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@cse.wustl.edu
Ron Cytron Robert Pless
Steven Brams Itai Sened
Jeremy Buhler Aaron Stump
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
2
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
3
Let’s vote!
45 voters 35 voters 20 voters
A B C (1st)
sincere
preferences C C B (2nd)
B A A (3rd)
4
Plurality voting
45 voters 35 voters 20 voters
A B C
sincere
ballots C C B
B A A
A: 45 votes
“zero-information”
result B: 35 votes
C: 20 votes
5
Plurality voting
45 voters 35 voters 20 voters
A B C
ballots
?
so far C C B
B A A
A: 45 votes
election
state B: 35 votes
C: 0 votes
6
Plurality voting
45 voters 35 voters 20 voters
A B C
strategic
ballots C C B insincerity!
B A A
B: 55 votes
final
[Gibbard ’73]
election A: 45 votes [Satterthwaite ’75]
state
C: 0 votes
7
Manipulation decision problem
45 voters 35 voters 20 voters
A B C
ballot
sets C C B
BV B A A BU
B: 55 votes
election
state A: 45 votes
C: 0 votes
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 0 1
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 BV BU?
• My generalization of problems from the literature:
[Bartholdi, Tovey & Trick ’89] [Conitzer & Sandholm ’02]
[Conitzer & Sandholm ’03]
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 0 1
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 BV BU?
• 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
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 0 1
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 BV BU?
• This problem is computationally easy (in P) for:
– plurality voting [Bartholdi, Tovey & Trick ’89]
– approval voting
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 0 1
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 BV BU?
• This problem is computationally infeasible (NP-hard) for:
– Hare (single-winner STV) [Bartholdi & Orlin ’91]
– Borda [Conitzer & Sandholm ’02]
12
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
13
Declared-Strategy Voting
[Cranor & Cytron ’96]
cardinal rational
preferences strategizer
ballot
election
outcome
state
14
Declared-Strategy Voting
[Cranor & Cytron ’96]
sincerity strategy
cardinal rational
preferences strategizer
ballot
election
outcome
state
• 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
15
What is a declared strategy?
A: 0.0
cardinal B: 0.6
preferences
C: 1.0 A: 0
declared voted
strategy
B: 1 ballot
current A: 45 C: 0
election B: 35
state
C: 0
• Captures thinking of a rational voter
16
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!)
17
Favorite vs. compromise, revisited
45 voters 35 voters 20 voters
A B C
ballots
?
so far C C B
B A A
A: 45 votes
election
state B: 35 votes
C: 0 votes
18
Approve both!
45 voters 35 voters 20 voters
A B C insincerity
strategic
avoided
ballots C C B
B A A
B: 55 votes
final
election A: 45 votes
state
C: 20 votes
19
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
20
Strands of research
number of outcome Area of research
alternatives
k=1 an approval Voters approve or disapprove a
rating single alternative. What is the
equilibrium approval rating?
k>1 m=1 Voters elect a winner by approval
winner voting. What DSV-style approval
strategies are most effective?
k>1 m≥1 Voters elect a set of alternatives
winners with approval ballots. Which set
most satisfies the least satisfied
voter? [Brams, Kilgour & Sanver ’04]
21
Strands of research
number of outcome Area of research
alternatives
k=1 an approval Voters approve or disapprove a
rating single alternative. What is the
equilibrium approval rating?
k>1 m=1 Voters elect a winner by approval
winner voting. What DSV-style approval
strategies are most effective?
k>1 m≥1 Voters elect a set of alternatives
winners with approval ballots. Which set
most satisfies the least satisfied
voter? [Brams, Kilgour & Sanver ’04]
22
Strands of research
number of outcome Area of research
alternatives
k=1 an approval Voters approve or disapprove a
rating single alternative. What is the
equilibrium approval rating?
k>1 m=1 Voters elect a winner by approval
winner voting. What DSV-style approval
strategies are most effective?
k>1 m≥1 Voters elect a set of alternatives
winners with approval ballots. Which set
most satisfies the least satisfied
voter? [Brams, Kilgour & Sanver ’04]
23
Approval ratings
24
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
25
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?
