The purpose of the Tourney is to preserve the Citizens' monopoly on Protonite.
Let's start by acknowledging that for a wealthy ruling class, it's a rather strange policy to share the full privileges of membership with anyone who can win a certain game. It's on a level with things like hereditary monarchy, or making crowning Arthur just because some watery tart handed him a sword, or making a king out of the 47th person to walk through the city wall (as happens in one of L. Frank Baum 's books).
However, the whole point is to promote Citizen solidarity against off-world businessmen. Remember, the Citizens are monopolists, and if they acted like rational profit-maximizers, they would be likely to lose their monopoly. They aren't too worried about anyone discovering Protonite on another planet (although surely they have a plan for what to do if it happens). So all they need is a way to keep individual Citizens from making business alliances with off-world groups. But if they allowed themselves to think like businessmen instead of aristocrats, someone surely would make these alliances in no time flat.
So what they need is something rather like Plato's noble lie. Which, in the original dialogues, consisted of telling everyone that a certain lottery reveals who is a superior philosopher-king and who isn't. That's where the idea of the Tourney comes in, because really, anyone who wins the Tourney is more like The Ace than anyone we know.
So once the Tourney (and inheritance) are the only ways of becoming a Citizen, how do you prevent the case of the person who wins the Tourney and proceeds to undermine the system? Well, the best way to win the Tourney is to practice for many years. You let the residents of Proton, the serfs, practice all they want; no off-worlder has this advantage. The Citizens probably use disinformation to make sure that the off-worlders are rather ignorant about what would really be involved in winning the game. So the Tourney is won by people who have been living on Proton for twenty years, and so it's won by the kind of person who would rather be a serf on Proton than live anywhere else in the galaxy. This psychology of commitment is the reason fraternity hazing works to create loyalty to the group.
The books explain that there are a very small number (24 to 1000) of off-worlders in each Tourney. This is to make the rule against off-worlders becoming Citizens seem less like a taboo and more like just rules of a game. If you just told off-worlders, "You can never gain a share of our privileges, no matter what you do," then this would seem closed-minded and unfair, and might lead some off-worlders to do everything in their power to change that. But if you tell them that they can pay a huge fee for a tiny chance at winning the contest, then it will seem like just the rules. The books do explain that that's precisely the way the serfs are encouraged to think about it.
But off-worlders almost never win the Tourney, and neither do the people who become serfs and try to win the Tourney after just one year. With only one winner a year, there's never been any seriously disruptive individual who entered the ranks of Citizens.
Finally, very little of this rationale is understood by the Citizens alive today, many of whom are mere UpperClass Twits. No, it was all invented by the same people who put the Oracle in Phaze, with the help of the Oracle itself. And like the existence of Phaze and the Oracle, it was thought that the Citizens are better off not understanding this reasoning.
- <slow claps>