1. Presented by: Katherine Goulde
Chapter 10: The Manipulability of Voting Systems Chapter 11: Weighted Voting Systems
Presented by: Katherine Goulde

2. 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

3. Manipulability & the Borda Count
The Borda count assigns point values to the candidates and the winner is the candidate with the most points
Voter Voter 2
A B
B C
C A
D D
Candidate A has a score of 4
Candidate B has a score of 5
Candidate C has a score of 3
Candidate D has a score of 0
Therefore… B wins!

4. What if Voter 1 wants to manipulate the election???
Voter Voter 2
A B
B C
C A
D D
D C
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?

5. 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

6. 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

7. 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 Voter 3
A B C
C C A
B A B
Voter 1 Voter Voter 3
A B C
B C A
C A B

8. 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

9. 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

10. 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 Voter 3
A C B
B A D
D B C
C D A

11. 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’

12. 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.

13. The Chair’s Paradox Chair You Me A B C B C A C A B
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 Me
A B C
B C A
C A B

14. The Chair’s Paradox
Chair You Me
A B C
B C A
C A B
The chair will vote for A. ‘Me’ will vote for C. ‘You’ will also vote for C.

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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.

22. 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

23. 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.

24. Calculating the Banzhaf Index
Win.
Coalit.
Weight
Extra votes A
(c.v) B
(c.v.) C
{A,B} 3 1
{A,C}
{A,B,C} 4
Totals
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

25. Calculating the Banzhaf Index
Block.
Weight
Extra
A (c.v)
B (c.v)
C (c.v)
{A} 2 1
{B,C}
{A,B} 3
{A,C}
{A,B,C} 4
Totals:

26. 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 6
B has an index of 2
C has an index of 2

27. 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

28. 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

29. 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.

30. 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

31. 3-Voter Systems & Minimal Winning Coalitions
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)

32. Discussion Chapter 10 Chapter 11 Homework:
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