Your question is interesting because it asks whether these are fundamentally separate problems or different manifestations of a more general process.
My impression is that the literature contains many pieces of what you’re describing, but not necessarily a single unifying model. As you noted, Akerlof’s adverse selection, Stigler’s economics of information, moral hazard, regulatory capture, rent seeking, public choice, principal-agent theory, and repeated games each describe part of the picture. They overlap, but each emphasizes a different mechanism.
For that reason, I’ve been approaching a similar problem from a somewhat different direction while developing a behavioral-economic model called the Productive Value-Productive Power (PV-PP) framework. It is still under development, so I certainly wouldn’t present it as an established alternative to the existing literature, but one of its motivations is precisely the kind of question you’re asking.
The framework treats money, political influence, information, institutional authority, reputation, and similar assets as different forms of Productive Power (PP) that affect future exchanges between participants. From that perspective, the common structure is not simply
capital → influence → rules → capital
but something more general:
Productive Power → influence over interaction structure → Productive Value exchanges → updated Productive Power
In this view, information asymmetry, regulatory capture, lobbying, unequal bargaining power, or preferential access to policymakers are not separate fundamental phenomena. They are mechanisms that alter the exchange process itself. Those altered exchanges redistribute productive power, which then changes the participants’ ability to shape future exchanges. The recursive nature of the process is what makes it self-reinforcing.
One implication is that the same architecture is not limited to money or politics. Similar feedback loops appear wherever accumulated productive power changes future interactions—for example through reputation, technological advantage, organizational control, scientific authority, or social influence.
Whether that perspective ultimately proves useful remains an open research question, but it has led me to view many of these literatures as examining different aspects of the same underlying recursive process rather than unrelated phenomena.
As for established literature, I would probably look most closely at work combining public choice theory, regulatory capture, institutional economics, principal-agent theory, repeated games, and information economics. Those fields collectively seem to come closest to the phenomenon you’re describing, even if they stop short of treating it as a single generalized interaction model.