### Mathematics and Democracy

In Mathematics and Democracy, I try to show how mathematics can be used to illuminate two essential features of democracy:

how individual preferences can be aggregated to give a social choice or election outcome that reflects the interests of the electorate; and

how public and private goods can be divided in a way that respects due process and the rule of law.

Whereas questions of aggregation are the focus of social choice theory, questions of division are the focus of fair division.

Democracy, as I use the term, will generally mean representative democracy, in which citizens vote for representatives, from a president on down. But I also analyze referendums, in which citizens vote directly on propositions, just as they did in assemblies in ancient Greece.

I focus on procedures, or rules of play, that produce outcomes. By making precise the properties that one wishes a voting or fair-division procedure to satisfy, and clarifying relationships among these properties, mathematical analysis can strengthen the intellectual foundations on which democratic institutions are built. But because there may be no procedure or institution that satisfies all the properties one might desire, I examine trade-offs among the properties. In the case of some procedures, I also consider practical problems of implementation and discuss experience with those that have been tried out.

2. Institutional design and engineering
The voting and fair-division procedures I analyze foster democratic choices by giving voters better ways of expressing themselves, by electing officials who are likely to be responsive to the electorate, and by allocating goods to citizens that ensure their shares are equitable or preclude envy. In some cases I criticize current procedures, but most of the analysis is constructive—I suggest how these procedures may be improved.

Designing procedures that satisfy desirable properties, or showing the limits of doing so, is sometimes referred to as institutional design or mechanism design. I present empirical examples to illustrate this approach, but the bulk of the analysis is theoretical.

The product of such analysis is normative: The prescription of new procedures or institutions that are superior, in terms of the criteria set forth, to ones that arose more haphazardly. Like engineering in the natural sciences, which translates theory (e.g., from physics) into practical design (e.g., a bridge), engineering in the social sciences translates theory into the design of political-economic-social institutions that better meet the criteria one deems important.

Mathematics and Democracy is divided into two parts:

Part 1. Voting procedures

One cornerstone of democracy is honest and periodic elections. Several of the voting procedures that are analyzed are relatively new and not well known, but they offer significant advantages over extant procedures. Common to many of them is approval balloting, whereby voters can approve of as many candidates of alternatives as they like without having to rank them.

Approval balloting may take different forms. Under approval voting, the candidate or alternative with the most votes wins. Under other methods of aggregating approval votes, different candidates or alternatives may win. These methods maximize different objective functions, or constrain outcomes in certain ways, in order to achieve certain ends, such as the proportional representation of different interests in the electorate.

Most social-choice analysis assumes rational individuals, who select the most effective or efficient means to satisfy their goals, and examines the implications of these individual choices on collective choices. Game theory is an important tool in such analysis, especially in identifying outcomes that are stable or in equilibrium, and institutions that support the equilibria one finds.

Part 2. Fair-division procedures

As central as elections are to the performance of a democracy, a democracy would be a sham if the politicians elected were not restricted by due process and the rule of law. Ideally, democracies treat all citizens the same way–at least when governed by a constitution or other laws–particularly with respect to their civil rights and certain freedoms, such as freedom of association and freedom of religion.

The equal treatment of citizens depends in part on their receiving fair shares of things that must be divided among them, not just on the efficiency of outcomes. Accordingly, I analyze different procedures of fair division–applicable to both divisible and indivisible goods–and study their distributional consequences.

Fairness requires that one take into account the different preferences or claims of players who have a stake in an outcome. Step-by-step rules or algorithms to implement the fair division of goods, which may be homogenous (like money) or heterogeneous (like land with different objects on it), are analyzed. Questions that relate to the fair division of people or groups include: What political parties are best suited to form a government? Which parties should get what cabinet ministries in the government?

3. Use of mathematics and scope
While a good mathematical background makes reading the theoretical parts of Mathematics and Democracy easier, several chapters are accessible to those with little mathematical training. When a topic in a chapter goes beyond the level of the rest of the chapter or is a digression from its main theme, I discuss it in an appendix to the chapter.

Of course, some chapters are inherently more analytic or mathematical than others, so the reader may want to skip those that cause difficulty. In fact, I encourage the selective reading of the chapters, most of which are relatively self-contained and can be read independently of others. A glossary at the end provides a quick reference to the most important concepts that I use in the book.

I have certainly not covered all institutions in the public sphere. For example, there is now a large literature on redistricting, or the drawing of district boundaries after a census; on auctions, which governments employ to sell such things as oil leases and parts of the electronic spectrum; and on matching algorithms, which are used in the selection of schools by children and parents, and hospital residencies by doctors. There is also a substantial qualitative literature on problems of implementing and evaluating democratic reforms.