Chapter 10: The Manipulability of Voting SystemsChapter 11: Weighted Voting Systems

Presented by: Katherine Goulde

Chapter 10 Outline

Introduction and example Majority Rule and Condorcet’s Method Voting Systems for 3 or more candidates

Borda Count Sequential Pairwise Voting Plurality Voting

Impossibility- The Gibbard- Satterthwaite Theorem The Chair’s Paradox

Manipulability & the Borda Count The Borda count assigns point values to the candidates

and the winner is the candidate with the most points

Voter 1 Voter 2 A B B C C A D D

Candidate A has a score of 4Candidate B has a score of 5Candidate C has a score of 3Candidate D has a score of 0

Therefore… B wins!

What if Voter 1 wants to manipulate the election???

Voter 1 Voter 2 A B B C C A D D

Voter 1 Voter 2 A B D C C A B D

Original Ballot where B wins the election

However, Voter 1 wants A to win. How can Voter 1 ensure that A wins?

In this second ballot, A has a Borda count of 4, B has 3, C has 3, and D has 2. Therefore A is the winner.

Is there any other way to obtain this result?

Unilateral Change- A change (in ballot) by a voter while every other voter keeps his or her ballot exactly as it was

- “single-voter manipulation”

A voting system is manipulable if

• there are two sequences of preference list ballots and a Voter so that

•Neither election results in a tie

•The only ballot change is by the Voter

•The Voter prefers the outcome of the second election to that of the first election.

•Take the two-candidate case with majority rule, and recall that it is monotone

•In this instance, nonmanipulability is the same thing as monotonicity

Majority Rule and Condorcet’s Method

May’s Theorem for Manipulability: Among all two-candidate voting systems that never

result in a tie, majority rule is the only one that treats all voters equally, treats both candidates equally, and is nonmanipulable

The Nonmanipulability of Condorcet’s Method: Condorcet’s method is nonmanipulable in the sense

that a voter can never unilaterally change an election result from one candidate to another candidate that her or she prefers

Cordorcet’s Voting Paradox and Manipulability

Election 1

In this example, C is the Condorcet winner

Election 2

In this example, there is no Condorcet winner at all

Voter 1 Voter 2 Voter 3 A B C

C C A B A B

Voter 1 Voter 2 Voter 3 A B C

B C A C A B

Back to the Borda Count

The Nonmanipulability of the Borda Count with exactly 3 candidates: With exactly 3 candidates, the Borda count cannot be

manipulated in the sense of a voter unilaterally changing an election outcome from one single winner to another single winner that he prefers

Why? Imagine B is the Borda winner, but you prefer A. Consider 3

cases: A > B > C C > A > B A > C > B

The Manipulability of the Borda Count with Four or More Candidates: With four or more candidates and two or more voters, the

Borda count can be manipulated in the sense that there exists an election in which a voter can unilaterally change the election outcome from one single winner to another single winner that he prefers

We’ve covered the example of 4 candidates and 2 voters. 1) Any candidates in addition to the 4 can be placed below those on

every ballot 2) The rest of the voters can be paired off with the members of each

pair holding ballots that rank the candidates in exactly opposite orders

Sequential Pairwise Voting

Assume we are able to set the order. Choose the ‘winner’, and place the candidate last Look for the others that would beat that candidate one

on one. Using this, we can arrange for any of the candidates to

be the winner.

Voter 1 Voter 2 Voter 3A C BB A DD B CC D A

Plurality Voting and Group Manipulability

Plurality voting cannot be manipulated by a single individual. However, it is group manipulable in the sense that there are elections in which a group of voters can change their ballots so that the new winner is preferred to the old winner by everyone in the group

Real-world election: third party candidate acts as a ‘spoiler’

Impossibility: the G-S Theorem Cordorcet’s theorm:

1) Elections never result in ties 2) Satisfies the Pareto condition 3) Is nonmanipulable 4) Isn’t a dictatorship

Can we extend this so that there is always a winner?? The Gibbard- Satterthwaite Theorem: With three or

more candidates and any number of voters, there doesn’t exist a voting system that always produces a winner, never has ties, satisfies the Pareto condition, is nonmanipulable, and is not a dictatorship.

for proof click here Weaker extension: Any voting system for 3 candidates

that agrees with Condorcet’s Method whenever there is a winner is manipulable.

