Constitutional Equilibrium

Every American president since 1869 has been a Democrat or a Republican. Third parties appear regularly and sometimes do well: Theodore Roosevelt’s Progressives won 27% of the popular vote in 1912, Ross Perot 19% in 1992. Then they fade or are absorbed. Most European parliaments look nothing like this: five to ten parties hold seats, and governments are coalitions negotiated after the election.

The standard explanation is that this has less to do with culture or temperament than with the rules that turn votes into seats. I wanted to see how far the mechanisms behind that explanation go on their own: give two systems the same voters, let parties and coalitions behave strategically, and see where each one settles.

The premise: Duverger’s law

The French political scientist Maurice Duverger set out the argument in Les partis politiques (1951; English translation Political Parties, 1954). In its usual form:

• Duverger’s law: plurality voting in single-member districts favors a two-party system. Each district elects one representative, and whoever gets the most votes wins, even without a majority. Duverger called this the closest thing to a true sociological law among his generalizations.
• Duverger’s hypothesis: proportional representation favors many parties. Seats are shared out in proportion to votes. William Riker (1982) later separated the two and called this second claim the weaker one.

Why plurality rules squeeze out third parties

Take a district where 60% of voters lean left and 40% lean right. The left is split between two parties, one with 35% and one with 25%. The right has one party with 40%. Under plurality rules the right wins the seat, although most voters preferred either left party to it.

Duverger identified two effects that follow from this:

• The mechanical effect. Votes do not translate into seats in proportion. A party that comes second or third in most districts wins few or no seats, however many votes it gets nationally. In 1992 Perot won 19% of the popular vote and not one electoral vote.
• The psychological effect. Voters and politicians learn from it. The 25% party’s supporters see that their vote helps elect the right, so many switch to the 35% party next time. Politicians see the same thing, so the two left parties merge, or one stops running, and ambitious newcomers join a major party instead of founding a new one.

Proportional representation weakens both effects. A party with 8% of the vote receives roughly 8% of the seats, so the penalty for backing a smaller party is much lower, especially in large districts with low thresholds. It does not disappear: a party polling near a legal threshold can still lose voters who fear wasting their vote, and coalition prospects still shape how people vote and whether parties join forces. The simulator models the threshold effect.

Is it correct?

Broadly, yes, at the level of the district. With qualifications that matter:

The electoral rule appears to matter more than the form of government. The United Kingdom is parliamentary but elects its House of Commons by plurality in single-member districts, and every British government since 1945 has been led by either the Conservatives or Labour. Brazil has a directly elected president, as the US does, but elects its Chamber of Deputies proportionally, and well over a dozen parties won seats there in 2022.

Rules shape the number of parties; they don’t fix it. Duverger’s mechanism works district by district, so a party whose voters are concentrated in one region can win seats there and survive nationally, as the Scottish National Party and the Bloc Québécois have. Clark and Golder (2006) show that electoral rules modify how a society’s divisions turn into parties: proportional representation permits many parties but does not create them where there are few cleavages to represent.

A presidency can add pressure, but not always. The US presidency requires a majority of electoral votes (failing that, the House chooses), which rewards building one broad coalition before the election. But Golder (2006) finds that presidential elections reduce legislative fragmentation only when there are few serious presidential candidates, and can increase it when there are many. Parties can also coordinate behind a joint presidential candidate without merging.

The US adds its own reinforcements. Primaries let insurgents capture an existing party rather than found a new one, and state ballot-access and “sore loser” laws make it hard for newcomers and defeated primary candidates to reach the general-election ballot.

The model

A single electorate (435 House districts in 50 states plus DC) lives under two rulebooks side by side. Voters, parties and coalition partners are identical agents in both worlds. Only the rules differ.

• United States: single-member plurality districts, a president chosen by the Electoral College, fixed terms.
• Proportional parliament: 150 seats elected from one national list (D’Hondt method, 2% threshold), and a cabinet that needs the confidence of a parliamentary majority.