26
Approval ratings
27
Average of ratings
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0.4, 0.7, 0.8, 0.8, 0.9
outcome: f avg (v ) 0.72
0.72
0 1
data from Metacritic.com: Videodrome (1983)
28
Average of ratings
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0, 0.7, 0.8, 0.8, 0.9
outcome: f avg (v ) 0.64
0.64
0 1
Videodrome (1983)
29
Another approach: Median
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0.4, 0.7, 0.8, 0.8, 0.9
outcome: f m ed (v ) 0.8
0.8
0 1
Videodrome (1983)
30
Another approach: Median
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0, 0.7, 0.8, 0.8, 0.9
outcome: f m ed (v ) 0.8
0.8
0 1
Videodrome (1983)
31
Another approach: Median
• Immune to insincerity [LeGrand ’08]
– voter i cannot obtain a better result by voting vi ri
– if f m ed (v ) vi , increasing vi will not change f m ed (v )
– if f m ed (v ) vi , decreasing vi will not change f m ed (v )
• Allows tyranny by a majority
– v 0, 0, 0,1,1,1,1
m ed (v ) 1
– f
– no concession to the 0-voters
32
Average with Declared-Strategy Voting?
• So Median is far from ideal—what now?
– try using Average protocol in DSV context
cardinal rational
preferences strategizer
ballot
election
outcome
state
• But what’s the rational Average strategy?
• And will an equilibrium always be found?
33
Equilibrium-finding algorithm
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0.4, 0.7, 0.8, 0.8, 0.9
0.72
0 1
Videodrome (1983)
34
Equilibrium-finding algorithm
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0, 0, 0, 0, 0
0
1
35
Equilibrium-finding algorithm
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0, 0, 0, 0,1
0 .2
0 1
36
Equilibrium-finding algorithm
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0, 0, 0,1,1
0 .4
0 1
37
Equilibrium-finding algorithm
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0, 0,1,1,1
0 .6
0 1
38
Equilibrium-finding algorithm
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0, 0.5,1,1,1
equilibrium!
0 .7
0 1
• Is this algorithm is guaranteed to find an equilibrium?
39
Equilibrium-finding algorithm
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0, 0.5,1,1,1
equilibrium!
0 .7
0 1
• Is this algorithm is guaranteed to find an equilibrium?
• Yes! [LeGrand ’08]
40
Expanding range of allowed votes
r 0.4, 0.7, 0.8, 0.8, 0.9
v 1, 1, 2, 2, 2
0.8
1 2
• These results generalize to any range [LeGrand ’08]
41
Multiple equilibria can exist
r 0.4, 0.7, 0.7, 0.8, 0.9
v 0, 0.5,1,1,1
v 0, 0.6, 0.9,1,1
v 0, 0.75, 0.75,1,1
outcome in each case:
f avg (v ) 0.7
• Will multiple equilibria will always have the same average?
42
Multiple equilibria can exist
r 0.4, 0.7, 0.7, 0.8, 0.9
v 0, 0.5,1,1,1
v 0, 0.6, 0.9,1,1
v 0, 0.75, 0.75,1,1
outcome in each case:
f avg (v ) 0.7
• Will multiple equilibria will always have the same average?
• Yes! [LeGrand ’08]
43
Average-Approval-Rating DSV
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0.4, 0.7, 0.8, 0.8, 0.9
outcome: f aveq (v , 0,1) 0.7
0 .7
0 1
Videodrome (1983)
44
Average-Approval-Rating DSV
r 0.4, 0.7, 0.8, 0.8, 0.9
v 0, 0.7, 0.8, 0.8, 0.9
outcome: f aveq (v , 0,1) 0.7
0 .7
0 1
• AAR DSV is immune to insincerity in general [LeGrand ’08]
45
Evaluating AAR DSV systems
• Expanded vote range gives wide range of AAR
DSV systems: a ,b (v ) 0 a 1 0 b 1
• If we could assume sincerity, we’d 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
46
Evaluating AAR DSV systems
b 0.5
RMSEa,0.5
a
minimum at a 0.3240 47
Evaluating AAR DSV systems: hill-climbing
b 0.4820
RMSEa,0.4820
a
minimum at a 0.3647 48
Evaluating AAR DSV systems: hill-climbing
a 0.3647
RMSE0.3647,b
b
minimum at b 0.4820 49
Evaluating AAR DSV systems
0.3647,0.4820(v )
f avg (v )
50
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
51
Strands of research
number of outcome Area of research
alternatives
k=1 an approval Voters approve or disapprove a
rating single alternative. What is the
equilibrium approval rating?
