Variations on the Shapley value

proved:
Theorem 14. There exists a unique probabilistic value ϕ π which is π-symmetric
and π-efficient, called the π-value, and it is given by:
(8.1) ϕ π v(j) = ϕv Bk (j) for j ∈ Bk.
Note that by (8.1), for i ∈ Bk, ϕ π v(i) depends only on the game v Bk . Hence,
the power of a player i ∈ Bk to negotiate with players outside Bk is not taken into
account when computing his share of v(Bk).
Myerson (1977) studied a value, called the fair allocation rule, where the social
structure is given by a graph with vertex set N. An arc (i, j) in graph g can be
thought of as signifying some social relation, say a neighborhood or a channel of
communication, between players i and j. For each coalition S, we denote by g|S
the restriction of the graph g to the vertices in S. By π g|S
of S into the connected components of g|S.
we denote the partition
The axiomatization of the fair allocation rule is accomplished by fixing the game
v, and varying the social structure. A function φ : GR → A, where GR is the set of
all graphs on N, is a fair allocation rule if it satisfies two properties: it is relatively
efficient, by which we mean that φg(S) = v(S) for every connected component of g;
and it is fair in the sense that for any two graphs g and h such that h = g ∪{(i, j)},
φh(i) − φg(i) = φh(j) − φg(j).
Thus, φ is fair if, by adding an arc to a graph, the players forming this arc equally
gain or lose.
Myerson (1977) proved:
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