Arrow's Impossibility Theorem

Arrow's impossibility theorem is a result in social choice theory that states that no voting system can satisfy a set of five desirable properties simultaneously. These properties are:

* * *Unanimity:* If every voter prefers A to B, then the voting system must also prefer A to B.
* * *Non-dictatorship:* No single voter should be able to dictate the outcome of the vote, regardless of the preferences of the other voters.
* * *Transitivity:* If a voter prefers A to B and B to C, then the voter must also prefer A to C.
* * *Independence of irrelevant alternatives:* The outcome of the vote should not be affected by the addition or removal of alternatives that are not voted on.
* * *Strong Pareto efficiency:* If every voter prefers A to B, then the voting system must also prefer A to B.

The theorem was first published by Kenneth Arrow in his 1951 book *Social Choice and Individual Values*. Arrow's theorem has important implications for the design of voting systems, as it shows that no voting system can be perfectly fair and democratic.

One way to understand Arrow's theorem is to think about a simple voting system in which voters are asked to choose between two candidates, A and B. If every voter prefers A to B, then Arrow's theorem tells us that the voting system must also prefer A to B. This is the property of unanimity.

However, Arrow's theorem also tells us that no voting system can be dictatorial. This means that no single voter should be able to dictate the outcome of the vote, regardless of the preferences of the other voters.

To see why this is true, imagine a voting system in which a single voter has the power to cast two votes. If this voter always votes for A, then the voting system will always prefer A to B, regardless of the preferences of the other voters. This is a violation of the property of non-dictatorship.

Arrow's theorem also tells us that no voting system can be transitive. This means that if a voter prefers A to B and B to C, then the voter must also prefer A to C.

To see why this is true, imagine a voting system in which voters are asked to rank three candidates, A, B, and C. If a voter ranks A first, B second, and C third, then the voting system could rank B first, C second, and A third. This is a violation of the property of transitivity.

Arrow's theorem also tells us that no voting system can be independent of irrelevant alternatives. This means that the outcome of the vote should not be affected by the addition or removal of alternatives that are not voted on.

To see why this is true, imagine a voting system in which voters are asked to choose between two candidates, A and B. If a third candidate, C, is added to the ballot, then the voting system could change its preference from A to B. This is a violation of the property of independence of irrelevant alternatives.

Finally, Arrow's theorem tells us that no voting system can be strongly Pareto efficient. This means that if every voter prefers A to B, then the voting system must also prefer A to B.

To see why this is true, imagine a voting system in which voters are asked to choose between two candidates, A and B. If every voter prefers A to B, then the voting system could still prefer B to A. This is a violation of the property of strong Pareto efficiency.

Arrow's impossibility theorem has important implications for the design of voting systems. It shows that no voting system can be perfectly fair and democratic. This means that when designing a voting system, it is important to consider the trade-offs between the different properties of voting systems.