Most advanced societies rely on democratic principles to set government policy. When a city is deciding between two locations to build a new park, for example, we have a simple way to choose The majority gets its way. Yet for most policy issues, the number of possible outcomes far exceeds two. A new park, for instance, could be placed in many possible locations. In this case, as the 18th-century French political theorist Marquis Condorcet famously noted, democracy might run into some problems trying to choose one of the outcomes.

For example, suppose there are three possible outcomes, labeled A, B, and C, and there are three voter types with the preferences shown in Table 1. The mayor of our town wants to aggregate these individual preferences into preferences for society as a whole. How should he do it At first, he might try some pairwise votes. If he asks voters to choose first between B and C, voter type 1 and 2 will vote for B, giving B the majority. If he then asks voters to choose between A and B, voter types 1 and 3 will vote for A, giving A the majority. Observing that A beats B, and B beats C. the mayor might conclude that A is the voters’ clear choice.

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