Arrow's Impossibility Theorem

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A fundamental result in social choice theory that shows the limitations of ranked voting systems.

Arrow's Impossibility Theorem is a fundamental result in social choice theory that shows the limitations of ranked voting systems.

It states that when there are more than two alternatives, there is no voting method that can satisfy all of the following criteria:
  • Unrestricted domain: Any preference order of the alternatives is allowed for each voter.
  • Non-dictatorship: There is no voter who can determine the outcome regardless of the preferences of the others.
  • Pareto efficiency: If all voters prefer X over Y, then the outcome should also prefer X over Y.
  • Independence of irrelevant alternatives: The outcome should depend only on the preferences between X and Y, and not on the preferences over other alternatives.

Kenneth Arrow used his PhD thesis as the foundation for his 1951 book Social Choice and Individual Values, which established the theorem. For his contributions to social choice theory and welfare economics, he was awarded the Nobel Prize in Economics in 1972.

Insinuating that there is no ideal solution to combine individual preferences into a collective preference, the theorem has significant consequences for democratic decision-making. Any voting process will either fail to meet one of the criteria or result in inconsistent or un-transparent results. A cycle of preferences could occur, for instance, when the group favors X over Y, Y over Z, and Z over X.

To understand why this is the case, let us consider a simple example with three alternatives: A, B and C. Suppose there are three voters with the following preferences:
  • Voter 1: A > B > C
  • Voter 2: B > C > A
  • Voter 3: C > A > B

In a majority rule system, where the choice that receives more than half of the votes wins, we encounter a paradox. No alternative can win a majority because just one voter favors each one. Nevertheless, if we compare two options at once, we arrive at a different conclusion. If we compare A and B, for instance, we observe that two voters (1 and 3) like A over B, hence A wins. Comparing B and C similarly reveals that B is preferred by two voters (1 and 2), leading to B's victory. Comparing C and A also reveals that two voters (2 and 3) favor C over A, which results in C winning.

As a result, it is impossible to rank the options consistently according to voter preferences. There is an ongoing cycle in which A beats B, B beats C, and C beats A. The transitivity property of a preference order states that if X is preferred over Y and Y is preferred over Z, then X should be preferred over Z. This action violates this property.

According to Arrow's theorem, this paradox is true for any voting system that satisfies the four requirements, not just majority rule. You can only prevent it by breaking one or more of the conditions. For instance, we may limit the range of acceptable orders in the domain of preferences to only include A > B > C or B > C > A. This would stop the cycle, but it would also restrict the voters' options. As an alternative, we may give each voter's preferences a predetermined weight, such as giving voters 1 twice as much weight as voters 2 and 3. This would guarantee a distinctive result, but it would also establish voter 1's tyranny.

The welfare economy and democratic decision-making are significantly impacted by Arrow's theorem. It demonstrates that there isn't a perfect solution to combine individual preferences into a group choice that reflects the will of the people and upholds their rights. It also calls into question the notion of social wellbeing as a clearly defined concept that can be assessed and improved upon. According to Arrow's Theorem, any social welfare function that satisfies a few prerequisites must either be a despotic or arbitrary system.

The theorem is also applicable in other contexts where preferences must be aggregated, such as welfare economics, decision theory, and game theory. The theorem demonstrates that no single criterion or function can adequately express the social welfare or utility of a group of people.

The theorem does not hold true in cardinal voting systems, where voters rate or score the options instead of ranking them. However, other issues like manipulation, strategic voting, or a lack of expressiveness may also be present in such systems.

Some possible ways to deal with the impossibility theorem are:
  • Easing up on or changing some of the requirements, like permitting ties or partial rankings.
  • Acknowledging that certain voting procedures may outperform others in particular circumstances or for particular preferences.
  • Utilizing different techniques for group decision-making, such as discussion, agreement, or randomization.