Annexations and Merging in Weighted Voting Games:The Extent of Susceptibility of Power Indices

by Ramoni Lasisi

Vicki Allan

Agenda

Weighted Voting Games(WVGs)

Power Indices : Shapley-Shubik, Banzhaf, & Deegan-Packel

Manipulation of WVGs :Annexations & Merging

• WVGs - mathematical abstractions of voting systems. Let V be the set of voters. A weight function is defined on V, w: V Q+.

• A coalition of agents C, wins in the game if the sum of their weights meets or exceeds a threshold called the quota q.

C is also called a winning coalition.

• Representation of WVG: G = [w1,w2,…, wn; q]

Weighted Voting Games (WVGs)

Example

Banzhaf wanted to prove the Nassau County board’s voting system (based on population) was unfair– Hempstead #1: 9– Hempstead #2: 9– North Hempstead: 7– Oyster Bay: 3– Glen Cove: 1– Long Beach: 1

This is 30 total votes, and a simple majority of 16 votes was required for a measure to pass.

[9, 9, 7, 3, 1, 1 :16 ]• Look at all possible winning coalitionsAB AC BC ABC ABD ABE ABF ACD ACE ACF BCD BCE BCF

ABCD ABCE ABCF ABDE ABDF ABEF ACDE ACDF ACEF BCDE BCDF BCEF ABCDE ABCDF ABCEF ABDEF ACDEF BCDEF ABCDEF

• Determine which voters are CRITICAL (underlined) to each coalition. Critical if not have enough votes without voter.

AB AC BC ABC ABD ABE ABF ACD ACE ACF BCD BCE BCF ABCD ABCE ABCF ABDE ABDF ABEF ACDE ACDF ACEF BCDE BCDF BCEF ABCDE ABCDF ABCEF ABDEF ACDEF BCDEF ABCDEF

Look at the number of swing votes

• 48 swing votes• Determine what proportion of the swing votes are held by each

voter• All power indices are a probability. [9, 9, 7, 3, 1, 1 :16 ]

– Hempstead #1 = 16/48– Hempstead #2 = 16/48– North Hempstead = 16/48– Oyster Bay = 0/48– Glen Cove = 0/48– Long Beach = 0/48

• Banzhaf argued that a voting arrangement that gives 0% of the power to 16% of the population is unfair.

How do we evaluate the strength of agents in WVGs?

Using Power Indices

Power Indicesa fraction of the power attributed to each voter

Power:• Not proportional to voting weight• Your ability to change the outcome with your

vote• The probability of having a significant role in

determining the outcome

• There are different definitions of ‘having a significant role’

Various Power indices

Banzhaf index: The number of winning coalitions in which an agent is critical. Denoted by ßi(G).

Critical – swing agent in a winning coalition is an agent that causes the coalition to lose when removed from it.

Considers all the marginal contributions of a player to all possible coalitions, without considering the order of the players

Normalize so add to one

The power index is the portion of coalitions in which the agent is critical

A

B

Quota

Banzhaf Power Index

A

A

B

C

C

[4,2,3:6]

What is critical?

A

B

Quota

Banzhaf Power Index

A

A

B

C

C

There are three winning coalitions- A is critical in all three- B is critical in only one- C is critical in only one 5 total swing votes

A = 3/(3 + 1 + 1); B = C = 1/(3 + 1 + 1)

[4,2,3:6]

Banzhaf Power Distribution

AB

C

[4,2,3:6]

Various Power indices

Shapley-Shubik index: Value added. The number of permutations of the set of agents for which an agent is critical. Denoted by φi(G).

– What would you be without me?– Well, what would YOU be without me?

– Solution – consider all possible orders.

A

B

C

Quota

How important is each voter?

Shapley ShubikLook 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?

A

A

A

A

A

B

B

B

B

B

C

C

C

C

C

A claims 2/3 of the power in this scheme

A

C

B

A

B

C

Quota

If quota changes, power shift.

A

A

A

A

A

B

B

B

B

B

C

C

C

C

C

Now, power is equal!

A

B

C

Various Power indices

Deegan-Packel index: Depends on the number of Minimal Winning Coalitions (MWCs) . Within each winning coalition, the credit is shared equally (as if any is removed, the coalition fails)

Thus, the size of each of the MWCs that include the agent is considered.

Denoted by γi(G).

A

Quota

Deegan-Packel Power Index

A

B

C

There are two minimal winning coalitions [4,2,3:6]-A is in both-B is in only one-C is in only one

A = (½)*(½ + ½) = ½;B = C = (½) * (½)=1/4

AB

C

Deegan-Packel Power Distribution

Quota

Deegan-Packel Another example

There are three minimal winning coalitions [4,2,3,1:4]- A is in one (of size 1)- B is in only one (of size two)- C is in two(of size two)- D is in one (of size two)

A = (1/3)*(1)= 1/3;B =(1/3) * (½)=1/6C=(1/3)* (½ + ½) = 1/3D = (1/3) *(1/2) = 1/6

CB

A

C D

Manipulation of WVGs

• Annexation Merging

• Susceptibility of Power Index to Manipulation

Let Φi(G) be the power of i in G. If there exist an altered game G’ such that Φi(G’)> Φi(G).

