Class 20: Elections and Review

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class  20: Elections, Review cs2102:  Discrete   Mathematics  |  F16 uvacs2102.github.io   0 David  Evans   University  of  Virginia Univac  predicts  big  win  for  Eisenhower,  1952
2. First  Presidential  Veto 1 George  Washington,  5  April  1792
3. First  Presidential  Veto 2 George  Washington,  5  April  1792
4. Cabinet  Battle  #1.5? 3
5. 4 Hamilton’s  Method  (vetoed  by  GW) Seats  =   US  "population"/30000 =   US  “population” State  Seats =   State  “population” D Distribute  remaining  seats  to  states   with  largest  remainders. Why  was  this  deserving  of  a  veto?
6. 5 Hamilton’s  Method  (vetoed  by  GW) =   US  “population” Number  of  Representatives State  Seats =   State  “population” D Distribute  remaining  seats  to  states   with  largest  remainders. Why  was  this  deserving  of  a  veto?
7. Hamilton’s  Method  [1790  census] 6 US  “Population”  =  3,615,920 Seats  =  120 D  =  30132.67 State  Seats =   State  “population” D
8. 7 State   Population Quota  =   / Floor(Quota) Remainder Final  Seats Pop/Rep Virginia 630,560 20.926 20 0.926 21 30,027   Massachusetts 475,327 15.774 15 0.774 16 29,708   Pennsylvannia 432,879 14.366 14 0.366 14 30,920   North  Carolina 353,523 11.732 11 0.732 12 29,460   New  York 331,589 11.004 11 0.004 11 30,144   Maryland 278,514 9.243 9 0.243 9 30,946   Conecticut 236,841 7.860 7 0.860 8 29,605   South  Carolina 206,236 6.844 6 0.844 7 29,462   New  Jersey 179,570 5.959 5 0.959 6 29,928   New  Hampshire 141,822 4.707 4 0.707 5 28,364   Vermont 85,533 2.839 2 0.839 3 28,511   Georgia 70,835 2.351 2 0.351 2 35,418   Kentucky 68,705 2.280 2 0.280 2 34,353   Rhode  Island 68,446 2.271 2 0.271 2 34,223   Delaware 55,540 1.843 1 0.843 2 27,770  
9. 8 State   Population Quota  =   / Floor(Quota) Remainder Final  Seats Pop/Rep Virginia 630,560 20.926 20 0.926 21 30,027   Massachusetts 475,327 15.774 15 0.774 16 29,708   Pennsylvannia 432,879 14.366 14 0.366 14 30,920   North  Carolina 353,523 11.732 11 0.732 12 29,460   New  York 331,589 11.004 11 0.004 11 30,144   Maryland 278,514 9.243 9 0.243 9 30,946   Conecticut 236,841 7.860 7 0.860 8 29,605   South  Carolina 206,236 6.844 6 0.844 7 29,462   New  Jersey 179,570 5.959 5 0.959 6 29,928   New  Hampshire 141,822 4.707 4 0.707 5 28,364   Vermont 85,533 2.839 2 0.839 3 28,511   Georgia 70,835 2.351 2 0.351 2 35,418   Kentucky 68,705 2.280 2 0.280 2 34,353   Rhode  Island 68,446 2.271 2 0.271 2 34,223   Delaware 55,540 1.843 1 0.843 2 27,770  
10. Jefferson’s  Algorithm 9 =   US  “population” Number  of  Representatives State  Seats =   State  “population” D   − Increase  ,  starting  from  0,  until  sum  of  all  state   seats  =  number  of  representatives.
11. Which  is  “better”? Hamilton Jefferson 10 State  “population” D   − Increase  ,  starting  from  0,   until  sum  of  all  state  seats  =   number  of  representatives. State  Seats =   State  “population” D Distribute  remaining  seats  to   states  with  largest  remainders.
12. Huntington-­‐Hill  Method   =  (1, … , 1) #  initially,  each  allocated  1 while  ∑ < :  #  more  to  allocate W ≔ YZY[\]^WZ_` a`(a`bc) find    such  that  ∀  . i ≥ W i = i + 1 11
13. 12 while  ∑ < : W ≔ YZY[\]^WZ_` a`(a`bc) find    such  that  ∀  . i ≥ W i = i + 1 Model  as  state  machine
14. 13 Model  as  state  machine =   = (c , l , … , _  |  W ∈  ℕ, W ≤ } while  ∑ < : W ≔ YZY[\]^WZ_` a`(a`bc) find    such  that  ∀  . i ≥ W i = i + 1 = →    sc + sl +  … + s_ < ,   ∃   ∈ 1, . .  s. t. ∀  . i ≥ W , = (c , l , … , ivc , i + 1, ibc , … , _ )}   x = (1, 1, … , 1)
15. 14 Preserved  Invariants =   = (c , l , … , _  |  W ∈  ℕ, W ≤ } = →    sc + sl +  … + s_ < ,   ∃   ∈ 1, . .  s. t. ∀  . i ≥ W , = (c , l , … , ivc , i + 1, ibc , … , _ )}  
16. 15 Preserved  Invariants =   = (c , l , … , _  |  W ∈  ℕ, W ≤ } = →    sc + sl +  … + s_ < ,   ∃   ∈ 1, . .  s. t. ∀  . i ≥ W , = (c , l , … , ivc , i + 1, ibc , … , _ )}  
17. 16 Prove  termination =   = (c , l , … , _  |  W ∈  ℕ, W ≤ } = →    sc + sl +  … + s_ < ,   ∃   ∈ 1, . .  s. t. ∀  . i ≥ W , = (c , l , … , ivc , i + 1, ibc , … , _ )}   x = (0, 0, … , 0)
18. 17 Prove  fairness Quota  property: for  all  terminating  states,  the  number  of  seats  allocated  to   every  state  is  either   YZY[\]^WZ_` y or   YZY[\]^WZ_` y + 1. =   = (c , l , … , _  |  W ∈  ℕ, W ≤ } = →    sc + sl +  … + s_ < ,   ∃   ∈ 1, . .  s. t. ∀  . i ≥ W , = (c , l , … , ivc , i + 1, ibc , … , _ )}   W ≔ W W (W + 1) x = (0, 0, … , 0)
19. 18 while     >  0:  #  more  to  allocate W ≔ YZY[\]^WZ_` a`(a`bc) find    such  that  ∀  . i ≥ W i = i + 1 Quota  property: for  all  terminating  states,  the  number  of  seats  allocated  to   every  state  is  either   YZY[\]^WZ_` y or   YZY[\]^WZ_` y + 1. Left  as  Challenge  Problem  (not  on  Exam  2). Prove  fairness
20. Math  Matters 19 2000  Election Bush Gore “Jefferson”  Apportionment 267 271 “Hamilton”  Apportionment 269 269 Huntington-­‐Hill  Method 271 266
21. 20 Exam  Review ü Defining  a  state  machine • Proving  a  Preserved  Invariant
27. Charge Exam  2  on   Thursday Next  week:   continuing  with   infinities! 26