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Computational Aspects of Approval Voting and

Declared-Strategy Voting

- Dissertation defense
- 17 April 2008

Rob LeGrand Washington University in St. Louis

Computer Science and Engineering legrand_at_cse.wustl

.edu

Robert Pless Itai Sened Aaron Stump

Ron Cytron Steven Brams Jeremy Buhler

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

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

Lets vote!

45 voters A C B

35 voters B C A

20 voters C B A

(1st) (2nd) (3rd)

sincere preferences

Plurality voting

45 voters A C B

35 voters B C A

20 voters C B A

sincere ballots

A 45 votes B 35 votes C 20 votes

zero-information result

Plurality voting

45 voters A C B

35 voters B C A

20 voters C B A

ballots so far

?

A 45 votes B 35 votes C 0 votes

election state

Plurality voting

45 voters A C B

35 voters B C A

20 voters C B A

strategic ballots

insincerity!

B 55 votes A 45 votes C 0 votes

final election state

Gibbard 73 Satterthwaite 75

Manipulation decision problem

45 voters A C B

35 voters B C A

20 voters C B A

ballot sets

BU

BV

B 55 votes A 45 votes C 0 votes

election state

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

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

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

Manipulation decision problem

- Existence of Probably Winning Coalition Ballots

(EPWCB) - INSTANCE Set of alternatives A and a

distinguished member a of A set of weighted

cardinal-ratings ballots BV the weights of a set

of ballots BU which have not been cast

probability - QUESTION Does there exist a way to cast the

ballots BU so that a has at least probability

of winning the election with the ballots

? - This problem is computationally infeasible

(NP-hard) for - Hare (single-winner STV) Bartholdi Orlin 91
- Borda Conitzer Sandholm 02

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

Declared-Strategy Voting

Cranor Cytron 96

rational strategizer

cardinal preferences

ballot

election state

outcome

Declared-Strategy Voting

Cranor Cytron 96

sincerity

strategy

rational strategizer

cardinal preferences

ballot

election state

outcome

- Separates how voters feel from how they vote
- Levels playing field for voters of all

sophistications - Aim a voter needs only to give sincere

preferences

What is a declared strategy?

A 0.0 B 0.6 C 1.0

cardinal preferences

A 0 B 1 C 0

declared strategy

voted ballot

A 45 B 35 C 0

current election state

- Captures thinking of a rational voter

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!)

Favorite vs. compromise, revisited

45 voters A C B

35 voters B C A

20 voters C B A

ballots so far

?

A 45 votes B 35 votes C 0 votes

election state

Approve both!

45 voters A C B

35 voters B C A

20 voters C B A

strategic ballots

insincerity avoided

B 55 votes A 45 votes C 20 votes

final election state

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

Strands of research

number of alternatives outcome Area of research

k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?

k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?

k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04

Strands of research

number of alternatives outcome Area of research

k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?

k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?

k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04

Strands of research

number of alternatives outcome Area of research

k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?

k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?

k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04

Approval ratings

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

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?

Approval ratings

Average of ratings

outcome

data from Metacritic.com Videodrome (1983)

Average of ratings

outcome

Videodrome (1983)

Another approach Median

outcome

Videodrome (1983)

Another approach Median

outcome

Videodrome (1983)

Another approach Median

- Immune to insincerity LeGrand 08
- voter i cannot obtain a better result by voting
- if , increasing will

not change - if , decreasing will

not change - Allows tyranny by a majority
- no concession to the 0-voters

Average with Declared-Strategy Voting?

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

rational strategizer

cardinal preferences

ballot

election state

outcome

Equilibrium-finding algorithm

Videodrome (1983)

Equilibrium-finding algorithm

Equilibrium-finding algorithm

Equilibrium-finding algorithm

Equilibrium-finding algorithm

Equilibrium-finding algorithm

- Is this algorithm is guaranteed to find an

equilibrium?

equilibrium!

Equilibrium-finding algorithm

- Is this algorithm is guaranteed to find an

equilibrium? - Yes! LeGrand 08

equilibrium!

