Apportionment and Voting

• Voting


  A process in choosing sides based on personal opinion
• Voting


  • the election
  • contest judging
  • proposal decisions
• Plurality Method


  The candidate with the most number of votes wins
• Majority Method


  The candidate with more than 50% of the votes wins
• None of the candidates in the election had a majority vote
• Al Gore had the plurality of popular votes
• Ranked-Choice Voting


  A ballot in which a voter is required to give an ordering of their preferences
• Ranked-Choice Voting


  1. Determine the majority

  2. Count the number of first place on each candidates

  3. Eliminate all those with no first place

  4. Reallocate the votes to the remaining candidates

  5. Count the first place each candidates

  6. Determine the winner
• Borda Count Method


  A type of ranked voting by issuing points per rank
• Borda Count Method


  1. Set the Borda score by getting the difference between the rank and the total number of option

  2. Determine the winner by the highest Borda score
• Apportionment


  The process of fairly dividing and distributing resources
• Part-to-part Ratio


  A larger number of population means a larger value of subjects
• The government will donate more vaccines to a much more populated city
• The ratios for each department are: S to C, C to T, S to T, F to P
• Standard Divisor


  The ratio of the population to the number of representative seats
• Standard Quota


  The ratio of state population over the standard divisor
• Hamilton's Method of Apportionment

  1. Find the standard divisor

  2. Find each state's standard quota

  3. Give each state the state's lower quota

  4. Give each remaining seat one at a time to the states with the largest fractional parts of their standard quotas

  5. Check the solution
• Jefferson's Method of Apportionment

  1. Find the standard divisor

  2. Find each state's quota

  3. Find the states' lower quotas and their sum

  4. If the sum equals the number of seats, the apportionment is complete. If less, reduce the standard divisor. If greater, increase the standard divisor.
• Adams's Method of Apportionment

  1. Find the standard divisor

  2. Find each state's quota

  3. Find the states' upper quotas and their sum

  4. If the sum equals the number of seats, the apportionment is complete. If less, reduce the standard divisor. If greater, increase the standard divisor.
• Graph


  Contains two sets: vertices and edges
• Vertex


  Represented by a single dot
• Edge


  Line segment connecting two vertices
• Graphs


  • Vertices do not necessarily connect to other vertices by edges, but edges need vertices at each end
  • A loop is a vertex connecting to itself
  • A simple graph contains no loop and pairs of vertices are connected with only one edge
  • A multigraph will contain a loop or a pair of vertices connected with two or more edges
• The vertices are v, w, x, y, and z. The edges are vw, vx, wx, wz, xy, and xz.