How to efficiently progress on a complex problem in research?
Write notes as you work. Brainstorm a list of questions, then try to write down answers to them as you investigate. Bonus points if some of them are questions that you would have asked before knowing the term game theory, or if some are ones a six-year-old would ask. It sounds like one of the questions on your list already is, "What conditions produce each equilibrium?"

You've already found the equilibria (do you know if you found all the equilibria?), but why is that interesting? How can you interpret that for someone? So what? (The questions that end up being most interesting or giving you the most to say should be your research questions, with their answers being your thesis statement(s).)



Below, I'm taking a lot of liberties guessing about what you're doing and what might be productive. Wherever I assume something wrong, try to use that as a jumping off point. ("That's completely silly! This wouldn't be a complex system because X!" or "Using an agent-based model wouldn't capture Y." or "Existing small world models don't account for Z.")

Depending on exactly what model you're using (e.g. an agent-based model for a repeated game) you may be creating and analyzing a complex system. There might be emergent properties you're trying to capture, and you may want to tune the model to better illustrates that emergence. If you just altered one variable at a time you might get a tipping point model ("Huh, turns out the dividing line for getting cooperative behavior is if everyone starts with an endowment of X.") but just cataloging those univariate effects doesn't necessarily capture emergent behaviors that involves interacting traits or evolving patterns of behavior. (E.g., an interesting emergent result might be: "Generally, increasing equality of endowments or increasing N leads to an equilibrium of cooperation. But when you have perfect equality of endowments and increase N, you paradoxically have higher rates of defection." Then you could illustrate how that typically plays out, turn by turn.)

Maybe mathematicians or others who work with complex systems could tell you about classes of functions or models that are thoroughly understood, and you could compare your system to that. My guess is that systematically recording the results of your playing around with the model (potentially trying to answer 6-year-old-style questions) will lead to better work.