I became curious about this, there is of course game semantics but that's not what you're looking for. I found a couple of interesting references about using categories to study game theory which I'm going to add to my reading list, Game Theory from the Category Theory Point of View [0] and Towards Compositional Game Theory [1].
[0] http://www.gtcenter.org/Archive/2015/Conf/Jimenez1880.pdf
[1] http://www.cs.ox.ac.uk/people/julian.hedges/papers/Thesis.pd...