The Chair’s Paradox The fact that with three voters and three candidates, the voter

with tie-breaking power (the ‘chair’) can, if all 3 voters act rationally in their own self-interest, end up with her or his least-preferred candidate as the election winner

Each voter gets to vote for one of the candidates. If a candidate gets 2 or more, he or she wins. If each candidate receives one vote, then whichever person the chair voted for wins.

Each voter will choose the best strategy given what the others might do.

Chair You MeA B CB C A C A B

The Chair’s Paradox

The chair will vote for A. ‘Me’ will vote for C. ‘You’ will also vote for C.

Chair You MeA B CB C A C A B

Chapter 11 Outline

Introduction and definitions The Shapley- Shubik Power Index

3 voters, 4 voters, a committee

The Banzhaf Power Index Critical voters, winning & blocking, combinations

Comparing Voting Systems 3 voters, using minimal winning coalitions

Introduction and Definitions Weighted voting system: a voting system in which

each participant is assigned a voting weight . A quota is specified, and if the sum of the voting weights of the voters supporting a motion is at least = the quota, the motion is approved

Weight: the number of votes assigned to a voter

Quota: the minimum number of votes necessary to pass a measure in a weighted voting system

Notation for Weighted voting systems [q: W1, W2, …, Wn] where there are n voters, q quota, and

voting weights W1, W2, …, Wn

Introduction and Definitions

Dictator: a participant who can pass or block any issue even if all other voters oppose it [10: 7, 13]

Dummy Voter: a participant who has no power, is never critical, and is never the pivotal voter [8: 5, 3, 1]

Veto power: had by a voter if no issue can pass without his vote. (a voter with veto power is a one-person blocking coalition) [6: 5, 3, 1] or [8: 5, 3, 1]

Power index: a numerical measure of an individual voter’s ability to influence a decision; the individual’s voting power

The Shapley-Shubik Power Index 1954- Lloyd Shapley and Martin Shubik This index is defined in terms of permutations (a

permutation of voters in an ordering of all of the voters in a voting system)

1) Voters are ordered in accordance with their commitment to an issue (from most favorable to those most against)

2)The first voter in a permutation who, when joined by those coming before her, would have enough voting weight to win is the pivotal voter in that particular permutation.

Examples: animal rights, environmentalism

The Shapley-Shubik Power Index This power index is computed by

1) counting the number of permutations in which that voter is pivotal

2) divide this number by the total number of possible permutations

If there are n voters, the total number of possible permutations is n!

Example: [6: 5, 3, 1]. Result: A= 4/6, B,C = 1/6

How to compute the S-S Power Index If all the voters have the same voting weight, then each

has the same share of power. If all but one of two voters have equal power, we can still

easily calculate the S-S power index Example: 7-person committee with the voting system [5:

3, 1, 1, 1, 1, 1, 1] CMMMMMM MCMMMMM MMCMMMM MMMCMMM MMMMCMM MMMMMCM MMMMMMC

How to compute the S-S Power Index

The chair is the pivotal voter 3 of the 7 times, so his S-S power index is 3/7.

The remaining 4/7 is split among the six other voters (since all have the same weight), so each has (4/7)/6 = 2/21 as their S-S power index.

