The Flaw in Elections: Arrow's Impossibility Theorem

Written by Emir Taha Macit

Has it ever occurred to you how most of the voting systems around the world are actually unfair in several ways? It may not seem flawed at first for many people, but this would all come to light when economist Kenneth J. Arrow proposed his idea of the Impossibility Theorem. This idea proposes that no voting system can satisfy ultimate fairness, highlighting the paradoxes inherent in creating the perfect voting system. The theory comes to the conclusion that there is no mathematical way to create such a system. Arrow’s Impossibility Theorem is an idea with various conditions that has defined people’s outlooks on social welfare functions and raised concerns as to how the world works.

Before we dissect the theory itself, we must first understand some other terms used in its analysis. The most prevalent term we will be seeing is the social welfare function. It is defined as the aggregation of individual utilities within a society. Elections are good examples of this, being aggregated by society to utilize government. Furthermore, there is an electoral system we will be using as an example that is used by most countries, called first-past-the-post, or FPTP. FPTP is a system in which the voters choose a candidate as their first preference, and the singular candidate who was preferred most wins. The candidate does not need to have a majority, more than 50% of the votes, contrary to some other systems. It has been used since the Middle Ages and is what first comes to mind when one thinks of elections. 

Now, we may begin by going over Arrow’s conditions of fairness, things that are essential for a perfect electoral system. The first condition suggests that there should be no dictatorship, meaning no single voter should have a vote that is more valuable than others, and more than one person must partake in the voting. The second condition, Pareto efficiency, proposes that if every voter prefers one candidate to another, then the preferred candidate should win, also called unanimity. The third condition, independence of irrelevant alternatives (IIA), brings forth the idea that if a candidate were to be removed from the pool, the rankings of the other candidates should not change, ensuring that the votes of a candidate who was set to lose do not mess up the results. The fourth condition, unrestricted domain, suggests that the voting must work and account for any set of personal preferences. The fifth and final condition, transitivity or social ordering, means that the candidate preferences of a voter should be able to be ranked, such as from best to worst. When all of these conditions are supposed to be implemented, we are faced with a paradox that stems from the violation of at least one of these conditions.

Suppose there are 3 candidates named A, B, and C. Ideologically, candidate A leans more to the left, while C leans to the right, and B is neutral. The vote percentages are as follows: A has 32%, B has 24%, and C has 44% of the total votes. If we were to remove either A or C, candidate B would most likely receive the removed candidate’s votes since B is ideologically closer. This leads to a major change in the results of the election, causing the least voted candidate to win in the end. This situation is an apparent violation of the third condition of the theory, IIA. While there are countless other examples of a violation of Arrow’s conditions encompassing other electoral systems as well, this is not to say these are the results of bad systematic design.

Arrow’s theory proves that these problems are mathematically unsolvable and that there is no way to create the perfect electoral system. As we can conclude from our example, any system that could be found has to violate at least one of the conditions. The problem being mathematical and not political makes it so we have to go with the system that seems most fair to us, and one that might lead to healthier political relationships within a nation. For example, the 2013 mayoral election for Minneapolis used the ranked choice voting system, in which candidates would take turns discussing their motivations and what they bring to the table, and voters would rank them according to their individual preferences. This election ended up becoming a very interesting day, in which the candidates were very kind and friendly to each other, opposed to the regular system of two enemy parties. The election was ended in singing and congratulating, painting a completely out-of-the-ordinary picture of politics.

In conclusion, Arrow’s Impossibility Theorem mathematically proves that there is no perfect system for elections. Across all systems that could be used, the one that should be chosen has the fewest possible flaws and embraces togetherness and peace.

References:

1. Daniel Liberto, “Understanding Arrow’s Impossibility Theorem: Definition, History, and Example”, Investopedia, 2025, https://www.investopedia.com/terms/a/arrows-impossibility-theorem.asp

2. Michael Morreau, “Arrow’s Theorem”, Stanford Encyclopedia of Philosophy, 2014, https://plato.stanford.edu/entries/arrows-theorem

3. Karen Boros, “With ranked-choice voting, Minneapolis mayoral race becomes Battle to Be Nice”, MinnPost, 2013 https://www.minnpost.com/politics-policy/2013/10/ranked-choice-voting-minneapolis-mayoral-race-becomes-battle-be-nice/