People accept the $10/$1000 situation all the time: it's often called "working on commission". Maybe I need to refresh my game theory, but I wouldn't call that a negative sum game, but a positive sum (+$1010) one, and refusing still leaves everyone in the status quo (+$0/+$0).
My recollection of a negative sum game is that the overall outcome is negative. For instance, paying $10 (to get access to the car park) to cause $1000 worth of damage (slashing tyres).
(I'm having difficulty imagining someone spending $1000 to make someone else worse off by $10, but can easily imagine a vindictive someone spending $1000 to make a much poorer someone else worse off by only $500.)
My recollection of a negative sum game is that the overall outcome is negative. For instance, paying $10 (to get access to the car park) to cause $1000 worth of damage (slashing tyres).
(I'm having difficulty imagining someone spending $1000 to make someone else worse off by $10, but can easily imagine a vindictive someone spending $1000 to make a much poorer someone else worse off by only $500.)