Electoral Systems

Top-level Outline

  • Election for a Single Office
    • Plurality
      • Approval
    • Majority
    • Ranked Choice Voting
  • Election of an Assembly
    • District Representation
    • At-Large Voting
      • Non-Proportional
        • Plurality
        • Cumulative
        • Single Vote and Limited
      • Proportional Representation
        • Party-List
        • Single Transferable Vote
        • Mixed Member
  • US Electoral Systems
    • House
    • Senate
    • Presidency
    • Logical Consequences
      • Distribution Dependency
      • One Person, Not One Vote
      • Gerrymandering

Outline

  1. Election for a Single Office (Single-winner Elections)
    1. Plurality (First Past the Post)
    2. Majority with Possible Runoff (Two-Round System)
    3. Ranked Choice Voting (Instant Runoff)
  2. Election of an Assembly (Multi-winner Elections)
    1. The Main Example I use
    2. District Representation
      1. Logical Consequences of District Representation
        1. Distribution Dependency
        2. One Person, Not One Vote
    3. At-Large Voting
      1. Non-Proportional
        1. Plurality-at-Large Voting (Plurality Block Voting)
        2. Cumulative Voting
        3. Single and Limited Voting
      2. Proportional Representation
        1. Voting for a Party: Party-List System
        2. Voting for Individual Candidates: Single Transferrable Voting
        3. Voting for both People and Parties: Mixed Member Voting
  3. US Electoral Systems
    1. House
    2. Senate
    3. Presidency
    4. Logical Consequences of US Electoral Systems
      1. Distribution Dependency in the House, Senate, and Electoral College
        1. Distribution Dependency in the House
        2. Distribution Dependency in the Senate
        3. Distribution Dependency in the Electoral College
      2. One Person, Not One Vote
        1. The Weight of a Vote
        2. The Weight of a Vote for a Representative in each Congressional District
        3. The Weight of a Vote for Senator in each State
        4. The Weight of a Vote for President in each State
      3. House is Subject to Gerrymandering
    5. Other US Electoral Systems
  4. Addenda
    1. Links to Good Sites on Kinds of Electoral Systems
    2. Redistribution of House Votes Across Congressional Districts
    3. Redistribution of Senate Votes Across States
    4. Redistribution of Electoral Votes Across States
    5. Democratization of Electoral Systems
    6. States’ Power over Elections
      1. States’ powers over elections derive from ArtIcle 1, Section IV, Clause 1 of the Constitution
      2. States’ Powers
        1. States decide who the “people” are
        2. States decide procedures for determining an election’s outcome
        3. States draw boundaries of congressional districts
        4. States decide procedures for distributing Electoral College votes
        5. States conduct elections
    7. Democratic Senators Represent More Voters than Republican Senators, per Brookings
    8. Small-state Bias of the Senate
    9. Senate Skews towards Rural Voters, per Nate Silver
    10. Why Rural Americans Vote Republican, per Munis and Jacobs

Election for a Single Office
(Single-winner Elections)

The simplest way of electing someone for a single office is by direct popular vote. The question is whether the winner should need a majority or a mere plurality.  The latter is easy to implement. The former can be accomplished by either ranked-choice voting or a straightforward vote with a possible runoff.

Plurality (First Past the Post)

  • In plurality voting, the candidate with more votes than any other candidate is elected.
  • The system is straightforward, easily understood, and widely used.
  • The downside is that, with more than two candidates, the winner may not get a majority of the votes.
  • A variant is Approval Voting, where voters can vote for any number of candidates, those they approve of

Majority with Possible Runoff (Two-Round System)

  • In plurality voting, the candidate with more than 50 percent of vote is elected.
  • If no candidate wins a majority (when there are more than two candidates), a runoff election between the top two vote-getters is held.

Ranked Choice Voting (Instant Runoff)

Example

  • Suppose the results of an election for mayor are: Al 400, Bob 350, Cora 250.
  • We could decide the winner by plurality, making Al the mayor.
  • We could have a runoff election between Al and Bob.
  • Or we could use Ranked Choice Voting, aka Instant Runoff Voting, which is like having a runoff election without an election.
  • In RCV voters vote for a candidate and backup candidates, for example:
  • In processing the ballots, the candidate with the fewest votes, Cora, is eliminated and her votes transferred to her voters’ second choices. If everyone who voted for Cora chose Bob as their backup, Cora’s 250 votes would to him, with the final result:
    • Al: 400
    • Bob: 350 + 250 = 600
  • Bob thus wins with a majority.

RCV’s Advantages

  • If there are only two candidates, Plurality, Majority, and RCV yield the same outcome.
  • If there are more than two candidates:
    • RCV is faster and less expensive than majority voting if a runoff election is required.
    • An advantage of RCV over plurality voting is that RCV eliminates the problem of vote-splitting. Suppose in the example above that Al is a liberal and Bob and Cora conservatives. In a plurality vote Al wins since the conservative vote is split. With RCV, in contrast, Bob or Cora can win if, say, conservatives voting for Cora make Bob their backup and conservatives voting for Bob make Cora their second choice.

Where RCV is Used

fairvote.org/our-reforms/ranked-choice-voting-information/#where-is-ranked-choice-voting-used

Procedure

  • Voters rank candidates 1, 2, 3, …
  • Algorithm:
    • Step 1: All number 1’s are tallied.
    • Step 2: If a candidate wins a majority of the number 1’s they win.
    • Step 3: If there’s no winner, the candidate with the fewest number 1’s is eliminated and his number 1’s are transferred to the candidates he ranked number 2.
    • Step 4: Go back to Step 1 and resume the cycle.