k>1 m=1 Voters elect a winner by approval
winner voting. What DSV-style approval
strategies are most effective?
k>1 m≥1 Voters elect a set of alternatives
winners with approval ballots. Which set
most satisfies the least satisfied
voter? [Brams, Kilgour & Sanver ’04]
52
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
53
DSV-style approval strategies
s [30, 25,15,10]
p [0, 1, 0.8, 0.3]
• Strategy Z: b [0, 1, 1, 0]
– Approve alternatives with higher-than-average cardinal
preference (zero-information strategy) [Merrill ’88]
54
DSV-style approval strategies
s [30, 25,15,10]
p [0, 1, 0.8, 0.3]
• Strategy Z: b [0, 1, 1, 0]
• Strategy T: b [0, 1, 0, 0]
– Approve favorite of top two vote-getters, plus all liked
more [Ossipoff ’02, Poundstone ’08]
– Simplest generalization of plurality DSV strategy
[Cranor & Cytron ’96]
55
DSV-style approval strategies
s [30, 25,15,10]
p [0, 1, 0.8, 0.3]
• Strategy Z: b [0, 1, 1, 0]
• Strategy T: b [0, 1, 0, 0]
• Strategy J: b [0, 1, 1, 0]
– Use strategy Z if it distinguishes between top two vote-
getters; otherwise use strategy T [Brams & Fishburn ’83]
56
DSV-style approval strategies
s [30, 25,15,10]
p [0, 1, 0.8, 0.3]
• Strategy Z: b [0, 1, 1, 0]
• Strategy T: b [0, 1, 0, 0]
• Strategy J: b [0, 1, 1, 0]
• Strategy A: b [0, 1, 1, 1]
– 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?
57
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?
58
Election-state-evaluation:
Merrill metric
• Estimate an alternative’s probability of winning to
be proportional to its current vote total raised to
some power x [Merrill ’88]
x
s
Wi k
i
s
j 1
x
j
59
Strategy comparison using the Merrill metric
Current election state s [ s1 , s2 , s3 ] s1 s2 s3
Focal voter’s preferences p [ p1 , p2 , p3 ]
p1 p2 p3 [1, 0, 0] (strategies A & T)
p1 p3 p2 [1, 0, 0] (A & T)
p2 p1 p3 [0, 1, 0] (A & T)
p2 p3 p1 [0, 1, 1] (A); [0, 1, 0] (T)
p3 p1 p2 [1, 0, 1] (A & T)
p3 p2 p1 [0, 1, 1] (A & T)
60
Strategy comparison using the Merrill metric
Current election state s [ s1 , s2 , s3 ] s1 s2 s3
Focal voter’s preferences p [ p1 , p2 , p3 ] p2 p3 p1
When p2 p3 p1 , A is better than T if and only if:
p1s1x p2 s2 1 p3 s3 1 p1s1x p2 s2 1 p3 s3
x x xx
s1 s2 1 s3 1 s1x s2 1 s3
x x x x x
or, equivalently:
x
p2 p3 s1 Intuitively, A always does better than T when:
s 1
• s1 is much larger than s2,
p3 p1 2 • x is large, or
• p3 is relatively close to p2 compared to p1
61
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
62
Further result for strategy A
More generally, it is true that if
– the election state is free of ties and near-ties:
s1 s2 1 s3 2 sk k 1
– and the focal voter’s cardinal preferences are tie-free:
pi p j when i j
– 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]
63
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
p1
p2 k p
iB
i 1
p2
64
Branching-probabilities metric: strategy A
It is true that if
– the election state is free of ties and near-ties:
s1 s2 1 s3 2 sk k 1
– and the focal voter’s cardinal preferences are tie-free:
pi p j when i j
– and the number of future ballots is taken to infinity
then strategy A dominates all other approval
strategies according to the branching-probabilities
metric [LeGrand ’08]
65
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?
66
Strands of research
number of outcome Area of research
alternatives
k=1 an approval Voters approve or disapprove a
rating single alternative. What is the
equilibrium approval rating?
k>1 m=1 Voters elect a winner by approval
winner voting. What DSV-style approval
strategies are most effective?
k>1 m≥1 Voters elect a set of alternatives
winners with approval ballots. Which set
most satisfies the least satisfied
voter? [Brams, Kilgour & Sanver ’04]
67
Electing a committee from approval ballots
approves of
k = 5 alternatives 11110 00011 alternatives
4 and 5
n = 6 ballots
01111 00111
10111 00001
•What’s the best committee of size m = 2?
