The context of politics
To get a good idea of Objectivist politics, we need to start with a discussion of the context. This will give us an idea of the problem that need to be understood, and the human needs that need to be satisfied, as well as the conditions that need to be present for a discussion of politics to make any sense.
To start, we need to remember that politics is an aspect of ethics. And what is the standard of ethics? Life. So the first goal of politics is to identify the human needs that need to be satisfied in order to live. We need an understanding of what life requires.
Man is a conceptual being who needs to identify his values in order to pursue them. The use of his mind is an integral part of his survival. Life is a series of choices based on what he believes will further his life, and putting those choices into action. To live and pursue his values, he need to be able to do both. He needs to be able to think, and to act on that thinking. This is the central need being addressed by politics. Man has to be allowed to pursue his values and consequently his life without other men getting in his way.
While on the subject of how man lives, there a crucial point that needs to be made. Mendon't just consume the values that promote their lives, they also produce them. An animal in the wild has to accept the conditions around it, looking for food that's available for it to consume. It eats well if it finds the right plants, or in the case of a carnivore, if it finds the right prey and manages to kill it.
Man works in a fundamentally different way. He actually produces wealth. He's not dependent on finding the right plant at the right time. Instead, he creates a garden, ensuring that the plant is available when he wants it. He doesn't have to hunt for his meat, hoping that a tasty animal crosses his path. Instead, he domesticates a cow, growing them on his farm. He feeds them and keeps them fenced in, and they're available for him when he eats.
More importantly, he breeds the cows. Man doesn't just consume from an already existing supply given to him by nature or luck. Instead, he adds to the supply. He produces wealth in order to consume it. By adding to the existing supply, he removes himself from a zero-sum world where everyone fights over the last little scraps, and instead expands the pot.
This fundamental fact of man's nature, that he produces the wealth he wants to consume, has an important side-effect. Instead of fighting with his fellow men over existing wealth, he creates wealth ending any necessary conflict of interest. And in fact, he finds that by cooperation, he's more easily able to produce wealth.
There are a number of reasons for this. There's the division of labor which allows a man to specialize in a particular field and produce more efficiently, knowing that he can trade the product of their efforts to others for supplies he hasn't produced. Related to this is the ability to have deep and specialized knowledge about a particular topic, instead of having to learn many. With the deeper knowledge, he can act more efficiently. There's the economy of scale which means he can produce a lot of a particular good and there will be enough buyers to make it worthwhile. That means fixed costs like a factory can be paid for by making slight profit on a lot of goods. And there are many other reasons why cooperation means more wealth.
So not only is there no necessary conflict of interest between men, but there is a harmony of interests. It makes sense to live amongst other men, and to live by trade and cooperation. It cannot be overstated how significant this factor is on a man's life.
But it is within this context that the need for politics arises. This social context creates the possibility that one man will prevent another man from living his life by preventing him from acting according to his own best judgment. The harmony of interest created by man's ability to produce and to cooperate is destroyed if men turn against each other and interfere with each other's ability to live.
Men aren't like animals though. Normal adults have the choice to either respect these needs of their fellow men, or to intentionally create a conflict between them. Because of our ability to reason and our free will, we can choose to live peacefully within society. And of course, we can choose not to. The harmony of interest exists only between those men that are will to live peacefully with one another.
So here we can see the critical elements important to a valid political theory. First, man lives by the use of his mind, and interference with his ability to act on his own best judgment is an attack on his life. Next, man produces wealth instead of merely consuming it, ending any necessary conflict of interest. In addition, since cooperation and trade allows for enhanced wealth production, there is an alignment of interests between people. But that harmony of interests only exist between men who choose to live peacefully with others.
Let's look at it from a slightly different perspective. You want to live your life. To do that, you need to be free to make your choices and do as you will. But you also want to live amongst other peaceful people so you can gain the amazing benefits of living in a society including massive increases in wealth. Objectivist politics is concerned with achieving the latter goal without compromising the more basic need. There is no necessary conflict between these goals, so we should be able to achieve both. And this is the fundamental problem in politics.