Flow Chart

Image Credit: rockthevote.org/ranked-choice-voting-an-explainer/

  • States using ranked choice voting Fairvote
    • Alaska:
      • Adopted in 2020 for all state and federal general elections and in the 2024 Presidential election. All uses except presidential election in “Top Four” form, with open ballot primary advancing 4 candidates to the general election with RCV. First being used in a special U.S. House election in August 2022, then all offices in November 2022.
    • Maine:
      • Adopted in 2016 and first used in 2018 for all state and federal primary elections and all general elections for Congress. Extended to apply to the general election for president beginning in 2020 and presidential primary elections beginning in 2024. Elections every even year.

Ranked Choice Voting in Alaska

  • Sarah Palin’s defeat in Alaska proves ranked-choice voting works  NYT Editorial
    • Ms. Peltola received 40 percent to Ms. Palin’s 31 percent in the first round of what’s also called an instant runoff process. Under that system, Ms. Palin’s fellow Republican Nick Begich III, who received 29 percent, was eliminated from contention, and his voters had their next choices tabulated. The result: Ms. Peltola beat Ms. Palin, 52 percent to 49 percent. The same three candidates will face off again in November for a full term.
  • How second-choice votes pushed a Democrat to victory in Alaska  NYT

Ranked Choice Voting in Maine

Bangor Daily News

Janet Mills won the Democratic gubernatorial nomination in a fourth and final round of retabulation.

View FairVote

Election of an Assembly
(Multi-winner Elections)

  • The are two main ways of electing members of an assembly, such as a legislature, city council, or school board,
    • In district representation a member of an assembly is elected by the electorate of the district she represents.
    • In proportional representation a member of an assembly is elected by the electorate at large and represents the portion of the electorate that voted for her.

The Main Example I use

  • An assembly has three seats, all up for election.
  • The electorate consists of 300 voters.
  • 165 voters nationwide prefer Party A. 135 voters prefer Party B.
  • Party A’s candidates for the three seats are A1, A2, and A3. B’s candidates are B1, B2, and B3.

District Representation

In district representation residents of an electoral district elect a member of the assembly to represent them. Representatives can be elected by a plurality, a majority with a possible runoff, or ranked-choice voting.

Logical Consequences of District Representation
  1. Distribution Dependency
    • The party breakdown of an assembly depends not only on the total votes each party receives but also on the distribution of those votes across districts.  Indeed, where the vote totals of two parities are relatively close, the party with more votes can win anywhere from every seat to only one seat, with the same number of total votes. Thus, the party breakdown of an assembly can differ, sometimes dramatically, from that of the electorate.
  2. One Person, Not One Vote
    • Some votes count more than others if district electorates differ in size.
Distribution Dependency
  • The party breakdown of an assembly depends not only on the total votes each party receives but also on the distribution of those votes across districts. 
  • Using the Main Example, assume the legislature has 3 members, each representing one of three 100-person electoral districts. Party A has 165 members nationwide, B 135. An election is held for all three seats.
  • Ballots look like this:
  • Depending on how the parties are distributed across the districts, the outcome of the election can vary anywhere from (a) partyA winning every seat to (b) B winning every seat but one.
  • Scenario (a):
    • Parties A and B are equally distributed throughout the state so that A beats B 55 to 45 in every district. Party A wins every seat.

Scenario (a)
A beats B 55 to 45 in every district

  • Scenario (b):
    • Party B beats A 55 to 45 in Districts 1 and 2.
    • A beats B 75 to 25 in District 3.
    • The totals work out:
      • Party B: 2 x 55 + 25 = 135
      • Party A 2 x 45 + 75 = 165
    • So party B wins two seats and A one.
      • (The concentration of party A in one district may be due to gerrymandering or because A‘s prefer living together.)

Scenario (b)
Party B beats A 55 to 45 in two districts
A beats B 75 to 25 in one district

  • In the example Party A has 165 members and B 135 members.
  • The minimum number of members Party A can have in the example and, depending on its distribution across districts, be capable of (a) winning all districts and also (b) losing all districts but one is 153
    • (a)  D1 = 51, D2 = 51, D3=51 (minimal way of winning)
    • (b)  D1 = 27, D2 = 26, D3 = 100 (among others)
  • The maximum number of members Party A can have and, depending on its distribution across districts, be capable of (a) winning all districts and also (b) losing all districts but one is 198
    • (a)  D1 = 66, D2 = 66, D3=66 (among others)
    • (b)  D1 = 49, D2 = 49, D3 = 100 (maximum way of losing)
  • In general, for d number of districts each consisting of n members,
    • the minimum number of party members = d (n/2 + 1)
    • the maximum number of party members = (n /2 – 1) (d – 1) + n
One Person, Not One Vote
  • A basic principle of fairness is One Person, One Vote, that every vote should count the same, i.e. have the same weight or voting power. No one’s vote should count more than another’s.
  • If districts have different electorate sizes, not everyone’s vote counts the same.
  • Suppose a state had four electoral districts with 100, 200, 500, and 1,000 voters. As Chief Justice Earl Warren put it in Reynolds v. Sims:
    • “The resulting discrimination against those individual voters living in disfavored areas is easily demonstrable mathematically. Their right to vote is simply not the same right to vote as that of those living in a favored part of the State. Two, five, or 10 of them must vote before the effect of their voting is equivalent to that of their favored neighbor. Weighting the votes of citizens differently, by any method or means, merely because of where they happen to reside, hardly seems justifiable.”
    • “Whatever the means of accomplishment, the overriding objective must be substantial equality of population among the various districts, so that the vote of any citizen is approximately equal in weight to that of any other citizen in the State.”
  • The weight of a single vote in a district = 1 / the population size of the district’s electorate. Thus the weights of votes in the four districts are: 1/100, 1/200, 1/500, and 1/1,000. A vote in the first district thus has twice the weight of a vote in the second: (1/100)/(1/200) = 2. And it has 5 times a vote in the third and 10 times a vote in the fourth.
  • The principle of One Person, One Vote is thus violated if electoral districts have different population sizes.
  • Proof that not everyone’s vote has the same weight if districts have different size electorates.
    • The weight of a single vote in an electoral district with an electorate of size n = 1 / n
    • If the electorates of two districts d1 and d2 have different sizes, s1 and s2, the weights of a vote in the districts are w1 = 1 / s1 and w2 = 1 / s2.
    • s1 ≠ s2
    • Therefore w1 ≠ w2

At-Large Voting

  • In At-Large Voting each member of an assembly is elected by the state’s entire electorate.
  • Non-district voting avoids the problem of violating One Person, One Vote, since there is only one electoral district.
Non-Proportional

In non-proportional voting proportional representation is not an objective.

