Problems with Voting by District

Table of Contents

  1. Electing a President
  2. Electing a Legislature
  3. Two Problems with Voting by Electoral District
    1. Districts with Different Populations
    2. Unproductive Votes
  4. Problem of Electoral Districts with Different Populations
    1. How Much a Vote Weighs
    2. The Weight of a Vote for a Representative, by Congressional District
    3. The Weight of a Vote for Senator, by State
    4. The Weight of a Vote for President, by State + DC
  5. Problem of Unproductive Votes
    1. Three Electoral Systems
      1. Direct Popular Vote
      2. Vote by Party
      3. Vote by District
    2. How the party breakdown of a state’s congressional delegation can differ from the party breakdown of the statewide vote for the House
      1. Gerrymandering
      2. Natural ‘Gerrymandering’
    3. How the party breakdown of the House can differ from the party breakdown of the nationwide vote for representatives
      1. Misrepresentation in the House of Representatives, 2017, Brookings
    4. How the party breakdown of the Senate can differ from the party breakdown of the nationwide vote for senators
      1. Democratic Senators Represent More Voters than Republican Senators
    5. How the party breakdown of the electoral votes can differ from the party breakdown of the nationwide presidential vote.
      1. Undemocratically Elected Presidents
      2. Example of 2016 Presidential Election
  6. Addenda
    1. Evolution of US Electoral Systems
    2. Plurality System
    3. Majority System
    4. Ranked Choice Voting (Instant Runoff)
      1. Ranked Choice Voting in Alaska
      2. Ranked Choice Voting in Maine
    5. Proportional Representation (Vote by Party)
  7. Not every vote counts the same: Quick Take
  8. Disparity in Party Makeup of the Legislature and the Electorate: Quick Take

Electing a President

The simplest, most democratic method of electing a president and vice president is by direct popular vote.

But in the US, residents of 51 electoral districts (the states plus DC) vote for 538 electors, a majority determining the winner.

Electing a Legislature

The simplest, most democratic method of electing a Legislature is by direct popular vote of each legislator.

  • But in the US:
    • Residents of 50 electoral districts (states) vote for 100 senators, two for each district.
    • Residents of 435 electoral districts (congressional districts) vote for 435 representatives, one for each district.

The US thus votes for representatives, senators, and presidents through electoral districts rather than directly.

Two Problems with Voting by Electoral District

There are two problems with voting for representatives, senators, and presidents through electoral districts rather than directly:

Districts with Different Populations

Districts may have different numbers of members, violating the principle of One Person, One Vote.

Unproductive Votes

Voting by district can result in one political party having significantly more ‘unproductive’ or ‘wasted’ votes, votes that don’t contribute to winning the election. This can result in a sizable disparity between (a) the party breakdown of the House, Senate, or Electoral College and (b) the party breakdown of the votes cast.

