Journal · Data

Voter turnout, simulated: what if nobody campaigned?

Election Day, 3 November 2026. Switch every campaign off, let each person decide alone, and drop simulated elections down a Galton board. Then the tests that tell chance from something more, and the game theory of why anyone votes at all.

Highland Willow · Boone, North Carolina · 3 November 2026

On Election Day, 3 November 2026, voter turnout is the number everyone quotes on the night. Here is a stranger question: what would voter turnout look like if nobody campaigned at all? No door knocking, no phone calls, no mailers, no ads, no rallies. Every eligible person decides alone, with the same chance of turning up. We built that world in code, at three sizes, and dropped thousands of simulated elections down a Galton board. The answer depends almost entirely on how many people are voting, and it tells you something real about the elections we actually have.

What is voter turnout, and which one?

Voter turnout is the share of people who vote out of the people who could have. The argument is about the bottom of that fraction.

Registered voters vs eligible voters

The turnout of registered voters divides ballots by the people on the rolls. It always comes out higher than the other measures, because everyone who never registered drops out of the bottom of the fraction. The voting-age population counts every adult resident, including people who cannot vote, such as non-citizens. The measure the University of Florida Election Lab uses, and this post uses, is the voting-eligible population: adult citizens, less those barred by state law, mostly people serving felony sentences, plus citizens living overseas. In 2024 that was 243,803,423 people, and 156,766,239 ballots were counted, a turnout of 64.3%.

Presidential turnout rates and midterm turnout rates

Presidential turnout rates over this century ran from 55.3% in 2000 to 66.0% in 2020; the 2008 presidential turnout was 62.2%. Midterm turnout is lower every time. The 2014 midterm turnout, 36.6%, was the low point of the period, and the 2018 midterm turnout, 50.1%, the high.

Primary turnout is lower again. Pew Research Center put 2016 presidential primary turnout at more than 57.6 million votes, 28.5% of voting-age citizens by its estimate, just short of the 2008 record of 30.4%. Pew left caucus states out, because caucus attendance is not reliably recorded.

Low turnout in local elections, and among young voters

Local elections are where turnout is lowest. Portland State University's 2016 study Who Votes for Mayor? looked at 85 mayoral elections in the 30 largest US cities. Turnout of eligible citizens was under 15% in 10 of them, and as low as 6.1% in Dallas and 6.5% in Fort Worth. The median age of the people who voted was 57. Timing matters too. As Zoltan Hajnal, Vladimir Kogan and G. Agustin Markarian summarise the research, city elections held on the November day of state and national races generally double local turnout. Their own 2022 study of California cities found that moving to that day also gives an electorate considerably more representative by race, age and party.

Young voter turnout lags too. In the Census Bureau's Current Population Survey, 47.7% of citizens aged 18 to 24 said they voted in 2024, against 74.7% of those 65 and over. In the 2022 midterms the figures were 27.6% and 66.8%. The survey is self-reported, so it runs higher than counted ballots, and the 65-and-over figure is our combination of the survey's 65 to 74 and 75-plus groups.

The simulation: switch every campaign off

The model is the simplest one there is. Every eligible person flips their own weighted coin: with chance p they vote, otherwise they stay home. Nobody influences anybody. An election is then a random sample of who happened to turn up, and statistics says exactly how far its turnout can stray from p by chance alone. The spread shrinks with the square root of the number of people.

We do not know what p would be with no campaigning at all. No country has run that experiment, and the field experiments further down only measure campaigning a piece at a time. So p is a slider, and the worked examples use 30%. The three electorates are made up too, except the last, which is the size of the 2024 eligible population.

Ten simulated no-campaign elections in the town came out at 29.5%, 30.8%, 28.3%, 30.0%, 29.8%, 29.4%, 30.2%, 30.9%, 28.8%, 30.1%. Ten in the country all landed between 29.996% and 30.004%. In a country, a world without campaigns would hold turnout within about a hundredth of a point, election after election.

The Galton board

Francis Galton built his board, which he called a quincunx, to show how many small chance events add up to a bell curve. The one in University College London's Galton Collection was made in 1873 by Tisley and Spiller to his design, and he described the idea in Natural Inheritance in 1889. Galton also coined the word eugenics, in 1883, and promoted it. That is part of his record too; this post borrows only his board. Each ball here is one election. Try the club, then the town, then the country, and watch the pile narrow until every ball lands in the same slot.

Then tick the box that switches campaigns, news and stakes back on. Before anyone decides, the whole electorate's chance moves together, by an amount sized to match how much real presidential turnout has varied since 2000. Now even the country spreads out, across roughly 22% to 34% in ten simulated elections. That shared swing is the only thing in the model that can make a big electorate's turnout move.

