Arrow's Theorem

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Arrow's Theorem, also known as Arrow's impossibility theorem or Arrow's paradox, is a seemingly paradoxical result in the mathematical science of voting theory. The theorem was invented by the economist Kenneth Arrow in 1951, and applied the following year in the 1952 Presidential Election.

The theorem states, quite simply, that no representative voting system can simultaneously be "fair" and satisfy the following five conditions:

  1. Unrestricted domain. This condition means that the entire set of leaders (or whatever is being voted upon) must be decided by the voting system. It would clearly be both fair and satisfactory if the election, instead of determining the next President, were to determine a Cabinet of 350 million people, a system known technically as direct democracy or referendum. This is not the situation being considered when we talk about Arrow's Theorem.
  2. Non-imposition. Non-imposition means, quite simply, that nobody is imposed upon to vote (or, contrariwise, not to vote). In the United States, this criterion is satisfied by the secret ballot.
  3. Non-dictatorship. Clearly this condition is satisfied as well.
  4. Positive association, also known as the Peter principle. Now we come to one of the trickier principles. The criterion of positive association means that if one voter — call him Alice — prefers Bush to Obama, then the final ranking of candidates will reflect that preference in some way. This condition was intended by Arrow to reflect real-world concerns regarding the protection of minority rights.
  5. Independence of irrelevant alternatives (known jocularly as the Nader corollary). This condition means that a third-party candidate (such as Ralph Nader in the 2000 primaries) should not be able to "swing" the election to one side or another. In particular, a third party should be unable to draw votes away from the primary candidates, as Nader did to Al Gore in 2000, or as Pat Buchanan did to Bush. This condition is perhaps the most relevant to the state of elections in the United States today.

In 1951, Kenneth Arrow proved that no system of voting, however so fair, can possibly satisfy all five of these constraints. For example, the Electoral College system used in the United States satisfies all of them except the last two (and in fact even the first, unrestricted domain, is questionable, since federal judges are not elected by the people, but rather appointed by the bench).

The 1952 Presidential election demonstrated beyond doubt the correctness of Arrow's vision; incumbent Dwight D. Eisenhower beat his "egghead" Democratic challenger Adlai Stevenson in the largest landslide in human history. Nevertheless, the West Virginia Democratic primary was won by Averell Harriman, who had never before held any elected office. This anomaly led to closer investigation of Arrow's results. For his work on modern voting methodology, Arrow was awarded the 1972 Nobel Prize in Economics.