Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington...
-
Upload
shauna-bishop -
Category
Documents
-
view
216 -
download
0
Transcript of Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington...
![Page 1: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/1.jpg)
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
![Page 2: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/2.jpg)
2
Let’s vote!
45 voters
A
C
B
sincere
preferences
(1st)
(2nd)
(3rd)
35 voters
B
C
A
20 voters
C
B
A
![Page 3: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/3.jpg)
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
![Page 4: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/4.jpg)
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
![Page 5: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/5.jpg)
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!
![Page 6: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/6.jpg)
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
![Page 7: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/7.jpg)
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
![Page 8: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/8.jpg)
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
![Page 9: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/9.jpg)
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
![Page 10: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/10.jpg)
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
![Page 11: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/11.jpg)
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
![Page 12: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/12.jpg)
12
Declared-Strategy Voting[Cranor & Cytron ’96]
electionstate
cardinal
preferences
rational
strategizer
ballot
outcome
![Page 13: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/13.jpg)
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
![Page 14: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/14.jpg)
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
![Page 15: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/15.jpg)
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.
![Page 16: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/16.jpg)
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
![Page 17: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/17.jpg)
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
![Page 18: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/18.jpg)
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
![Page 19: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/19.jpg)
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]
![Page 20: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/20.jpg)
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]
![Page 21: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/21.jpg)
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]
![Page 22: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/22.jpg)
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
![Page 23: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/23.jpg)
23
One approach: Average
9.,6.,2.,1.,0v
9.,6.,2.,1.,0r
0 136.
outcome: 36.)( vfavg
![Page 24: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/24.jpg)
24
One approach: Average
0 144.
9.,1,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 44.)( vfavg
![Page 25: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/25.jpg)
25
Another approach: Median
0 12.
9.,6.,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 2.)( vfmed
![Page 26: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/26.jpg)
26
Another approach: Median
0 12.
9.,1,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 2.)( vfmed
![Page 27: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/27.jpg)
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
![Page 28: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/28.jpg)
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?
![Page 29: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/29.jpg)
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 )(
![Page 30: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/30.jpg)
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)
![Page 31: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/31.jpg)
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
![Page 32: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/32.jpg)
32
Average-approval-rating DSV
0 14.
9.,6.,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 4.)( vfaveq
![Page 33: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/33.jpg)
33
Average-approval-rating DSV
0 14.
9.,1,2.,1.,0v
9.,6.,2.,1.,0r
outcome: 4.)( vfaveq
![Page 34: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/34.jpg)
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
![Page 35: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/35.jpg)
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
![Page 36: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/36.jpg)
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
![Page 37: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/37.jpg)
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
![Page 38: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/38.jpg)
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
![Page 39: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/39.jpg)
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]
![Page 40: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/40.jpg)
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
![Page 41: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/41.jpg)
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:
![Page 42: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/42.jpg)
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:
![Page 43: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/43.jpg)
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:
![Page 44: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/44.jpg)
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:
![Page 45: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/45.jpg)
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?
![Page 46: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/46.jpg)
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]
![Page 47: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/47.jpg)
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)
![Page 48: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/48.jpg)
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:
![Page 49: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/49.jpg)
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:
![Page 50: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/50.jpg)
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
![Page 51: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/51.jpg)
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:
![Page 52: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/52.jpg)
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.
![Page 53: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/53.jpg)
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
![Page 54: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/54.jpg)
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
![Page 55: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/55.jpg)
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.
![Page 56: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/56.jpg)
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]
![Page 57: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/57.jpg)
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
![Page 58: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/58.jpg)
58
Sum of Hamming distances
11110 00011
00111
0000110111
01111 110004 5
2 4
4 3 sum = 22
m = 2 winners
![Page 59: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/59.jpg)
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
![Page 60: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/60.jpg)
60
Maximum Hamming distance
11110 00011
00111
0000110111
01111 000112 1
4 0
2 1 sum = 10max = 4
m = 2 winners
![Page 61: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/61.jpg)
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]
![Page 62: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/62.jpg)
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)
?
![Page 63: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/63.jpg)
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)
![Page 64: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/64.jpg)
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
![Page 65: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/65.jpg)
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
![Page 66: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/66.jpg)
66
Approximating FSM
00111
00001
10111
01111
00011
11110
00111
m = 2 winners
choosea ballot
arbitrarily
![Page 67: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/67.jpg)
67
Approximating FSM
00111
00001
10111
01111
00011
11110
0010100111coerce to
size m
m = 2 winners
choosea ballot
arbitrarily
outcome =m-completed ballot
![Page 68: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/68.jpg)
68
Approximation ratio ≤ 3
00111
00001
10111
01111
00011
11110
00110
2
2
1
3
2
2
≤ OPT
optimalFSM set
OPT = optimal maxscore
![Page 69: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/69.jpg)
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
![Page 70: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/70.jpg)
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)
![Page 71: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/71.jpg)
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
![Page 72: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/72.jpg)
72
Local search approach for FSM
1. Start with some c {0,1}k of weight m
010014
![Page 73: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/73.jpg)
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
![Page 74: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/74.jpg)
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
![Page 75: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/75.jpg)
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
![Page 76: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/76.jpg)
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
![Page 77: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/77.jpg)
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
![Page 78: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/78.jpg)
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)
![Page 79: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/79.jpg)
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
![Page 80: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/80.jpg)
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
![Page 81: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/81.jpg)
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
![Page 82: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/82.jpg)
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
![Page 83: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/83.jpg)
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?
![Page 84: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/84.jpg)
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 :
![Page 85: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/85.jpg)
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
![Page 86: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/86.jpg)
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
![Page 87: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/87.jpg)
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
![Page 88: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/88.jpg)
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
![Page 89: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/89.jpg)
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)
![Page 90: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/90.jpg)
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
![Page 91: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/91.jpg)
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
![Page 92: Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649d125503460f949e5e62/html5/thumbnails/92.jpg)
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)