Wei_Dai comments on Fair Division of Black-Hole Negentropy: an Introduction to Cooperative Game Theory - Less Wrong

26 Post author: Wei_Dai 16 July 2009 04:17AM

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Comment author: cousin_it 17 July 2009 01:28:37PM *  0 points [-]

I'm gonna ramble a bit to clear up my own thoughts, apologies if this sounds obvious...

The Shapley value is the only value operator defined on all coalitional games that satisfies some intuitive axioms. (They're all very natural and don't include "Equal Impact".) Proportional allocation isn't defined on all coalitional games, only a subset of them where you arbitrarily chose some numbers as players' "contributions". (A general coalitional game doesn't come with that structure - it's just a set of 2^N numbers that specifies the payoff for each possible cooperating clique.) After this arbitrary step, proportional allocation does seem to satisfy the same natural axioms that the Shapley value does. But you can't expand it to all coalitional games coherently, because otherwise the intuitive axioms would force your "contribution" values to become Shapley values.

In general, I see no reason to use proportional allocation over the Shapley value. If each player suffers a loss of utility proportional to their individual contribution, or any other side effect with an arbitrary cost function, just include it in the SV calculation.

Comment author: Wei_Dai 17 July 2009 02:43:04PM 1 point [-]

According to Moulin, there are several different sets of axioms that can be used to uniquely derive the Shapley Value, and Equal Impact is among them (it can be used to derive Shapley Value by itself, if I understand correctly).

The problem with all of those sets of axioms is that each set seems to include at least one axiom that isn't completely intuitive. For example, using the terminology in the Wikipedia article, we can use Symmetry, Additivity and Null Player, and while Symmetry and Null Player seem perfectly reasonable, I'm not so sure about Additivity.