There is nothing explicitly preventing more than two parties from existing in the US (they do).

The problem is that people tend to become tribal because of collectivist philosophy (most don't even know they're afflicted with the disease, which makes it even more lethal).

Having more parties isn't really going to fix anything if people are still tribalists.

In fact, parties wouldn't matter all that much in a rational society. Many of the US Founders were actually against the concept of parties because they feared what the collectivism and tribalism would lead to (they were right).

Having more of the cancer doesn't seem like a good solution to me. You'll still be ruled by gang warfare. There will just be more smaller gangs. So maybe tyrannical rule by 31% instead of 51%. That isn't much of an improvement. 

 Not at all. When there is one president, how can a new party get as big to be a real competitive parto to go for presidency? But when there are like 10 people to act together as presidents, there is much more opportunity for more minorities to be represented in the government. And they have to at least argue and convice eachother to execute a new order. It is slower. But more based. 

 Tribalism also tends to centralize anyway, just as it did to give us the two major parties we have now.

The incentive will always be for the smaller gangs to consolidate to gain more power. Four will just become two again once people realize that winning on the big issues is more important than losing for the minor differences between a given set of smaller gangs. Purity tests return, as they always do with tribalists, and we are right back to trying to have more gangs to fix the problem.

Parties and collectivism are the actual problems in my opinion. 

 But this is a well proven concept, that with one president to vote for it leads to two parties. This has nothing to do how people act or think.  

 Besides that, I don't think issues of culture and philosophy can be fixed with politics. Politics is downstream of culture. A sick culture leads to sick politics. We have to fix culture. Politics can't be fixed before that. Any measure like this will just be erroded for this reason. The cancer will always grow again because it was never fully killed. Ayn Rand wrote a lot of good material on this issue. 

 You can think this. But there is just reality whefe anyone can discover that the world does not behave like this. The problems that has the US are nothing we have in Switzerland. So there you need to find the difference. 

 One of her best points in my opinion is that you also can't escape politics by simply not participating. You'll still suffer the consequences and they will likely be worse if those with principles and virtue yield to those who have no virtues or principles.