This book was written to teach an accessible and authentic mathematics course to a general audience. Games and elections are ideal topics for such a course.The motivating examples are familiar and engaging： from the 2000 U.S.presidential election， to economic pricing wars， to gambling games such as Poker and Blackjack. The elementary theory of these topics involves manageable computations and emphasizes analytic reasoning skills. Most important， the study of games and elections has produced a wealth of beautiful theorems， many within the last century. Games and elections offer a wide selection of elegant results that can be approached without extensive technical background. A course on these topics provides a view of mathematics as a powerful and thriving modern discipline.
　　There are excellent texts covering aspects of the theory of games and elections，such as those of Straffin [481 and Taylor and Pacelli [49]. The topics are also
　　frequently covered as chapters in liberal arts mathematics texts. This text is pitched at a level between these two types of treatments. The book is designed to be more mathematically ambitious than the general textbook but to allow for a less demanding course than one offered using a more specialized text.
　　A novelty of this book is the integrative approach taken to the three subjects：
　　probability theory， game theory， and social choice theory. This approach highlights the mix of ideas occurring in seminal results on games and elections such as the Jury Theorem， the Minimax Theorem， and the Gibbard-Satterthwaite Theorem. On a practical level， the integrative approach allows for a more gradual development of material. Rather than taking a steep descent in one area， the chapters follow a spiraling path from chance to strategy to social choice and back again， exploring examples and developing techniques and intuitions in all areas while delving
　　deeper into the theory of games and elections.
　　Structure of the Book
　　This book is divided into three parts. Part Ⅰ introduces the main devices used in the text. These are the probability tree， the game tree， the payoff matrix， and the preference table. Expected values， dominated strategies， backward induction， and rational play are then defined for the various types of games. The idea of preference ballots is motivated with examples from sports and politics， and the basic voting methods are introduced.
　　Part Ⅰ includes three chapters exploring two of the three notions of chance，strategy， and choice together. The paradox of the chair， the Gibbard-Satterthwaite Theorem， swing states in the Electoral College， the Jury Theorem， and a simplified version of Poker are introduced in these chapters. The final chapter of Part Ⅰ presents Nash equilibria as a unifying concept， tying together several examples in the categories of zero-sum， partial-conflict， mixed-strategy， and electoral games.
　　Part Ⅱ focuses more directly on the development of mathematical theory. A
　　chapter on logic prepares the way， with topics from the first part of the book serving as representative examples of conditional and universal statements. The method of proof by induction for game trees is illustrated with a version of Zermelo's Theorem. Conditionals are explored further in the probability setting， while fairness criteria for voting methods provide exercises with universal statements and an exposure to the dichotomy between proofs and counterexamples in mathematics. Counting techniques are developed for computing the power index of a weighted voting system. The problem of determining weights for a yes-no
　　system and the notion of a mathematical invariant are introduced， leading up to a statement of the Taylor-Zwicker Theorem.
　　Part n also includes a series of chapters exploring the various types of games. Counting techniques are applied again to compute probabilities with card hands in a chapter on gambling games. The game Craps shows the power of introducing a conditional， whereas a problem in Poker motivates Bayes'
　　formula. A chapter on zero-sum games contains the proof of the 2x2 Minimax Theorem and introduces techniques for the 2 x n case. A chapter on partial-confiict games explores backward induction through a variety of examples， including the Iterated Prisoner's Dilemma， the Chain Store Game， the Dollar Auction， and Bram's Theory of Moves. Next， a chapter on takeaway games introduces the Sprague-Grundy numbers and includes a proof of Bouton's beautiful solution of the game Nim. The final chapter in Part Ⅱ is a return to the question of fairness for voting methods. Fairness criteria for social welfare methods including Arrow's famous Independence criteria are introduced. May's Theorem， a simplified version
　　of Arrow's Theorem， and Sen's Impossibility Theorem are proved in this chapter.
　　Part Ⅲ consists of five independent chapters offering possible enhancements to abasic course. A chapter on probability explores famous paradoxes ranging from the
　　Monte Hall Problem to Bertrand's Paradox in geometry. Nim Misere， Chomp， and Hex are analyzed in a chapter on combinatorial games. The debate over fairness between Borda and Condorcet is explored， with the arguments for both sidesinterlaced with three theorems： a version of the Jury Theorem， a modern result of
　　McGarvey's on the realization of social preference graphs， and Borda's Theorem from his original paper introducing the Borda count. The final two chapters focus on proofs of two beautiful results on games and elections. The Sprague-Grundy Theorem for combinatorial games is proved， introducing the notion of Nim sums.