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−4
x 10
k=0.2
k=0.5
k=0.9
1
0.8
0.6
0.4
0.2
0
1000
0.002
0.001
500
0
0
Quota
Shapley value
Fig. 5.
A small player's Shapley value and uncertainty for a varying quota
k=0.1
k=0.5
k=0.9
0.025
0.02
0.015
0.01
50
0.15
40
0.1
0.05
30
0
m
Shapley value
Fig. 6.
A large player's Shapley value and uncertainty for a varying number of players
Although a player's Shapley value and its uncertainty vary with
j
,
k
,
q
,and
m
,the
uncertainty was always found to increase with Shapley value. More specifically, for the
constraints defined above, if the number of players is large (i.e.,
m
20)andthere
are multiple large and multiple small players, we found the following relation. For each
player, the uncertainty increases as its Shapley value increases (see Figures 2 to 7).
To begin, consider Figure 2. For
m
= 200 and
q
= 200, this is a plot of a large
player's Shapley value and uncertainty for differing weights (i.e., different values for
j
). For each value of
j
, the figure shows the Shapley value and uncertainty for all
k
between 0
.
1 and 0
.
9. Likewise, Figures 3 is a plot for each small player.
For
m
= 200 and
j
=5, Figure 4 is a plot for a large player's Shapley value and
uncertainty for a varying quota. For each quota (i.e.,
q
), the figure shows the Shapley
value and uncertainty for all
k
between 0
.
1 and 0
.
9. Figure 5 is a plot for each small
player.
Consider Figure 6. For
j
=5and
q
=25, this is a plot of a large player's Shapley
value and uncertainty for a varying number of players. For each
m
, the figure shows the
Shapley value and uncertainty for
k
between 0
.
1 and 0
.
9. Likewise, Figures 7 is a plot
for each small player.
Thus, for two player types and variations of
j
,
k
,
q
,and
m
that satisfy constraints
C
1
and
C
2
, the uncertainty for each player (large or small) increases as its Shapley value
increases. It is worth noting that in all the above figures, the number of players is at
least 20, and there is more than one player of each type (i.e., 0
.
1
≥
0
.
9). For such
games, the Shapley value of each player is less than 0
.
5. Thus, the relation between the
≤
k
≤
