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Ramoni Lasisi and Vicki Allan Utah State University. A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES. by. A Weighted Voting Game (WVG). Consists of a set of agents Each agent has a weight A game has a quota A coalition wins if - PowerPoint PPT Presentation
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A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES
Ramoni Lasisi and Vicki Allan
Utah State University
by
A Weighted Voting Game (WVG) Consists of a set of agents
Each agent has a weight
A game has a quota
A coalition wins if
In a WVG, the value of a coalition is either (i.e., ) or (i.e., )
Notation for a WVG :
WVG Example Consider a WVG of three agents with quota =5
3 3 2Weight
Any two agents form a winning coalition. We attemptto assign power based on their ability to contribute to a winning
coalition
Annexation and Merging
Annexation Merging
C
Annexation and Merging
Annexation Merging
The focus of this talk:To what extent or by how much can agents improve their
power via annexation or merging?
Power Indices
The ability to influence or affect the outcomes of decision-making processes
Voting power is NOT proportional to voting weight
Measure the fraction of the power attributed to each voter
Two most popular power indices are Shapley-Shubik index Banzhaf index
A
B
C
Quota
Shapley-Shubik Power Index
Looks at value added. What do I add to the existing group?
Consider the group being formed one at a time.
[4,2,3: 6]
A
B
C
Quota
How important is each voter?
AA
AA
AB
B
BB
B
CC
CC
C
A claims 2/3 of the power, but look at what happens when
the quota changes.
A
C
B
AB
QuotaBanzhaf Power Index
AA
B
CC
There are three winning coalitions : {4,2}, {4,2,3},{4,3}-A is critical three times-B is critical once-C is critical once5 total swing votesA = 3/(3 + 1 + 1) = 3/5; B = C = 1/(3 + 1 + 1) = 1/5
[4,2,3:6]
Banzhaf Power Distribution
ABC
Consider annexing and merging
We expect annexing to be better
as you don’t have to split the power With merging, we must gain
more power than is already in the agents individually.
Consider Shapley Shubik1
2
3
4
5
6
Yellow 2 3 4 4 3 2
Blue 2 3 1 1 3 2
White 2 0 1 1 0 2
Consider merging yellow/white To understand effect, remove all
permutations where yellow and white are not together
1 x
2
3 x
4
5
6
Remove permutations that are redundant
1 x
2
3 x
4 x
5
6 x
Merge 1/2 1/2 1 1 1/2 1/2
Orig 2/3 1/2 5/6 5/6 1/2 2/3
Annex 1/2 1/2 1 1 1/2 1/2
Orig 1/3 1/2 2/3 2/3 1/2 1/3
Merging can be harmful. Annexing cannot.
[6, 5, 1, 1, 1, 1, 1;11] Consider player A (=6) as the annexer. We expect annexing to be non-harmful,
as agent gets bigger without having to share the power.
Bloc paradox Example from Aziz, Bachrach, Elkind, &
Paterson
Consider Banzhaf power index with annexing
Original GameShow onlyWinning coalitions
A = critical 33B = critical 31C = critical 1D = critical 1E = critical 1F = critical 1G = critical 1
1 A B C D E F G
2 A B C D E F G
3 A B C D E F G
4 A B C D E F G
5 A B C D E F G
6 A B C D E F G
7 A B C D E F G
8 A B C D E F G
9 A B C D E F G
10 A B C D E F G
11 A B C D E F G
12 A B C D E F G
13 A B C D E F G
14 A B C D E F G
15 A B C D E F G
16 A B C D E F G
17 A B C D E F G
18 A B C D E F G
19 A B C D E F G
20 A B C D E F G
21 A B C D E F G
22 A B C D E F G
23 A B C D E F G
24 A B C D E F G
25 A B C D E F G
26 A B C D E F G
27 A B C D E F G
28 A B C D E F G
29 A B C D E F G
30 A B C D E F G
31 A B C D E F G
32 A B C D E F G
33 A B C D E F G
Power A =33/(33+31+5)= .47826
Paradox Total number of winning coalitions shrinks as
we can’t have cases where the members of bloc are not together.
If agent A was critical before, since A got bigger, it is still critical.
If A was not critical before, it MAY be critical now.
BUT as we delete cases, both numerator and denominator are changing
Surprisingly, bigger is not always better
num den
A Org C D E F G x 1 2
A B C D E F G x 1 2
A B C D E F G x 1 2
A B C D E F G x 1 2
A B C D E F G x 1 2
A B C D E F G
A B C D E F G x 1 2
A B C D E F G x 1 2
A B C D E F G x 1 2
A B C D E F G
A B C D E F G x 1 2
A B C D E F G x 1 2
A B C D E F G
A B C D E F G x 1 2
A B C D E F G
A B C D E F G
A B C D E F G x 1 2
A B C D E F G x 1 2
A B C D E F G
A B C D E F G x 1 2
A B C D E F G
A B C D E F G
A B C D E F G x 1 2
A B C D E F G
A B C D E F G
A B C D E F G
A B C D E F G
A B C D E F G x 1 2
A B C D E F G
A B C D E F G
A B C D E F G
A B C D E F G
A B C D E F G 1 1
n total agentsd in [1,n-1]1/d0/d
In this example, we only see cases of1/21/1
In EVERY line youeliminate, SOMETHINGwas critical!
