Negative sum vs zero sum (vs positive sum, in fact) depend on defining some “default state”, against which the outcome is compared. A negative sum game can become a positive sum game if you just give all the “players” a fixed bonus (ie translate the default state). Default states are somewhat tricky and often subjective to define.
Now, you said “the best states for one of the rewards are bad for the other”. “Bad” compared with what? I’m taking as a default something like “you make no effort to increase (or decrease) either reward”.
So, my informal definition of “zero sum” is “you may choose to increase either R1 or R2 (roughly) independently of each other, from a fixed budget”. Weakly positive sum would be “the more you increase R1, the easier it gets to increase R2 (and vice versa) from a fixed budget”; strongly positive sum would be “the more you increase R1, the more R2 increases (and vice versa)”.
Negative sum would be the opposite of this (“easier”->”harder” and “increases”->”decreases”).
The reason I distinguish weak and strong, is that if we add diminishing returns, this reduces the impact of weak negative sum, but can’t solve strong negative sum.
Negative sum vs zero sum (vs positive sum, in fact) depend on defining some “default state”, against which the outcome is compared. A negative sum game can become a positive sum game if you just give all the “players” a fixed bonus (ie translate the default state). Default states are somewhat tricky and often subjective to define.
Now, you said “the best states for one of the rewards are bad for the other”. “Bad” compared with what? I’m taking as a default something like “you make no effort to increase (or decrease) either reward”.
So, my informal definition of “zero sum” is “you may choose to increase either R1 or R2 (roughly) independently of each other, from a fixed budget”. Weakly positive sum would be “the more you increase R1, the easier it gets to increase R2 (and vice versa) from a fixed budget”; strongly positive sum would be “the more you increase R1, the more R2 increases (and vice versa)”.
Negative sum would be the opposite of this (“easier”->”harder” and “increases”->”decreases”).
The reason I distinguish weak and strong, is that if we add diminishing returns, this reduces the impact of weak negative sum, but can’t solve strong negative sum.
Does this help, or add more confusion?