Plurality-at-Large Voting (Plurality Block Voting)
  • In plurality-at-large voting for n seats, voters vote for n different candidates or fewer.
    • They can’t vote for the same candidate more than once.
  • The n candidates with the most votes win.

Main Example Scenario

  • Scenario
    • Candidates A1, A2, A3, B1, B2, and B3 run for three seats of an assembly. 300 citizens vote in the election.
  • Election
    • Party A voters vote for A1, A2, A3
    • Party B voters vote for B1, B2, B3
    • So  each of A1, A2, A3 gets 165 votes and each of B1, B2, B3 gets 135 votes.
    • The three candidates with the most votes win.
    • So A1, A2, A3 win all three three seats
Cumulative Voting
  • In Cumulative Voting you distribute your votes how you want, including voting for the same candidate more than once.
  • The candidates with the most votes win the seats.
Single and Limited Voting
  • In Single Vote you vote for only one candidate, no matter how many seats.
  • The candidates with the most votes win the seats.
  • In Limited Vote you vote for fewer candidate than there are seats.
  • The candidates with the most votes win the seats.
Proportional Representation
  • The objective of proportional representation is to elect an assembly such that
    • its members represent groups of voters rather than electoral districts
    • the percentage of seats representing a group of voters equals the percentage of the electorate that belongs to that group.
  • One way of achieving proportional representation is by the electorate voting for groups of voters rather than for individual candidates. This is known as the Party-List System.
  • A second way is by
    • the electorate voting for individual candidates and additionally specifying their second choices
    • transferring “wasted” votes to second choices.
  • This is called the Single Transferrable Vote.
Voting for a Party: Party-List System
  • In a party-list system each political party sets forth a list of candidates for all seats up for election, its “party-list.”  Every citizen then votes for one of the party-lists.
  • Using the Main Example the three seats of the assembly are up for election.for all the seats. The parties set forth their party-lists:
    • A‘s party-list: A1, A2, A3
    • B‘s party-list: B1, B2, B3
  • Citizens vote for one of the two party-lists.
  • The outcome is:
    • 165 votes for the party-list A.
    • 135 votes for party party-list B.
  • The party-list candidates are assigned seats accordingly.
    • Party A: 165 / 300  x 3 = 1.65, rounding to 2 seats
    • Party B: 135 / 300  x 3 = 1.35, rounding to 1 seat

Proportional Representation by Party-List
Party A beats B 165 to 135
So A gets 55% of the seats and B 45%

Voting for Individual Candidates: Single Transferrable Voting
  • The Party-List System is a straightforward way of achieving proportional representation: the electorate votes directly for groups of voters. Single Transferrable Voting, developed in 19th century Britain, is an ingenious way of achieving proportional representation without predefining groups of voters, e.g. political parties. The core idea is transferring “wasted” votes to second choices.
  • I first discuss the concept of a wasted vote and then explain how STV works.

Wasted Votes

  • Wasted votes are those that don’t contribute to winning. Suppose that candidate W beats candidate L in an election 55 votes to 45.  The 45 votes for L are wasted because L lost.  The 9 votes for W beyond the 46 needed to win are also wasted. (In practice it’s easier to subtract 45 from 55 giving 10.)  Wasted votes are thus either losing votes or surplus winning votes.
  • In the example below (from Scenario (b) above)
    • Party A has 45 losing-wasted votes in District 1, 45 losing-wasted votes in District 2, and 50 surplus-wasted votes in District 3, for a total of 140 wasted votes.
    • Party B has 10 surplus-wasted votes in District 1, 10 surplus-wasted votes in District 2, and 25 losing-wasted votes in District 3, for a total of 45 wasted votes.
  • Gerrymandering, Ranked Choice Voting, and Single Transferable Voting all use the concept of wasted votes.
  • In partisan gerrymandering one political party weakens the voting power of a second by drawing electoral boundaries so that the latter wastes significantly more votes than the former.
  • In Ranked Choice Voting losing-wasted votes are transferred to voters’ second choices.
  • In Single Transferable Voting, as we’ll see:
    • losing-wasted votes are transferred to voters’ second choices
    • surplus-wasted votes are also transferred to voters’ second choices.

The Idea of STV

  • The idea of ranked choice voting and single transferable voting is that no vote goes to waste.
  • In ranked choice voting (for a single office) losing (wasted) votes are transferred to voters’ second choices.
  • In single transferable voting (for seats of an assembly):
    • losing (wasted) votes are transferred to voters’ second choices.
    • surplus winning (wasted) votes are also transferred to voters’ second choices.
      • This step is not needed in RCV since processing stops once there’s a winner.

