An Analysis of the Shapley Value and Its Uncertainty for the Voting Game - Agent-Mediated Electronic Commerce - page 90

for our voting game, a player's marginal contribution to a coalition has only two
possible values: zero or one.
Consider the large party. This party can join a coalition as the i th member where
i satisfies 1 ( m +1). However, the marginal contribution of the large

party is one if it joins a coalition as the i th member where i satisfies the condition
( q

≤ i ≤

q . In all the remaining cases, its marginal contribution is zero.
Thus, out of the total ( m +1)possible cases, its marginal contribution is one in j
cases. Hence, the Shapley value of the large party is:

− j +1) ≤ i ≤

ϕ l = j/ ( m +1) (2)

Consider a small player. For a small player, the marginal contribution is one in
two cases. First, if it joins a coalition (that already has the large party in it) as the
( q j +1)th member. Out of the ( m +1)!possible permutations, the number of

permutations that satisfy this condition is ( q

−

− j )( m − 1)!. Second, if it joins a

coalition (consisting of q 1 small players) as the q th member. The number of

permutations that satisfy this condition is ( m

−

− q +1)( m − 1)!. Hence, the Shapley

value of each small party is:

ϕ s =( m − j +1) /m ( m +1) (3)

Using Definition 2, we get the uncertainty for the large party as:

ϕ l ) ϕ l 2 + ϕ l (1 ϕ l ) 2 β l =(1 − − (4)

For each small party, the uncertainty is:

ϕ s ) ϕ s 2 + ϕ s (1 ϕ s ) 2 β s =(1 − − (5)

2. Consider q>m . As before, the quota satisfies the relation j +1 ≤ q ≤ m + j − 1.

Also, 2 1). Consider the large party. As before, this party can join

a coalition as the i th member where 1

≤ j ≤ ( q −

( m +1). However, its marginal
contribution is one only if it joins as the i th member where ( q

≤ i ≤

q .
Thus, out of all ( m +1)possible cases, its marginal contribution is one in j cases.
Hence the Shapley value of the large party is:

− j +1) ≤ i ≤

ϕ l = j/ ( m +1) (6)
Consider a small player. Since q>m , a small player's marginal contribution is one
in only one case: if it joins a coalition (that already has the large party in it) as the
( q

j +1)th member. Out of the ( m +1)!possible permutations, the number of
permutations that satisfy this condition is ( q

−

− j )( m − 1)!. Hence the Shapley value

of each small party is:

ϕ s =( q − j ) /m ( m +1)

(7)

We get the uncertainty for the large party as:

ϕ l ) ϕ l 2 + ϕ l (1 ϕ l ) 2 β l =(1 − −

(8)

For each small party, the uncertainty is:
β s =(1 − ϕ s ) ϕ s 2 + ϕ s (1 − ϕ s ) 2

(9)