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Computational Aspects of Approval Voting
and Declared-Strategy Voting
Rob LeGrandWashington University in St. Louis
Computer Science and [email protected]
A Dissertation Proposal15 March 2007
2
Let’s vote!
45 voters
A
C
B
sincere
preferences
(1st)
(2nd)
(3rd)
35 voters
B
C
A
20 voters
C
B
A
3
Plurality voting
A: 45 votes
B: 35 votes
C: 20 votes
sincere
ballots
45 voters
A
C
B
35 voters
B
C
A
20 voters
C
B
A
“zero-information”
result
4
Plurality voting
ballots
so far
election
state
45 voters
A
C
B
35 voters
B
C
A
A: 45 votes
B: 35 votes
C: 0 votes
?
20 voters
C
B
A
5
Plurality voting
B: 55 votes
A: 45 votes
C: 0 votes
strategic
ballots
final
election
state
45 voters
A
C
B
35 voters
B
C
A
20 voters
C
B
A
[Gibbard ’73] [Satterthwaite ’75]
insincerity!
6
What is manipulation?
B: 55 votes
A: 45 votes
C: 0 votes
45 voters
A
C
B
35 voters
B
C
A
20 voters
C
B
A BUBV
ballot
sets
election
state
7
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]
UV BB
10
8
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
UV BB
10
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 ?
• This problem is computationally easy (in P) for:– plurality voting [Bartholdi, Tovey & Trick ’89]
– approval voting
UV BB
10
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 ?
• This problem is computationally infeasible (NP-hard) for:– Hare [Bartholdi & Orlin ’91]
– Borda [Conitzer & Sandholm ’02]
UV BB
10
11
What can we do about manipulation?
• 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
12
Declared-Strategy Voting[Cranor & Cytron ’96]
electionstate
cardinal
preferences
rational
strategizer
ballot
outcome
13
Declared-Strategy Voting[Cranor & Cytron ’96]
electionstate
cardinal
preferences
rational
strategizer
ballot
outcome
• Separates how voters feel from how they vote• Levels playing field for voters of all sophistications• Aim: a voter needs only to give honest preferences
sincerity manipulation
14
What is a declared strategy?
A: 0.0
B: 0.6
C: 1.0
A: 45
B: 35
C: 0
cardinal
preferences
current
election
state
declared
strategy
A: 0
B: 1
C: 0
voted
ballot
• Captures thinking of a rational voter
15
Can DSV be hard to manipulate?
I propose to show that DSV can be made to be NP-hard to manipulate (in the EPWCB sense) if a particular declared strategy is imposed on the voters.
16
Favorite vs. compromise, revisited
ballots
so far
election
state
45 voters
A
C
B
35 voters
B
C
A
A: 45 votes
B: 35 votes
C: 0 votes
?
20 voters
C
B
A
17
Approval voting[Ottewell ’77] [Weber ’77] [Brams & Fishburn ’78]
strategic
ballots
45 voters
A
C
B
35 voters
B
C
A
20 voters
C
B
A
B: 55 votes
A: 45 votes
C: 20 votes
final
election
state
insincerityavoided
18
Themes of research
• Approval voting systems• Susceptibility to insincere manipulation
– encouraging sincere ballots
• Effectiveness of various strategies• Internalizing insincerity
– separating manipulation from the voter
• Complexity issues– complexity of manipulation– complexity of calculating the outcome
19
Strands of proposed 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 > 1 m = 1 winner
Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k > 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]
20
Strands of proposed 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 > 1 m = 1 winner
Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k > 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]
21
Strands of proposed 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 > 1 m = 1 winner
Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k > 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
Approval ratings
• Voters are asked about one alternative: Approve or disapprove?– like a Presidential approval rating– typically, average is reported
• Why not allow votes between 0 (full disapproval) and 1 (full approval) and then average them?– like metacritic.com
• Let’s see what happens when voters are strategic
23
One approach: Average
9.,6.,2.,1.,0v
9.,6.,2.,1.,0r
0 136.
outcome: 36.)( vfavg
24
One approach: Average
0 144.
9.,1,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 44.)( vfavg
25
Another approach: Median
0 12.