Basic Procedure

  • Basic Algorithm
    • Step 1:
      • Calculate the quota, the number of votes needed to win a seat.
    • Step 2:
      • See if any candidates beat the quota. If so, declare them winners. If all seats have been filled, stop. Otherwise transfer their surplus votes (number of votes – quota) to their voters’ next choices.
    • Step 3:
      • If no candidates beat the quota, eliminate the candidate with the fewest votes and transfer his votes to his voters’ next choices. Then return to Step 2.
  • Quota
    • The quota is the number of votes needed to win a seat.
    • The Droop Quota is the widely used among competing calculations.
      • Droop Quota = ( total votes / (number of seats + 1) ) + 1.
    • The Droop Quota is a natural extension of a majority in an election for a single office. If there are 100 votes, the minimum majority is 51, which is what the Droop formula yields:
      • Droop Quota = ( 100 / (1 + 1) ) + 1 = 51.
    • A minimum majority for 2 seats would then be (100 / 3) + 1 = 34.33
      • Droop Quota = ( 100 / (2 + 1) ) + 1 = 34.33
    • And so on.

Main Example Using STV

Candidates A1, A2, A3, B1, B2, and B3 run for three seats of an assembly. 300 citizens vote in a single transferable vote election.

Ballots:

  • Ballot Tally
    • 165 votes for {A1, A2, A3}
    • 135 votes for {B1, B2, B3}
  • Determining Winners
    • round 1 = {{165, A1, A2, A3}, {135, B1, B2, B3}}
      • quota = ( total votes / (number of seats + 1) ) + 1 = 76
      • A1 wins round 1 and his 89 surplus votes (165 – quota) are transferred to A2
    • round 2 = {{89, A2}, {135, B1, B2, B3}}
      • B1 wins round 2 and his 69 surplus votes (145 – quota) are transferred to B2
    • round 3 = {{89, A2}, {69, B2}}
      • A2 wins round 3 and procedure stops
    • Results
      • A1, A2 and B1 are elected.

Summary of Outcomes for Main Example

  • Main Example
    • An assembly has three seats, all up for election.
    • The electorate consists of 300 voters.
    • Party A has 165 members statewide. Party B 135 members.
      • That’s 55% A’s versus 45% B’s.
  • Outcomes
    • District Representation
      • Outcomes range from A winning 3 seats to B winning 2 seats, depending on how the parties are distributed across districts
    • Non-District Voting
      • Plurality at Large
        • A wins all 3 seats
      • Proportional Representation
        • Party-List
          • A wins 2 seats and B one.
        • STV
          • A wins 2 seats and B one
Voting for both People and Parties: Mixed Member Voting
  • Mixed-Member Voting is a hybrid system combining plurality district voting and party-list voting.
  • Suppose a state is divided into five electoral districts and there are ten seats in an assembly, all up for election.
  • Five of the seats are filled by candidates representing electoral districts. The other five are filled from party-lists.
  • A ballot for district n looks like this:
  • A voter from District n votes for An, Bn, or Cn to represent the district. He also votes for a party.
  • Suppose A1, B2, B3, C1, and C2 win the five district seats.
  • And suppose the party-list voting breaks down
    • Party A 40%
    • Party B 40%
    • Party C 20%.
  • And the five candidates from the party-lists are A6, A7, B8, B9, and C10.
  • The final winners are then: A1, A6, A7, B2, B3, B8, B9, C1, C2 and C10.

US Electoral Systems

House

  • The House has 435 seats, per the 1929 Permanent Apportionment Act
  • Apportionment
    • Each state is apportioned a number of House seats, approximately corresponding to its share of the aggregate population of the 50 states, with each state guaranteed at least one seat.  
    • Reapportionment takes effect three years after the census
    • The algorithm is explained in Computing Apportionment.
    • California has the most representatives, 53. Seven states have 1 representative.
  • All states except four use plurality voting for their representatives
  • One Person, One Vote
    • One Person, One Vote is the principle (not in the Constitution) that every person’s vote should count the same, i.e. should have the same weight or voting power.
    • In Wesberry v. Sanders, the Court ruled in 1964 that One Person, One Vote applied to congressional districts within a state, meaning that the districts had to be approximately equal in population.
  • Contiguity and Compactness
    • 34 states require that their congressional districts be contiguous, i.e. a person can travel from any point in the district to any other point without having to cross a different district.
    • 31 states require that their congressional districts be compact, i.e. not spread out, more like a circle than an elongated ellipse.
  • Representation in the House is subject to partisan gerrymandering.

Senate

  • Senate representation results from a compromise at the Constitutional Convention
    • Failed Virginia (Large State) Plan
      • Congress should be bicameral, with the number of a state’s representatives in each chamber being proportional to its population.
    • Failed New Jersey (Small State) Plan
      • Congress should have one chamber, with the states having the same number of representatives.
    • Grand (Connecticut) Compromise
      • Virginia Plan for the House of Representatives
      • New Jersey Plan for the Senate
  • Like the House, all states except four use plurality voting for their Senators
  • Senators were initially chosen by state legislatures. Since 1913 they are elected by the people.
    • Constitution: Article I, Section 3, Clause 1
      • “The Senate of the United States shall be composed of two Senators from each state, chosen by the legislature thereof, for six years; and each Senator shall have one vote.”
    • Seventeenth Amendment (1913)
      • “The Senate of the United States shall be composed of two Senators from each State, elected by the people thereof, for six years; and each Senator shall have one vote.”
  • The electoral system for the Senate fails to satisfy the principle of One Person, One Vote.
  • The party breakdown of the Senate can differ from the party breakdown of the voters.

