How to Avoid Gerrymandering ANew Algorithmic Solution

Document Sample
How to Avoid Gerrymandering ANew Algorithmic Solution Powered By Docstoc
					                                    Re-Districting: A . . .
                                    How to Avoid . . .
                                    Toward Formulation of . . .

  How to Avoid                      Relation to Clustering
                                    Towards a More . . .


 Gerrymandering:                    Example
                                    Resulting Iterative . . .
                                    Conclusion

A New Algorithmic                   Acknowledgments



     Solution                                Title Page


        Gregory B. Lush
   Department of Electrical and
     Computer Engineering
                                           Page 1 of 16
        Esteban Gamez
       Vladik Kreinovich                      Go Back

  Department of Computer Science            Full Screen


   University of Texas at El Paso              Close
     El Paso, TX 79968, USA
                                                Quit
     contact vladik@utep.edu
                                                                   Re-Districting: A . . .

1.     Re-Districting: A Practical Problem                         How to Avoid . . .

                                                                   Toward Formulation of . . .
     • The notion of electoral districts: in the USA and in
                                                                   Relation to Clustering
       other countries, voting is done by electoral districts.
                                                                   Towards a More . . .

     • First objective: equal representation.                      Example


     • How: all voting districts of the same level (federal,       Resulting Iterative . . .

       state, city, etc.) contain same number of voters.           Conclusion

                                                                   Acknowledgments
     • Problem: in time, demography changes – some districts
                                                                           Title Page
       lose voters, some gain them.
     • Solution: re-districting.
     • Main objective of re-districting: best represent geo-
       graphic regions.                                                   Page 2 of 16



     • Additional objective: represent rural and city areas, mi-            Go Back


       norities, border areas, flooded areas, etc.                          Full Screen


     • Open problem: how to take all these factors into ac-                   Close

       count?
                                                                               Quit
                                                                     Re-Districting: A . . .

2.     Gerrymandering: A Problem                                     How to Avoid . . .

                                                                     Toward Formulation of . . .
     • How re-districting is done: in most states, it is voted
                                                                     Relation to Clustering
       upon by the legislature.
                                                                     Towards a More . . .

     • Drawback: in the next elections, the representation           Example

       may be unfairly biased towards a party in power.              Resulting Iterative . . .


     • Phenomenon behind bias: for parties A and B, A votes          Conclusion


       in B-majority district are “lost” – this district votes for   Acknowledgments

       B anyway.                                                             Title Page



     • Gerrymandering – how: a party in power (A)
        – divides all the A voters into A-majority districts,
          and                                                               Page 3 of 16


        – attaches, to each A-district, many B-voters (w/out                  Go Back

          violating A-majority).                                             Full Screen

     • Result: many B votes are lost, while no A votes are                      Close
       lost.
                                                                                 Quit
                                                                                Re-Districting: A . . .

3.     Gerrymandering: Example                                                  How to Avoid . . .

                                                                                Toward Formulation of . . .
     • Example: 18 towns on similar size:
                                                                                Relation to Clustering

        – A1 , . . . , A10 vote for A,                                          Towards a More . . .

        – B1 , . . . , B8 vote for B.                                           Example

                                                                                Resulting Iterative . . .
     • Objective: form 6 electoral districts.
                                                                                Conclusion
                                   10        10
     • Fair representation: A gets    ·6 =      = 3.33 . . . votes              Acknowledgments
                                   18         3                                         Title Page
       – i.e., 3 or 4.
     • Example of gerrymandering: divide 18 towns into the
       following 6 districts:
        – {A1 , A2 , B1 }, {A3 , A4 , B2 }, {A5 , A6 , B3 }, {A7 , A8 , B4 },          Page 4 of 16


          and {A9 , A10 , B5 } vote for A;                                               Go Back

        – only 1 district out of 6, {B6 , B7 , B8 } votes for B.                        Full Screen


     • Result: A gets 5 votes out of 6.                                                    Close


                                                                                            Quit
                                                                     Re-Districting: A . . .