9.,6.,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 2.)( vfmed
26
Another approach: Median
0 12.
9.,1,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 2.)( vfmed
27
Another approach: Median
• Nonmanipulable– 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
ii rv imed vvf )(
imed vvf )( iv
iv
1,1,1,1,0,0,0v
1)( vfmed
)(vfmed
)(vfmed
28
Average with Declared-Strategy Voting?
electionstate
cardinal
preferences
rational
strategizer
ballot
outcome
• So Median is far from ideal—what now?– try using Average protocol in DSV context
• But what’s the rational Average strategy?
29
Rational Average strategy
• 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
ni 1iv
ir
)1),0,min(max(
ij jii vnrviavg rvf )(
ij jii vnrv
iavg rvf )(
1iv
0iviavg rvf )(
iavg rvf )(
30
Multiple equilibria are possible
outcome in each case:
5.)( vfavg
1,1,5.,0,0v 8.,5.,5.,3.,2.r
1,9.,6.,0,0v
1,75.,75.,0,0v
Multiple equilibria always have same average(proof in written proposal)
31
An equilibrium always exists?
• At equilibrium, must satisfy
I propose to prove that, given a vector , at least one equilibrium exists.
• If an equilibrium always exists, then average at equilibrium can be defined as a function, .
• Applying to instead of gives a new system, Average-approval-rating DSV.
)1),0,min(max()(
ij jii vnrviv
r
)(rfaveq
v
r
aveqf
32
Average-approval-rating DSV
0 14.
9.,6.,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 4.)( vfaveq
33
Average-approval-rating DSV
0 14.
9.,1,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 4.)( vfaveq
34
• AAR DSV could be manipulated if some voter i could gain an outcome closer to ideal by voting insincerely ( ).
I propose to show that Average-approval-rating DSV cannot be manipulated by insincere voters.
ii rv
Average-approval-rating DSV
35
• AAR DSV could be manipulated if some voter i could gain an outcome closer to ideal by voting insincerely ( ).
I propose to show that Average-approval-rating DSV cannot be manipulated by insincere voters.
• Intuitively, if , increasing will not change .
ii rv
iaveq vvf )(
iv)(vfaveq
Average-approval-rating DSV
36
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
1d
1010:, 2 yxyx
37
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]
1:, 222 yxyx
38
Higher-dimensional outcome space
• Average-approval-rating DSV can be generalized– optimal strategy moves the result as close to sincere
ideal as possible (by Euclidean distance)
I propose to find the optimal strategy for Average in the case and determine whether the resulting DSV system is rotationally invariant and/or nonmanipulable by insincere voters.
1:, 222 yxyx
39
Strands of proposed 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 > 1 m = 1 winner
Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k > 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]
40
Approval strategies for DSV
• Rational plurality strategy has been well explored [Cranor & Cytron, ’96]
• But what about approval strategy?• If each alternative’s 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
• Let’s look at several approval strategies and approaches to evaluating their effectiveness
41
DSV-style approval strategies
• Strategy Z [Merrill ’88]:– Approve alternatives with higher-than-average cardinal
preference (zero-information strategy)
]10,15,25,30[s
]3.,8.,1,0[p
]0,1,1,0[b Z recommends:
42
DSV-style approval strategies
• Strategy T [Ossipoff ’02]:– Approve favorite of top two vote-getters, plus all liked
more
]10,15,25,30[s
]3.,8.,1,0[p
]0,0,1,0[b T recommends:
43
DSV-style approval strategies
• Strategy J [Brams & Fishburn ’83]:– Use strategy Z if it distinguishes between top two vote-
getters; otherwise use strategy T
]10,15,25,30[s
]3.,8.,1,0[p
]0,1,1,0[b J recommends:
44
DSV-style approval strategies
• Strategy A:– Approve all preferred to top vote-getter, plus top vote-
getter if preferred to second-highest vote-getter
But how to evaluate these strategies?
]10,15,25,30[s
]3.,8.,1,0[p
]1,1,1,0[bA recommends:
45
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 alternative’s probability of eventually winning
• How to calculate those probabilities?
