[Question] What is a decision theory as a mathematical object?

If we want a space of all decision theories, what mathematical objects does it contain? For example, if a decision theory is a function, what are its domain and codomain?

The only approach I’m familiar with is to view expected utility maximizing decision theories as ways of building counterfactuals (section 5 in the FDT paper). A decision theory could then be described as a function that takes in a state and an action and spits out a distribution over world states that result from counterfactually taking action in state .

But EDT, CDT and FDT require different amounts and kinds of structure in the description of the state they take as input (pure probability distributions, causal models and logical models respectively), so this approach only works if there is some kind of structure that is sufficient for all decision theories we might come up with at some point.