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The Math

With so many voting methods from which to choose, why do most U.S. elections use the plurality method? Likely, the answer has to do with tradition—the first elections used the plurality method, and there was never a compelling reason to switch. Indeed, there are many who have argued that a different method should be used, but which one?

As the previous page shows, different voting methods can give very different results. Consequently, discussions about choosing a new voting method become politically charged. In an attempt to settle the debate, economic theorist Kenneth J. Arrow tried to identify a voting system that would be most fair.

Unfortunately, what he found is that the collective choice of a society cannot be determined by aggregating the votes of individuals. He proved that no system of voting is fair, if the following four criteria are considered:

• Every possible choice by individuals must lead to an aggregate choice.
• If all individuals make the same orderings then this must also be the aggregate ordering.
• The aggregate ordering must not depend solely on the choice of a single individual.
• The aggregate ordering must not depend on individuals in any way other than in respect to their ordering.

It may seem unlikely, but every voting method violates at least one of these criteria. Because this seems so counterintuitive, his result became known as Arrow’s Paradox.

For an autobiography of Kenneth Arrow, visit the Nobel Prize web site.

The following books have chapters that discuss various voting methods:

• Tannenbaum, P., & Arnold, R. (1997). Excursions in Modern Mathematics. Upper Saddle River, NJ: Prentice Hall.
• Consortium for Math and Its Applications (COMAP). (2006). For All Practical Purposes: Mathematical Literacy in Today’s World, 7th ed. New York: W. H. Freeman.

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