Mathematics and Voting:

More Than Just Counting Votes

By Michael A. Jones

Sparta beat Athens to the first democracy
by approximately 150 years.

Voting has been used as a way to aggregate information since at least the 7th century B.C., when the first recorded democracy in Sparta included in its constitution a way for a general assembly to vote on bills.  Nearly 2700 years later, people are still voting, using a variety of election procedures.   Perhaps surprisingly, once votes are cast, a change in the procedure can result in a change in the outcome!  In such cases, how we tally votes is just as important as how we rank the candidates.  Mathematics is used to determine how and how often such outcomes occur, to analyze properties of procedures, and to design election procedures to satisfy specific desirable properties.

Take for example the 1860 election for President of the United States.  Such a highly regarded president as Abraham Lincoln, the first U.S. president to appear on a U.S. coin, would have lost the election to Democratic candidate Stephen A. Douglas under most election procedures.  Not only would this have changed the face of U.S. currency, but many historians agree that if Douglas had been elected he would have helped the U.S. avoid the Civil War.

The debate over election procedures is very much alive.  Recently, Voter Choice Act-House Resolution 2690 (in the U.S. House of Representatives) included a provision to use the Instant Runoff Procedure for all single-winner federal elections.  And, voting is not limited to the political arena: Major League Baseball votes on its Most Valuable Players, the Academy of Motion Picture Arts and Sciences votes to award its Oscars™, and companies vote to choose between business alternatives. 

The myriad of procedures is a testament to Kenneth Arrow, who proved that there is no “best” election procedure, a result known as Arrow's Impossibility Theorem for which Arrow received, in part, the 1972 Nobel Prize in Economics.  Because not all desirable properties are available in one procedure, it is paramount for voters and designers of election procedures to understand which properties their procedure satisfies.

Click the links to the right to learn more about specific election procedures, their applications, and how mathematics is used to analyze their properties.


About the Author

Michael A. Jones, Associate Professor, Montclair State University

Michael A. Jones is an Associate Editor for Mathematical Reviews, American Mathematical Society, Ann Arbor, MI.  His research in game theory, voting theory, and fair division includes applications of mathematics combinatorics, functional analysis, and dynamical systems) to the social sciences (political science, economics, and psychology).