4.     How to Avoid Gerrymandering: Known Approaches                 How to Avoid . . .

                                                                     Toward Formulation of . . .
     • Typical idea: limit the “weirdness” of the district shapes.
                                                                     Relation to Clustering

     • Example: make them as round as possible, with the             Towards a More . . .

       smallest possible length of the separation lines.             Example


     • Limitations:                                                  Resulting Iterative . . .

                                                                     Conclusion
        – these approaches only take into account geograph-          Acknowledgments
          ical closeness;                                                    Title Page

        – they may not give adequate representation to mi-
          norities or rural population.
     • Remaining problem: take into account
                                                                            Page 5 of 16

        – not only geographical closeness and differences,
                                                                              Go Back
        – but also other types of closeness and difference.
                                                                             Full Screen


                                                                                Close


                                                                                 Quit
                                                                            Re-Districting: A . . .

5.      Toward Formulation of the Problem in Precise Math-                  How to Avoid . . .

        ematical Terms                                                      Toward Formulation of . . .

                                                                            Relation to Clustering
     • Representing voters: by parameters x1 , . . . , xn :
                                                                            Towards a More . . .

         –   geographic location (xi are geographic coordinates),           Example

         –   income,                                                        Resulting Iterative . . .

         –   rural vs. urban status,                                        Conclusion

         –   number of children, etc.                                       Acknowledgments

                                                                                    Title Page
     • Representatives are described by v1 , . . . , vn .
     • Utility approach: in decision making, preferences are
       represented by utility functions:
                       u = u(x1 − v1 , . . . , xn − vn ).                          Page 6 of 16


                                                                                     Go Back
     • Different characteristics are usually independent: it is
       known that then u(d1 , . . . , dn ) = u1 (d1 ) + . . . + un (dn ).           Full Screen


     • Differences are small: representatives are close to the                          Close

                                def
       voters, so differences di = ri − vi are small.                                    Quit
                                                                                       Re-Districting: A . . .

6.     Toward Formulation of the Problem in Precise Math-                              How to Avoid . . .

       ematical Terms (cont-d)                                                         Toward Formulation of . . .

                                                                                       Relation to Clustering
     • Taylor expansion: since differences di = ri − vi are
                                                                                       Towards a More . . .
       small,
                                                                                       Example
                                                u (0) 2
              ui (di ) = ui (0) + ui (0) · di +      · di + . . .                      Resulting Iterative . . .
                                                  2
                                                                                       Conclusion
     • Analysis: the largest possible utility is when a repre-
                                                                                       Acknowledgments
       sentative is a perfect match, i.e., di = 0.                                             Title Page

     • Conclusion: ui (di ) ≈ ui (0) − wi ·               d2
                                                           i   and
                                 n                  n                n
         u(d1 , . . . , dn ) =         ui (di ) =         ui (0) −         wi · d2 .
                                                                                 i
                                 i=1                i=1              i=1                      Page 7 of 16

     • Thus, maximizing utility is equivalent to minimizing                                     Go Back
       the disutility
                                          n                                                    Full Screen

                         ρ(x, v) =             wi · (ri − vi )2 .                                 Close
                                         i=1
                                                                                                   Quit
                                                                   Re-Districting: A . . .

7.     Resulting Formulation of the Problem                        How to Avoid . . .

                                                                   Toward Formulation of . . .
     • We have:
                                                                   Relation to Clustering

        – an integer n (number of characteristics),                Towards a More . . .

        – positive real numbers w1 , . . . , wn (weights);         Example

                                               (k)      (k)
        – n-dimensional vectors x(k) = (x1 , . . . , xn )          Resulting Iterative . . .


          (1 ≤ k ≤ N ) describing voters;                          Conclusion

                                                                   Acknowledgments
        – number c of voting districts.
                                                                           Title Page

     • Objective:
        – subdivide N voters into c groups D1 , . . . , Dc , and
        – select a vector v(1), . . . , v(c) within each group            Page 8 of 16

        – so as to minimize the overall disutility
                                                                            Go Back
                            c
                                      ρ(x(k) , v(j)).                      Full Screen


                           j=1 k∈Dj                                           Close


                                                                               Quit
                                                                     Re-Districting: A . . .

8.     Relation to Clustering                                        How to Avoid . . .

                                                                     Toward Formulation of . . .
     • Informal objective: a voter is closer to other voters from
                                                                     Relation to Clustering
       his/her district than to voters from other districts.
                                                                     Towards a More . . .

