a search-based approach to annexation and merging in weighted voting games

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



Aver

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fact

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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



Ave

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fact

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[1,10] [1,20] [1,30] [1,40] [1,50]0.81.01.21.41.61.8



Aver

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crem

ent

[1,10] [1,20] [1,30] [1,40] [1,50]0.81.01.21.41.61.8



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fact

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[1,10] [1,20] [1,30] [1,40] [1,50]0.81.01.21.41.61.8



Aver

age

fact

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f in-

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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



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[1,10] [1,20] [1,30] [1,40] [1,50]0.0

20.040.060.080.0

100.0120.0140.0



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[1,10] [1,20] [1,30] [1,40] [1,50]0.0

20.040.060.080.0

100.0120.0140.0



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[1,10] [1,20] [1,30] [1,40] [1,50]0.0

40.080.0

120.0160.0200.0240.0280.0



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a search-based approach to annexation and merging in weighted voting games

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