A chi-square goodness of fit example, run on the board itself

Is the board fair? A chi-square goodness of fit test compares how many balls landed in each slot with how many the coin-flip maths expects. We dropped 1,000 club elections at 30%: the statistic was 29.4 on 17 degrees of freedom, p = 0.031. By the usual 5% rule that rejects a board we built to be fair. It happens: we ran 1,000 more fair boards of 1,000 elections each, and 52 of them failed the same way, about the one in twenty the test promises. Press Clear, choose the club at 30% and press "Add 1,000 at once", and the board replays exactly this run, so you will see the same number. A single p-value below 0.05 is not proof; it is a reason to look again.

Five statistical tests, and what they tell us

1. Is real turnout just coin flips? A dispersion test

If each person decided alone, turnout in different elections would differ only by the coin-flip spread. Pearson's dispersion test checks that: it adds up how far each election sits from the average, measured in coin-flip units, and asks whether the total is bigger than chance allows. On our ten simulated town elections the statistic was 6.8 on 9 degrees of freedom, p = 0.654. Nothing unusual, as it should be, since they were coin flips.

On the seven real presidential elections from 2000 to 2024 the statistic is 7,055,655 on 6 degrees of freedom, and the p-value has about 1,532,103 zeros after the decimal point. Do not read much into that number. The test rejects one thing only: that everyone had the same, unchanging chance of voting in every election. Several things break that at once. Stakes and campaigns change from one election to the next. The electorate itself changes: it grew by 49.5 million eligible people between 2000 and 2024. And the published figures carry their own error: two versions of the same series from the same lab differ by up to about a point, hundreds of times the coin-flip spread. The test cannot tell these apart.

The plain observation needs no test. Real presidential turnout has a standard deviation of 3.6 points; independent coin flips at an unchanging chance would give about 0.0031 points across the 2024 eligible population, roughly 1,200 times less. So the chance of voting itself moves between elections. What moves it, this test cannot say.

2. Did turnout really change? A two-proportion z test

Say the town's turnout rose from 30.0% (720 of 2,400) to 32.5% (780). Is that a real change? A two-proportion z test, also called a z test for proportions, divides the difference by the coin-flip spread you would expect between two samples that size. Here z = 1.87 and p = 0.062: suggestive, not conclusive. In a town this size a change needs to be about 2.6 points before the test calls it unlikely to be chance. In the country, 0.008 points is enough, so a rise of one hundredth of a point gives z = 2.41.

That is the trap at national scale. With hundreds of millions of people, every difference is "significant", and the test stops answering the question you care about. The useful question becomes how big the change is and what caused it. At local scale the test still earns its keep: a two-point move in a small town's turnout can be noise.

3. Is this club unusual? A binomial test

In the club of 50, everyone has a 30% chance of voting, so 15 should turn up. One night 24 do. An exact binomial test adds up the chance of every result at least that unlikely: p = 0.008. A result that unlikely is a reason to ask what else happened that night. It is not, on its own, the chance that something did. With small numbers, use the exact test rather than a normal approximation; it costs nothing on a computer and it does not mislead when counts are small.

4. Does canvassing work? A permutation test

This is the question the no-campaign world dodges, and it has real answers. In 1998 Alan Gerber and Donald Green ran a field experiment with about 30,000 registered voters in New Haven, Connecticut, assigning them at random to canvassing visits, mailers, phone calls or nothing before the November election. Turnout rose substantially with canvassing, slightly with mail and not at all with phone calls. J-PAL's summary of the study puts the effect of a successful canvassing contact at 8.7 points, and of each mailing at 0.5 points.

We simulated an experiment in the same spirit. Forty precincts of 400 eligible people, each with its own no-campaign turnout scattered around 30%. Twenty precincts are chosen at random for canvassing. Canvassers reach 30% of the people there, and each person reached becomes 8.7 points more likely to vote, so the true effect on a precinct is 2.6 points. The 8.7 points come from the experiment above; the share reached, the precincts and everything else are invented.

A permutation test asks: if canvassing did nothing, how often would shuffling the "canvassed" labels at random produce a gap as big as the one we saw? We shuffled 10,000 times. The canvassed precincts beat the others by 4.8 points, and p = 0.052. The effect is real, because we put it there, and by the usual 5% rule the test still misses it. Repeating the whole experiment 400 times, 40 precincts found the effect 24% of the time, 120 precincts 56%, and 400 precincts 97%. With no effect at all, the three sizes "found" one 7%, 3% and 5% of the time, close to the 5% the test allows. A small study that finds nothing has often just been too small to look.