Presidency

  • The President and Vice President are elected by the 538 electors of the Electoral College rather than directly by the people. Each elector has one vote.
  • Article II, Section 1 of the Constitution:
    • Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress
    • Thus,
      • a state gets n electors, n being the total number of its Senators and Representatives
      • a state’s legislature can choose its electors any way it wants.
  • The 23rd Amendment (1961) allocates to the District of Columbia the number of electors equal to the number of the least-populated state, meaning 3 electors.
  • The total number of electors is therefore 100 + 435 + 3 = 538
  • To win, a candidate needs a majority of the electoral votes, 270.
  • The states gradually transitioned from their legislatures choosing electors to election of electors by popular vote. (Wikipedia):
    • Initially, state legislatures chose the electors in most states.
    • By 1824, there were six states whose legislatures selected electors.
    • By 1832, only South Carolina selected electors by legislature vote.
    • Since 1864, electors in every state have been chosen based on the popular vote.
  • The winner of a state’s popular vote gets all its electoral votes, with two exceptions.
    • Maine and Nebraska allocate one electoral vote to the winner of each House district and two electoral votes to the statewide winner.
  • The electoral system for the Presidency contravenes the principle of One Person, One Vote.
  • The percentages of Democrats, Republicans, and other parties in the Electoral College can differ from their percentages in the electorate.
    • Four presidents were elected with fewer popular votes than their opponents
      • Rutherford B. Hayes in 1876
      • Benjamin Harrison in 1888
      • George W. Bush in 2000
      • Donald Trump in 2016.

More on Electoral College.

Logical Consequences of US Electoral Systems

Distribution Dependency in the House, Senate, and Electoral College
  • In District Representation the party breakdown of the members of an assembly depends not only on the total votes each party receives but also on the distribution of those votes across districts.
  • The House, Senate, and Electoral College use District Representation
    • The electoral districts of the House are congressional districts. Those of the Senate and Electoral College are states
  • Therefore, the party breakdowns of the House, Senate, and Electoral College depend not only on the total votes each party receives but also on the distribution of those votes across congressional districts and states.
  • Therefore, the party breakdowns of the House, Senate, and Electoral College can differ, sometimes dramatically, from the party breakdown of the electorate.
Distribution Dependency in the House
  • The number of House seats each party wins in an election depends not only on the nationwide popular vote but also on the distribution of those votes across congressional districts.
  • Indeed, given a fixed number of total popular votes for each party, it’s theoretically possible (within certain limits) for the party with most votes to win anywhere from (a) all 435 seats to (b) a small minority.
  • Consider the 2020 House election.
    • The Democrats won 222 seats and Republicans 213.
    • The breakdown of the national popular vote was:
  • The Democrats 77,134,868 votes, the GOP’s 72,622,513 and Other’s  2,965,959 can be redistributed across congressional districts so that, with each district total unchanged:
    • (a) Democrats win all 435 seats
  • The votes can also be redistributed so that:
    • (b) The Republicans win 405 seats and Democrats 30.
  • (a)  Democrats win all 435 seats
    • In each district Democrats win 50.51 percent of the vote, Republicans 47.55 percent, and Other 1.94 percent.
    • The nationwide popular vote remains the same, except for rounding.
    • The overall state totals remain the same, except for rounding. For example:
      • Actual Votes
      • Redistributed Votes with Democrats winning all seats
  • (b) Republicans win 405 seats and Democrats 30.
    • In every district, except for 30, Trump wins 51.37 percent of the vote, Other gets 1.94 percent, and the Democrats get what’s left. Republicans win no seats in VA 5-11, VT, WA, WI, WV and WY.
    • The nationwide popular vote remains the same, except for rounding.
    • The overall state totals remain the same, except for rounding. For example:
      • Actual Votes
      • Redistributed Votes with Republicans winning 405 seats
  • See complete redistribution
  • The percentages of House seats won by the political parties can differ, sometimes dramatically, from their percentages in the electorate
Distribution Dependency in the Senate
  • The number of Senate seats each party wins in an election depends not only on the nationwide popular vote but also on the distribution of those votes across states.
  • Indeed, given a fixed number of total popular Senate votes for each party, it’s theoretically possible (within certain limits) for the party with the most votes to win anywhere from (a) all 100 seats to (b) a small minority.
  • Consider the 2020 Senate based on the Senate elections of 2016, 2018, and 2020
    • The Democrats had 48 seats, Republicans 50, and Independents 2.
    • The breakdown of the national popular vote over the three elections was:
  • The Democrats’ 147,600,858 votes over these elections, the GOP’s 120,605,927 votes, Independents’ 528,288 and Other’s  10,848,783 can be redistributed across Senators so that, with each Senator’s total unchanged:
    • (a) Democrats win all 100 seats
  • The votes can also be redistributed so that:
    • (b) Republicans win 95 seats and Democrats 5.
  • (a) Democrats win all 100 seats
    • For the Senate elections of 2016, 2018, and 2020 Democrats win 52.79 percent of the vote of each Senator, Republicans 43.14 percent, Independents 0.19 percent, and Other 3.88 percent.
    • The nationwide popular vote remains the same, except for rounding.
    • The overall Senator totals remain the same, except for rounding. For example:
      • Actual Votes
      • Redistributed Votes with Democrats winning all seats
  • (b) Republicans win 95 seats and Democrats 5.
    • In each senate race, except five, Republicans win 53.31 percent of the vote, Independents 0.19 percent, Other 3.88 percent, and Democrats what’s left. Republicans win zero votes in elections for:
      • CA 2016 (1 seat)
      • CA 2018 (1 seat)
      • TX 2020 (1 seat)
      • GA 2020 (2 seats)
    • The nationwide popular vote remains the same, except for rounding.
    • The overall Senator totals remain the same, except for rounding. For example:
      • Actual Votes
      • Redistributed Votes with Republicans winning 95 seats
  • The percentages of Senate seats won by the political parties can differ, sometimes dramatically, from their percentages in the electorate
Distribution Dependency in the Electoral College
  • The total number of electoral votes each candidate receives depends not only on the nationwide popular vote but also on the distribution of those votes across states.
  • Indeed, given a fixed number of total popular votes for each candidate, it’s theoretically possible (within certain limits) for the candidate with most votes to win anywhere from (a) all 538 electoral votes to (b) a small minority.
  • Consider the 2020 presidential election.
    • Biden beat Trump 306 electoral votes to 232.
    • The breakdown of the national popular vote was:
  • Biden’s 81,283,501 votes, Trump’s  74,223,975, and Other’s  2,922,155 can be redistributed across states so that, with each state’s total unchanged:
    • (a) Biden wins all 538 electoral votes.
  • The votes can also be redistributed so that:
    • (b) Trump wins 483 electoral votes, losing only California’s 55 votes.
  • (a)  Biden wins all 538 electoral votes
    • In each state Biden wins 51.31 percent of the votes, Trump 46.85 percent, and Other 1.84 percent. For example:
    • The nationwide popular vote remains the same, except for rounding.
    • The overall state totals remain the same, except for rounding. For example:
      • Actual Votes
      • Redistributed Votes with Biden the winner
    • See complete redistribution
  • (b) Trump wins 483 electoral votes, losing only California’s 55 votes.
    • In every state except California Trump wins with 52.67 percent of the vote, Other gets 1.84 percent, and Biden gets what’s left.  Trump gets zero in California.
    • The nationwide popular vote remains the same, except for rounding.
    • The overall state totals remain the same, except for rounding. For example:
      • Actual Votes
      • Redistributed Votes with Trump the winner
    • See complete redistribution
  • The percentages of electoral votes won by the candidates can differ, sometimes dramatically, from their percentages in the electorate
  • Four presidents were elected with fewer popular votes than their opponents
    • Rutherford B. Hayes in 1876
    • Benjamin Harrison in 1888
    • George W. Bush in 2000
    • Donald Trump in 2016.

