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					Computational Aspects of Approval Voting
     and Declared-Strategy Voting

          A Dissertation Proposal
                   15 March 2007




                Rob LeGrand
           Washington University in St. Louis
           Computer Science and Engineering
                legrand@cse.wustl.edu
                          Let’s vote!

              45 voters    35 voters    20 voters

                 A            B            C        (1st)
  sincere
preferences      C            C            B        (2nd)

                 B            A            A        (3rd)




                                                            2
                         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
                                                         3
                  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
                                                  4
                   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
                                                                      5
                What is manipulation?

              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
                                                           6
              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]

                                                                   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 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
                                                                   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?

• This problem is computationally easy (in P) for:
   – plurality voting [Bartholdi, Tovey & Trick ’89]
   – approval voting
                                                                   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?

• This problem is computationally infeasible (NP-hard) for:
   – Hare [Bartholdi & Orlin ’91]
   – Borda [Conitzer & Sandholm ’02]
                                                               10
       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




                                                           11
        Declared-Strategy Voting
              [Cranor & Cytron ’96]



  cardinal            rational
preferences         strategizer

                                      ballot

                     election
 outcome
                      state




                                               12
                 Declared-Strategy Voting
                        [Cranor & Cytron ’96]

          sincerity                             manipulation

          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 honest preferences
                                                               13
              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

                                                          14
      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.




                                               15
          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
                                                      16
                            Approval voting
           [Ottewell ’77]    [Weber ’77]   [Brams & Fishburn ’78]

                 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
                                                                             17
                 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


                                             18
               Strands of proposed 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]
                                                                  19
               Strands of proposed 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]
                                                                  20
               Strands of proposed 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
                    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



                                                         22
    One approach: Average

     
     r  0, .1, .2, .6, .9
     
     v  0, .1, .2, .6, .9
                       
     outcome:   f avg (v )  .36

           .36
0                                  1




                                       23
    One approach: Average

     
     r  0, .1, .2, .6, .9
     
     v  0, .1, .2, 1, .9
                       
     outcome:   f avg (v )  .44

                .44
0                                  1




                                       24
    Another approach: Median

      
      r  0, .1, .2, .6, .9
      
      v  0, .1, .2, .6, .9
                        
      outcome:   f med (v )  .2

      .2
0                                  1




                                       25
    Another approach: Median

       
       r  0, .1, .2, .6, .9
       
       v  0, .1, .2, 1, .9
                        
      outcome:   f med (v )  .2

      .2
0                                  1




                                       26
              Another approach: Median

• Nonmanipulable
  – voter i cannot obtain a better result by voting vi  ri 
              
  – if f med (v )  vi , increasing vi will not change f med (v )
                                                              
  – if f med (v )  vi , decreasing vi will not change f med (v )


• Allows tyranny by a majority
    
  – v  0, 0, 0,1,1,1,1
            
  –  f med (v )  1
  – no concession to the 0-voters



                                                                    27
      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?

                                                         28
                Rational Average strategy

• 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 , 0),1)

• After voter i uses this strategy, one of these is true:
           
  – f avg (v )  ri and vi  1
           
  – f avg (v )  ri
           
  – f avg (v )  ri and vi  0

                                                             29
   Multiple equilibria are possible
         
         r  .2, .3, .5, .5, .8
           
           v  0, 0, .5, 1, 1
          
          v  0, 0, .6, .9, 1
         
         v  0, 0, .75, .75, 1

            outcome in each case:
                       
                f avg (v )  .5

Multiple equilibria always have same average
           (proof in written proposal)
                                               30
              An equilibrium always exists?
                  
• At equilibrium, v must satisfy
  (i) vi  min(max(ri n   j i v j , 0),1)
                                          
  I propose to prove that, given a vector r , at least
  one equilibrium exists.

• If an equilibrium always exists, then average at
                                                    
  equilibrium can be defined as a function, f aveq (r ) .
                                 
• Applying f aveq to v instead of r gives a new
  system, Average-approval-rating DSV.
                                                            31
    Average-approval-rating DSV

        
        r  0, .1, .2, .6, .9
        
        v  0, .1, .2, .6, .9
                           
       outcome:    f aveq (v )  .4

                  .4
0                                     1




                                          32
    Average-approval-rating DSV

        
        r  0, .1, .2, .6, .9
        
        v  0, .1, .2, 1, .9
                           
       outcome:    f aveq (v )  .4

                  .4
0                                     1




                                          33
           Average-approval-rating DSV

• AAR DSV could be manipulated if some voter i
  could gain an outcome closer to ideal by voting
  insincerely ( vi  ri ).

  I propose to show that Average-approval-rating
  DSV cannot be manipulated by insincere voters.




                                                    34
              Average-approval-rating DSV

• AAR DSV could be manipulated if some voter i
  could gain an outcome closer to ideal by voting
  insincerely ( vi  ri ).

    I propose to show that Average-approval-rating
    DSV cannot be manipulated by insincere voters.
                            
•   Intuitively, if f aveq (v )  vi , increasing vi will not
                       
    change f aveq (v ).


                                                                35
         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




                                                     36
           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]

                                                              37
         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  x, y    2 : x 2  y 2  1 case and determine
       
  whether the resulting DSV system is rotationally
  invariant and/or nonmanipulable by insincere
  voters.