Problem of Electoral Districts with Different Populations

How Much a Vote Weighs
  • A basic principle of democracy is one person, one vote, that every vote should have the same weight.   This means that electoral districts should be the same same size.  As Chief Justice Earl Warren said in Reynolds v. Sims about congressional districts within a state:
    • “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.”
  • Thus districts with equal populations have votes with equal weight.  Conversely, votes in districts with unequal populations have different weights. A vote in a district with 100 members, for instance, has 10 times the weight of a vote in a district with 1000 members.
  • The underlying principle is:
    • The weight of a vote in a district = 1 / population of the district
  • The weight of a vote in a district with 100 members is 1/100.  The weight of a vote with 1000 members is 1/1000.  (1/100) / (1/1000) = 10. Therefore the weight of a vote in the smaller district is ten times the weight of a vote in the larger.
  • We thus have a method for weighing people’s votes for senators and president across states and for representatives across congressional districts: divide the number of elected officials by the population.
The Weight of a Vote for a Representative, by Congressional District
  • The graph 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.
  • Here’s how it works.  Take the rightmost vertical bar, for example, representing Wyoming’s single district. The absolute weight of a vote in the district is 1 / 581,024, since Wyoming’s population is 581,024.
  • If all 435 congressional districts had the same number of members, every vote would have the same weight of 435 / 330,000,000.
    • The District of Columbia has no representatives.
  • So the weight of a vote in Wyoming’s single district relative to the ideal equal weight is:
    • (1 / 581,024 ) / (435 / 330,000,000) = 1.30566, as shown on the graph.
  • The relative weights are pretty close to 1. This is due 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, nevertheless, e.g. a vote in RI 2nd has twice the weight of a vote in Montana’s single district.  Such anomalies are due to the algorithm for apportioning house seats to the states, explained in Computing Apportionment.
The Weight of a Vote for Senator, by State
  • The graph 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.
  • The weight of votes for Senators differ substantially from state to state.  For example, a vote in WY has 68 times the weight of a vote in CA.  This is because state populations vary greatly, though each state has the same number of senators.
  • The absolute weight of a vote for senator in a particular state equals 2 divided by the state’s population. So the absolute weight of a vote in WY equals 2 / 577,719.
  • The hypothetical equal weight of a vote for senator is 100 / 330,000,000.
    • The District of Columbia has no senators.
  • Thus the relative weight of a vote in WY is
    • (1/577719) / (50/330000000) = 11.42, as shown on the graph.

The Senate is said to have a small-state bias. Here’s what that looks like:

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 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.
The Weight of a Vote for President, by State + DC
  • The graph shows the weight of a vote for president in each of the 51 states plus DC, relative to a baseline of 1, where each vote has the same weight.
  • The weight of a vote for president differs from state to state, but not nearly as much for Senators.  For example, a vote in WY has 3.5 times the weight of a vote in CA.  This is because, while senators have an expanding effect on the difference in electoral votes, representatives have a moderating effect.
    • The number of a state’s electors equals the number of the state’s representatives plus the number of senators. 
  • The absolute weight of a vote for president in a particular state equals the number of the state’s electoral votes divided by the state’s population. So the absolute weight of a vote in WY is 3 / 577,719.
  • The hypothetical equal weight of a vote for president is 538 / 330,000,000.
    • 538 electoral votes = 435 representatives + 100 senators + 3 D.C.electoral votes.
  • Thus the relative weight of a vote in WY is
    • (3 / 577,719) / (538 / 330,000,000.) = 3.1852
  • 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.

Problem of Unproductive Votes

Three Electoral Systems
  • A nation of 300 people wants to elect three people to run the government. Here are three ways of electing them:
    • Direct popular vote
    • Vote by party
    • Vote by district
  • Each of the 300 citizens casts one vote for Seat 1, a second vote for Seat 2, and a third vote for Seat 3.
Vote by Party
  • There are two political parties, Green and Pink. Both submit slates of three candidates.   People vote for one or the other of the slates.
  • An election is held. The results are: Green 190, Pink 110. That’s about 2/3 versus 1/3.  So Green wins 2 seats and Pink 1. (There are algorithms for calculating this result.)
  • The party breakdowns are pretty close:
    • Legislature: Green 67%, Pink 33%
    • Electorate: Green 63%, Pink 37%
Vote by District
  • The nation is divided into three electoral districts, each with 100 residents who vote for one representative.
  • The party composition of the electorate is 63% Green and 37% Pink, as above.
  • But the party composition of the Legislature is determined not only by the party makeup of the electorate, but also by the distribution of the parties across the districts.
  • Here are two scenarios.