     • Relation to clustering: such groups are called clusters.      Example


     • Iterative clustering: we start with some representations      Resulting Iterative . . .

       v(1), . . . , v(c), then repeat the following 2 steps:        Conclusion

                                                                     Acknowledgments
        – each voter x(k) is assigned to the group Dj for which
                                                                             Title Page
          the disutility ρ(x(k) , v(j)) is the smallest;
        – after that, for each group Dj , we re-calculate v(j)
          as the average of all x(k) (k ∈ Dj ).
     • Limitation of this approach: unequal clusters.                       Page 9 of 16


     • Example: a population consisting of a (larger) city and                Go Back

       a (smaller) rural area.                                               Full Screen


     • Resulting clusters: a larger all-city cluster and a smaller              Close

       all-rural cluster.
                                                                                 Quit
                                                                       Re-Districting: A . . .

9.     Towards a More Adequate Solution                                How to Avoid . . .

                                                                       Toward Formulation of . . .
                                                             N
     • Main idea: groups D1 , . . . , Dc must have equal size .        Relation to Clustering
                                                             c
                                                                       Towards a More . . .
     • Technical challenge: when we have the representatives
                                                                       Example
       v(j), how can we get districts?
                                                                       Resulting Iterative . . .
     • Technical result: in the optimal districting, the division      Conclusion
       between Di and Dj is determined by a threshold tij for          Acknowledgments
                         def ρ(x, v(i))
       the ratio rij (x) =              :                                      Title Page
                             ρ(x, v(j))
        – if rij (x) < tij , then x ∈ Di ;
        – if rij (x) > tij , then x ∈ Dj .
                                                                             Page 10 of 16
     • Conclusion: for some weights αj , a voter x is assigned
       to the class Dj for which αj · ρ(x(k) , v(j)) is the largest.            Go Back


                                                                               Full Screen
     • Remaining problem: find the weights αj for which the
       resulting districts Dj are of equal size.                                  Close


                                                                                   Quit
                                                               Re-Districting: A . . .

10.    Towards an Algorithm                                    How to Avoid . . .


                       (0)           (0)                       Toward Formulation of . . .
 • We start: with α1 = . . . = αc = 1.
                                                               Relation to Clustering
                                            (p)          (p)
 • Start iteration: with some values       α1 , . . . , αc .   Towards a More . . .


 • Main part of the iteration: for each j, we find βj for       Example


   which there are exactly N/c points x for which              Resulting Iterative . . .

                                    (p)                        Conclusion
                  βj · ρ(x, v(j)) ≤ αk · ρ(x, v(k))
                                                               Acknowledgments
      for all k = j.                                                   Title Page


 • How to find βj : bisection
       – if we get < N/c points, we decrease βj ;
       – if we get > N/c points, we increase βj .                    Page 11 of 16

 • Observation: if we multiply all the values αj by the                 Go Back
   same constant, we get the same classes.
                                                                       Full Screen
                                                αj   βj
 • Last part of the iteration: find αj for which    ≈ (p)
                                                αk  αk
                                                                          Close


   for all j = k.                                                          Quit
                                                           Re-Districting: A . . .

11.   Analysis of the Auxiliary Problem                    How to Avoid . . .

                                                           Toward Formulation of . . .
                                        αj   βj
 • Problem – reminder: find αj for which    ≈ (p) for       Relation to Clustering
                                        αk  αk
                                                           Towards a More . . .
   all j = k.
                                                           Example
 • Difficulty: this problem is non-linear in unknowns αj .   Resulting Iterative . . .
                                def               def
 • Idea: turn to logarithms Aj = ln(αj ), Bj = ln(βj ),    Conclusion

         (p) def   (p)
   and Aj = ln(αj ).                                       Acknowledgments

                                                                   Title Page
 • New problem: find Aj for which
                                            (p)
                   Aj − Ak ≈ Bj − Ak .
 • Least Squares solution:
                                                                 Page 12 of 16
                      1       (p)
                Aj = · (Bj + Aj ) + const.                          Go Back
                      2
 • Resulting value of αj :                                         Full Screen


                                      (p)                             Close
                      αj =   βj · αj .
                                                                       Quit
                                                              Re-Districting: A . . .

