Experimental Examples

1
Experimental Examples

• Selection of a pair of trajectories for NASA
  spacecrafts.
• Consider four methods of aggregating teams
  preferences.
• Two types of utilities considered
• Ordinal utilities Only rank counts
• Cardinal utilities Teams assign numbers
  indicating strength of preference

2
Four Methods
31

• Sum of Ordinal Rank The Borda Rule
• Rank, not only first preference, matters
• Sum of Cardinal Rank Bentham
• Maximize Social Welfare
• Multiply Cardinal Utility Nash
• Protect the losers 0.1111 vs. .5.5.50.125
• Pairwise Comparison Condorcet

26
31
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3
Simple Majority Decision

• Different Rules can lead to Different Outcomes
  Chapter 2
• Is there a Superior Rule ?
• If there is, it is Simple Majority Rule
• When there are only two alternatives it works
  without a problem
• Problem Choices normally not dichotomous ! ?
  Primaries, for example

4
What is so great about Simple Majority Rule ???

• Definitions
• A set Xx,y of alternatives
• A set N1,2,3,,n citizens/voters
• Pi Individual is strict preference over X
• Not complete No strict preference possible
• Asymmetric xPiy implies not yPix
• Ii Individual is indifference over X
• Not complete if xPiy then not xIiy or yIix
• Symmetric if xIiy then yIix

5
What is so great about Simple Majority Rule ???

• Definitions
• Ri Individual is weak preference over X
• If xPiy and xIiy then xRiy
• Complete Either xRiy or yRix
• A Social Choice Rule C(X)
• If xPiy then say Di 1
• Then preferences of voters can be summarized
  D(D1, D2, D3,Dn)

6
What is so great about Simple Majority Rule ???

• Definitions
• D?D. D is the set all profiles
• Two person society D(1,1),(1,0),(0,1),(0,0),(0,
  -1),(-1,0),(-1,-1)
• Define a function F(D) equivalent to C(X)
• F(D)1 ? C(X)x
• Indecisive Rule F(D)0 for all D?D
• Imposed Rule F(D)1 (or 1) for all D?D
• Simple Majority Function Plurality
• Absolute Majority Function gt50
• Special Majority Function gtz, zgt50

7
Outcomes under SMR

• SMR with equal weights Find outcome sum over
  D gt 0 ? x is chosen.

8
Three qualities of SMR

• Monotonicity, Undifferentiatedness,
  Neutrality.
• Monotonicity If some voters switch from x to y
  then xs outcome cannot worsen
• Formally D, D ? D. If Di ? Di for all i?N
  then F(D) ? F(D).
• Strong monotonicity A change breaks ties

9
Monotonicity Unanimity

• Unanimity If everyone favours x then x wins
• Weak Unanimity If everyone favours y then y
  does not win
• Strong Monotonicity ? Unanimity
• Monotonicity ? Weak Unanimity
• Does Unanimity imply monotonicity ?

10
Violations of Monotonicity

• Musical Contest Decision Rule
• Judges award contestants 1-25 points
• Median determined, outliers eliminated
• Points summed, highest score wins
• Assume three judges, two contestants
• Does this make sense?

11
Musical Contest

• Performer A Performer B
• Judge 1 15 10
• Judge 2 16 10
• Judge 3 24 15
• Total 55 35

12
Musical Contest

• Performer A Performer B
• Judge 1 15 10
• Judge 2 16 10
• Judge 3 25 15

Total 31 35
Median 16
Median 8 24
13
Musical Contest

• Performer A Performer B
• Judge 1 15 10
• Judge 2 16 10
• Judge 3 25 15

Total 31 35
What else is bad about this system ?
14
Monotonicity with more than Two Alternatives

• STV V voters electing S candidates
• Calculate quota q(V/S1)1
• If candidate gets q votes he is elected and
  surplus votes are distributed to second ranked
  candidate of his voters
• Repeat until no one has q votes
• Start dropping candidates with fewest votes and
  redistribute votes.
• Repeat until someone has q votes.