First Scenario: The 190 Green voters and 110 Pink voters are spread equally and randomly across the districts, looking like this:

  • An election is held. The results are:
    • District 1: Pink 37, Green 63
    • District 2: Pink 37, Green 63
    • District 3: Pink 36, Green 64.
  • Green thus wins all 3 districts
  • There’s a significant disparity between party breakdowns:
    • Legislature: Green 100%, Pink 0%
    • Electorate: Green 63%, Pink 37%
  • The results look like this:

Second Scenario: Half of Green voters are concentrated around a city in District 3, looking like:

  • The results of an election:
    • District 1: Pink 52, Green 48
    • District 2: Pink 53, Green 47
    • District 3: Pink 5, Green 95.
  • So Pink wins 2 seats and Green 1.
  • There’s a significant disparity between party breakdowns:
    • Legislature: Green 33%, Pink 67%
    • Electorate: Green 63%, Pink 37%
  • Here’s how the results look:
  • Unlike Voting by Party, Voting by District can result in a significant disparity between party breakdowns of the legislature and the electorate. Though the voters are split 63% green and 37% pink, the geographic spread of the parties results in legislative breakdowns of 100% green vs 0% pink in the first scenario and 33% green vs 67% pink in the second.
  • The effect of geographic spread on the party composition of districts can be quantified by enumerating productive and unproductive votes.
  • A productive vote is one that contributes to winning. An unproductive vote is either a losing or excess vote. For example, take the election result for District 3 in the second scenario:
    • District 3: Pink 5, Green 95
  • Green won by 90 votes, but only needed 6. So 89 of Green’s 95 votes were unproductive excess baggage. Pink’s 5 votes were also unproductive, but for a different reason: they were losers.
  • Here are the calculations for the two scenarios.

Productivity Analysis of First Scenario

  • Election Results
    • District 1: Pink 37, Green 63
    • District 2: Pink 37, Green 63
    • District 3: Pink 36, Green 64.
    • Thus Green won all 3 seats.
  • Pink Votes
    • Losing Votes = 37 + 37 + 36 = 110
    • Excess Votes = 0
    • Unproductive Votes = 110
    • Productive Votes = 300 – 110 = 190
  • Green Votes
    • Losing Votes = 0
    • Excess Votes = (63 – 37) + (63 – 37) + (64 – 36) = 80
    • Unproductive Votes = 80
    • Productive Votes = 300 – 80 = 220
  • Green thus had 30 more productive votes than Pink.

Productivity Analysis of Second Scenario

  • Election Results
    • District 1: Pink 52, Green 48
    • District 2: Pink 53, Green 47
    • District 3: Pink 5, Green 95.
    • Thu Pink won 2 seats and Green 1.
  • Pink Votes
    • Losing Votes = 5
    • Excess Votes (52 – 48) + (53 – 47) = 10
    • Unproductive Votes = 15
    • Productive Votes = 300 – 15 = 285
  • Green Votes
    • Losing Votes = 48 + 47 = 95
    • Excess Votes = (95 -5) = 90
    • Unproductive Votes = 90 + 95 = 185
    • Productive Votes = 300 – 185 = 115
  • Thus Pink had 170 more productive votes than Green
How the party breakdown of a state’s congressional delegation can differ from the party breakdown of the statewide vote for the House
Gerrymandering
  • Gerrymandering is drawing electoral boundaries to maximize a targeted voting bloc’s unproductive votes, i.e. those votes that don’t contribute to winning.
    • You can make the targeted bloc waste votes by arranging districts so that
      • you win districts by slim margins, thus minimizing your unproductive excess votes and maximizing the targeted voting bloc’s unproductive losing votes
      • you lose districts by large margins, thus minimizing your unproductive losing votes and maximizing the targeted voting bloc’s unproductive excess votes
  • Partisan gerrymandering is where the the targeted bloc is a political party.
    • In Rucho v. Common Cause, the Supreme Court ruled that partisan gerrymandering is beyond the purview of federal courts.
  • Racial gerrymandering is where the targeted bloc is a racial minority.
    • The Voting Rights Act prohibits racial gerrymandering.

