Who moves first, and who moves last? In a game between the monetary authority and the fiscal authority, the order of moves matters. Just like in Stackelberg duopoly, where the effect of an increase in one firm's quantity will depend on whether the other firm is the leader and has already chosen its quantity, or is the follower and will adjust its quantity in response.
But there is also a Short Run vs Long Run distinction between the order of moves. In the Short Run game, stabilisation of Aggregate Demand is what's at stake. In the Long Run game, satisfying the Long Run Government Budget Constraint is what's at stake.
When I examine my own implicit assumptions, this is what I normally assume:
Short Run. The fiscal authority moves first, and the monetary authority moves last. I think this assumption is right for Canada. If the government changes fiscal policy, it knows the Bank of Canada will adjust monetary policy in response to keep inflation at the 2% target. We saw this happen in 1995.
Long Run. The monetary authority moves first, and the fiscal authority moves last. I think this assumption is right for Canada. If the Bank of Canada adjusts the nominal interest rate or monetary base to keep inflation at the 2% target, or if it changes that monetary policy target, it knows that this will affect the flow of seigniorage profits it pays to the government, and it knows that the government will sooner or later adjust taxes or expenditure so that the Long Run Government Budget Constraint remains satisfied. We saw this happen in 1995.
When I examine my assumptions, they do look weird. Because I have the exact opposite assumption about the order of moves in the Short Run and in the Long Run. But I think my weird assumptions are right, for Canada. If I were talking about a different country, like Zimbabwe in the hyperinflation, where central bank independence was just a fiction, I would need to change my assumptions.
You might disagree with my assumptions about the order of moves. But you might nevertheless agree with me that the order of moves matters, and that different assumptions about the order of moves is at the root of the disagreement between monetarists and fiscalists. And by "fiscalists" I here include the Fiscal Theory of the Price Level.