In cases you do NOT eliminate, you could havereduced the total number
So what is happening? Let k=1Consider all original winning coalitions.Since all coalitions are considered originally, there are
no additional winning coalitions created.The original set of coalitions to too large. Remove any
winning coalitions that do not include the bloc.Notice:If both of the merged agents were critical, only one is
critical (decreasing numerator/denominator)If only one was in the block, you could remove many
critical agents from the total count of critical agents.If neither of the agents was critical, the bloc could be (increasing numerator/denominator)
Original GameShow onlyWinning coalitions
A = critical 17B = critical 15C = critical 1D = critical 1E = critical 1F = critical 1
1 A B C D E F
2 A B C D E F
3 A B C D E F
4 A B C D E F
5 A B C D E F
6 A B C D E F
7 A B C D E F
8 A B C D E F
9 A B C D E F
10 A B C D E F
11 A B C D E F
12 A B C D E F
13 A B C D E F
14 A B C D E F
15 A B C D E F
16 A B C D E F
17 A B C D E F
Power A =17/(17+15+4)= .47222
Suppose my original ratio is 1/3
Suppose my decreasing ratio is ½.I lose
Suppose my decreasing ratio is 0/2.I improve
Suppose my increasing ratio is 1/1.I improve
Win/Lose depends on the relationship between the original ratio and the new ratioand whether you are increasing or decreasing by that ratio.
Pseudo-polynomial Manipulation Algorithms
Merging
The NAÏVE approach checks all subsets of agents to find the best merge – EXPONENTIAL!
. . . Our idea sacrifices optimality for “good”
mergeWe limit the size of the coalition to constant using the following assumptions: Manipulators prefer smaller-sized coalitions – easier to form
and manage Intra-coalition coordination, communication, other overheads
increase with coalition size
1 2 n
Pseudo-polynomial Manipulation Algorithms…
Merging Note that computing Shapley-Shubik
and Banzhaf Index is NP-Hard We need to search only coalitions for
good merge By considering the possibilities in a
reasonable order, we can often prune less likely candidates.
Is that all?NO!
The problem remains NP-hard even with limitation on coalition size
Also, coalitions may be large to search in practice
So, we employ informed heuristic search strategy to improve the search.
Experimental Results-Merging
[1,10] [1,20] [1,30] [1,40] [1,50]0.80.91.01.11.21.31.41.51.61.71.81.9
Manipulation via merging with n = 10 and k = 5A
vera
ge fa
ctor
of i
ncre
men
t
[1,10] [1,20] [1,30] [1,40] [1,50]0.80.91.01.11.21.31.41.51.61.71.81.9
Manipulation via merging with n = 20 and k = 5
The distribution of agents' weights in WVGs
Aver
age
fact
or o
f inc
rem
ent
Experimental Results-Annexation
[1,10] [1,20] [1,30] [1,40] [1,50]0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
Manipulation via annexation with n = 10 and k = 5
Aver
age
fact
or o
f inc
rem
ent
[1,10] [1,20] [1,30] [1,40] [1,50]0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
Manipulation via annexation with n = 20 and k = 5
The distribution of agents' weights in WVGs
Aver
age
fact
or o
f inc
rem
ent
Conclusions We present two search-based Pseudo-
polynomial manipulation algorithms We complement the algorithms with
informed heuristic search strategies to improve performance
Our manipulation algorithm for annexation improves annexer’s benefit by more than
Our manipulation algorithm for merging improves manipulators’ benefits between to
Thanks
Experimental Results-Merging
[1,10] [1,20] [1,30] [1,40] [1,50]0.8
1.0
1.2
1.4
1.6
1.8
Manipulation via merging with n = 10 and k = 5
The distribution of agents' weights in WVGs
Ave
rage
fact
or o
f in-
crem
ent
[1,10] [1,20] [1,30] [1,40] [1,50]0.81.01.21.41.61.8
Manipulation via merging with n = 10 and k = 10
The distribution of agents' weights in WVGs
Aver
age
fact
or o
f in-
crem
ent
[1,10] [1,20] [1,30] [1,40] [1,50]0.81.01.21.41.61.8
Manipulation via merging with n = 20 and k = 5
The distribution of agents' weights in WVGs
Aver
age
fact
or o
f in-
crem
ent
[1,10] [1,20] [1,30] [1,40] [1,50]0.81.01.21.41.61.8
Manipulation via merging with n = 20 and k = 10
The distribution of agents' weights in WVGs
Aver
age
fact
or o
f in-
crem
ent
(a) (b)
(c) (d)
Experimental Results-Annexation
(a) (b)
(c) (d)
[1,10] [1,20] [1,30] [1,40] [1,50]0.0
20.040.060.080.0
100.0120.0140.0
Manipulation via annexation with n = 10 and k = 5
The distribution of agents' weights in WVGs
Aver
age
fact
or o
f in-
crem
ent
[1,10] [1,20] [1,30] [1,40] [1,50]0.0
20.040.060.080.0
100.0120.0140.0
Manipulation via annexation with n = 10 and k = 10
The distribution of agents' weights in WVGs
Aver
age
fact
or o
f in-
crem
ent
[1,10] [1,20] [1,30] [1,40] [1,50]0.0
20.040.060.080.0
100.0120.0140.0
Manipulation via annexation with n = 20 and k = 5
The distribution of agents' weights in WVGs
Aver
age
fact
or o
f in-
crem
ent
[1,10] [1,20] [1,30] [1,40] [1,50]0.0
40.080.0
120.0160.0200.0240.0280.0
Manipulation via annexation with n = 20 and k = 10
The distribution of agents' weights in WVGs
Aver
age
fact
or o
f in-
crem
ent