This recent game theory question got me thinking (this is a tangent, of course): Is it possible to efficiently optimize a personal strategy for choosing research questions to work on using game theory?
In order to head towards a formalization of the question, I'll make the following (informally stated) assumptions:
- I equally "enjoy" any particular problem available for me to work on (in order to avoid the "soft" (and correct!) answer of "Do what you like!").
- I may or may not be successful at finding an answer to any given problem I choose to work on. For any given problem, I have some estimate of the probability of how good I will be at solving a problem (after investing time in it).
- My goal is to maximize my payoff when being evaluated down the line (applying for a job, applying for tenure, applying for a fellowship, etc.), which is a function of how many problems I solve and how important or hard the problems are. I do not have a clear idea of the exact payoffs per problem, but I can make a reasonable estimate.
- There is a loose inverse relationship between problem payoff and problem difficulty. Another statement of my goal is to "game" the differences (i.e. look for "low-hanging fruit").
- An instance of this overall problem is specified by a list of research questions (possibly infinite in number), to which I firmly attach (at no computational cost; it's given as input) an estimate of the question's worth and the question's difficulty. I am playing this game against an adversary (the person evaluating me); nature decides, given the probability of me solving a given problem, whether I solve it successfully after I choose to attempt it.
In an effort to really formalize what's going on (and sidestep uninteresting or argumentative/discussion-type responses), I will view this problem as an extensive-form game with incomplete information with an infinite action set.
Question: I assume games of this type are not efficiently computable. However, is there a polynomial time algorithm to approximately maximize my payoff? What about a PTAS?
Or, alternatively, is there a more accurate game-theoretic model for this problem? If so, the same question holds: Can I (approximately) maximize my payoff efficiently? If so, how?