68
Sum of Hamming distances
m = 2 winners 11110 00011
2 4
4 5
01111 11000 00111
4 3 sum = 22
10111 00001
•What if we elect alternatives 1 and 2?
69
Fixed-size minisum
m = 2 winners 11110 00011
4 0
2 1
01111 00011 00111
2 1 sum = 10
10111 00001
•Minisum elects winner set with smallest HD sum
•Easy to compute (pick alternatives with most approvals)
70
Maximum Hamming distance
m = 2 winners 11110 00011
4 0
2 1
01111 00011 00111
2 1 sum = 10
max = 4
10111 00001
•One voter is quite unhappy with minisum outcome
71
Fixed-size minimax
[Brams, Kilgour & Sanver ’04]
m = 2 winners 11110 00011
2 2
2 1
01111 00110 00111
2 3 sum = 12
max = 3
10111 00001
•Minimax elects winner set with smallest maximum HD
•Harder to compute?
72
Complexity
Endogenous minimax Bounded-size minimax Fixed-size minimax
= EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m)
NP-hard NP-hard
?
[Frances & Litman ’97] (generalization of EM)
73
Complexity
Endogenous minimax Bounded-size minimax Fixed-size minimax
= EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m)
NP-hard NP-hard NP-hard
[Frances & Litman ’97] (generalization of EM) [LeGrand ’04]
74
Approximability
Endogenous minimax Bounded-size minimax Fixed-size minimax
= EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m)
has a PTAS* no known PTAS no known PTAS
[Li, Ma & Wang ’99]
* Polynomial-Time Approximation Scheme: algorithm
with approx. ratio 1 + ε that runs in time polynomial in
the input and exponential in 1/ε
75
Approximability
Endogenous minimax Bounded-size minimax Fixed-size minimax
= EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m)
has a PTAS* no known PTAS; no known PTAS;
has a 3-approx. has a 3-approx.
[Li, Ma & Wang ’99]
[LeGrand, Markakis & [LeGrand, Markakis &
Mehta ’06] Mehta ’06]
* Polynomial-Time Approximation Scheme: algorithm
with approx. ratio 1 + ε that runs in time polynomial in
the input and exponential in 1/ε
76
Susceptibility to insincerity
Endogenous minimax Bounded-size minimax Fixed-size minimax
= EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m)
insincere voters insincere voters insincere voters
can benefit can benefit can benefit
[LeGrand, Markakis & [LeGrand, Markakis & [LeGrand, Markakis &
Mehta ’06] Mehta ’06] Mehta ’06]
But our 3-approximation for FSM is
immune to insincere strategy!
77
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?
78
Rational [m,M]-Average strategy
• Allow votes between m 0 and M 1
• For 1 i n, voter i should choose vi to move
outcome as close to ri as possible
• Choosing vi ri n j i v j would give f avg (v ) ri
• Optimal vote is vi min(max( ri n j i v j , m), M )
• After voter i uses this strategy, one of these is true:
– f avg (v ) ri and vi M
– f avg (v ) ri
– f avg (v ) ri and vi m
79
What happens at equilibrium?
• The optimal strategy recommends that no voter
change
• So (i ) v ri vi 1
• And (i ) v ri vi 0
– equivalently, (i) vi 0 v ri
• Therefore any average at equilibrium must satisfy
two equations:
– (A) v n i : v ri
– (B) i : v ri v n
80
Proof: Only one equilibrium average
A( ) n i : ri
B( ) i : ri n
• Theorem:
A(1 ) B(1 ) A(2 ) B(2 ) 1 2
• Proof considers two symmetric cases:
– assume 1 2
– assume 2 1
• Each leads to a contradiction
81
Proof: Only one equilibrium average
case 1: 1 2
(i) 2 ri 1 ri
i : 2 ri i : 1 ri
i : 2 ri i : 1 ri
2 n i : 2 ri A(2 )
i : 1 ri 1n B(1 )
2 n i : 2 ri i : 1 ri 1n
2n 1n
2 1 , contradicting 1 2
82
Proof: Only one equilibrium average
Case 1 shows that 1 2
Case 2 is symmetrical and shows that 2 1
Therefore 1 2
Therefore, given r , the average at equilibrium is unique
83
An equilibrium always exists?
