VIEWS: 10 PAGES: 16 CATEGORY: Guides POSTED ON: 1/28/2010 Public Domain
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