ARROW’S IMPOSSIBILITY THEOREM

Since political theorists first noticed Condorcet’s paradox, they have spent much energy studying voting systems and proposing new ones. For example, as an alternative to pairwise majority voting, the mayor of our town could ask each voter to rank the possible outcomes. For each voter, we could give 1 point for last place, 2 points for second to last, 3 points for third to last, and so on. The outcome that receives the most total points wins. With the preferences in Table 1, outcome B is the winner. (You can do the arithmetic yourself.) This  voting method is called a Borda count for the 18th-century French and political theorist who devised it. It is often used in. polls that rank sports teams.

Is there a perfect voting system? Economist Kenneth Arrow took up this question in his 1951 book Social Choice and Individual Values. Arrow started by defining what a perfect voting system would be. He assumes that individuals in society have preferences over the various possible outcomes: A, B, C, and so on. He then assumes that society wants a voting system to choose among these outcomes that satisfies several properties:

• Unanimity: If everyone prefers A to B, then A should beat B.

• Transitivity: If A beats B, and B beats C, then A should beat C.

• Independence of irrelevant alternatives: The ranking between any two outcomes A and B should not depend on whether some third outcome C is also available.

• No dictators: There is no person that always gets his way, regardless of everyone else’s preferences.

These all seem like desirable properties for a voting system to have. Yet Arrow proved, mathematically and incontrovertibly, that no voting system can satisfy all of these properties. This’ amazing result is called Arrow’s impossibility theorem The mathematics needed to prove Arrow’s theorem is beyond the scope of this book, but we can get some sense of why the theorem is true from a couple of examples. We have already seen the problem with the method of majority rule. The Condorcet paradox shows that majority rule fails to produce a ranking of outcomes that always satisfies transitivity As another example, the Borda count fails to satisfy the independence of irrelevant alternatives. Recall that, using the preferences in Table I, outcome B wins with a Borda count. But suppose that suddenly C disappears as an alternative. If the Borda count method is applied only to outcomes A and B, then A wins. (Once again, you can do the arithmetic on your own.) Thus, eliminating alternative C changes the ranking between A and B. The reason for this change is that the result of the Borda count depends on the number of points that A and B receive, and the number of points depends on whether the irrelevant alternative, C, is also available Arrow’s impossibility theorem is a deep and disturbing result. It doesn’t say that we should abandon democracy as a form of government. But it does say that, no matter what voting system society adopts for aggregating the preferences of its members, in some way it will be flawed as a mechanism for social choice.

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