This book focuses on the use of mathematics to illuminate two essential features of democracy: a) how individual preferences can be aggregated to give an election outcome that reflects interests of the electorate; and b) how public or private goods can be divided in a way that respects the rule of the law. It analyzes procedures, or rules of play, that produce outcomes. The book is divided into two parts.

The first part is on voting procedures. The emphasis in this part is on approval balloting, whereby voters can approve as many candidates or alternatives as they like without having to rank them. The author discusses different forms that approval balloting can take. The mathematical analysis mainly uses elementary combinatorics and game theory. The author considers different ways of selecting the election outcome, discusses possibilities of manipulation of the result by various interest groups and stability of the election outcome. The theoretical discussion is supplemented by examples of approval balloting adopted by some organizations. The second part is on fair division procedures. Various procedures of fair division applicable to both divisible and indivisible goods are discussed together with their distributional consequences.