5. Does the board match the maths? Goodness of fit

The fifth is the chi-square goodness of fit test from the board section. In club mode the readout under the board reruns it after every drop, so you can watch a fair board pass, and now and then fail.

Your vote, in numbers

Your vote decides an election only if everyone else splits exactly evenly. When each of the other voters is a fair coin, that chance is 11.2% with 50 other voters, 1.63% with 2,400, and about 1 in 20,000 with 243.8 million. Tilt every coin to 51% for one side and 50 voters barely notice (11.1%), 2,400 drop to 1.01%, and for 243.8 million the chance becomes a number with 21,185 zeros after the decimal point.

Real elections sit between those extremes, because nobody knows the tilt in advance. Andrew Gelman, Nate Silver and Aaron Edlin, working from state forecasts made about a week before the 2008 presidential election, estimated that the average American voter had a 1 in 60 million chance of being decisive, rising to about 1 in 10 million in New Mexico, Virginia, New Hampshire and Colorado. Looking back over US elections, Casey Mulligan and Charles Hunter found that about 1 vote in 89,000 in congressional races, and 1 in 15,000 in state legislative races, was cast for a candidate who tied or won by a single vote.

Ties do happen in small electorates. In 2017 the race for Virginia's 94th House of Delegates district ended tied at 11,608 votes each, after a recount and a court's ruling on one disputed ballot. On 4 January 2018 the State Board of Elections settled it by lot: the two names went into film canisters in a ceramic bowl, and the first one drawn, David Yancey's, won the seat over Shelly Simonds.

Game theory: why does anyone vote?

Game theory in politics starts with a puzzle. If your vote almost never decides anything, and voting costs time and effort, a purely self-interested person should stay home. Anthony Downs set this out in 1957, and it is known as the Downs paradox. It is also called the paradox of voting, a name social choice theorists also give to Condorcet's cycle, further down. William Riker and Peter Ordeshook wrote it as a sum in 1968: you vote when the chance your vote is decisive, times how much you care about the result, plus the satisfaction of voting itself, outweighs the cost.

The cost-benefit part can be played as a game, as Thomas Palfrey and Howard Rosenthal showed in the 1980s. Ours is a simplified version of that idea. Everyone weighs their own cost against their chance of deciding the result, and that chance depends on how many others turn out. We solved for the equilibrium, the turnout at which nobody would change their mind, with invented values for how much winning is worth.

The club votes in full in both columns. The town's answer swings with the invented prize, so read nothing into its level. The country, on cost and benefit alone, would get 0.6% turnout, or 1.9% if the win were worth a hundred times the largest cost of voting. To reach the 64% who really voted in 2024, this game needs a win worth about 20,000 times the largest cost of voting. Either people value the national result that much, or they vote for reasons the sum leaves out. Riker and Ordeshook's answer was their extra term, the duty or satisfaction of voting; other researchers have proposed other answers, and this model cannot choose between them.

The same arithmetic explains rational ignorance. If one vote is almost never decisive, the hours needed to study every race buy very little, so many voters sensibly rely on shortcuts: party labels, endorsements, a trusted friend. That is a description of incentives, not a judgement of voters.

Where candidates stand: Hotelling's law and the median voter

In 1929 Harold Hotelling asked why two shops on a street end up side by side in the middle. In the Hotelling model, competitors for the same customers drift toward each other, and that drift is called Hotelling's law. Downs built the same idea into his theory of party competition: with two candidates and voters spread along one left-to-right line, each candidate gains by moving toward the voter in the middle, the median voter, until both stand close together. Real candidates also face primaries, money and the need to turn out their own side, which this one-line model leaves out.

Why two parties: first past the post, Duverger's law and the spoiler effect

Many American races are first past the post: the candidate with the most votes wins, even without a majority. Maurice Duverger observed, in Les Partis politiques in 1951, that this rule tends to produce two main parties, now called Duverger's law. He gave two reasons. Smaller parties have difficulty winning seats, a mechanical effect; and voters and parties coordinate around the candidates who look able to win, a psychological one. The spoiler effect is the voter's side of that: a third candidate who splits one side's vote can hand the race to the other, so supporters of small parties may switch to the larger party nearest to them.

Same ballots, four winners

How you count matters as much as who votes. Here are 100 made-up ranked ballots for four candidates, A to D.

• Plurality, first preferences only: A wins with 40 votes, short of a majority.
• Instant runoff: drop the last-placed candidate and pass their ballots on. B goes first with 11, then C with 25, and D beats A 56 to 44.
• Borda count, three points for a first choice down to none for a last: C wins with 174, ahead of B on 166, D on 136 and A on 124.
• Condorcet method, every pair head to head: B beats A 60 to 40, C 51 to 49 and D 55 to 45.

