Imagine a situation where a group of *n* people have to make a decision by simple majority. For the sake of the argument, let’s call this group of people a *parliament*.

Now let’s consider an individual person *P* who is a member of the parliament. Let *d* be the number of decisions which the parliament is going to make while *P* is a member, disregarding any decisions of which *P* does not care about the outcome.

Let us assume that *P* is inherently selfish and therefore wants to maximise the number of these *d* decisions which are decided in *P*'s favour. A good strategy for *P* is to rally together a group of *n/2+1* members of the parliament with similar opinions to *P*. Let us call this group a *party*. If this party were to agree to always vote together in the parliament for an outcome decided in a fair manner by the party, this would be likely to increase the number of decisions which the parliament decides in *P*'s favour.

All that remains then, is for the party to settle on what constitutes a “fair manner” for deciding which option the party will vote for in the parliament. The obvious answer is of course for the party to make a decision by simple majority.

So we now have a situation where a group of *n/2+1* people have to make a decision by simple majority. A good strategy for *P* is to rally together a group of *(n/2+1)/2+1* members of the party with similar opinions to *P*. Let us call this group a *faction*. If this faction were to agree to always vote together in the party for an outcome decided by simple majority by the faction, this would be likely to increase the number of decisions which the party decides in *P*'s favour and therefore increase the number of decisions which the parliament decides in *P*'s favour.

So we now have a situation where a group of *(n/2+1)/2+1* people have to make a decision by simple majority. A good strategy for *P* is to rally together a group of *((n/2+1)/2+1)/2+1* members of the faction with similar opinions to *P*. Let us call this group a *subfaction*. If this subfaction were to agree to always vote together in the faction for an outcome decided by simple majority by the subfaction, this would be likely to increase the number of decisions which the faction decides in *P*'s favour and therefore increase the number of decisions which the parliament decides in *P*'s favour.

This concludes my demonstration.

## 3 Responses to

On Factions and Fractions