Arrow’S Impossibility Theorem

Published Oct 25, 2023

Definition of Arrow’s Impossibility Theorem

Arrow’s Impossibility Theorem, formulated by economist Kenneth Arrow, is a mathematical theorem that highlights the limitations of democratic decision-making systems. It states that there is no perfect voting method for ranking preferences that satisfies a set of desirable criteria simultaneously. In other words, it is impossible to design a voting system that accurately reflects the preferences of individuals in a fair and consistent way.

Example

To understand Arrow’s Impossibility Theorem, let’s consider a simple example. Imagine a group of five friends who have three options for dinner: pizza, burgers, and sushi. Each friend has their own preferences, and they submit their ranked choices as follows:

– Friend A: Pizza > Burgers > Sushi
– Friend B: Sushi > Burgers > Pizza
– Friend C: Burgers > Sushi > Pizza
– Friend D: Pizza > Sushi > Burgers
– Friend E: Burgers > Pizza > Sushi

Now, let’s assume we want to determine the group’s overall preference by using a voting method. One common method is a majority vote, where each option is ranked against the others. In this case, we would consider pairwise comparisons:

– Pizza vs. Burgers: 3 friends prefer pizza, 2 friends prefer burgers (Pizza wins)
– Pizza vs. Sushi: 3 friends prefer pizza, 2 friends prefer sushi (Pizza wins)
– Burgers vs. Sushi: 3 friends prefer burgers, 2 friends prefer sushi (Burgers win)

Based on the majority vote method, pizza would be the group’s overall preference. However, if we consider transitivity (if A is preferred to B and B is preferred to C, then A should be preferred to C), we can see a problem. Friend B prefers sushi over pizza, and friend E prefers burgers over sushi. So, according to transitivity, friend B would prefer burgers over pizza. This creates an inconsistency in the ranking.

This example illustrates how Arrow’s Impossibility Theorem shows that it is impossible to design a voting method that satisfies all desirable criteria, such as transitivity and non-dictatorship, simultaneously.

Why Arrow’s Impossibility Theorem Matters

Arrow’s Impossibility Theorem has significant implications for political systems and decision-making processes. It challenges the idea of finding a perfect democratic system that accurately represents the preferences of individuals without any flaws. The theorem suggests that trade-offs and compromises are inevitable in any voting system. Policymakers and researchers use Arrow’s theorem to better understand the limitations and complexities of democratic decision-making, prompting discussions on how to design fairer voting methods and systems.