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I am beginning to acquaint myself with the fair division literature. So far, I've always encountered the assumption that the value functions $V_i$ are normalized, such that $V_i([0,1])=1$.

But it seems like there are natural cases where we may want to relax this assumption, such as when the good in question, in its entirety, is more valuable to some of the players. In the extreme case, there is even an individual $i$ such that $V_i([0,1])=0$ ("doesn't want cake").

I am wondering about what happens to the fairness definitions (proportionality, envy-freeness, equitable) when the value functions are not assumed to be normalized and it could be that $V_i([0,1]) > V_j([0,1])$. Clearly they need some adjustments (with the possible exception of envy-freeness, which could still make sense as-is). I would greatly appreciate any references where this has been studied / discussed as well as any additional input. Thanks!

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