Arrow’s Impossibility Theorem is an important mathematical result in the field of collective choice and welfare economics. It is a sub-field of economics and deals with how decisions are made on a collective level. The theorem comes with some important consequences for democratic processes like voting.
Arrow’s Impossibility Theorem states that clear community-wide ranked preferences cannot be determined by converting individuals’ preferences from a fair ranked-voting electoral system. The theorem is a study in social choice and is also known as “The General Possibility Theorem” or “Arrow’s Paradox.” It is named after economist Kenneth Arrow, who demonstrated it in his paper, “A Difficulty in the Concept of Social Welfare.”
Arrow’s Impossibility Theorem states that a ranked-voting electoral system cannot reach a community-wide ranked preference by converting individuals’ preferences while meeting all the conditions of a fair voting system.
The conditions for a reasonably fair voting electoral system include non-dictatorship, unrestricted domain, independence of irrelevant alternatives, social ordering, and Pareto efficiency.
The theorem does not cover cardinal-voting electoral systems.
Understanding the Arrow’s Impossibility Theorem
Arrow’s impossibility theorem is a social choice theory that studies the combining of preferences, welfares, and opinions from individuals to reach asocial welfare or community-wide decisions. It discusses the flaws of a ranked-voting electoral system.
According to the impossibility theory, when there are more than two options, it is impossible for a ranked-voting system to reach a community-wide order of preferences by collecting and converting individuals’ preferences orders while meeting a set of conditions. The conditions are the requirements for a reasonably fair voting procedure and will be further discussed in the next section.
For a better understanding of the theorem, here is an example that explains why individuals’ preference orders cannot be converted to be a society-wide order. Let’s assume there are three alternatives (options) in ranked voting: X, Y, and Z. The following table shows the voting results from 100 voters:
Based on the results, option X will win since the order of X>Y>Z garners the most votes (45 voters prefer Y over Z and prefer X over Y). The order with option Z as the top preference shows the fewest number of votes, with only 20 voters preferring Z over the other two alternatives. However, if option Y is no longer an available alternative, the result will be reversed.
The total number of votes for Z over X will be 55 (combining the votes for the orders of Y>Z>X and Z>X>Y), and the votes for X over Z is still 45. It result means that Z is socially ranked above X. The conflicting result is proof of Arrow’s impossibility theorem.
Conditions in Arrow’s Impossibility Theorem
As mentioned above, there is a set of conditions (criteria) for a reasonably fair electoral procedure. It includes non-dictatorship, unrestricted domain, independence of irrelevant alternatives, social ordering, and Pareto efficiency.
Non-dictatorship means that a single voter and the voter’s preference cannot represent a whole community. The social welfare function needs to consider the wishes of multiple voters.
2. Unrestricted Domain
Unrestricted domain requires all the preferences of every voter to be counted, which conveys a complete ranking of social preferences.
3. Independence of Irrelevant Alternatives (IIA)
The independence of irrelevant alternatives condition requires that when individuals’ rankings of irrelevant alternatives of a subset change, the social ranking of the subset should not be impacted.
The example mentioned in the section above violates the condition. To meet the IIA condition, the result should remain the same (option X should still be socially ranked above option Z) when option Y is removed.
4. Social Ordering
The social ordering condition requires that voters should be able to order their choices in a connected and transitive relation, i.e., from better to worse.
5. Pareto Efficiency
For Pareto efficiency, the unanimous preferences of individuals must be respected. The order of social preferences must agree with that of individual preferences if every voter strictly prefers one of the alternatives over another. The result should not be sensitive to the preference profile.
Cardinal Voting vs. Ranked Voting
Arrow’s impossibility theorem only applies to a ranked-voting electoral system, but not to a cardinal-voting electoral system. In ranked voting, voters give ranked ballots and rank their choices on an ordinal scale. In cardinal voting, voters give rated ballots and can rate each choice independently.
Numerical scores can be assigned to options in cardinal voting. Compared with ranked voting, cardinal voting provides more information, which makes it possible for a cardinal-voting system to convert the preference orders of individuals into a social preference order.
“Ways Out” of Arrow’s Impossibility Theorem
Some attempts are made to escape the impossibility theorem and investigate possibilities. Such attempts can be classified into two major categories. One includes the approaches that draw every preference profile into an alternative or social preference. The approaches try to weaken or eliminate one or more of the conditions for a fair electoral system. One example is pairwise voting, which limits the number of alternatives to two.
The other category includes approaches that investigate other rules. A cardinal-voting electoral system, which conveys more information, is an example. Thus, the cardinal utility is regarded as a more reliable tool to show social welfare.
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