Your argument would at most prove that you can't have a positive sum. But it doesn't say anything about not having a negative sum.

We CAN needlessly increase entropy without that benefiting anyone. It's easy.

The sum doesn't have to be zero.

And, of course, once you agree that the sum can go negative. Then we can work on trying to avoid that. Game theory doesn't actually care all that much about any finite offset. Whether the maximum we can reach is 0 or ten quadrillion, it's all the same to the theory.