46
Election-state-evaluation:Merrill metric
k
jjs
is
x
iw
1
• Estimate an alternative’s probability of winning to be proportional to its current vote total raised to some power x [Merrill ’88]
47
Strategy comparison using the Merrill metric
],,[ 321 ssss
321 sss ],,[ 321 pppp
Current election state
Focal voter’s preferences
321 ppp 231 ppp
312 ppp
132 ppp
213 ppp
123 ppp
[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)
48
Strategy comparison using the Merrill metric
],,[ 321 ssss
321 sss ],,[ 321 pppp
Current election state
Focal voter’s preferences 132 ppp
[0, 1, 1] (A)
[0, 1, 0] (T) xxx
xxx
sss
spspspV
321
332211]0,1,0[
1
1
xxx
xxx
sss
spspspV
11
11
321
332211]1,1,0[
expected values of possible next election states:
49
Strategy comparison using the Merrill metric
],,[ 321 ssss
321 sss ],,[ 321 pppp
Current election state
Focal voter’s preferences 132 ppp
xxx
xxx
xxx
xxx
sss
spspsp
sss
spspsp
321
332211
321
332211
1
1
11
11
so T is better than A only when:
x
s
s
pp
pp
12
1
13
32
or, equivalently:
50
Strategy comparison using the Merrill metric
],,[ 321 ssss
321 sss ],,[ 321 pppp
Current election state
Focal voter’s preferences 132 ppp
xxx
xxx
xxx
xxx
sss
spspsp
sss
spspsp
321
332211
321
332211
1
1
11
11
so T is better than A only when:
x
s
s
pp
pp
12
1
13
32
or, equivalently:
Intuitively, T does better than A only when:
• s1 and s2 are relatively close
• x is relatively small
• p3 is relatively close to p1 compared to p2
51
Strategy comparison using the Merrill metric
],,[ 321 ssss
321 sss ],,[ 321 pppp
Current election state
Focal voter’s preferences 132 ppp
Corollaries:– When x is taken to infinity and , strategy A
dominates strategy T– When
, strategy A dominates strategy T
121 ss
221
3
ppp
x
s
s
pp
pp
12
1
13
32T is better than A only when:
52
Approval strategy evaluation
I propose to extend this 3-alternative result to strategy pairs A vs. J, T vs. J and A vs. Z.
I propose to extend this result to strategy pairs A vs. T and A vs. J in the 4-alternative case.
53
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 voter’s cardinal preferences are tie-free:
when– and the Merrill-metric exponent x is taken to infinity
then strategy A dominates all other strategies according to the Merrill metric
• (proof in written proposal)
121 321 kssss k
ji pp ji
54
Election-state-evaluation:Branching-probabilities metric
• Estimate an alternative’s 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
1Bi
ip1p
kp22p
55
Approval strategy evaluation
I propose to extend the Merrill-metric results to strategy pairs A vs. T, A vs. J, T vs. J and A vs. Z in the 3-alternative case using the branching-probabilities metric.
I propose to determine whether strategy A dominates all others in the near-tie-free case using the branching-probabilities metric as the number of future ballots goes to infinity.
56
Strands of proposed 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 > 1 m = 1 winner
Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?
k > 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]
57
Electing a committee from approval ballots
11110 00011
00111
0000110111
01111
•What’s the best committee of size m = 2?
approves ofcandidates
4 and 5k = 5 candidates
n = 6 ballots
58
Sum of Hamming distances
11110 00011
00111
0000110111
01111 110004 5
2 4
4 3 sum = 22
m = 2 winners
59
Fixed-size minisum
11110 00011
00111
0000110111
01111 00011
•Minisum elects winner set with smallest sumscore•Easy to compute (pick candidates with most approvals)
2 1
4 0
2 1 sum = 10
m = 2 winners
60
Maximum Hamming distance
11110 00011
00111
0000110111
01111 000112 1
4 0
2 1 sum = 10max = 4
m = 2 winners
61
Fixed-size minimax
•Minimax elects winner set with smallest maxscore•Harder to compute?
11110 00011
00111
0000110111
01111 001102 1
2 2
2 3 sum = 12max = 3
m = 2 winners
[Brams, Kilgour & Sanver ’04]
62
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)
?