More on Electoral College.

One Person, Not One Vote
The Weight of a Vote
  • A basic principle of fairness is One Person, One Vote, that every vote should count the same, i.e. have the same weight or voting power. No one’s vote should count more than another’s.
  • One Person, One Vote requires that electoral districts have the same population size. 
  • Suppose, for example, a state had four electoral districts with 100, 200, 500, and 1,000 residents. As Chief Justice Earl Warren put it in Reynolds v. Sims:
    • “The resulting discrimination against those individual voters living in disfavored areas is easily demonstrable mathematically. Their right to vote is simply not the same right to vote as that of those living in a favored part of the State. Two, five, or 10 of them must vote before the effect of their voting is equivalent to that of their favored neighbor. Weighting the votes of citizens differently, by any method or means, merely because of where they happen to reside, hardly seems justifiable.”
    • “Whatever the means of accomplishment, the overriding objective must be substantial equality of population among the various districts, so that the vote of any citizen is approximately equal in weight to that of any other citizen in the State.”
  • The weight of a single vote in a district = 1 / the population size of the district’s electorate. Thus the weights of votes in the four districts are: 1/100, 1/200, 1/500, and 1/1,000. A vote in the first district has twice the weight of a vote in the second: (1/100)/(1/200) = 2; and 5 times a vote in the third and 10 times a vote in the fourth.
  • Thus in an election of an assembly using district representation, if the districts’ electorates differ in population size, not everyone’s vote counts the same and the election violates the principle of One Person, One Vote.
The Weight of a Vote for a Representative in each Congressional District
  • The graph below shows the weight of a vote for each of the 435 representatives, relative to a baseline of 1, where each vote has the same weight.
  • Calculation
    • The nationwide average weight of a congressional vote is 435 / 324,005,000 = 1 / 744,839.
    • For a district with a population of say 1,000,000:
      • The weight of a vote for its Representative is 1 / 1,000,000.
      • The weight of a vote for its Representative relative to the nationwide average baseline is therefore (1 / 1,000,000) / (1 / 744,839) = 0.745
  • Take Wyoming’s single district with a population of 581,024. The weight of a vote for the Wyoming Representative is 1 / 581,024. The weight of the House vote in Wyoming relative to the baseline is therefore (1/581,024) /( 1/744,839) = 1.28, represented by the rightmost vertical bar in the chart.
  • The relative weights are pretty close to 1.0, thanks to the 1964 Supreme Court ruling in Wesberry v Sanders that “as nearly as practicable one man’s vote in a congressional election is to be worth as much as another’s.”
  • There are anomalies, of course, e.g. a vote in RI 2nd has twice the weight of a vote in Montana’s single district, 1.42 versus 0.71. Such anomalies are due to the algorithm for apportioning house seats to the states, set forth in Computing Apportionment.
  • Statistics for relative weights
    • Max = 1.423 (Rhode Island 2nd)
    • Min = 0.709 (Montana At-Large)
    • Ratio of Max to Min = 2.007
    • Mean = 1.005
    • Standard Deviation = 0.069
The Weight of a Vote for Senator in each State
  • The graph below shows the weight of a vote for senator in each of the 50 states relative to a baseline of 1, where each vote has the same weight.
  • Calculation
    • The nationwide average weight of a vote for Senator is 50 / 332,616,000 = 1 / 6,652,315.
    • For a state with a population of say 1,000,000:
      • The weight of a vote for Senator is 1 / 1,000,000.
      • The weight of a vote for Senator relative to the baseline is (1 / 1,000,000) / (1 / 6,652,315) = 6.65
  • Take a state like Wyoming with a population of 581,381. The weight of a vote for Senator in Wyoming is 1 / 581,381. The weight of a vote in Wyoming relative to the baseline is therefore (1 / 581,381) / (1 / 6,652,315) = 11.44, as represented in the graph.
  • By contrast, the relative weight of a vote for Senator in California (population 39,029,342) is 0.17.
  • Thus a vote for Senator in Wyoming has 67 times the weight of a vote in California, 11.44 / 0.17. Put another way, a vote for Senator in Wyoming (11.44) is equivalent to 67 votes in California (67 x 0.17).
  • Statistics for relative weights:
    • Max = 11.44 (Wyoming)
    • Min = 0.17 (California)
    • Ratio of Max to Min = 67.1321
    • Mean = 2.65415
    • Standard Deviation = 2.78379
The Weight of a Vote for President in each State
  • The graph below shows the weight of a vote for president in each state plus DC, relative to a baseline of 1, where each vote has the same weight.
  • In casting a presidential vote, you’re voting for a set of electors representing your state in the electoral college, where each elector’s vote has the same weight. To calculate the weight of a presidential vote in a state we have to consider both the size of the state’s population and the number of electors. The weight of a vote = (1 / state’s population size) x number of electors. If a state’s population is 100 and there are 5 electors, the weight of a vote is 5/100 = 1/20. If there were 10 electors the weight of a vote is double: 10/100 = 1/10 = 2/20.
  • Take Rhode Island, which has 4 electoral votes and a population of 1,093,734. The weight of a presidential vote in RI is 4 / 1,093,734 = 1 / 273,434.
  • The nationwide average weight of a presidential vote is 538 / 333,288,000 = 1 / 619,494.
  • So the weight of a presidential vote in RI relative to the national average baseline is (1 / 273,434) / (1 / 619,494) = 2.26561, as shown on the graph.
  • (The calculation assumes a state is winner-take-all, that is, the winner gets all the state’s electoral votes.
  • Two states split their electoral votes. Maine and Nebraska allocate one electoral vote to the winner of each House district and two electoral votes to the statewide winner. For a voter in House District 1, say, the weight of a vote = ( 2 / state population) + (1 / population of District 1).)
  • Statistics for the relative weights:
    • Max = 3.19667 (WY)
    • Min = 0.783919 (TX)
    • Ratio of Max to Min = 4.078
    • Mean = 1.34337
    • Standard Deviation = 0.589
  • Note on DC:
    • The District of Columbia has 3 electoral votes, thanks to the 23rd Amendment, which allocates to D.C. the number of electoral votes equal to the number of the least-populated state (meaning 3). Not being a state, DC has no senators or representatives.

