It is not just amatter opf a piece of paper, polygamy gives rights of ownership and inheritance to all thwe wives, where as being just lovers does not. Children in polygamous marraiges according to islamic law are legitimate.The son of so and so. even the children ahve shares in the inheritance. Islam's polygamy is to do with responsibilty and the betterment of the society not just a bit of sex on the side like what would happen in countries that condemn polygamy there is a very good book called"Rather wives than mistersses" In one chapter it deals with all these world leaders who although they weer against polygamy had mistresses on the side.President miterrand of france is one example cited, where there were two women at the funeral in black, his wife and another, and everyone else questioned who was the other it was his mistress for years.he was cheating on his wife yet she was unaware for years until he died.