Expanding range of allowed votes

- These results generalize to any range LeGrand

08

Multiple equilibria can exist

- Will multiple equilibria will always have the

same average?

outcome in each case

Multiple equilibria can exist

- Will multiple equilibria will always have the

same average? - Yes! LeGrand 08

outcome in each case

Average-Approval-Rating DSV

outcome

Videodrome (1983)

Average-Approval-Rating DSV

- AAR DSV is immune to insincerity in general

LeGrand 08

outcome

Evaluating AAR DSV systems

- Expanded vote range gives wide range of AAR DSV

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

Evaluating AAR DSV systems

minimum at

Evaluating AAR DSV systems hill-climbing

minimum at

Evaluating AAR DSV systems hill-climbing

minimum at

Evaluating AAR DSV systems

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

Strands of research

number of alternatives outcome Area of research

k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?

k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?

k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04

Approval strategies for DSV

- Rational plurality strategy has been well

explored Cranor Cytron 96 - But what about approval strategy?
- If each alternatives probability of winning is

known, optimal strategy can be computed Merrill

88 - But what about in a DSV context?
- have only a vote total for each alternative
- Lets look at several approval strategies and

approaches to evaluating their effectiveness

DSV-style approval strategies

- Strategy Z
- Approve alternatives with higher-than-average

cardinal preference (zero-information strategy)

Merrill 88

DSV-style approval strategies

- Strategy Z
- Strategy T
- Approve favorite of top two vote-getters, plus

all liked more Ossipoff 02, Poundstone 08 - Simplest generalization of plurality DSV strategy
- Cranor Cytron 96

DSV-style approval strategies

- Strategy Z
- Strategy T
- Strategy J
- Use strategy Z if it distinguishes between top

two vote-getters otherwise use strategy T Brams

Fishburn 83

DSV-style approval strategies

- Strategy Z
- Strategy T
- Strategy J
- Strategy A
- Approve all preferred to top vote-getter, plus

top vote-getter if preferred to second-highest

vote-getter - LeGrand 02
- . . . but how to evaluate these strategies?

Election-state-evaluation approaches

- Evaluate a declared strategy by evaluating the

election states that are immediately obtained - Calculate expected value of an election state by

estimating each alternatives probability of

eventually winning - How to calculate those probabilities?

Election-state-evaluationMerrill metric

- Estimate an alternatives probability of winning

to be proportional to its current vote total

raised to some power x Merrill 88

Strategy comparison using the Merrill metric

Current election state Focal voters preferences

1, 0, 0 (strategies A T) 1, 0, 0 (A

T) 0, 1, 0 (A T) 0, 1, 1 (A) 0, 1, 0

(T) 1, 0, 1 (A T) 0, 1, 1 (A T)

Strategy comparison using the Merrill metric

Current election state Focal voters preferences

When , A is better than

T if and only if

or, equivalently

- Intuitively, A always does better than T when
- s1 is much larger than s2,
- x is large, or
- p3 is relatively close to p2 compared to p1

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

Further result for strategy A

- More generally, it is true that if
- the election state is free of ties and near-ties
- and the focal voters cardinal preferences are

tie-free - when
- and the Merrill-metric exponent x is taken to

infinity - then strategy A dominates all other approval

strategies according to the Merrill metric

LeGrand 08

Election-state-evaluationBranching-probabilities

metric

- Estimate an alternatives probability of winning

by looking ahead - Assume that the probability that alternative a is

approved on each future ballot is equal to the

proportion of already-voted ballots that approve

a

Branching-probabilities metric strategy A

- It is true that if
- the election state is free of ties and near-ties
- and the focal voters cardinal preferences are

tie-free - when
- and the number of future ballots is taken to

infinity - then strategy A dominates all other approval

strategies according to the branching-probabilitie

s metric LeGrand 08

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?

Strands of research

number of alternatives outcome Area of research

k 1 an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating?

k gt 1 m 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective?

k gt 1 m 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? Brams, Kilgour Sanver 04

Electing a committee from approval ballots

approves of alternatives 4 and 5

11110

00011

k 5 alternatives n 6 ballots

00111

01111

00001

10111

- Whats the best committee of size m 2?

Sum of Hamming distances

11110

00011

m 2 winners

2

4

4

5

00111

01111

11000

4

3

sum 22

00001

10111

- What if we elect alternatives 1 and 2?

Fixed-size minisum

11110

00011

m 2 winners

4

0

2

1

00111

01111

00011

2

1

sum 10

00001

10111

- Minisum elects winner set with smallest HD sum
- Easy to compute (pick alternatives with most

approvals)

Maximum Hamming distance

11110

00011

m 2 winners

4

0

2

1

00111

01111

00011

2

1

sum 10 max 4

00001

10111

- One voter is quite unhappy with minisum outcome

Fixed-size minimax

Brams, Kilgour Sanver 04

11110

00011

m 2 winners

2

2

2

1

00111

01111

00110

2

3

sum 12 max 3

00001

10111

- Minimax elects winner set with smallest maximum

HD - Harder to compute?

