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Title: Computational Aspects of Approval Voting and Declared-Strategy Voting


1
Computational Aspects of Approval Voting and
Declared-Strategy Voting
  • Dissertation defense
  • 17 April 2008

Rob LeGrand Washington University in St. Louis
Computer Science and Engineering legrand_at_cse.wustl
.edu
Robert Pless Itai Sened Aaron Stump
Ron Cytron Steven Brams Jeremy Buhler
2
Themes of research
  • Approval voting systems
  • Susceptibility to insincere strategy
  • encouraging sincere ballots
  • Evaluating effectiveness of various strategies
  • Internalizing insincerity
  • separating strategy from indication of
    preferences
  • Complex voting protocols
  • complexity of finding most effective ballot
  • complexity of calculating the outcome

3
What is manipulation?
  • Broadly, effective influence on election outcome
  • Election officials can . . .
  • exclude/include alternatives Nurmi 99
  • exclude/include voters Bartholdi, Tovey Trick
    92
  • choose election protocol Saari 01
  • Alternatives may be able to . . .
  • drop out to avoid a vote-splitting effect
  • Voters can . . .
  • find the ballot that is likeliest to optimize the
    outcome
  • This last sense is what we mean

4
Lets vote!
45 voters A C B
35 voters B C A
20 voters C B A
(1st) (2nd) (3rd)
sincere preferences
5
Plurality voting
45 voters A C B
35 voters B C A
20 voters C B A
sincere ballots
A 45 votes B 35 votes C 20 votes
zero-information result
6
Plurality voting
45 voters A C B
35 voters B C A
20 voters C B A
ballots so far
?
A 45 votes B 35 votes C 0 votes
election state
7
Plurality voting
45 voters A C B
35 voters B C A
20 voters C B A
strategic ballots
insincerity!
B 55 votes A 45 votes C 0 votes
final election state
Gibbard 73 Satterthwaite 75
8
Manipulation decision problem
45 voters A C B
35 voters B C A
20 voters C B A
ballot sets
BU
BV
B 55 votes A 45 votes C 0 votes
election state
9
Manipulation decision problem
  • Existence of Probably Winning Coalition Ballots
    (EPWCB)
  • INSTANCE Set of alternatives A and a
    distinguished member a of A set of weighted
    cardinal-ratings ballots BV the weights of a set
    of ballots BU which have not been cast
    probability
  • QUESTION Does there exist a way to cast the
    ballots BU so that a has at least probability
    of winning the election with the ballots
    ?
  • My generalization of problems from the
    literature
  • Bartholdi, Tovey Trick 89
    Conitzer Sandholm 02
  • Conitzer Sandholm 03

10
Manipulation decision problem
  • Existence of Probably Winning Coalition Ballots
    (EPWCB)
  • INSTANCE Set of alternatives A and a
    distinguished member a of A set of weighted
    cardinal-ratings ballots BV the weights of a set
    of ballots BU which have not been cast
    probability
  • QUESTION Does there exist a way to cast the
    ballots BU so that a has at least probability
    of winning the election with the ballots
    ?
  • These voters have maximum possible information
  • They have all the power (if they have smarts too)
  • If this kind of manipulation is hard, any kind is

11
Manipulation decision problem
  • Existence of Probably Winning Coalition Ballots
    (EPWCB)
  • INSTANCE Set of alternatives A and a
    distinguished member a of A set of weighted
    cardinal-ratings ballots BV the weights of a set
    of ballots BU which have not been cast
    probability
  • QUESTION Does there exist a way to cast the
    ballots BU so that a has at least probability
    of winning the election with the ballots
    ?
  • This problem is computationally easy (in P) for
  • plurality voting Bartholdi, Tovey Trick 89
  • approval voting