15
The Single Transferable Vote

• Example Xx,y,z,w, n26, S2
• q(26/21)1 ? 9
• 9 voters w ? z ? x ? y
• 6 voters x ? y ? z ? w
• 2 voters y ? x ? z ? w
• 4 voters y ? z ? x ? w
• 5 voters z ? x ? y ? w

i) w wins first seat
ii) No one has q
iii) z eliminated
iv) x wins second seat
16
The Single Transferable Vote

• Example Xx,y,z,w, n26, S2
• q(26/21)1 ? 9
• 9 voters w ? z ? x ? y
• 6 voters x ? y ? z ? w
• 2 voters x ? y ? z ? w
• 4 voters y ? z ? x ? w
• 5 voters z ? x ? y ? w

i) w wins first seat
ii) No one has q
iii) y eliminated
iv) z wins second seat
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The Majority Run-off

• Xx,y,z, N1,2,3,,100

31 voters x ? y ? z 30 voters z ? x ? y 29
voters y ? z ? x 10 voters y ? z ? x
X 31 votes Y 39 votes Z 30 votes
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The Majority Run-off

• Xx,y,z, N1,2,3,,100

31 voters x ? y ? z 30 voters z ? x ? y 29
voters y ? z ? x 10 voters y ? z ? x
X 61 votes Y 39 votes
19
The Majority Run-off

• Xx,y,z, N1,2,3,,100

31 voters x ? y ? z 30 voters z ? x ? y 29
voters y ? z ? x 10 voters y ? z ? x
20
The Majority Run-off

• Xx,y,z, N1,2,3,,100

31 voters x ? y ? z 30 voters z ? x ? y 29
voters y ? z ? x 10 voters x ? y ? z
X 41 votes Y 29 votes Z 30 votes
21
The Majority Run-off

• Xx,y,z, N1,2,3,,100

31 voters x ? y ? z 30 voters z ? x ? y 29
voters y ? z ? x 10 voters x ? y ? z
X 41 votes Z 59 votes
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Undifferentiatedness

• Not equality but sufficient.
• Undifferentiatedness allows anonymity and
  anonymity allows equality.
• Violations of anonymity Weighted voting some
  votes count more than others, e.g., U.K. in the
  good old days.
• U.N. Security Council

23
Undifferentiatedness

• Formally, any permutation of the vote vector
  results in the same outcome
• 3 voters. The vectors (1,1,0), (1,0,1) and
  (0,1,1) all result in the same choice.
• Undifferentiatedness vs. Anonymity
• Possibility of coercion
• Legislative Voting

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Demand Revealing Processes

• Necessarily violate undifferentiatedness
• Argument for
• Strength of Preferences should be taken into
  account
• Argument against
• Wasteful
• Unfair Wealth not evenly distributed
• Possibility of Coercion

25
Neutrality

• A Neutral Method favours no alternative
• Formally, if preferences are switched around
  then outcome changes
• If (1,1,-1) 1 then (-1,-1,1) -1

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Neutrality

• Many rules violate
• Super-majority rules
• Simple Majority Rule with a Tie-breaker
• Minority Decisions Rules (Agenda Setting)
• Some that satisfy
• The Jury Rule Tie mistrial
• SMR without a tiebreaker (or a coin-flip)
• Rules for gt2 alternatives Condorcet, Borda,
  Bentham, Plurality

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Summary

• Three qualities of SMR with Xx,y
• Undifferentiatedness votes equal
• Neutrality alternatives treated equal
• Monotonicity Non-arbitrary
• What is fair ?
• Demand revealing processes strength of
  preferences.

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Summary

• SMR with Xx,y the only rule that satisfies
  the conditions of monotonicity, neutrality and
  undifferentiatedness (Mays Theorem).
• Is SMR with Xx,y superior ?
• X is rarely x,y naturally.
• Any process of narrowing choices will violate
  some of the conditions.

