Arrow's impossibility theorem facts for kids

Arrow's impossibility theorem, also known as Arrow's theorem or Arrow's paradox, is a big idea from social choice theory. It was named after an economist named Kenneth Arrow, who first wrote about it in 1950.

Imagine a group of people voting on at least three different choices. Each person ranks their choices from their favorite to their least favorite. Arrow's theorem says that it's impossible to combine all these individual rankings into one fair group ranking, while also following some simple rules.

Arrow showed this idea in his PhD paper. He made it famous in his 1951 book, Social Choice and Individual Values. His first paper was called "A Difficulty in the Concept of Social Welfare."

What is Arrow's Impossibility Theorem?

Simply put, Arrow's theorem states that no voting system that uses ranked choices can always meet three "fairness" rules at the same time. These rules are about how a group makes decisions when people have different opinions.

The Three Fairness Rules

For a voting system to be considered fair, it should ideally follow these three rules:

• Rule 1: If Everyone Prefers X over Y, So Does the Group.

* This means if every single voter likes option X more than option Y, then the final group decision must also show that X is preferred over Y. This seems like a very basic and fair rule.

• Rule 2: Changes in Other Choices Don't Matter for X vs. Y.

* If voters' opinions about X versus Y stay the same, then the group's preference between X and Y should also stay the same. This is true even if people change their minds about other options, like X versus Z, or Y versus Z. This rule helps keep decisions stable.

• Rule 3: No Dictator.

* There should not be one single voter who always gets to decide what the group prefers, no matter what anyone else thinks. In a fair system, everyone's vote should count.

Why It's "Impossible"

Arrow's theorem shows that it's impossible to design a voting system that always satisfies all three of these rules when there are three or more options to choose from. This means that any voting system will have some kind of flaw or compromise. It might sometimes go against one of these fairness rules.

This theorem is important because it tells us that making group decisions can be very tricky. Especially when people have many different ideas and ways of ranking their choices. It helps us understand the challenges of creating truly fair and democratic voting systems.

See also

In Spanish: Paradoja de Arrow para niños