What subjects do I study if I want to learn about non-gameable systems?

Incentives structures are important to learn about, because people often find it useful to have well-defined goals to work towards, and company or government policies can provide those goals.

A software security mindset can be useful, because in the same way that software attackers theorize about how some code could be exploited, or about the way that a system may behave, people attempting to game a system are looking for flaws that they can benefit from.

Statistics, research and analytical skills can be useful to practice, because for any given {game/policy situation}, it's likely that there exist many similar examples throughout history and the world at-large. It's often easier to find out how similar systems have worked rather than attempting to design your own from scratch.

Good luck!

Game theory? Seems you have something to say about your understanding of subject.

Sybil Attacks, System's Thinking

Mechanism design.