• At equilibrium, v must satisfy
(i) vi min(max( ri n j i v j , m), M )
I proposed to prove that, given a vector r , at least
one equilibrium exists.
A particular algorithm will always find an equilibrium
for any r . . .
84
An equilibrium always exists!
Equilibrium-finding algorithm:
• sort r so that (i j ) ri rj
• for i = 1 up to n do
vi min(max( ri n k i vk (n i)m, m), M )
(full proof and more efficient algorithm in dissertation)
• Since an equilibrium always exists, average at
equilibrium is a function, f aveq (r , m, M ) .
• Applying f aveq to v instead of r gives a new
system, Average-Approval-Rating DSV.
85
Average-Approval-Rating DSV
• What if, under AAR DSV, voter i could gain an
outcome closer to ideal by voting insincerely
( vi ri )?
I proposed to prove that Average-Approval-Rating
DSV is immune to strategy by insincere voters.
• Intuitively, if f aveq (v , m, M ) vi, increasing v i
will not change f aveq (v , m, M ) .
86
AAR DSV is immune to strategy
• If f aveq (v , m, M ) vi ri,
– increasing v i will not change f aveq (v , m, M ).
– decreasing v i will not increase f aveq (v , m, M ) .
• If f aveq (v , m, M ) vi ri,
– increasing v i will not decrease f aveq (v , m, M ) .
– decreasing v i will not change f aveq (v , m, M ) .
(complete proof in dissertation)
• So voting sincerely ( v r ) is guaranteed to
i i
optimize the outcome from voter i’s point of view
87
Parameterizing AAR DSV
• [m,M]-AAR DSV can be parameterized nicely using
a and b, where 0 a 1 and 0 b 1:
1 m
a b
M m 1 M m
b 1 b
m b M b
a a
b 1 b
a ,b (v ) lim f aveq v , b , b
x a x x
88
Parameterizing AAR DSV
• For example:
1,b (v ) f aveq (v , 0,1)
1 1 (v ) f aveq v , 1, 2
,
3 2
1 1 (v ) f aveq v , 10,11
,
21 2
1 (v ) f m ed v
0,
2
0,0 (v ) maxv
0,1 (v ) min v 89
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
0 a 1 0 b 1
SEa,b v a,b v f avg v
2
v SEa,b v
RMSEa ,b V vV
v
vV 90
Higher-dimensional outcome space
• What if votes and outcomes exist in d 1
dimensions?
• Example: x, y 2 : 0 x 1 0 y 1
• If dimensions are independent, Average, Median
and Average-approval-rating DSV can operate
independently on each dimension
– Results from one dimension transfer
91
Higher-dimensional outcome space
• But what if the dimensions are not independent?
– say, outcome space is a disk in the plane:
x, y : x2 y2 1
2
• 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]
92
Strategy comparison using the Merrill metric
Current election state s [ s1 , s2 , s3 ] s1 s2 s3
Focal voter’s preferences p [ p1 , p2 , p3 ] p2 p3 p1
expected values of possible next election states:
p1s1x p2 s2 1 p3 s3 1
x x
V[ 0,1,1] [0, 1, 1] (A)
s1x s2 1 s3 1
x x
p1s1x p2 s2 1 p3 s3x
x
V[ 0,1,0 ] [0, 1, 0] (T)
s1x s2 1 s3x
x
93
Strategy comparison using the Merrill metric
Current election state s [ s1 , s2 , s3 ] s1 s2 s3
Focal voter’s preferences p [ p1 , p2 , p3 ] p2 p3 p1
so T is better than A only when:
p1s1x p2 s2 1 p3 s3 1 p1s1x p2 s2 1 p3 s3
x x x x
s1 s2 1 s3 1 s1x s2 1 s3
x x x x x
or, equivalently:
x
p2 p3 s1
s 1
p3 p1 2
94
Strategy comparison using the Merrill metric
Current election state s [ s1 , s2 , s3 ] s1 s2 s3
Focal voter’s preferences p [ p1 , p2 , p3 ] p2 p3 p1
x
T is better than A only when:
p2 p3 s1
s 1
p3 p1 2
Corollaries:
– When x is taken to infinity and s s 1 strategy A
,
1 2
dominates strategy T
– When p1 p2
p3 , strategy A dominates strategy T
2
95
Further result for strategy A
V[ 0, 0,...0] p1 , p2 , pk
s1 , s2 , sk
s1 s2 sk
• just a weighted average of pi values
• assume p1 p2
• as x , V[ 0,0,...0] p1 from below
• so maximized when weights of those pi p1 are
