The word that is translated here as "purpose" is from the Green word γνώμη gnṓmē, (pronounced: gno'-may) (Strongs# G1097). It is from the root γινώσκω (G1097) which means to know. It means: view, judgment, or opinion. It basically means that they think alike. They have the same mind. They have the same ideals, goals and purpose.
It says that they all have the same purpose. What is this purpose? To give power and authority to the Antichrist. They were given authority over a region for one purpose only, to give it to the Antichrist. They may have been elected or come to power for one reason, but that reason is only a front. They worship the beast as god and live for one purpose, to give him control of the entire world.