Voting theory seems pretty easy to Google, find interesting reference material on, etc. But I'm currently interested in going a bit beyond voting theory, to broader questions of governance. For example, what are the real advantages of parliamentary systems vs presidential systems? What elements of constitutions lead to actual good governance? Questions like these are less clear-cut and more highly politicized, so I expect it to be much more difficult to filter out BS. Wikipedia is full of good historical information, but articles on specific government styles tend to only contain vague statements of advantages and disadvantages, rather than rigorous assessment.

Are there good subfields of political science that would offer powerful predictive models, or at least mathematically rigorous analysis?

Are there terms I should Google? Areas with good textbooks? Other book recommendations? Specific papers?

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It depends a bit on what you are searching for. Many interesting topics are covered in Persson/Tabellini's Political Economics (they also have a book on constitutions, I cannot say much about that) and Drazen's Political Economy in Macroeconomics. Or maybe you start by having a look at Acemoglu's lecture notes. And there's "Principled Agents?" by Besley, basically applying simple principle-agent models to governments. Which of these models are convincing is a different matter. Having spent a lot of time with Grossman & Helpman's Lobby models, I think they don't tell us as extremely much about lobbyism or politics.

The book on constitutions looks very close to what I was looking for, thanks! 

Great, then maybe Daron Acemoglu's review of the book is the right place to start.
So here's something that is not fomal theory, but may interest you: The Economist, March 27th, has a review of a new book on the history of constitutions called "The Gun, the Ship and the Pen" by Linda Colley.

There's Selectorate Theory. I read The Logic of Political Survival and I've been working on and off on a post distilling the theory. The theory seems very good to me (At the very least a great start), but I wasn't satisfied with the resources about it.

The Logic of Political Survival is a long and somewhat difficult book, and the Dictator's Handbook, though I only read parts of it, seems to not give a full coverage of the theory. one of the motivations for my post is to have a resource that's more accessible but touches on all aspects of the theory. I hope to get back to it after I finish the post I'm currently focused on.

I'm also eagerly awaiting the full version of Nassim Taleb's Principia Politica, of which there's currently a short draft version online.

I look forward to your post! I have read Dictator's Handbook, and thought it was pretty good, but I did think The Logic of Political Survival looked way better in terms of providing actual models.

2Yoav Ravid2y
The post is finally published! I'm not sure if it will quite satisfy your desire for the actual model because I didn't get into the math or the specifics of how the game is constructed, because I think it deserves its own post and I would like to reproduce the model in code before I do that, but I still describe the model more deeply than The Dictator's Handbook did. Hope you like it! :)