Decisions of groups

Decision-making in groups often suffers from what is known as the Arrow’s impossibility problem. Named after the American Economist Kenneth Arrow, this theory says that if a decision is made by a group of individuals who are not run by a dictator, through sincere voting, they may reach a state of non-transitive preference even if they are all rational.

The statement sounds very complicated. Let’s look at each of those words. First, something about the group – they are free and have clear preferences. The second one is non-transitive preference. To understand this, we must understand what transitive preferences are.

Transitive preference

Rational decision-makers have transitive preferences. That means if a decision-maker prefers A over B, then B over C, it must be that she prefers A over C. A sort of mathematical consistency. But this is so if the decision-maker is one person. What can happen if there is more than one? Take this example of three members of a local committee, Mrs Anna, Mr Brown and Miss Carol. The following represent their choices on what they prefer to build for the local community this year.

The committee votes for pair-wise comparison. The first is school vs library. Anna and Brown vote for their first choices, the school and the library, respectively, and Carol for school (because the library is her least preferred ). The school won 2-1.

The school vs playground happens next. The votes go through a similar process, and the playground wins this time, thanks to Mr Brown; the school was his least preferred option.

You may conclude that the committee should build a playground because it beat the school that defeated the library. But before that, they have to do the final voting- the library vs the playground. Since it was sincere voting, as you expected, the library won by 2 to 1 as Anna, the decider, broke the tie. And we ended up in a non-transitive situation.