• Factor of Increment

Φi(G’) /Φi(G)

• Domination of Manipulability Let Φ and θ be two power indices. If the increment of Φ w.r.t G

and G’ is greater than θ. Then Φ dominates θ, i.e., more susceptible.

Important Terms in the Paper

An Example

• Let G =[5,8,3,3,4,2,4;18] be a WVG of seven agents with agent 1 an annexer. The Deegan-Packel index of the agent 1 in G is 0.1722.

• Annexation implies that the agent combines with another agent who relinquishes its claim on the power.

• Suppose the annexer annexes another agent with weight 4. We have a new game G’=[9,8,3,3,2,4;18].

• The new Deegan-Packel index of this agent is 0.2604 > 0.1722, and the factor of increment is 1.51.

Motivation for using Power Indices

• WVGs have many applications: Economics, Political Science, Neuro Science, Distributed Systems, Multi agent Systems.

• It is important that games adopt power index which motivates truth telling in order to eliminate the appeal of participating in manipulations.

• When truth telling is dominant, it provides some assurance of fairness in the games – to the degree that the original power index is fair.

State of the Art

WVGs Manipulations

False Name Manipulationsbreak into pieces

Annexations and Merging

Bachrach and Elkind 2008

Azeez and Paterson 2009

Lasisi and Allan 2010

Machover and Felsenthal 2002

Azeez and Paterson 2009

About the Paper

• We consider agents engaging in Annexation and Merging in WVGs.

• We evaluate agents’ power using Shapley-Shubik, Banzhaf, and Deegan-Packel Indices.

• A WVG in which there is a single winning coalition and every agent is critical to the coalition is a Unanimity WVG. All voters must be present to form a winning coalition.

• We consider Unanimity and Non Unanimity WVGs.

Original Contributions-Unanimity WVGs

Annexation increases the power of other agents (that are not annexed) by the same factor of increment as the annexer. In other words, if any agent annexes, all benefit as they are all equally critical and there are fewer agents to split the power. The annexer also incurs annexation costs, thus reducing its benefits.

All of the indices are affected by annexation. However, the manipulability of any one type of power index does not dominate the manipulability of other types of indices.

Given that there are n agents in the original game, the upper bound on the extent to which a strategic agent may gain is at most n times the power of the agent in the original game.

Experiment

• We have 15 agents.• We annex anywhere from 1-10 other agents

(termed the bloc size). The block size is randomly generated as are the agents which are annexed.

• The weight of the annexer is the sum of its original weight and the weight of the annexed agents.

Original Contributions– Non Unanimity WVGs Annexation

0 1 2 3 4 5 6 7 8 9 10 110.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0Average Factor of Increment vs Bloc Size

Deegan-Packel Banzhaf Shapely-Shubik

Bloc Size

Avera

ge F

act

or

of

Incr

em

ent

Figure: Susceptibility to Manipulation via Annexation

Original Contributions– Non Unanimity WVGs Merging

Figure: Susceptibility to Manipulation via Merging

1 2 3 4 5 6 7 8 9 10 110.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2 Average Factor of Increment vs Bloc Size

Deegan-Packel Banzhaf Shapely-Shubik

Bloc Size

Avera

ge F

act

or

of

Incr

em

ent

Interpretation

• Only Shapley-Shubik appears to be susceptible to manipulation via merging.

• There appears to be no correlation between block size and factor of increment, so would be manipulators would need to use trial and error.

• The relative susceptibility between the indices is clear.

• The highest average factor of increment is less than 1.

Original Contribution– Non Unanimity WVGs Merging

Figure: Percentage of Advantageous and Disadvantageous Games for Manipulation via Merging

Deegan-Packel Index Banzhaf Index Shapley-Shubik Index

0%

20%

40%

60%

80%

100%

Advantageous Games Disadvantageous Games

Conclusions

• The three indices show various degrees of susceptibility to manipulations via annexation and merging with Shapley-Shubik being the most susceptible for both annexation and Merging

• For unanimity WVGs of n agents, the upper bound on the extent to which a strategic agent may gain is n times its power in the original game.

• For non unanimity WVGs, the games are less vulnerable to manipulation via merging, while they are extremely vulnerable to manipulation via annexation.

• Finally, in relation to Lasisi and Allan 2010, the situation where splitting by a strategy is disadvantageous corresponds to situation where it is advantageous for several strategic agents to merge.

Future Work

• We have assumed in the paper that for the case of merging in non Unanimity WVGs, agents can easily distribute their gains in a fair and stable manner.

• We plan to investigate the assumption using Game-theoretic approach if there exists such stable and fair ways of distributing the gains.