding voting system. As Taylor and Zwicker show, acyclicity of such a relation approximates weightedness--the more sensitive the relation, the closer the approximation. A trade is an exchange of players among coalitions, and robustness under such trades is equivalent to weightedness of the game. Robustness under trades that fit some restrictive exchange pattern typically characterizes a wider class of simple games--for example, games for which some particular desirability order is acyclic. Finally, one can often describe these wider classes of simple games by weakening the total additivity of a weighting to obtain what is called a pseudoweighting. In providing such uniform explanations for many of the structural properties of simple games, this book showcases numerous new techniques and results.

Honesty in voting, it turns out, is not always the best policy. Indeed, in the early 1970s, Allan Gibbard and Mark Satterthwaite, building on the seminal work of Nobel laureate Kenneth Arrow, proved that with three or more alternatives there is no reasonable voting system that is non-manipulable; voters will always have an opportunity to benefit by submitting a disingenuous ballot. The ensuing decades produced a number of theorems of striking mathematical naturality that dealt with the manipulability of voting systems. This 2005 book presents many of these results from the last quarter of the twentieth century, especially the contributions of economists and philosophers, from a mathematical point of view, with many new proofs. The presentation is almost completely self-contained, and requires no prerequisites except a willingness to follow rigorous mathematical arguments. Mathematics students, as well as mathematicians, political scientists, economists and philosophers will learn why it is impossible to devise a completely unmanipulable voting system.

Since the publication of Roger Fisher and William Ury's highly influential book, Getting to Yes, it has been widely recognized that there is a middle ground between winning and losing in negotiation. Yet, while Getting to Yes was long on motivation, it was short on technique. What you really want to know is on which issues you will win, on which you will lose, and on which you will have to compromise. To this question, Steven J. Brams and Alan D. Taylor bring a patented procedure that not only is fair but also actually guarantees that both parties walk away with as much of the "win-win" potential as possible. "One can hire a lawyer and spend years and thousands of dollars fighting in a divorce], or one can make use of a neat new formula devised by Steven Brams and Alan Taylor." The New Yorker"

Fair Division, unlike most research on fairness in the social sciences and mathematics, is devoted solely to the analysis of constructive procedures for actually dividing things up and resolving disputes, including indivisible items or issues, such as the marital property in a divorce or sovereignty in an international dispute.

Honesty in voting, it turns out, is not always the best policy. Indeed, in the early 1970s, Allan Gibbard and Mark Satterthwaite, building on the seminal work of Nobel laureate Kenneth Arrow, proved that with three or more alternatives there is no reasonable voting system that is non-manipulable; voters will always have an opportunity to benefit by submitting a disingenuous ballot. The ensuing decades produced a number of theorems of striking mathematical naturality that dealt with the manipulability of voting systems. This 2005 book presents many of these results from the last quarter of the twentieth century, especially the contributions of economists and philosophers, from a mathematical point of view, with many new proofs. The presentation is almost completely self-contained, and requires no prerequisites except a willingness to follow rigorous mathematical arguments. Mathematics students, as well as mathematicians, political scientists, economists and philosophers will learn why it is impossible to devise a completely unmanipulable voting system.

What is the best way to divide a 'cake' and allocate the pieces among some finite collection of players? In this book, the cake is a measure space, and each player uses a countably additive, non-atomic probability measure to evaluate the size of the pieces of cake, with different players generally using different measures. The author investigates efficiency properties (is there another partition that would make everyone at least as happy, and would make at least one player happier, than the present partition?) and fairness properties (do all players think that their piece is at least as large as every other player's piece?). He focuses exclusively on abstract existence results rather than algorithms, and on the geometric objects that arise naturally in this context. By examining the shape of these objects and the relationship between them, he demonstrates results concerning the existence of efficient and fair partitions.