The gain of a particular player’s individual, single move is the number of points by which it shifts the count in their favour. In combinatorial game theory, the “incentive” is a more general form of gain, expressed as a difference game.
PJT: Robert, it sounds as though you intend the gain to be a expressed as an abstract game, but the page count seems to suggest that that is calculated by averaging, and thus yields a number. It sounds as though we have a number of different usages to reconcile. Can you clarify?
RobertJasiek: A gain is a number, although I am not sure about CGT whether it might be a game there.
PJT: If it is a number, why do you prefer to call it an amount? (“Number of points” sounds a lot more natural in English.) If it can have an infinitesimal component (in CGT), then surely it must be a game there!
RobertJasiek: number of points is also fine.