# How to efficiently progress on a complex problem in research?

I am writing my master thesis in game theory and realized one type of situation where I am often stuck for a long time and feel frustration creeping in. It is, to no surprise, when facing a complex situation: one with too many variables or different cases to look at.

Example setting: I build a model and look at some equilibrium points as results. The different equilibria are conditioned on many parameters such as the income of people, their social ties, their level of empathy etc. My task is to understand these conditions in order to be able to generalize some of the results.

Example being stuck: I a-priori think that the plan to crack this issue is to look at each equilibrium individually and understand the conditions for that case and then move on and in the end put everything together, I know it is going to be annoying but at some point it will just work. Unfortunately, I often fail to stick to the plan when the conditions for a certain equilibrium point are hard to understand, my mind starts to go back and forth, loosing focus and in the end failing to understand the situation at hand. I usually succeed by repeatedly going the same way again and again, but it is very costly.

I know this isn't something specific to me, this happens to many/most people when the problem contains too many cases, variables or causal links.

Do you have tips or resources on how to best handle problems like these?

• Not really an answer, but there apparently is a saying: "How to eat an elephant?" – "Just like anything else, piece by piece." Mar 27, 2018 at 19:54
• One presumes you would also like effectiveness? Mar 27, 2018 at 23:24
• @OlegLobachev as I mentioned this was my way of looking at things but the point is that there's a lot of possibilities of how and which pieces to eat, and there lies the fun. Maybe even taking a few pieces at once and eating them simultaneously would work better. Mar 28, 2018 at 9:32
• @JonCuster exactly, I know that I can crack anything by just trying to go this direct path of piece by piece but at what costs, maybe even time that I don't have any more. Mar 28, 2018 at 9:33

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.

• I like the suggestion to write questions down, I think that is something I do a lot. Being systematic is also a goal, which I am doing my best for. How do you exactly mean it with "Wherever I assume something wrong, try to use that as a jumping off point"? Wrong as counter-intuitive? Or wrong as actually proven wrong? Mar 28, 2018 at 9:50
• @FrantišekKaláb - I meant "Wherever I assume..." as meta-commentary: below the horizontal line, I was making assumptions about the details of your Masters project content. So the most likely way I would be wrong is that what I'm saying doesn't match the reality of the problem you're solving. And in fact, I'm probably overstepping the Academia Stack Exchange boundaries by trying to make suggestions related to the specific content of your research rather than sticking to general approaches. ;) Mar 28, 2018 at 10:05

Near the bottom of your thesis draft, add a section along the following lines.

Future Work

This investigation suggests several interesting avenues of exploration for future research. In particular:

• Does Theorem 3 still hold if the assumption of a perfectly spherical cow is relaxed to allow for ellipsoid cows?

• Does the albedo of the cow affect the maximum time in orbit?

* Is anybody willing to hire me to research these questions, preferably for more money than I am currently earning as a Master's student?

Every time you have a new idea for something that should be investigated, ask yourself whether it really needs to be explored within your Master's thesis, or if you would do better to put it in the "Future Work" section. Think about how valuable it would be to your thesis and to the field vs. how much work it would require, and prioritise accordingly.

If you're doing interesting research, your work may well open as many new questions as it answers. You don't have to answer them all, or at least not yet! Just ensure that what you do cover is enough of a contribution for a solid Master's thesis, and de-scope the rest of it for now.

If your work has an applied focus, I would also advise asking yourself which of these questions really need to be answered in order to solve the problem at hand.

For example, my PhD project involved some issues of fluid dynamics. Modelling them in detail would have required a great deal of work to learn the fluid dynamics, and very significant computational difficulties (modelling a non-Newtonian fluid of poorly-known physical properties reacting to a poorly-characterised force, in a grid likely to require thousands of cells, using mid-90s technology.) I figured out that a static approximation would provide answers which were good enough for practical applications, so I was able to avoid a very long digression.

Above all, check in with your supervisor about how much they expect you to cover.

• +1000 for "check in with your supervisor about how much they expect you to cover." Mar 28, 2018 at 6:45