In their great variety, from contests of global significance such as a championship match or the election of a president down to a coin flip or a show of hands, games and elections share one common feature: each game or election offers the possibility of a final, decisive result obtained according to well-established rules, a public outcome. Mathematics offers a similar possibility. In mathematics, the rules are founded in the laws of logic and represent a formalization of our basic common sense. An outcome in mathematics, to pursue the analogy, is a theorem, a statement that can be proven true. The outcome of a game or election may be surprising or expected. Similarly, a theorem can either defy intuition or confirm a well-evidenced conjecture. Just as a game or an election separates winners from losers, a proven theorem distinguishes true statements from false ones, creating a new fact-of-the-matter about mathematics.

This book is an introduction to the mathematical theory of games and elections. We pursue the analogy between analyzing a game or election and developing a mathematical theory somewhat further before turning to our main topics. First, just as a game or an election creates a new language of specialized terminology, a mathematical theory begins with the formulation of definitions. A mathematical definition is a precise and verifiable description of an object of study. We adopt the following convention for definitions: when we define a new term, we use boldface. Thus the terms “outcome,” “theorem,” and “definition” were defined earlier. (Of course, subsequent definitions will be more technical than these.) We use italics to indicate we are mentioning a term that has not been defined yet but will be defined later. For example, we mentioned the term “definition” before subsequently defining it earlier.

The best way to learn to play a new game is often to just give it a try. In mathematics, the corresponding hands-on method of learning is the study of examples. By an example, we mean a specific instance of a definition or, alternately, a particular consequence of a theorem. A great advantage of our chosen topics is the wealth of examples.