Four counts, four winners. B, the Condorcet winner, is the candidate a majority prefers to each rival, and B is the first one instant runoff throws out, for having the fewest first preferences. Sometimes there is no Condorcet winner at all. With three voters ranking A over B over C, B over C over A, and C over A over B, A beats B, B beats C and C beats A, each by two votes to one. That cycle is the Condorcet paradox, which the Marquis de Condorcet described in 1785.

Kenneth Arrow proved in 1951 that this is not a failure of cleverness. Arrow's impossibility theorem shows that no way of turning rankings of three or more options into a group ranking can meet a short list of reasonable fairness conditions at once, unless one voter decides everything. The Gibbard–Satterthwaite theorem, proved separately by Allan Gibbard in 1973 and Mark Satterthwaite in 1975, adds that every rule for choosing one winner from three or more candidates, with no dictator and a real chance for each candidate to win, sometimes rewards voting tactically rather than honestly.

Types of elections, and what is on the 2026 ballot

Primaries, runoffs and the rest

A general election chooses who holds the office. A primary chooses each party's candidate for it, by secret ballot. In a closed primary only voters registered with that party may take part; in an open primary voters do not have to be registered with a party to vote in its primary. A top-two primary puts every candidate on one ballot and sends the two with the most votes to the general election, whatever their party. A runoff is a second round, held when nobody reached the required share in the first. A special election fills a seat that fell vacant mid-term. Many city and county races are nonpartisan, with no party label on the ballot at all.

Caucus vs primary

A primary is a secret ballot run like any other election. A caucus is a meeting run by a political party at the county, district or precinct level, where supporters gather and are counted. A caucus takes an evening and asks you to be there in person, which a ballot does not.

Referendum vs initiative

Both put a question to voters directly. In a referendum, voters approve or reject a law or amendment: one the legislature puts to them, or, in 23 states, a law already passed that citizens gather signatures to put to a vote. An initiative starts with citizens: enough signatures put their own proposed law or amendment on the ballot without the legislature. In all, 26 states have an initiative or popular referendum process at the statewide level.

The federal calendar

Federal Election Day is the Tuesday after the first Monday in November, which in 2026 is 3 November. Congress fixed that day for choosing presidential electors in 1845, extended it to House elections in 1872 and to Senate elections in 1914. Today the House rule is 2 U.S.C. §7, and the presidential one is the definition of election day in 3 U.S.C. §21. Every seat in the House of Representatives is on the ballot every two years, because the Constitution gives representatives two-year terms. So how many representatives are up for reelection? Every one who runs again, since all 435 seats are on the ballot each time; members who retire are not up for anything. Senators serve six-year terms in three staggered classes, so about a third of the Senate faces the voters each time. On 3 November 2026 that is 35 seats: the 33 Class 2 seats, plus special elections in Ohio and Florida for the rest of two terms that fell vacant in 2025.

Why do incumbents win reelection so often?

Most incumbents who run, win. OpenSecrets puts the reelection rate for incumbents in the House at 96.6% in 2024 and 93.2% in 2022, and in the Senate at 88% and 100%. That congressional reelection rate has not fallen below 85% for the House in any election since 1964; the Senate's has dipped as low as 55%, in 1980.

OpenSecrets points to wide name recognition and an advantage in campaign cash. In the terms of this post, a sitting member has in effect been campaigning for years. Selection may matter too: incumbents already won once, so they tend to be strong candidates in districts that suit them, and those who expect to lose may retire instead of running, which keeps them out of the count. The incumbent reelection rate measures all of that together, and on its own cannot separate one cause from another.

How the count is checked

Ballots are counted by scanners or by hand. After election night, officials check and total the results, a process called the canvass, and then certify them. Close races can trigger a recount under state law.

Statistics offers a cheaper check than recounting everything, with a stated limit on the chance of missing a wrong result. Risk-limiting audits grew out of work by the statistician Philip Stark from 2008. The simplest kind draws paper ballots at random and keeps drawing until the evidence that the reported winner really won is strong enough. If it never gets strong enough, the audit escalates to a full hand count. The ballot-polling method simulated below, called BRAVO, was published by Mark Lindeman, Philip Stark and Vanessa Yates in 2012. The "risk limit" is the largest chance that a wrong outcome survives the audit. In the simplest ballot-polling version, the number of ballots needed depends mostly on the margin, not on how many ballots were cast.

A landslide is confirmed by a few dozen ballots. A race won by two points needs thousands, and a near-tie needs a full count, which is what an audit of a near-tie becomes.