12
Manipulation decision problem
  • Existence of Probably Winning Coalition Ballots
    (EPWCB)
  • INSTANCE Set of alternatives A and a
    distinguished member a of A set of weighted
    cardinal-ratings ballots BV the weights of a set
    of ballots BU which have not been cast
    probability
  • QUESTION Does there exist a way to cast the
    ballots BU so that a has at least probability
    of winning the election with the ballots
    ?
  • This problem is computationally infeasible
    (NP-hard) for
  • Hare (single-winner STV) Bartholdi Orlin 91
  • Borda Conitzer Sandholm 02

13
What can we do to make manipulation hard?
  • One approach tweaks Conitzer Sandholm 03
  • Add an elimination round to an existing protocol
  • Drawback alternative symmetry (fairness) is
    lost
  • What if we deal with manipulation by embracing
    it?
  • Incorporate strategy into the system
  • Encourage sincerity as advice for the strategy

14
Declared-Strategy Voting
Cranor Cytron 96
rational strategizer
cardinal preferences
ballot
election state
outcome
15
Declared-Strategy Voting
Cranor Cytron 96
sincerity
strategy
rational strategizer
cardinal preferences
ballot
election state
outcome
  • Separates how voters feel from how they vote
  • Levels playing field for voters of all
    sophistications
  • Aim a voter needs only to give sincere
    preferences

16
What is a declared strategy?
A 0.0 B 0.6 C 1.0
cardinal preferences
A 0 B 1 C 0
declared strategy
voted ballot
A 45 B 35 C 0
current election state
  • Captures thinking of a rational voter

17
Can DSV be hard to manipulate?
  • DSV can be made to be NP-hard to manipulate in
    the EPWCB sense. LeGrand 08
  • Proof by reduction
  • Simulate Hare by using particular declared
    strategy in DSV
  • Hare is NP-hard to manipulate Bartholdi Orlin
    91
  • If this DSV system were easy to manipulate, then
    Hare would be
  • DSV can be made NP-hard to manipulate
  • So why use tweaks? (DSV is better!)

18
Favorite vs. compromise, revisited
45 voters A C B
35 voters B C A
20 voters C B A
ballots so far
?
A 45 votes B 35 votes C 0 votes
election state
19
Approve both!
45 voters A C B
35 voters B C A
20 voters C B A
strategic ballots
insincerity avoided
B 55 votes A 45 votes C 20 votes
final election state
20
Approval voting
Ottewell 77 Weber 77 Brams
Fishburn 78
  • Allows approval of any subset of alternatives
  • Single alternative with most votes wins
  • Used historically Poundstone 08
  • Republic of Venice 1268-1789
  • Election of popes 1294-1621
  • Used today Brams 08
  • Election of UN secretary-general
  • Several academic societies, including
  • Mathematical Society of America
  • American Statistical Association

21
Strands of research
number of alternatives outcome Area of research
k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?
k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04
22
Strands of research
number of alternatives outcome Area of research
k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?
k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04
23
Strands of research
number of alternatives outcome Area of research
k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?
k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04
24
Approval ratings
25
Approval ratings
  • Aggregating film reviewers ratings
  • Rotten Tomatoes approve (100) or disapprove
    (0)
  • Metacritic.com ratings between 0 and 100
  • Both report average for each film
  • Reviewers rate independently

26
Approval ratings
  • Online communities
  • Amazon users rate products and product reviews
  • eBay buyers and sellers rate each other
  • Hotornot.com users rate other users photos
  • Users can see other ratings when rating
  • Can these voters benefit from rating
    insincerely?