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) ?

Complexity

Endogenous minimax EM BSM(0, k) Bounded-size minimax BSM(m1, m2) Fixed-size minimax FSM(m) BSM(m, m)

NP-hard Frances Litman 97 NP-hard (generalization of EM) NP-hard LeGrand 04

Approximability

Endogenous minimax EM BSM(0, k) Bounded-size minimax BSM(m1, m2) Fixed-size minimax FSM(m) BSM(m, m)

has a PTAS Li, Ma Wang 99 no known PTAS no known PTAS

Polynomial-Time Approximation Scheme algorithm

with approx. ratio 1 e that runs in time

polynomial in the input and exponential in 1/e

Approximability

Endogenous minimax EM BSM(0, k) Bounded-size minimax BSM(m1, m2) Fixed-size minimax FSM(m) BSM(m, m)

has a PTAS Li, Ma Wang 99 no known PTAS has a 3-approx. LeGrand, Markakis Mehta 06 no known PTAS has a 3-approx. LeGrand, Markakis Mehta 06

Polynomial-Time Approximation Scheme algorithm

with approx. ratio 1 e that runs in time

polynomial in the input and exponential in 1/e

Susceptibility to insincerity

Endogenous minimax EM BSM(0, k) Bounded-size minimax BSM(m1, m2) Fixed-size minimax FSM(m) BSM(m, m)

insincere voters can benefit LeGrand, Markakis Mehta 06 insincere voters can benefit LeGrand, Markakis Mehta 06 insincere voters can benefit LeGrand, Markakis Mehta 06

But our 3-approximation for FSM is immune to

insincere strategy!

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?

Rational m,M-Average strategy

- Allow votes between and
- For , voter i should choose to

move outcome as close to as possible - Choosing would give
- Optimal vote is
- After voter i uses this strategy, one of these is

true - and
- and

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)

Proof Only one equilibrium average

- Theorem
- Proof considers two symmetric cases
- assume
- assume
- Each leads to a contradiction

Proof Only one equilibrium average

case 1

, contradicting

Proof Only one equilibrium average

Case 1 shows that

Case 2 is symmetrical and shows that

Therefore

Therefore, given , the average at equilibrium

is unique

An equilibrium always exists?

- At equilibrium, must satisfy
- I proposed to prove that, given a vector , at

least one equilibrium exists. - A particular algorithm will always find an

equilibrium for any . . .

An equilibrium always exists!

- Equilibrium-finding algorithm
- sort so that
- for i 1 up to n do
- Since an equilibrium always exists, average at

equilibrium is a function,

. - Applying to instead of gives a new

system, Average-Approval-Rating DSV.

(full proof and more efficient algorithm in

dissertation)

Average-Approval-Rating DSV

- What if, under AAR DSV, voter i could gain an

outcome closer to ideal by voting insincerely

( )? - I proposed to prove that Average-Approval-Rating

DSV is immune to strategy by insincere voters. - Intuitively, if

, increasing will not change

.

AAR DSV is immune to strategy

- If ,
- increasing will not change

. - decreasing will not increase

. - If ,
- increasing will not decrease

. - decreasing will not change

. - So voting sincerely ( ) is guaranteed

to optimize the outcome from voter is point of

view

(complete proof in dissertation)

Parameterizing AAR DSV

- m,M-AAR DSV can be parameterized nicely using a

and b, where and

Parameterizing AAR DSV

- For example

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

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

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

Strategy comparison using the Merrill metric

Current election state Focal voters preferences

expected values of possible next election states

0, 1, 1 (A) 0, 1, 0 (T)

Strategy comparison using the Merrill metric

Current election state Focal voters preferences

so T is better than A only when

or, equivalently

Strategy comparison using the Merrill metric

Current election state Focal voters preferences

T is better than A only when

- Corollaries
- When x is taken to infinity and ,

strategy A dominates strategy T - When
- , strategy A

dominates strategy T

Further result for strategy A

- just a weighted average of values
- assume
- as , from

below - so maximized when weights of those

are maximized, which is done by approving only

alternatives i where - case is similar approve i where
- only strategy A always does this