12.   Example                                                 How to Avoid . . .

                                                              Toward Formulation of . . .
 • Situation: uniform population distribution on [0, 1].
                                                              Relation to Clustering

 • Starting reps: v(1) = 0, v(2) = 0.5, and v(3) = 1.0.       Towards a More . . .


 • Simple clustering: D1 = [0, 0.25], D2 = [0.25, 0.75],      Example


   D3 = [0.75, 1].                                            Resulting Iterative . . .

                                                              Conclusion
 • Problem: D2 is twice larger than D1 or D3 .                Acknowledgments

 • 1st iteration: find β1 for which β1 · x2 ≤ (0.5 − x)2 for           Title Page

   exactly 1/3 of points.
 • Computing βj : β1 = 0.25, similarly β2 = 4, β3 = 0.25.
                    (2)            (1)   √                          Page 13 of 16
 • Computing αj : α1 = β1 · α1 = 0.25 · 1 = 0.5;
               (2)        (1)   √             (2)                      Go Back
   similarly, α2 = β2 · α2 = 4 · 1 = 2, α3 = 0.5.
                                                                      Full Screen
 • Result: 1/3 of voters in the first district, for which
                                                                         Close
   α1 · ρ(x, v(1)) → min, and 1/3 each in D2 and D3 .
                                                                          Quit
                                                                  Re-Districting: A . . .

13.   Resulting Iterative Algorithm                               How to Avoid . . .

                                                                  Toward Formulation of . . .
 • Start: representations v(1), . . . , v(c) (e.g., representa-
                                                                  Relation to Clustering
   tives of the existing districts).
                                                                  Towards a More . . .

 • Main idea: iterations of the 2-step process:                   Example


      – subdivide voters into c equal groups corr. to v(j);       Resulting Iterative . . .


      – re-calculate v(j) as the average of the j-th group.       Conclusion

                                                                  Acknowledgments
 • 1st step starts with an iterative process:                             Title Page
                                (1)            (1)
      – start with weights     = ... =
                               α1            = 1;
                                              αc
      – use bisection to find βj for which there are exactly
        N/c points x(k) for which, for all l = j:
                                                                        Page 14 of 16
               βj · ρ(x(k) , v(j)) ≤ α(p) · ρ(x(k) , v(l));
                                                                           Go Back
                   (p+1)                (p)
      – compute   αj       =    βj ·   αj .                               Full Screen

 • 1st step ends: assign each voter x to to the group Dj                     Close
   for which αj · ρ(x(k) , v(j)) is the smallest.
                                                                              Quit
                                                               Re-Districting: A . . .

14.   Conclusion                                               How to Avoid . . .

                                                               Toward Formulation of . . .
 • What we propose: a new algorithm for dividing an area
                                                               Relation to Clustering
   into voting districts.
                                                               Towards a More . . .

 • New features: we take take into account not only geo-       Example

   graphic closeness, but also common interests of voters.     Resulting Iterative . . .


 • Necessary input: weights wi of different factors:            Conclusion

                                                               Acknowledgments
      – geographic location (xi are geographic coordinates),           Title Page

      – income,
      – rural vs. urban status,
      – number of children,
                                                                     Page 15 of 16
      – etc.
                                                                        Go Back


                                                                       Full Screen


                                                                          Close


                                                                           Quit
15.   Acknowledgments                                      Re-Districting: A . . .
                                                           How to Avoid . . .

This work was supported in part by:                        Toward Formulation of . . .
                                                           Relation to Clustering

 • by National Science Foundation grants HRD-0734825,      Towards a More . . .
                                                           Example
   EAR-0225670, and EIA-0080940,                           Resulting Iterative . . .
                                                           Conclusion
 • by Texas Department of Transportation grant             Acknowledgments
   No. 0-5453,
 • by the Japan Advanced Institute of Science & Technol-            Title Page


   ogy (JAIST) Int’l Joint Research Grant 2006-08, and
 • by the Max Planck Institut f¨r Mathematik.
                               u
                                                                 Page 16 of 16


                                                                     Go Back


                                                                   Full Screen


                                                                      Close


                                                                       Quit