27
Approval ratings
28
Average of ratings
outcome
data from Metacritic.com Videodrome (1983)
29
Average of ratings
outcome
Videodrome (1983)
30
Another approach Median
outcome
Videodrome (1983)
31
Another approach Median
outcome
Videodrome (1983)
32
Another approach Median
  • Immune to insincerity LeGrand 08
  • voter i cannot obtain a better result by voting
  • if , increasing will
    not change
  • if , decreasing will
    not change
  • Allows tyranny by a majority
  • no concession to the 0-voters

33
Average with Declared-Strategy Voting?
  • So Median is far from idealwhat now?
  • try using Average protocol in DSV context
  • But whats the rational Average strategy?
  • And will an equilibrium always be found?

rational strategizer
cardinal preferences
ballot
election state
outcome
34
Equilibrium-finding algorithm
Videodrome (1983)
35
Equilibrium-finding algorithm
36
Equilibrium-finding algorithm
37
Equilibrium-finding algorithm
38
Equilibrium-finding algorithm
39
Equilibrium-finding algorithm
  • Is this algorithm is guaranteed to find an
    equilibrium?

equilibrium!
40
Equilibrium-finding algorithm
  • Is this algorithm is guaranteed to find an
    equilibrium?
  • Yes! LeGrand 08

equilibrium!
41
Expanding range of allowed votes
  • These results generalize to any range LeGrand
    08

42
Multiple equilibria can exist
  • Will multiple equilibria will always have the
    same average?

outcome in each case
43
Multiple equilibria can exist
  • Will multiple equilibria will always have the
    same average?
  • Yes! LeGrand 08

outcome in each case
44
Average-Approval-Rating DSV
outcome
Videodrome (1983)
45
Average-Approval-Rating DSV
  • AAR DSV is immune to insincerity in general
    LeGrand 08

outcome
46
Evaluating AAR DSV systems
  • Expanded vote range gives wide range of AAR DSV
    systems
  • If we could assume sincerity, wed use Average
  • Find AAR DSV system that comes closest
  • Real film-rating data from Metacritic.com
  • mined Thursday 3 April 2008
  • 4581 films with 3 to 44 reviewers per film
  • measure root mean squared error

47
Evaluating AAR DSV systems
minimum at
48
Evaluating AAR DSV systems hill-climbing
minimum at
49
Evaluating AAR DSV systems hill-climbing
minimum at
50
Evaluating AAR DSV systems
51
AAR DSV Future work
  • Website ratingsbyrob.com
  • Users can rate movies, books, each other, etc.
  • They can see current ratings without being
    tempted to rate insincerely
  • Find more strategy-immune rating systems
  • Richer outcome spaces
  • Hypercube like rating several films at once
  • Simplex dividing a limited resource among
    several uses
  • How assumptions about preferences are generalized
    is important

52
Strands of research
number of alternatives outcome Area of research
k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?
k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04
53
Approval strategies for DSV
  • Rational plurality strategy has been well
    explored Cranor Cytron 96
  • But what about approval strategy?
  • If each alternatives probability of winning is
    known, optimal strategy can be computed Merrill
    88
  • But what about in a DSV context?
  • have only a vote total for each alternative
  • Lets look at several approval strategies and
    approaches to evaluating their effectiveness

54
DSV-style approval strategies
  • Strategy Z
  • Approve alternatives with higher-than-average
    cardinal preference (zero-information strategy)
    Merrill 88

55
DSV-style approval strategies
  • Strategy Z
  • Strategy T
  • Approve favorite of top two vote-getters, plus
    all liked more Ossipoff 02, Poundstone 08
  • Simplest generalization of plurality DSV strategy
  • Cranor Cytron 96

56
DSV-style approval strategies
  • Strategy Z
  • Strategy T
  • Strategy J
  • Use strategy Z if it distinguishes between top
    two vote-getters otherwise use strategy T Brams
    Fishburn 83

57
DSV-style approval strategies
  • Strategy Z
  • Strategy T
  • Strategy J
  • Strategy A
  • Approve all preferred to top vote-getter, plus
    top vote-getter if preferred to second-highest
    vote-getter
  • LeGrand 02
  • . . . but how to evaluate these strategies?

58
Election-state-evaluation approaches
  • Evaluate a declared strategy by evaluating the
    election states that are immediately obtained
  • Calculate expected value of an election state by
    estimating each alternatives probability of
    eventually winning
  • How to calculate those probabilities?