The Banzhaf Power Index Based on the count of coalitions in which a voter is

critical

Coalition: a set of voters who are prepared to vote for, or to oppose, a motion. Winning coalition: favors the motion & has enough votes to

pass it Blocking coalition: opposes the motion & has enough votes to

block it Losing coalition: set of voters that does not have the

votes to have its way

Critical voter: a member of the winning (or blocking) coalition whose vote is essential for the coalition to win (or block) a measure

The Banzhaf Power Index To determine the B. Power index of voter A, count all the

possible winning and blocking coalitions of which A is a member and casts a critical vote The weight of a winning coalition must be great than or equal to

q (where q is the quota) The weight of a blocking coalition must be big enough to block

the ‘yes’ voters the q votes they need to win. So it must be at least n-q+1 (where n is number of voters)

Extra Votes Principle: A winning coalition with total weight w has w-q ‘extra votes’. A

blocking coalition with weight w has w-(n-q +1) extra votes. The critical voters are those whose weight is more than the coalition’s extra votes. These are the voters the coalition can’t afford to lose.

Calculating the Banzhaf Index

Take the voting system [3:2,1,1]

Winning coalition- have a weight of

3 or 4

A has 3 critical votes,

B and C both have1

Win.

Coalit.

Weight Extra votes

A

(c.v)

B

(c.v.)

C

(c.v)

{A,B} 3 0 1 1 0{A,C} 3 0 1 0 1{A,B,C} 4 1 1 0 0

Totals 3 1 1

Calculating the Banzhaf Index

Block. Weight Extra A (c.v) B (c.v) C (c.v)

{A} 2 0 1 0 0

{B,C} 2 0 0 1 1

{A,B} 3 1 1 0 0

{A,C} 3 1 1 0 0

{A,B,C} 4 2 0 0 0

Totals: 3 1 1

Calculating the Banzhaf Index

In the blocking coalitions, A is critical in 3 and B and C are both critical in 1 each

So, taking the blocking coalitions and winning coalitions together,A has an index of 6B has an index of 2C has an index of 2

Comparing Voting Systems

Two voting systems are equivalent if there is a way for all of the voters of the first system to exchange places with the voters of the second system and preserve all winning coalitions. [50: 49, 1] and [4: 3, 3] - unanimous support [2: 2, 1] and [5: 3, 6] – dictator

Every 2-voter system is equivalent to a system with a dictator or one that needs consensus

Minimal winning coalition: a winning coalition in which each voter is a critical voter

Minimal Winning Coalitions

Take the voting system [6: 5, 3, 1] where the respective voters are A, B, C.

The 3 winning coalitions are {A,B}, {A,C} and {A,B,C}.

Which coalitions are minimal?

Only {A,B} and {A,C}, but not {A,B,C} since only A is critical

Minimal Winning Coalitions Instead of using weights and quotas to describe a voting

system, one can describe it by using its minimal winning coalitions.

The following conditions must be satisfied 1) The list can’t be empty (otherwise there is no way to

approve a motion) 2) There can’t be one minimal coalition that contains

another one 3)Every pair of coalitions in the list must overlap- otherwise

two opposing motions could pass.

3-Voter Systems & Minimal Winning Coalitions

Make a list of all voting systems with 3 voters The 3 voters are A, B, C 1) Suppose the M.W.C is {A}

Dictatorship 2) Suppose the M.W.C is {A,B,C}

Consensus rule 3) Suppose the M.W.C. is {A,B}

A clique where C is the dummy voter 4) Suppose the M.W.C. are {A,B} and {A,C}

A has veto power- the chair veto All 2-member coalitions are M.W.C

Majority rule

3-Voter Systems & Minimal Winning Coalitions

System Min. W. Coaltions

Weights Banzhaf Index

Dictator {A} [3: 3, 1, 1] (8, 0, 0)

Clique {A, B} [4: 2, 2, 1] (4, 4, 0)

Majority {A,B} {A,C}, {B,C}

[2: 1, 1, 1] (4, 4, 4)

Chair Veto {A,B} {A, C} [3: 2, 1, 1] (6, 2, 2)

Consensus {A, B, C} [3: 1, 1, 1] (2, 2, 2)

Discussion Chapter 10

Where do we see manipulation of voting systems? Are there any political elections that stand out in your mind?

Chapter 11 What are some applications of weighted voting systems? How would you describe a jury as a weighted voting system? What might advantages/disadvantages of certain types of

weighted voting systems?

Homework: Chapter 10 pg 387 # 9 Chapter 11 pg 425 # 7