                                                            38
               Strands of proposed 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]
                                                                  39
               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


                                                       40
             DSV-style approval strategies

• Strategy Z [Merrill ’88]:
   – Approve alternatives with higher-than-average cardinal
     preference (zero-information strategy)

                    
                    s  [30, 25,15,10]
                      
                      p  [0, 1, .8, .3]
                        Z recommends:
                       b  [0, 1, 1, 0]


                                                              41
             DSV-style approval strategies

• Strategy T [Ossipoff ’02]:
   – Approve favorite of top two vote-getters, plus all liked
     more

                    
                    s  [30, 25,15,10]
                      
                      p  [0, 1, .8, .3]
                       T recommends:
                      b  [0, 1, 0, 0]


                                                                42
             DSV-style approval strategies

• Strategy J [Brams & Fishburn ’83]:
   – Use strategy Z if it distinguishes between top two vote-
     getters; otherwise use strategy T

                    
                    s  [30, 25,15,10]
                      
                      p  [0, 1, .8, .3]
                       J recommends:
                      b  [0, 1, 1, 0]


                                                                43
           DSV-style approval strategies

• Strategy A:
  – Approve all preferred to top vote-getter, plus top vote-
    getter if preferred to second-highest vote-getter

                  
                  s  [30, 25,15,10]
                    
                    p  [0, 1, .8, .3]
                      A recommends:
                     b  [0, 1, 1, 1]
  But how to evaluate these strategies?
                                                               44
      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?




                                                     45
             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 
                   w      i
                              
                    i     k
                        sj 
                       j 1 

                                                        46
   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)

                                                           47
    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)
                    s1 
                      x
                         s2  1x  s3  1x
                p1s1x  p2 s2  1  p3 s3x
                                      x
  V[ 0,1,0]                                        [0, 1, 0] (T)
                    s1x  s2  1  s3x
                                  x




                                                                    48
   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   


                                                                   49
   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               Intuitively, T does better than A only when:
           s 1
                           • s1 and s2 are relatively close
  p3  p1  2                • x is relatively small
                              • p3 is relatively close to p1 compared to p2

                                                                          50
   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 s1  s2  1, strategy A
     dominates strategy T
   – When         p1  p2
           p3            , strategy A dominates strategy T
                         2
                                                               51
         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.




                                                      52
              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 strategies
  according to the Merrill metric

• (proof in written proposal)

                                                                53
              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
                                             iB
                                                   i   1
                                p2




                                                               54
         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.

                                                      55
               Strands of proposed 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]
                                                                  56
      Electing a committee from approval ballots

                                                      approves of
k = 5 candidates        11110       00011             candidates
                                                        4 and 5
n = 6 ballots


                01111                         00111




                        10111       00001



  •What’s the best committee of size m = 2?
                                                                57
                 Sum of Hamming distances


m = 2 winners           11110                   00011

                                    2       4

                          4                         5
                01111                   11000           00111

                                4               3           sum = 22


                        10111                   00001




                                                                       58
                        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 sumscore
 •Easy to compute (pick candidates with most approvals)
                                                                       59
                Maximum Hamming distance


m = 2 winners           11110                   00011

                                    4       0

                          2                         1
                01111                   00011           00111

                                2               1           sum = 10
                                                             max = 4
                        10111                   00001




                                                                       60
                        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 maxscore
 •Harder to compute?
                                                                       61
                           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)




                                                                    62
                           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)      (proof in written
                                                        proposal)




                                                                         63
                         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
                                                                      64
                         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]
                             (proof in written          (proof in written
                                proposal)                  proposal)

  * Polynomial-Time Approximation Scheme: algorithm
  with approx. ratio 1 + ε that runs in time polynomial in
  the input and exponential in 1/ε
                                                                            65
             Approximating FSM


11110                            m = 2 winners

00011

00111
                      00111
00001    choose
         a ballot
10111
        arbitrarily
01111




                                                 66
             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



                                                            67
                   Approximation ratio ≤ 3

                         optimal
       11110
                   2     FSM set
       00011   2

       00111 1
                         00110
               3
       00001
               2
       10111
               2
       01111
               ≤ OPT


OPT = optimal maxscore
                                             68
                   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
                                                   69
                   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
                                                                         70
                   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




                                                           71
            Local search approach for FSM

1.   Start with some c  {0,1}k
     of weight m


                                   01001
                                     4




                                            72
            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




                                                          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
                                  01010
                                    4




                                            74
            Local search approach for FSM

1.   Start with some c  {0,1}k
     of weight m
2.   In c, swap up to r 0-bits
     with 1-bits in such a way
                                   01010
     that minimizes the              4
     maxscore of the result




                                            75
            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



                                                          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
                                            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



                                                    77
                     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


                                                                       78
                   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                                   79
                      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
                                                                      80
         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




                                                    81
                       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          Completed: comparison of A and T in 3-alt. case,
approval           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



                                                                        82
                           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?




                                                     83
             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   vn

                                                      84
        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

                                                   85
         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 
2n  i : 2  ri             A(2 )
i : 1  ri   1n            B (1 )
2n  i : 2  ri   i : 1  ri   1n
2 n  1n
2  1 , contradicting 1  2
                                               86
      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




                                                            87
                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




                                                            88
                    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)

                                                              89
     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

                                                         90
     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

                                                         91
           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)




                                                                92

				
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