The relative weight of a presidential vote is higher in less populated states, the smallest state (Wyoming) with a relative weight of 3.2 and the largest (California) with a relative weight of 0.87.

House is Subject to Gerrymandering

Gerrymandering is drawing electoral boundaries to reduce a targeted group’s effect on the outcome of elections

  • The ability of state legislatures to gerrymander their congressional districts follows from two propositions:
    • The House of Representatives uses district representation.
    • State legislatures have the power to draw district boundaries.
  • Proof:
    • In district representation, the party breakdown of a legislature depends, not just on the overall vote count, but also on the distribution of parties across districts.
    • The House of Representatives uses district representation.
    • State legislatures have the power to redraw the boundaries of congressional districts, thereby altering the distribution of the parties across districts
    • Therefore state legislatures have the power to affect the party breakdown of the House of Representatives.
    • Therefore, state legislatures have the power to gerrymander their congressional districts.

More on Gerrymandering

Other US Electoral Systems

Addenda

Redistribution of House Votes Across Congressional Districts

The Democrats 77,134,868 votes, the GOP’s 72,622,513 and Other’s 2,965,959 are redistributed across congressional districts so that, with each district total unchanged, Democrats win all 435 seats

redistribution-of-house-seats-with-democrats-winning-all-435Download

The Democrats 77,134,868 votes, the GOP’s 72,622,513 and Other’s  2,965,959 are redistributed across congressional districts so that, with each district total unchanged, Republicans win 405 seats and Democrats 30

redistribution-of-house-seats-with-republicans-winning-405Download

Redistribution of Senate Votes Across States

The popular vote by political party in the 2020 Senate, based on the elections of 2016, 2018, and 2020

senate161820-2Download

The Democrats’ 147,600,858 votes over three elections, the GOP’s 120,605,927 votes, Independents’ 528,288 and Other’s  10,848,783 are redistributed across Senators so that, with each Senator’s total unchanged, Democrats win all 100 seats

senate161820dems-1Download

The Democrats’ 147,600,858 votes over three elections, the GOP’s 120,605,927 votes, Independents’ 528,288 and Other’s  10,848,783 are redistributed across Senators so that, with each Senator’a total unchanged, Republicans win 95 seats and Democrats 5

senate161820gop-1Download

Redistribution of Electoral Votes Across States

The popular votes of Biden, Trump and Others by state in the 2020 presidential election

2020preselectionDownload

Biden’s 81,283,501 votes, Trump’s  74,223,975, and Other’s  2,922,155 are redistributed across states so that, with each state total unchanged, Biden wins all 538 electoral votes

bidenwinsallDownload

Biden’s 81,283,501 votes, Trump’s  74,223,975, and Other’s  2,922,155 are redistributed across states so that, with each state total unchanged, Trump wins 483 electoral votes, losing only California’s 55 votes

trumpwinsmostDownload

Democratization of Electoral Systems

House

  • Wesberry v. Sanders (1964)
    • Congressional districts are required to have roughly equal populations

Senate

  • Senate representation results from a compromise at the Constitutional Convention
    • Failed Virginia (Large State) Plan
      • States should be represented in proportion to their populations
    • Failed New Jersey (Small State) Plan
      • States should be represented by the same number of representatives
    • Grand (Connecticut) Compromise
      • Virginia Plan for the House of Representatives
      • New Jersey Plan for the Senate
  • Constitution: Article I, Section 3, Clause 1
    • The Senate of the United States shall be composed of two Senators from each State, chosen by the Legislature thereof, for six Years; and each Senator shall have one Vote.
  • Seventeenth Amendment (1913)
    • The Senate of the United States shall be composed of two Senators from each State, elected by the people thereof, for six years; and each Senator shall have one vote. 

