What's the name for this type of non zero sum game? Many times you hear Economics is a "non zero sum game" implying that anything that is earnt by one person or group isnt lost by other. This might not be the definition of non zero sum game, but that's how it is used many times. What I believe it happens it's part of what is earnt by a person or group is lost by other and other part isnt. 

In the other hand, you have "zero sum game", where everything earnt by someone is lost by other. 

Is there a name for a game where part of what a person or group earns is part of what other person or group loses, and other part of what this first person or group earns isnt part of what other person or group loses?
As far as I know the definition is yet to be extended beyond two person games, but you seem to be looking for some notion of antagonism.

A two player game is antagonistic if given any two outcomes A and B, if player one prefers outcome A then and only then player two prefers outcome B.

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