59
Election-state-evaluationMerrill metric
  • Estimate an alternatives probability of winning
    to be proportional to its current vote total
    raised to some power x Merrill 88

60
Strategy comparison using the Merrill metric
Current election state Focal voters preferences
1, 0, 0 (strategies A T) 1, 0, 0 (A
T) 0, 1, 0 (A T) 0, 1, 1 (A) 0, 1, 0
(T) 1, 0, 1 (A T) 0, 1, 1 (A T)
61
Strategy comparison using the Merrill metric
Current election state Focal voters preferences
When , A is better than
T if and only if
or, equivalently
  • Intuitively, A always does better than T when
  • s1 is much larger than s2,
  • x is large, or
  • p3 is relatively close to p2 compared to p1

62
Strategy comparison using the Merrill metric
  • Also compared other strategy pairs LeGrand 08
  • As x goes to infinity (3 alternatives)
  • Strategy A dominates strategy T
  • Strategy A dominates strategy J
  • Strategy A dominates strategy Z
  • Neither strategy T nor strategy J dominates the
    other
  • As x goes to infinity (4 alternatives)
  • Strategy A dominates strategy T

63
Further result for strategy A
  • More generally, it is true that if
  • the election state is free of ties and near-ties
  • and the focal voters cardinal preferences are
    tie-free
  • when
  • and the Merrill-metric exponent x is taken to
    infinity
  • then strategy A dominates all other approval
    strategies according to the Merrill metric
    LeGrand 08

64
Election-state-evaluationBranching-probabilities
metric
  • Estimate an alternatives probability of winning
    by looking ahead
  • Assume that the probability that alternative a is
    approved on each future ballot is equal to the
    proportion of already-voted ballots that approve
    a

65
Branching-probabilities metric strategy A
  • It is true that if
  • the election state is free of ties and near-ties
  • and the focal voters cardinal preferences are
    tie-free
  • when
  • and the number of future ballots is taken to
    infinity
  • then strategy A dominates all other approval
    strategies according to the branching-probabilitie
    s metric LeGrand 08

66
Approval DSV strategies Future work
  • Consider different strategy-evaluation metrics
  • Study strategy-A equilibria
  • How good are the outcomes?
  • How often are strong Nash equilibria found?
  • How strategy-vulnerable is Approval DSV with
    strategy A?
  • How often will submitting insincere preferences
    benefit a voter?

67
Strands of research
number of alternatives outcome Area of research
k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?
k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04
68
Electing a committee from approval ballots
approves of alternatives 4 and 5
11110
00011
k 5 alternatives n 6 ballots
00111
01111
00001
10111
  • Whats the best committee of size m 2?

69
Sum of Hamming distances
11110
00011
m 2 winners
2
4
4
5
00111
01111
11000
4
3
sum 22
00001
10111
  • What if we elect alternatives 1 and 2?

70
Fixed-size minisum
11110
00011
m 2 winners
4
0
2
1
00111
01111
00011
2
1
sum 10
00001
10111
  • Minisum elects winner set with smallest HD sum
  • Easy to compute (pick alternatives with most
    approvals)

71
Maximum Hamming distance
11110
00011
m 2 winners
4
0
2
1
00111
01111
00011
2
1
sum 10 max 4
00001
10111
  • One voter is quite unhappy with minisum outcome

72
Fixed-size minimax
Brams, Kilgour Sanver 04
11110
00011
m 2 winners
2
2
2
1
00111
01111
00110
2
3
sum 12 max 3
00001
10111
  • Minimax elects winner set with smallest maximum
    HD
  • Harder to compute?