President and Vice President

  • Article II, Section 1, of the Constitution stipulates that
    • a state gets n electors, n being the total number of its Senators and Representatives
    • a state can choose its electors any way it wants
  • The 23rd Amendment (1961) allocates to the District of Columbia the number of electoral votes equal to the number of the least-populated state, meaning three.
  • The total number of electoral votes = 538 = 100 + 435 + 3.
  • Electoral College, Wikipedia
    • Initially, state legislatures chose the electors in most states.
    • In 1824, there were six states whose legislatures selected electors.
    • By 1832, only South Carolina selected electors by legislature vote.
    • Since 1864, electors in every state have been chosen based on the popular vote.
  • Except for Maine and Nebraska, the winner of the state’s popular vote gets all its electoral votes. Maine and Nebraska allocate one electoral vote to the winner of each House district and two electoral votes to the statewide winner.

States’ Power over Elections

States’ powers over elections derive from ArtIcle 1, Section IV, Clause 1 of the Constitution
  • “The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the Places of chusing Senators.”
States’ Powers
States decide who the “people” are
  • States decide who the “people” are, i.e. who can vote (within the constraints imposed by the Constitution and federal law).
  • Constitutional amendments and federal law have put constraints on states’ power to decide who can vote.
States decide procedures for determining an election’s outcome
  • Systems for determining the outcome of an election: A candidate wins if they get:
    • A plurality of votes
    • A majority of votes, using either
      • ranked choice voting (instant runoff)
        • or
      • possible runoff election
    • A supermajority of votes
    • A unanimity of votes
  • Georgia, Louisiana, and Maine elect officials by majority
    • Georgia and Louisiana conduct runoff elections if needed
    • Maine uses Instant Runoff Voting (Ranked Choice Voting)
States draw boundaries of congressional districts
  • In Rucho v. Common Cause the Supreme Court ruled that partisan gerrymandering does not violate the Constitution.

View Gerrymandering

States decide procedures for distributing Electoral College votes
  • The winner takes all except in Maine and Nebraska.

View Electoral College

States conduct elections
  • Voter Registration
    • Specifying how and when voters can register, e.g. online
  • Vote Casting
    • Designating polling locations and hours
    • Specifying the kind of ID required
    • Making rules for early voting and voting by mail
  • Counting Votes
    • Specifying the procedure
    • Announcing the results

Democratic Senators Represent More Voters than Republican Senators, per Brookings

  • The challenge to democracy—overcoming the small state bias, July 2022 Brookings
    • With the even split in the current Senate, the 50 Democratic senators represent 56.5% of the voters, while the 50 Republican senators represent just 43.5% of the voters. In 2018, the Democrats won nearly 18 million more votes for Senate than the Republicans, but the Republicans still gained two seats.

Small-state Bias of the Senate

Senate Skews towards Rural Voters, per Nate Silver

The Senate has a major skew towards rural voters, Nate Silver FiveThirtyEight

  • At FiveThirtyEight, our favorite way to distinguish between urban and rural areas is based on using census tracts to estimate how many people live within a 5-mile radius of you. Based on this, we can break every person in the country down into four buckets:
    • Rural: Less than 25,000 people live within a 5-mile radius of you;
    • Exurban or small town: Between 25,000 and 100,000 people within a 5-mile radius;
    • Suburban or small city: Between 100,000 and 250,000 people within a 5-mile radius;
    • Urban core or large city: More than 250,000 people within a 5-mile radius.
  • As it happens, the overall U.S. population (including Washington D.C. and Puerto Rico) is split almost exactly evenly between these buckets:
    • 25 percent rural,
    • 23 percent exurban/small town,
    • 27 percent suburban/small city, and
    • 25 percent urban core/large city.
  • But what does representation look like in the Senate? Since each state has the same number of senators, this is simple to calculate. We can take the urban/rural breakdown for each state and average the 50 states together, as in the table below:
Image Credit FiveThirtyEight
  • Because there are a lot of largely rural, low-population states, the average state — which reflects the composition of the Senate — has 35 percent of its population in rural areas and only 14 percent in urban core areas, even though the country as a whole — including dense, high-population states like New York, Texas and California — has about 25 percent of the population in each group. That’s a pretty serious skew. It means that the Senate, de facto, has two or three times as much rural representation as urban core representation … even though there are actually about an equal number of voters in each bucket nationwide.

Why Rural Americans Vote Republican, per Munis and Jacobs

Why resentful rural Americans vote Republican, Analysis by Kal Munis and Nicholas Jacobs WaPo

  • Over the past 25 years, rural areas have increasingly voted Republican while cities have increasingly voted Democratic — a dividing line that has replaced the North/South divide as the nation’s biggest source of political friction. That divide will influence which party takes control of Congress in January.
  • But why are rural and urban voters so sharply divided? Some scholars and pundits argue that it comes down to who lives where: that the disproportionately White, older, more religious, less affluent and less highly educated voters who live in rural areas are more likely to hold socially conservative views generally championed by Republicans. Meanwhile, urban areas are filled with younger, more racially diverse, more highly educated and more affluent people who hold the more socially liberal views generally championed by Democrats.
  • While all that matters, our new research shows that place itself also matters. Unlike Republican voters in suburbs and the cities, rural voters care about what we might call “geographic inequity” — the idea that rural areas receive less than their fair share from the government, are ignored by politicians, and are mocked and derided in popular culture. Without these beliefs, the urban-rural political divide would not be as vast as it is today.