View Gerrymandering

Natural ‘Gerrymandering’
  • Unintentional Gerrymandering: Political Geography and Electoral Bias in Legislatures, by Jowei Chen and Jonathan Rodden, Quarterly Journal of Political Science, 2013
  • Abstract
    • While conventional wisdom holds that partisan bias in U.S. legislative elections results from intentional partisan and racial gerrymandering, we demonstrate that substantial bias can also emerge from patterns of human geography.
    • We show that in many states, Democrats are inefficiently concentrated in large cities and smaller industrial agglomerations such that they can expect to win fewer than 50% of the seats when they win 50% of the votes. To measure this ‘’unintentional gerrymandering,’’ we use automated districting simulations based on precinct-level 2000 presidential election results in several states.
    • Our results illustrate a strong relationship between the geographic concentration of Democratic voters and electoral bias favoring Republicans.
How the party breakdown of the House can differ from the party breakdown of the nationwide vote for representatives
Misrepresentation in the House of Representatives, 2017, Brookings
  • Despite its name, the House of Representatives is not so representative.
  • As the chart to the right shows, the total vote differential between the two parties for elections to the House in 2016 was 1.2 percent. But the difference in the number of seats is 10.8 percent, giving a total of 21 extra seats to Republicans
How the party breakdown of the Senate can differ from the party breakdown of the nationwide vote for senators
Democratic Senators Represent More Voters than Republican Senators
  • 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.
How the party breakdown of the electoral votes can differ from the party breakdown of the nationwide presidential vote.
Undemocratically Elected Presidents
  • A candidate can win the presidency but lose the popular vote, which has happened four times.
    • Rutherford B. Hayes in 1876
    • Benjamin Harrison in 1888
    • George W. Bush in 2000
    • Donald J Trump in 2016
  • This happens because the winning party, though receiving fewer popular votes, has more “productive” votes, i.e. votes that actually help produce electoral votes.
Example of 2016 Presidential Election
  • Hilary Clinton won the popular vote by 2,868,691.
    • 65,853,516 – 62,984,825
  • But lost the electoral vote by 77
    • 304 – 227
  • This happened because:
    • The Republicans had more productive votes, i.e. votes that helped generate electoral votes. And fewer unproductive votes.
  • Unproductive votes are of two kinds: losing votes and excess votes.
  • Unproductive Losing Votes
    • Trump won Florida’s 29 electoral votes by winning the popular vote 4,617,886 to 4,504,975. The 4.5 million Democratic votes helped Clinton win the nationwide popular vote but contributed zilch to winning the electoral vote.
  • Unproductive Excess Votes
    • Clinton won California’s 55 electoral votes by winning the popular vote 8,753,788 to 4,483,810. Winning 55 electoral was a good thing for the Democrats. But they didn’t need 8,753,788 votes to win the state. All they needed was 4,483,810 + 1. The excess 4,269,977 votes were unproductive.

Here’s a breakdown of the Democratic and Republican productive and unproductive votes nationwide.

  • The Democrats had more total votes, but the Republicans more productive votes, generating more electoral votes.

These charts show unproductive votes by state for Democrats and Republicans. Democrats had more losing votes. Republicans had few excess votes.

View Electoral College

View Why the Founders Chose the Electoral College

Addenda

Evolution of US Electoral Systems

View Evolution of US Electoral Systems

Plurality System
  • Britannica 
    • Plurality system, electoral process in which the candidate who polls more votes than any other candidate is elected. Election by a plurality is the most common method of selecting candidates for public office.
    • Advantages of the plurality system are that it is easily understood by voters, provides a quick decision, and is more convenient and less costly to operate than other methods. The main argument against it is that in an election with more than two candidates, it may result in the election of a candidate who has received only a minority of the votes cast: for example, in a closely contested election with four candidates, the total required to win by a plurality could be as little as 25 percent of the total vote plus one.
Majority System
  • Britannica
    • Under the majority system, the party or candidate winning more than 50 percent of the vote in a constituency is awarded the contested seat. A difficulty in systems with the absolute-majority criterion is that it may not be satisfied in contests in which there are more than two candidates.
  • Runoff Systems
    • Runoff Election (Georgia, Louisiana)
    • Instant Runoff / Ranked Choice Voting (Maine)
Ranked Choice Voting (Instant Runoff)
  • 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.
  • Instant Runoff Voting Wiki
    • A voting method used in single-seat elections with more than two candidates. Instead of voting for a single candidate, voters rank candidates in order of preference. Ballots are counted for each elector’s top choice, losing candidates are eliminated, and ballots for losing candidates are redistributed until one candidate wins a majority of the votes.
  • How RCV works Fairvote
    • For a single office, like for a mayor or governor, RCV helps to elect a candidate who reflects a majority of voters in a single election even when several viable candidates are in the race. Ranked choice voting is a way to ensure elections are fair for all voters. It allows voters the option to rank candidates in order of preference: one, two, three, and so forth.
    • If your vote cannot help your top choice win, your vote counts for your next choice.
    • In races where voters select one winner, if a candidate receives more than half of the first choices, that candidate wins, just like in any other election. However, if there is no majority winner after counting first choices, the race is decided by an “instant runoff.” The candidate with the fewest votes is eliminated, and voters who picked that candidate as ‘number 1’ will have their votes count for their next choice. This process continues until there’s a majority winner or a candidate won with more than half of the vote.
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.

