First of all, I preemptively apologize if my question happens to be naive, I am no expert of CGT (or general game theory, for that matter).
Now the question:
**is there such a thing as the study of resource-aware combinatorial games?
By this I mean the following: there are two players, each with an assigned amount of resources (say a heap of money).
The two players play a game in the sense of CGT, but each move is preassigned a cost. A player loses iff he loses in the standard sense (gets to some terminal state) OR he/she runs out of money. Using the graph terminology, this would amount to playing on a weighted graph, the weight being the cost of that particular move**
NOTE: If I remember correctly, there has been some work on categorical CGT, and even on game semantics for resource-aware logics (Linear Logic and the like), so I would think that the scenario I informally described has been envisioned. If so, I would appreciate any pointers you may have.