73
Complexity
Endogenous minimax EM BSM(0, k) Bounded-size minimax BSM(m1, m2) Fixed-size minimax FSM(m) BSM(m, m)
NP-hard Frances Litman 97 NP-hard (generalization of EM) ?
74
Complexity
Endogenous minimax EM BSM(0, k) Bounded-size minimax BSM(m1, m2) Fixed-size minimax FSM(m) BSM(m, m)
NP-hard Frances Litman 97 NP-hard (generalization of EM) NP-hard LeGrand 04
75
Approximability
Endogenous minimax EM BSM(0, k) Bounded-size minimax BSM(m1, m2) Fixed-size minimax FSM(m) BSM(m, m)
has a PTAS Li, Ma Wang 99 no known PTAS no known PTAS
Polynomial-Time Approximation Scheme algorithm
with approx. ratio 1 e that runs in time
polynomial in the input and exponential in 1/e
76
Approximability
Endogenous minimax EM BSM(0, k) Bounded-size minimax BSM(m1, m2) Fixed-size minimax FSM(m) BSM(m, m)
has a PTAS Li, Ma Wang 99 no known PTAS has a 3-approx. LeGrand, Markakis Mehta 06 no known PTAS has a 3-approx. LeGrand, Markakis Mehta 06
Polynomial-Time Approximation Scheme algorithm
with approx. ratio 1 e that runs in time
polynomial in the input and exponential in 1/e
77
Susceptibility to insincerity
Endogenous minimax EM BSM(0, k) Bounded-size minimax BSM(m1, m2) Fixed-size minimax FSM(m) BSM(m, m)
insincere voters can benefit LeGrand, Markakis Mehta 06 insincere voters can benefit LeGrand, Markakis Mehta 06 insincere voters can benefit LeGrand, Markakis Mehta 06
But our 3-approximation for FSM is immune to
insincere strategy!
78
Fin
  • Thanks to
  • my advisor, Ron Cytron
  • Steven Brams
  • members of my committee
  • co-authors Vangelis Markakis and Aranyak Mehta
  • Morgan Deters and the rest of the DOC Group
  • Questions?

79
Rational m,M-Average strategy
  • Allow votes between and
  • For , voter i should choose to
    move outcome as close to as possible
  • Choosing would give
  • Optimal vote is
  • After voter i uses this strategy, one of these is
    true
  • and
  • and

80
What happens at equilibrium?
  • The optimal strategy recommends that no voter
    change
  • So
  • And
  • equivalently,
  • Therefore any average at equilibrium must satisfy
    two equations
  • (A)
  • (B)

81
Proof Only one equilibrium average
  • Theorem
  • Proof considers two symmetric cases
  • assume
  • assume
  • Each leads to a contradiction

82
Proof Only one equilibrium average
case 1
, contradicting
83
Proof Only one equilibrium average
Case 1 shows that
Case 2 is symmetrical and shows that
Therefore
Therefore, given , the average at equilibrium
is unique
84
An equilibrium always exists?
  • At equilibrium, must satisfy
  • I proposed to prove that, given a vector , at
    least one equilibrium exists.
  • A particular algorithm will always find an
    equilibrium for any . . .

85
An equilibrium always exists!
  • Equilibrium-finding algorithm
  • sort so that
  • for i 1 up to n do
  • Since an equilibrium always exists, average at
    equilibrium is a function,
    .
  • Applying to instead of gives a new
    system, Average-Approval-Rating DSV.

(full proof and more efficient algorithm in
dissertation)
86
Average-Approval-Rating DSV
  • What if, under AAR DSV, voter i could gain an
    outcome closer to ideal by voting insincerely
    ( )?
  • I proposed to prove that Average-Approval-Rating
    DSV is immune to strategy by insincere voters.
  • Intuitively, if
    , increasing will not change
    .