Approximating FSM

11110

m 2 winners

00011

00111

00111

00001

choose a ballot arbitrarily

10111

01111

Approximating FSM

11110

m 2 winners

00011

00111

coerce to size m

00101

00111

00001

choose a ballot arbitrarily

10111

01111

outcome m-completed ballot

Approximation ratio 3

optimal FSM set

11110

2

00011

2

1

00111

00110

3

00001

2

10111

2

01111

OPT

OPT optimal maxscore

Approximation ratio 3

optimal FSM set

chosen ballot

11110

2

00011

2

1

00111

1

00110

00111

3

00001

2

10111

2

01111

OPT

OPT

OPT optimal maxscore

Approximation ratio 3

optimal FSM set

chosen ballot

m-completed ballot

11110

2

00011

2

1

00111

1

1

00110

00111

00011

3

00001

2

10111

2

01111

OPT

OPT

OPT

(by triangle inequality)

3OPT

OPT optimal maxscore

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

Local search approach for FSM

- Start with some c ? 0,1k of weight m

01001

4

Local search approach for FSM

- Start with some c ? 0,1k of weight m
- In c, swap up to r 0-bits with 1-bits in such a

way that minimizes the maxscore of the result

11000

10001

5

4

01001

01100

00101

4

4

4

01010

00011

4

4

Local search approach for FSM

- Start with some c ? 0,1k of weight m
- In c, swap up to r 0-bits with 1-bits in such a

way that minimizes the maxscore of the result

01010

4

Local search approach for FSM

- Start with some c ? 0,1k of weight m
- In c, swap up to r 0-bits with 1-bits in such a

way that minimizes the maxscore of the result

01010

4

Local search approach for FSM

- Start with some c ? 0,1k of weight m
- In c, swap up to r 0-bits with 1-bits in such a

way that minimizes the maxscore of the result - Repeat step 2 until maxscore(c) is unchanged k

times - Take c as the solution

11000

10010

5

4

01010

01100

00110

4

4

3

01001

00011

4

4

Local search approach for FSM

- Start with some c ? 0,1k of weight m
- In c, swap up to r 0-bits with 1-bits in such a

way that minimizes the maxscore of the result - Repeat step 2 until maxscore(c) is unchanged k

times - Take c as the solution

00110

3

Heuristic evaluation

- Parameters
- starting point of search
- radius of neighborhood
- Ran heuristics on generated and real-world data
- All heuristics perform near-optimally
- highest approx. ratio found 1.2
- highest average ratio lt 1.04
- The fixed-size-minisum starting point performs

best overall (with our 3-approx. a close second) - When neighborhood radius is larger, performance

improves and running time increases

(maxscore of solution found) (maxscore of exact

solution)

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)

Specific FSM heuristics

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

Average approx. ratios found

radius 1 radius 2

fixed-size minimax 1.0012 1.0000

3-approx. 1.0017 1.0000

random set 1.0057 1.0000

highest-maxscore 1.0059 1.0000

performance on GTS 03 election data k 24

candidates, m 12 winners, n 161 ballots

Largest approx. ratios found

radius 1 radius 2

fixed-size minimax 1.0714 1.0000

3-approx. 1.0714 1.0000

random set 1.0714 1.0000

highest-maxscore 1.0714 1.0000

performance on GTS 03 election data k 24

candidates, m 12 winners, n 161 ballots

Conclusions from all experiments

- All heuristics perform near-optimally
- highest ratio found 1.2
- highest average ratio lt 1.04
- When radius is larger, performance improves and

running time increases - The fixed-size-minisum starting point performs

best overall (with our 3-approx. a close second)

Manipulating FSM

m 2 winners

00110

00011

2

0

2

1

00111

01111

00011

2

1

max 2

00001

10111

- Voters are sincere
- Another optimal solution 00101

Manipulating FSM

00110

m 2 winners

11110

00011

0

2

2

2

1

00111

01111

00110

2

3

max 3

00001

10111

- A voter manipulates and realizes ideal outcome
- But our 3-approximation for FSM is nonmanipulable

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

Progress so far

Area of research State of progress

Approval rating Completed rational Average strategy, equality of average at equilibria To do equilibrium always exists, strategy-immunity of AAR DSV, evaluation of AAR DSV systems

DSV-style approval strategies Completed Merrill-metric comparison of A and T in 3-alt. case, domination of A as To do comparisons of other pairs, analysis using branching-probabilities metric

Fixed-size minimax Completed NP-hardness proof, 3-approximation, heuristic evaluation, manipulability analysis