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Lessons

• Arguments favouring a Responsible Party System
  misguided
• England Disciplined Parties offering real
  choices
• Two counter-arguments
• Dahl Observes that policy is made after
  elections building coalitions out of many
  minorities.

30
Lessons

• Two counter-arguments (cont.)
• Downs Proposes a model of two candidate/party
  competition. Party platforms should converge on
  median.

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Three or More Alternatives

• SMR Some great qualities but rarely applicable
• Choice between two alternative
• Modify definitions
• X x,y,z,.
• Di complete and transitive
• xy xPiy, yx yPix, or (xy) xIiy
• Indifferent voters dont vote

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Majoritarian Methods

• Condorcet Rule
• Often no best alternative the Paradox of
  Voting
• Policy Cycles can occur

33
Condorcet Rule

• Xw,x,y,z, N1,2,3,4

D1 w ? y ? x ? z D2 x ? w ? z ? y D3 y ? z ?
w ? x D4 x ? w ? y ? z
w wIx, wPy, wPz x xIw, xIy, xPz
Choice?
34
Majoritarian Rules

• The Amendment Procedure
• The goal is to find a Condorcet Winner
• X t,w,x,y,z,s
• t Motion
• w Amendment
• x Amendment to Amendment
• y Substitute Amendment
• z Amendment to Substitute
• s Status quo

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The Amendment Procedure
w vs. t
D D1, D2 s t w D3, D4 t s w D5
w s t
w wins
t wins
w vs. s
t vs. s
w
s
t
s
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The Amendment Procedure
w vs. t
D D1, D2 t w s D3 s t w
D4 s w t D5 w s t
w wins
t wins
w vs. s
t vs. s
w
s
t
s
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The Amendment Procedure

• Xw,x,y,t,s, N1,2,3

Step 1 x vs. w ? w Step 2 w vs. y ? y Step 3 y
vs. t ? t Step 4 t vs. s ? t
D1 w ? x ? t ? y ? s D2 y ? w ? x ? t ? s D3
s ? x ? t ? y ? w
Yet x is unanimously preferred to t !!!
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The Successive Procedure

• Pick a Candidate and pit against every other
  Candidate in X
• A Voter votes for the Candidate (x) if he is
  preferred to all other candidates in X\x
• Problems
• Sometimes no winner (POV)
• Condorcet winner may lose

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The Successive Procedure

• Xw,x,y,z, N1,2,3

D1 w ? x ? y ? z D2 y ? x ? w ? z D3 z ? x ?
y ? w
The Successive Procedure fails to select a winner
Yet x is a Condorcet winner !!!
40
The Majority Runoff

• Again the goal is to select the Condorcet Winner
  or at least a large plurality winner.
• Need not find Condorcet Winner

41
The Majority Runoff Condorcet Winners

• Xx,y,z, N1,2,3,4,5

D1,D2 x ? z ? y D3,D4 y ? z ? x D5 z ? x
? y
X 2 votes Y 2 votes Z 1 votes
X 3 votes Y 2 votes
Yet z is a Condorcet winner !!!
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Some other rules

• Copeland
• Schwartz Top cycles
• Kemmeny

43
Positional Methods

• Majoritarian Methods generally use binary
  comparisons
• Positional Methods aim at using all the
  information we have about the voters preferences

44
Borda Rule

• Xx,y,a,b,c, N1,2,3,4,5

4 3 2 1 0
D1 x ? y ? a ? b ? c D2 y ? a ? c ? b ? x D3
c ? x ? y ? a ? b D4 x ? y ? b ? c ? a D5 y ? b
? a ? x ? c
x 12 y 16 a 8 b 7 c 7
But x is the Condorcet winner !
45
The 1912 Presidential Election

• XWilson, Roosevelt, Taft, ?, N?