maximized, which is done by approving only alternatives i
where pi p1
• p2 p1 case is similar: approve i where pi p1
• only strategy A always does this
96
Approximating FSM
11110 m = 2 winners
00011
00111
00111
00001 choose
a ballot
10111
arbitrarily
01111
97
Approximating FSM
11110 m = 2 winners
00011
00111
coerce to
00111 00101
00001 size m
choose
a ballot
10111
arbitrarily
01111
outcome =
m-completed ballot
98
Approximation ratio ≤ 3
optimal
11110
2 FSM set
00011 2
00111 1
00110
3
00001
2
10111
2
01111
≤ OPT
OPT = optimal maxscore
99
Approximation ratio ≤ 3
optimal chosen
11110
2 FSM set ballot
00011 2
00111 1
1
00110 00111
3
00001
2
10111
2
01111
≤ OPT ≤ OPT
OPT = optimal maxscore
100
Approximation ratio ≤ 3
optimal chosen m-completed
11110
2 FSM set ballot ballot
00011 2
00111 1
1 1
00110 00111 00011
3
00001
2
10111
2
01111
≤ OPT ≤ OPT ≤ OPT
(by triangle inequality)
OPT = optimal maxscore ≤ 3·OPT
101
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
102
Local search approach for FSM
1. Start with some c {0,1}k
of weight m
01001
4
103
Local search approach for FSM
1. Start with some c {0,1}k
of weight m
11000 10001
2. In c, swap up to r 0-bits 5 4
with 1-bits in such a way
01100 01001 00101
that minimizes the 4 4 4
maxscore of the result
01010 00011
4 4
104
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
01010
4
105
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
01010
that minimizes the 4
maxscore of the result
106
Local search approach for FSM
1. Start with some c {0,1}k
of weight m
11000 10010
2. In c, swap up to r 0-bits 5 4
with 1-bits in such a way
01100 01010 00110
that minimizes the 4 4 3
maxscore of the result
01001 00011
3. Repeat step 2 until 4 4
maxscore(c) is
unchanged k times
4. Take c as the solution
107
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
00110
that minimizes the 3
maxscore of the result
3. Repeat step 2 until
maxscore(c) is
unchanged k times
4. Take c as the solution
108
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 (maxscore of solution found)
– highest average ratio < 1.04 (maxscore of exact solution)
• 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
109
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)
110
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
111
Average approx. ratios found
radius = 1 radius = 2
fixed-size 1.0012 1.0000
minimax
3-approx. 1.0017 1.0000
random 1.0057 1.0000
set
highest- 1.0059 1.0000
maxscore
performance on GTS ’03 election data
k = 24 candidates, m = 12 winners, n = 161 ballots
112
Largest approx. ratios found
radius = 1 radius = 2
fixed-size 1.0714 1.0000
minimax
3-approx. 1.0714 1.0000
random 1.0714 1.0000
set
highest- 1.0714 1.0000
maxscore
performance on GTS ’03 election data
k = 24 candidates, m = 12 winners, n = 161 ballots
113
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)
114
Manipulating FSM
00110 00011 m = 2 winners
2 0
2 1
01111 00011 00111
2 1
max = 2
10111 00001
•Voters are sincere
•Another optimal solution: 00101 115
Manipulating FSM
00110
11110 00011 m = 2 winners
0 2 2
2 1
01111 00110 00111
2 3
max = 3
10111 00001
•A voter manipulates and realizes ideal outcome
•But our 3-approximation for FSM is nonmanipulable
116
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
117
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 Completed: Merrill-metric comparison of A and T in 3-
approval alt. case, domination of A as x
strategies To do: comparisons of other pairs, analysis using
branching-probabilities metric
Fixed-size Completed: NP-hardness proof, 3-approximation,
minimax heuristic evaluation, manipulability analysis
118