Mathematics and Democracy: The Design of Institutions

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The focus of this course is on two essential features of democracy: (1) how individual preferences can be aggregated to give a social choice or election outcome that reflects as accurately as possible the preferences of the electorate; (2) how public and private goods can be divided in a way that is fair, respects due process, and is consonant with the rule of law. Democracy is taken to mean representative democracy, in which citizens vote for representatives, from a president on down, who represent different constituencies. But referendums, in which citizens vote directly for leaders or in support or opposition to propositions—just as they did in assemblies in ancient Greece—will also be analyzed. So will special institutions, like the Electoral College, and practices like gerrymandering. Emphasis will be given to procedures or algorithms—the rules of play of games—that produce outcomes. By making precise properties that one wishes a voting or fair-division procedures 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, tradeoffs among the properties will be studied. In the case of procedures that have actually been used, practical problems of implementation will be discussed.