What this post is, and is not

It is a set of models with stated, invented parameters, run with fixed random seeds so that every number above can be reproduced. It is not a forecast, and it says nothing about any party, candidate or race on the 2026 ballots. The real figures are labelled with their sources below. The one observation that does not depend on any invented number is the simplest: real national turnout moves by points between elections, while people deciding independently at an unchanging chance would move it by thousandths of a point. So the chance of voting itself changes from one election to the next. Some of that is who is eligible and how ballots are counted; field experiments show that campaigns are part of it too, a door at a time.

If you liked the statistics here, our post on Benford's law shows another pattern that turns up in counts of all kinds.

Questions people ask

What is voter turnout?

Voter turnout is the share of people who vote out of the people who could have. The University of Florida Election Lab, the source used here, divides ballots counted by the voting-eligible population: adult citizens, less those barred by state law, plus citizens overseas. On that measure US turnout was 64.3% in 2024 and 46.1% in the 2022 midterms.

What would voter turnout look like with no campaigning?

Nobody knows the level, because no country has run that experiment. The spread is knowable: if every person decided alone at an unchanging chance, a town's turnout would wobble by a point or two between elections and a country's by under a hundredth of a point. Real national turnout moves by several points, so the chance of voting itself changes between elections, and campaigns are among the reasons.

How many seats are up for reelection in the Senate?

About a third of the Senate's 100 seats in any general election. On 3 November 2026 that is 35 seats: the 33 Class 2 seats, plus special elections in Ohio and Florida for the rest of two terms that fell vacant in 2025.

How many senators are up for reelection every two years?

One class of the Senate, 33 or 34 seats, plus any special elections for seats that fell vacant. Senators serve six-year terms, staggered so a third of the seats are on the ballot every two years. Senators who retire are not up for reelection.

How many House members are up for reelection?

All 435 House seats are on the ballot every two years. Every member who runs again is up for reelection; members who retire are not.

Why are all House seats up for reelection every two years?

The Constitution sets a two-year term for the House of Representatives, so the whole House is elected at once, in every even-numbered year.

What percentage of incumbents win reelection?

Most of them. By OpenSecrets' count, 96.6% of House incumbents and 88% of Senate incumbents seeking reelection won in 2024. The House rate has not been below 85% in any election since 1964.

Caucus vs primary: what is the difference?

A primary is a secret-ballot election. A caucus is a meeting run by a political party at the county, district or precinct level, and it asks you to be there in person.

Referendum vs initiative: what is the difference?

In a referendum, voters approve or reject a law or amendment: one the legislature puts to them, or, in some states, a law already passed that citizens petition to put to a vote. An initiative starts with citizens, who gather enough signatures to put their own proposal on the ballot.

What is the Downs paradox?

Anthony Downs's 1957 observation that one vote is almost never decisive, so a purely self-interested person should not bother to vote, yet millions do. It is also called the paradox of voting, a name also used for Condorcet's voting cycle.

What is Duverger's law?

Maurice Duverger's observation that first-past-the-post elections, where the most votes wins, tend to produce two main parties. He gave two reasons: smaller parties have difficulty winning seats, and voters and parties coordinate around the candidates who look able to win.

What are risk-limiting audits?

Post-election checks that draw paper ballots at random until there is strong statistical evidence that the reported winner really won, and escalate to a full hand count if that evidence never arrives. The risk limit is the largest chance a wrong outcome survives.

What are the odds that one vote decides an election?

It depends on how many people vote and how close the race is expected to be. For the 2008 presidential election, Gelman, Silver and Edlin estimated 1 in 60 million on average and about 1 in 10 million in the closest states. In small local races, ties and one-vote margins are rare but real.

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35. USA.gov. Primary elections and caucuses. https://www.usa.gov/primaries-caucuses
36. Wikipedia. Initiatives and referendums in the United States. https://en.wikipedia.org/wiki/Initiatives_and_referendums_in_the_United_States
37. Stark. Conservative statistical post-election audits. Annals of Applied Statistics 2(2), 2008, the work risk-limiting audits grew from. https://arxiv.org/abs/0807.4005
38. Lindeman, Stark and Yates. BRAVO: Ballot-polling Risk-limiting Audits to Verify Outcomes. EVT/WOTE 2012, USENIX. https://www.usenix.org/conference/evtwote12/s6-paper-title-tbd
39. Highland Willow. Simulations in election_model.js and election_figures.js, fixed seeds, run 3 November 2026. Every simulated number above is printed by that run; the electorates, the chance of voting, the shared swing, the experiment's precincts and the ranked ballots are invented.