Economist here (but not a game theorist): Repeated games are actually really hard to find closed form equilibria for in general. Additionally, many repeated games have multiple equilibria which makes every bit of follow on analysis that much more annoying. Only very specific categories of repeated game have known nice unique solutions like you are hoping for, and even then usually only under some idealized information set structure. But as other commenters have said, this is an active area of research in Economics and many grad classes are offered on it.