Information Technology Reference
In-Depth Information
k=0.1
k=0.5
k=0.9
x 10
−4
9
8
7
6
5
4
3
50
40
0.025
0.02
0.015
30
0.01
0.005
m
Shapley value
Fig. 7.
A small player's Shapley value and uncertainty for a varying number of players
Shapley value and its uncertainty is the same as that for the left half of the curve of
Figure 1 (i.e., for
ϕ<
0
.
5).
7
More Than Two Player Types
Consider a voting game with more than two types of players. Let
w
i
denote the weight of
player
i
. Thus, for
m
players and for quota
q
thegameisoftheform
.
Consider a player population in which each individual player's weight has a
standard
normal distribution
3
-
q
;
w
1
,w
2
,...,w
m
N
(0
,
1). Since this distribution allows negative weights, we trans-
form this to
(4
,
1) in order to get positive weights. We know from Definition 1, that
the Shapley value for a player is the expectation (
E
) of its marginal contribution to a
coalition that is chosen randomly. Thus, in order to determine the Shapley value for the
above population of players (i.e.,
N
(4
,
1)), we use the following rule from Sampling
Theory (see [2] p417) that holds good for a normal distribution.
From a normal distribution (with mean
μ
and variance
ν
), if a sample of size
m
is
drawn, then the sum of the weights of all
m
players in the sample has the distribu-
tion
N
(4
,
1)) we defined above, the sum of
the weights of the players in a random sample of size
m
is given by the distribution
N
N
(
mμ, mν
). Thus, for the distribution (
N
(4
m, m
). We use this rule to determine the Shapley value as follows.
A Randomised Method for the Shapley Value.
For player
i
with weight
w
i
,let
ϕ
i
de-
note the Shapley value and
β
i
its uncertainty. Let
X
denote the size of a large random
sample drawn from a population in which individual player weights have the distribu-
tion
(4
,
1). The marginal contribution of player
i
to this random sample is one, if the
total weight of the
X
players in the sample is greater than or equal to
q
N
w
i
but less
than
q
. Otherwise, its marginal contribution is zero. Thus, the expected marginal con-
tribution of player
i
(denoted
Δ
i
) to the sample coalition is the area under the curve
−
3
Note that in Section 6 when we dealt with two player types, there was no restriction on how the
population was distributed. But for more than two player types, we assume that the population
has a normal distribution. Thus, while the results of Section 6 are valid for two player types
with any population distribution, the results of this section are valid for more than two player
types that have a normal distribution.