Proportional Representation (Vote by Party)
  • “Systems of proportional representation have been adopted in many countries, including Belgium, Denmark, Finland, Greece, Hungary, Israel, Italy, Luxembourg, Norway, Russia, Spain, Sweden, and Switzerland.”
  • List of 87 countries using PR (Wiki)
  • How proportional representation would work in Texas
    • Texas has 36 representatives, each representing a congressional district.  Residents of a district vote for their representative.
    • Suppose instead everyone in Texas voted for a political party: Democratic, Republican, Libertarian,  Suppose the results are: Democratic 40%, Republicans 55%, Libertarian 5%. Then the allocation of seats would be: Democrats 14 seats, Republicans 20 seats, Libertarians 2 seats.
    • A benefit is no gerrymandering problems.
  • britannica.com/topic/proportional-representation
    • Proportional representation, electoral system that seeks to create a representative body that reflects the overall distribution of public support for each political party. Where majority or plurality systems effectively reward strong parties and penalize weak ones by providing the representation of a whole constituency to a single candidate who may have received fewer than half of the votes cast (as is the case, for example, in the United States), proportional representation ensures minority groups a measure of representation proportionate to their electoral support.
  • wikipedia.org/wiki/Proportional_representation
    • Proportional representation characterizes electoral systems by which divisions in an electorate are reflected proportionately in the elected body. If n% of the electorate support a particular political party, then roughly n% of seats will be won by that party. The essence of such systems is that all votes contribute to the result: not just a plurality, or a bare majority, of them.
    • Under the party-list system, the elector votes not for a single candidate but for a list of candidates. Each list generally is submitted by a different party, though an individual can put forward his own list.
    • Each party gets a share of the seats proportional to its share of the votes. There are various alternative rules for achieving this; the two principal ones are the largest-remainder rule and the highest-average rule.
    • Under the highest-average rule, seats are assigned one at a time to the party with the highest total. After each seat is assigned, the winning party’s total is adjusted: the original vote total is divided by the number of seats it has won plus one.

Not every vote counts the same: Quick Take

  • Not every vote counts the same. For example, because of the difference in populations, a vote for senator in WY has 68 times the weight of a vote in CA
  • The population difference also biases the presidential vote. A vote for president in WY has 3.5 times the weight of a vote in CA. Thus, Bush in 2000 and Trump in 2016 won with fewer popular votes.
  • Differences in state populations also indirectly impact the Supreme Court. Trump, for instance, though losing the popular vote, appointed three SC justices.

View Problem of Electoral Districts with Different Populations

Disparity in Party Makeup of the Legislature and the Electorate: Quick Take

  • The US elects representatives, senators, and presidents through electoral districts rather than directly:
    • Representatives by way of congressional districts
    • Senators by way of states
    • Electoral votes by way of states+DC.
  • The problem is that voting by district can result in a significant disparity between (a) the party breakdown of the House, Senate, and Electoral College and (b) the party breakdown of the votes cast.

View Problem of Disparity in the Party Makeup of the Legislature and the Electorate