42 Wilson ? Roosevelt ? Taft 27 Roosevelt ?
Taft ? Wilson 24 Taft ? Roosevelt ? Wilson 7
Other
46
Approval Voting

• Xx,y,z, N1,2,3,,101

D1-D61 x ? y ? z D62- D81 y ? x ? z
D82-D101 z ? y ? x
Assume all voters cast two votes.
x 81 y 101 z 20
Condorcet winner ?
47
Borda Count

• Ranking may reverse if an alternative is removed
• Alternative y may win in X, but lose in all
  proper subsets of X.

48
Utilitarian Methods of Voting

• Majoritarian Methods rely on pairwise
  comparisons of alternatives
• Positional Methods rely on comparisons of rank
  of all alternatives
• Neither takes strength of preference into
  account.
• Utilitarian Methods incorporate intensity of
  preference

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Utility

• A utility is a measure of preference
• It is not obvious how to assign utilities
• Von Neumann-Morgenstern utilities
• Obtain a rank ordering (say x, y, z)
• Assign 1 to top, 0 to bottom
• Generate a lottery over x and z pu(x)
  (1-p)u(z)
• Find a p such that i is indifferent between
  lottery and z.

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Utilitarian Methods

• Already seen
• Bentham Social Welfare
• Demand Revealing Truthful
• Nash Consistent and Protective

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Criteria for Judging Methods

• Undifferentiatedness
• Monotonicity
• Neutrality
• Condorcet Criterion A majority winner?
• Consistency Divide X if y wins in both
  subsets it should also win in X
• Independence of Irrelevant Alternatives Whether
  x or y wins should not depend on other
  alternatives

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Violations

• Certain types of violations can be associated
  with different methods
• Majoritarian Consistency
• Positional Condorcet
• Borda also IIA
• Approval Undifferentiatedness
• Utilitarian Methods IIA

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Copeland Consistency
N1
N2
17 x ? y ? z 8 y ? z ? x 5 z ? x ?
y
14 x ? z ? y 16 y ? x ? z 15 z ? y ? x
x 2 points y 0 points z -2 points
x 0 points y 0 points z 0 points
54
Copeland Consistency
N
N1
N2
?
17 x ? y ? z 8 y ? z ? x 5 z ? x ?
y
14 x ? z ? y 16 y ? x ? z 15 z ? y ? x
x 0 points y 2 points z -2 points
55
Borda IIA
D1 D2 D3 Total
D1 a ? b ? c D2 c ? a ? b D3 c ? a ? b
a 2 1 1 4 b 1 0 0 1 c 0
2 2 4
a and c tie
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Borda IIA
D1 a ? b ? c D2 c ? a ? b D3 c ? a ? b
57
Borda IIA
D1 D2 D3 Total
D1 a ? b ? c D2 c ? b ? a D3 c ? a ? b
a 2 0 1 3 b 1 1 0 2 c 0
2 2 4
c wins !
58
Approval Vote Undifferentiatedness

• Assumptions about how many votes a voter casts
  these depend preferences
• Rikers example assumes that the rule determines
  the number of votes
• Skip example but need to know Approval Rule

59
Bentham IIA
D1 D2 Total
D1 x ? y ? z D2 y ? x ? z
x 1.0 0.6 1.6 y 0.5 1.0 1.5 z 0
0 0
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Bentham IIA
D1 D2 Total
D1 x ? y ? z D2 y ? x ? z
x 1.0 0.5 1.5 y 0.6 1.0 1.6 z 0
0 0
Why is this a violation of IIA ?
Because of the way we construct the utilities
they are in essence relative measures
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What to do ?

• No fair way of aggregating preferences
• No fair voting methods
• Moreover, no Philosopher Kings.
• Even if we know everyones preferences fair
  choices can not always be made
• Many additional criteria of fairness

62
Rikers Recommendations

• Different voting rules for different situations
  because choices must be made
• Legislatures Amendment, Borda, Kemeny
• Elections of Officials Plurality
• Primaries Approval
• Economic Planning Demand Revealing

63
Protecting the Two-Party System

• In a previous chapter Riker demonstrates the
  hopelessness of a Responsible Party System
• Why is Riker so protective of the two-party
  system?