i am an ib student and need to write an extended essay on math. Basically extended essay is a 4000 word limited article which focuses on a problem, however I need some ideas to write about in my extended essay. I have a interest in game theory and real analysis, and want to write about that topic but I need a specific problem just like a prisoners dilemma. I am open to any ideas, if you have an idea that is not related to game theory, please mention and i can take into consideration but remember that i am only a high school student.
An interesting question in game theory would be the ever-elusive definition of what it means for an agent/person/player to be 'rational.'
To motivate, imagine a world with one person in it, trying to do his or her best in the face of outside forces. Within such a context, it is not unreasonable to suppose that the person either knows the true probabilities with which events may happen, or at the least has some subjective beliefs about this. In such a world then, given these beliefs about what events they are facing, it makes sense to deem a person rational if they act in such a way as to maximize their expected utility (where the expectation is taken with regard to their beliefs/the true probabilities over events).
Consider now game theory, where multiple (even just two) equally clever people are matched up against each other. Even in the simplest possible context when the payoffs/structure of the game and the rationality of both players is common knowledge (either formally or informally construed), what it means to be rational is much more difficult and elusive thing, and indeed arguments over its definition have at least implicitly been the driving force behind the small cottage industry of refinements to the basic Nash equilibrium.
If you already have a background in basic game theory, depending upon your mathematical level you might find it interesting to discuss the various motivations for Nash equilibrium versus sub-game perfect Nash equilibrium, and perhaps, depending upon your mathematical background, things such as trembling-hand perfect equilibrium or sequential equilibrium.