63
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
(proof in written proposal)
64
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 + ε that runs in time polynomial in the input and exponential in 1/ε
65
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.
(proof in written proposal)
no known PTAS;
has a 3-approx.
(proof in written proposal)
* Polynomial-Time Approximation Scheme: algorithm with approx. ratio 1 + ε that runs in time polynomial in the input and exponential in 1/ε
66
Approximating FSM
00111
00001
10111
01111
00011
11110
00111
m = 2 winners
choosea ballot
arbitrarily
67
Approximating FSM
00111
00001
10111
01111
00011
11110
0010100111coerce to
size m
m = 2 winners
choosea ballot
arbitrarily
outcome =m-completed ballot
68
Approximation ratio ≤ 3
00111
00001
10111
01111
00011
11110
00110
2
2
1
3
2
2
≤ OPT
optimalFSM set
OPT = optimal maxscore
69
Approximation ratio ≤ 3
00111
00001
10111
01111
00011
11110
00110 00111
2
2
1
3
2
2
1
≤ OPT ≤ OPT
optimalFSM set
chosenballot
OPT = optimal maxscore
70
Approximation ratio ≤ 3
00111
00001
10111
01111
00011
11110
00110 00111 00011
2
2
1
3
2
2
1 1
≤ OPT ≤ OPT ≤ OPT
≤ 3·OPT
optimalFSM set
chosenballot
m-completedballot
OPT = optimal maxscore (by triangle inequality)
71
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
72
Local search approach for FSM
1. Start with some c {0,1}k of weight m
010014
73
Local search approach for FSM
1. Start with some c {0,1}k 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
01001
11000 10001
01100
01010 00011
001014
44
4
5
4
4
74
Local search approach for FSM
1. Start with some c {0,1}k 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
010104
75
Local search approach for FSM
1. Start with some c {0,1}k 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
010104
76
Local search approach for FSM
1. Start with some c {0,1}k 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
01010
11000 10010
01100
01001 00011
001104
44
4
5
3
4
77
Local search approach for FSM
1. Start with some c {0,1}k 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
001103
78
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 < 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)
79
Manipulating FSM
00110 00011
00111
0000110111
01111 00011
•Voters are sincere
•Another optimal solution: 00101
2 1
2 0
2 1
max = 2
m = 2 winners
80
Manipulating FSM
11110 00011
00111
0000110111
01111 00110
•A voter manipulates and realizes ideal outcome
•But our 3-approximation for FSM is nonmanipulable
2 1
2 2
2 3
00110
0
max = 3
m = 2 winners
81
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
82
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, nonmanipulability of AAR DSV, analysis of Average in planar disk
DSV-style approval strategies
Completed: 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
x
83
Fin
Thanks to– my adviser, Ron Cytron– Morgan Deters and the rest of the DOC Group– co-authors Vangelis Markakis and Aranyak Mehta– my committee
Questions?
84
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)
1)( ii vrvi
ii rvvi 0)(0)( ii vrvi
irvinv : nvrvi i :
85
Proof: Only one equilibrium average
irinA :)( nriB i :)(
212211 )()()()( BABA
• Theorem:
• Proof considers two symmetric cases:– assume– assume
• Each leads to a contradiction
21
12
86
Proof: Only one equilibrium average
21 case 1:
ii rri 12)( ii riri 12 :: ii riri 12 ::
irin 22 : nri i 11:
nririn ii 1122 :: nn 12
12 21 , contradicting
)( 2A)( 1B
87
Proof: Only one equilibrium average
21 Case 1 shows that
Case 2 is symmetrical and shows that 12
21 Therefore
Therefore, given , the average at equilibrium is uniquer
88
Specific FSM heuristics
• Two parameters:– where to start vector c:
1. a fixed-size-minisum solution
2. a m-completion of a ballot (3-approx.)
3. a random set of m candidates
4. a m-completion of a ballot with highest maxscore– radius of neighborhood r: 1 and 2
89
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 paper)
90
Average approx. ratios found
radius = 1 radius = 2fixed-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
91
Largest approx. ratios found
radius = 1 radius = 2fixed-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
92
Conclusions from all experiments
• All heuristics perform near-optimally– highest ratio found: 1.2– highest average ratio < 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)