87
AAR DSV is immune to strategy
  • If ,
  • increasing will not change
    .
  • decreasing will not increase
    .
  • If ,
  • increasing will not decrease
    .
  • decreasing will not change
    .
  • So voting sincerely ( ) is guaranteed
    to optimize the outcome from voter is point of
    view

(complete proof in dissertation)
88
Parameterizing AAR DSV
  • m,M-AAR DSV can be parameterized nicely using a
    and b, where and

89
Parameterizing AAR DSV
  • For example

90
Evaluating AAR DSV systems
  • Real film-rating data from Metacritic.com
  • mined Thursday 3 April 2008
  • 4581 films with 3 to 44 reviewers per film

91
Higher-dimensional outcome space
  • What if votes and outcomes exist in
    dimensions?
  • Example
  • If dimensions are independent, Average, Median
    and Average-approval-rating DSV can operate
    independently on each dimension
  • Results from one dimension transfer

92
Higher-dimensional outcome space
  • But what if the dimensions are not independent?
  • say, outcome space is a disk in the plane
  • A generalization of Median the Fermat-Weber
    point Weber 29
  • minimizes sum of Euclidean distances between
    outcome point and voted points
  • F-W point is computationally infeasible to
    calculate exactly Bajaj 88 (but
    approximation is easy Vardi 01)
  • cannot be manipulated by moving a voted point
    directly away from the F-W point Small 90

93
Strategy comparison using the Merrill metric
Current election state Focal voters preferences
expected values of possible next election states
0, 1, 1 (A) 0, 1, 0 (T)
94
Strategy comparison using the Merrill metric
Current election state Focal voters preferences
so T is better than A only when
or, equivalently
95
Strategy comparison using the Merrill metric
Current election state Focal voters preferences
T is better than A only when
  • Corollaries
  • When x is taken to infinity and ,
    strategy A dominates strategy T
  • When
  • , strategy A
    dominates strategy T

96
Further result for strategy A
  • just a weighted average of values
  • assume
  • as , from
    below
  • so maximized when weights of those
    are maximized, which is done by approving only
    alternatives i where
  • case is similar approve i where
  • only strategy A always does this


97
Approximating FSM
11110
m 2 winners
00011
00111
00111
00001
choose a ballot arbitrarily
10111
01111
98
Approximating FSM
11110
m 2 winners
00011
00111
coerce to size m
00101
00111
00001
choose a ballot arbitrarily
10111
01111
outcome m-completed ballot
99
Approximation ratio 3
optimal FSM set
11110
2
00011
2
1
00111
00110
3
00001
2
10111
2
01111
OPT
OPT optimal maxscore
100
Approximation ratio 3
optimal FSM set
chosen ballot
11110
2
00011
2
1
00111
1
00110
00111
3
00001
2
10111
2
01111
OPT
OPT
OPT optimal maxscore
101
Approximation ratio 3
optimal FSM set
chosen ballot
m-completed ballot
11110
2
00011
2
1
00111
1
1
00110
00111
00011
3
00001
2
10111
2
01111
OPT
OPT
OPT
(by triangle inequality)
3OPT
OPT optimal maxscore
102
Better in practice?
  • So far, we can guarantee a winner set no more
    than 3 times as bad as the optimal.
  • Nice in theory . . .
  • How can we do better in practice?
  • Try local search

103
Local search approach for FSM
  1. Start with some c ? 0,1k of weight m

01001
4
104
Local search approach for FSM
  1. Start with some c ? 0,1k of weight m
  2. In c, swap up to r 0-bits with 1-bits in such a
    way that minimizes the maxscore of the result

11000
10001
5
4
01001
01100
00101
4
4
4
01010
00011
4
4
105
Local search approach for FSM
  1. Start with some c ? 0,1k of weight m
  2. In c, swap up to r 0-bits with 1-bits in such a
    way that minimizes the maxscore of the result

01010
4
106
Local search approach for FSM
  1. Start with some c ? 0,1k of weight m
  2. In c, swap up to r 0-bits with 1-bits in such a
    way that minimizes the maxscore of the result

