Half a Byte

Four-dimensional addendum to rational sharing.

Intuitively, fair allocations should be in the middle of the core, and for convex games, that is precisely where the Shapley value is. If a game is non-convex but is at least superadditive, the Shapley value will still be individually rational; however, failing even that, there are situations where both fairness and rationality are not achievable at the same time; some friends are just not worth keeping.

The interactive example above plots the normalized Banzhaf value (black) and the Shapley value (white), as it is rather difficult to appreciate the extra dimension of the standard Banzhaf value on a flat screen. For the same reason, please also keep in mind that angles do not look correct.

Other solution concepts.

Some friends might be pretty disappointed with their fair slice. A concept that minimizes the worst-case disappointment is the Nucleolus. If instead of sharing, we wish to measure the value of a person in a group, there is the Banzhaf value – a concrete use-case that measures the voting power of individuals is known as the Banzhaf index. Some of the concepts mentioned here may not have a solution in every game, but the least-core and the pre-nucleolus will, and the list does not end there.