Information Technology Reference
In-Depth Information
then the merging is
disadvantageous
. The agents in a bloc formed by merging are as-
sumed to be working cooperatively and have transferable utility. For the sake of sim-
plicity in our analysis, we also refer to the factor of increment as power gain or benefit.
Example 2 illustrates manipulation by merging in a weighted voting game.
Example 2.
Merging in Weighted Voting Game.
Let
G
=[12
,
16
,
18
,
19
,
23
,
26
,
33
,
40
,
45
;155]
be a WVG. The last four agents in the
game are designated as would-be manipulators. The Banzhaf indices of these agents
are:
β
6
(
G
)=0
.
116
,
β
7
(
G
)=0
.
142
,
β
8
(
G
)=0
.
174
,and
β
9
(
G
)=0
.
200
.So,
j
=6
β
j
(
G
)=0
.
632
. Suppose the agents decide to merge their weights. A bloc of
weight
144
is formed in the new game
G
=[12
,
16
,
18
,
19
,
23
,
144
;155]
. The Banzhaf
power index of the bloc
β
6
(
G
)=0
.
861
>
0
.
632
. The manipulators gain from the
merging and increase their power indices by a factor of
0
.
861
0
.
632
=1
.
36
.
3
Visual Description of Manipulation by Annexation and Merging
in Weighted Voting Games
We provide visual description of manipulation by annexation and merging in WVGs to
further explain the intricases of what goes on during manipulation. We use the Shapley-
Shubik power index for this illustration. Consider a WVG of three agents denoted by
the following patterns: Agent 1 ( ), Agent 2 ( ), and Agent 3 ( ). The weight of
each agent in the game is indicated by the associated length of the pattern. A box in
the pattern corresponds to a unit weight. Suppose all permutations (or ordering) of the
three agents are given as shown in Figure 1 where we vary the values of the quota of
the game from
q
=1
to
q
=6
. The Shapley-Shubik indices of the three agents are also
computed from the figure and shown in the associated table of the figure. These power
indices for the agents in the game correspond to using various values of the quota for
the same weights of the agents in the game.
Consider the case of manipulation by merging where Agent 1 and Agent 3 merge
their weights to form a new agent, say Agent X. In this situation, Agent 1 and Agent
3 cease to exist since they have been assimilated by Agent X. Thus, we have only two
agents (Agent X and Agent 2) in the altered WVG. Figure 2 shows the results of the
merging between Agent 1 and Agent 3. Consider the cases when the quota of the game
is
1
or
6
, the power of the assimilated agents for Agent X from Figure 1 shows that
Agent
1
and Agent
3
each has a power of
3
for a total power of
3
. Whereas, the power
of Agent X which assimilates these two agents in the two cases is each
2
<
3
.Onthe
other hand, the power of the manipulators stays the same for the cases where the quota
is either
2
or
5
. Specifically, the sum of the powers of Agent
1
and Agent
3
is
1
1
2
for
these cases. This is also true of Agent X for these cases. Finally, for the cases where the
quota of the game is either
3
of
4
, the power of Agent X is
1
which is greater than
5
6
,
the sum of the powers of Agent
1
and Agent
3
in the original game. Note the difficulty
of predicting what will happen when manipulators engage in merging.
Now, suppose Agent
1
annexes Agent
3
instead of the merging between the two
agents as described earlier, then Agent
3
ceases to exist as shown in Figure 3. In this
case, Agent
1
's power does not decrease. Felsenthal and Machover [12] have already