01010
4
107
Local search approach for FSM
  1. Start with some c ? 0,1k of weight m
  2. In c, swap up to r 0-bits with 1-bits in such a
    way that minimizes the maxscore of the result
  3. Repeat step 2 until maxscore(c) is unchanged k
    times
  4. Take c as the solution

11000
10010
5
4
01010
01100
00110
4
4
3
01001
00011
4
4
108
Local search approach for FSM
  1. Start with some c ? 0,1k of weight m
  2. In c, swap up to r 0-bits with 1-bits in such a
    way that minimizes the maxscore of the result
  3. Repeat step 2 until maxscore(c) is unchanged k
    times
  4. Take c as the solution

00110
3
109
Heuristic evaluation
  • Parameters
  • starting point of search
  • radius of neighborhood
  • Ran heuristics on generated and real-world data
  • All heuristics perform near-optimally
  • highest approx. ratio found 1.2
  • highest average ratio lt 1.04
  • The fixed-size-minisum starting point performs
    best overall (with our 3-approx. a close second)
  • When neighborhood radius is larger, performance
    improves and running time increases

(maxscore of solution found) (maxscore of exact
solution)
110
Heuristic evaluation
  • Real-world ballots from GTS 2003 council election
  • Found exact minimax solution
  • Ran each heuristic 5000 times
  • Compared exact minimax solution with heuristics
    to find realized approximation ratios
  • example 15/14 1.0714
  • maxscore of solution found 15
  • maxscore of exact solution 14
  • We also performed experiments using ballots
    generated according to random distributions (see
    dissertation)

111
Specific FSM heuristics
  • Two parameters
  • where to start vector c
  • a fixed-size-minisum solution
  • a m-completion of a ballot (3-approx.)
  • a random set of m candidates
  • a m-completion of a ballot with highest maxscore
  • radius of neighborhood r 1 and 2

112
Average approx. ratios found
radius 1 radius 2
fixed-size minimax 1.0012 1.0000
3-approx. 1.0017 1.0000
random set 1.0057 1.0000
highest-maxscore 1.0059 1.0000
performance on GTS 03 election data k 24
candidates, m 12 winners, n 161 ballots
113
Largest approx. ratios found
radius 1 radius 2
fixed-size minimax 1.0714 1.0000
3-approx. 1.0714 1.0000
random set 1.0714 1.0000
highest-maxscore 1.0714 1.0000
performance on GTS 03 election data k 24
candidates, m 12 winners, n 161 ballots
114
Conclusions from all experiments
  • All heuristics perform near-optimally
  • highest ratio found 1.2
  • highest average ratio lt 1.04
  • When radius is larger, performance improves and
    running time increases
  • The fixed-size-minisum starting point performs
    best overall (with our 3-approx. a close second)

115
Manipulating FSM
m 2 winners
00110
00011
2
0
2
1
00111
01111
00011
2
1
max 2
00001
10111
  • Voters are sincere
  • Another optimal solution 00101

116
Manipulating FSM
00110
m 2 winners
11110
00011
0
2
2
2
1
00111
01111
00110
2
3
max 3
00001
10111
  • A voter manipulates and realizes ideal outcome
  • But our 3-approximation for FSM is nonmanipulable

117
Fixed-size Minimax contributions
  • BSM and FSM are NP-hard
  • Both can be approximated with ratio 3
  • Polynomial-time local search heuristics perform
    well in practice
  • some retain ratio-3 guarantee
  • Exact FSM can be manipulated
  • Our 3-approximation for FSM is nonmanipulable

118
Progress so far
Area of research State of progress
Approval rating Completed rational Average strategy, equality of average at equilibria To do equilibrium always exists, strategy-immunity of AAR DSV, evaluation of AAR DSV systems
DSV-style approval strategies Completed Merrill-metric comparison of A and T in 3-alt. case, domination of A as To do comparisons of other pairs, analysis using branching-probabilities metric
Fixed-size minimax Completed NP-hardness proof, 3-approximation, heuristic evaluation, manipulability analysis
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