"Very well, then--what are they functions of?"
Indeed, a classic problem. I am unlikely to be the bearer of the long-sought answer. My humble opinion is this: When faced with the possibility of ordering that pizza or not ordering it, it is entirely possible (quite likely, really) that both alternatives are the product of pre-existing inputs into the "system" that is my brain. The decision can go either way, and neither outcome is uncaused. That seems to take care of part of your question (the "no inputs" part).
As for the actual decision ("pizza or no pizza") ... there's the deepest question. It may be undetermined, as with so many QM experiments. But because we are conscious of the functioning of our brains, we "watch" the decision being made, and we have the feeling that we actually made the decision -- when in reality (whatever that means) all we did was go along for the ride.
I don't believe this. I think we really do make decisions. But I don't know how to demonstrate it. The Objectivists take free will as an axiom, so they don't need to worry about proving it. Perhaps they're on the right track.
This is not an example of a decision. If you have to ask, the answer is "pizza."
The solution to your conundrum is found by assuming that the PIZZA makes the decision, not you. The pizza has free will; it decides whether or not it wants to be eaten by you. You are just along for the ride.
;-)