Open the simulator

Either world can also run a generic rulebook (presidential, Westminster, proportional parliament, or a president with a proportionally elected congress) or the rules of a specific country: the United States, United Kingdom, France, Germany, Switzerland, the Netherlands, Israel, Australia, New Zealand and Brazil. The country rules include two-round runoffs, ranked-choice ballots, France’s semi-presidential executive, Germany’s constructive vote of no confidence and Switzerland’s collegial Federal Council. Compare real countries runs every country and sets the results beside the observed effective number of parties from Michael Gallagher’s election indices. A literature review with a full bibliography is at the bottom of the simulator’s page.

Holding the electorate constant isolates the effect of the rules within the model. It cannot by itself say how much of the real difference between countries is due to institutions rather than geography, history or social divisions. That depends on how well the simulated voters and parties represent real ones.

Results

Averages over 20 random seeds of 80 simulated years each, starting from six parties, with the first 20 years discarded as burn-in. “Effective parties” is the Laakso–Taagepera index, which weights parties by size: two equal parties give 2.0, and one dominant party with a few small ones gives a little over 1. The range is the lowest and highest seed. The last column is explained in point 6 below.

These are outcomes of the model under its assumptions, not measurements of the world. Here is what I think they do and don’t support.

1. Under plurality rules, the model settles near two parties, and that result is robust. Starting from six parties, US and Westminster rules consolidate to about two; starting from two, they stay there (2.01 and 1.95). After a third-party surge or a party split, splinters merge back and newcomers fade or replace a weakened major party. No seed under either rulebook ended with more than four parties alive. Raising the model’s cap on the number of parties (below) does not change these results at all.

By “equilibrium” I mean a persistent pattern, not a proven game-theoretic equilibrium: with random shocks every month, the party count keeps returning to the same narrow range. A stable party count is also not the same as stable parties. A newcomer can replace a major party and the count stays at two.

2. Under proportional rules, the model sustains many more parties, but the exact number is set by the model’s ceiling. The simulator allows at most eight parties (one color each), and founders stop at seven. Every run of the proportional parliament ended with seven parties, at that limit. With the cap raised to 16 and everything else unchanged, the proportional parliament rises from 6.2 to 8.8 effective parties (7.6–11.1 across seeds), and the presidential system with a proportional congress from 5.4 to 6.5. So “about six” is not a finding. What holds up is the contrast: plurality near two, proportional representation well above it.

3. In this model, party mergers do more of the work than strategic voters. Switching mechanisms off one at a time under US rules:

Removing strategic voting does not raise the party count; removing mergers does. Even with both gone, plurality rules stay below three parties, against six or more under proportional rules. What is left is the seat formula itself and the founders’ entry rule: a party is started only if its founder expects to win at least 5% of seats. Whether real two-party systems rest more on voters or on elites is an empirical question this doesn’t settle; it shows what this model’s mechanisms can produce.

4. Larger districts relax the pressure to consolidate. Holding a 150-seat parliament fixed and changing only the number of seats per district (M), the effective number of parties rises from 1.9 (M = 1) to 3.0 (M = 2), 4.1 (M = 3) and 4.4 (M = 5). This is in the direction Gary Cox’s work predicts, but it is not a test of his M + 1 rule. That rule is an upper bound on the number of viable competitors within a district, under specific assumptions, and the bound need not be reached. The figures here are national seat counts, a different quantity, and the leveling off near six at large magnitudes is the party ceiling again.

5. The difference in government durability is largely built in. Proportional cabinets last 2.5 years on average and 38% fall early; presidents always serve their term. But in the model, only coalition and minority cabinets face the monthly crises that can bring them down. Single-party majority cabinets are exempt, and presidential terms are fixed by construction. Cabinet duration and early termination were also among the targets used to tune the model. So the narrower, defensible statement is: under these crisis and coalition-exit rules, cabinets that depend on parliament’s confidence turn over more often than executives with fixed terms. That says nothing about policy volatility or the quality of government. A cabinet can also change while the prime minister and most of the program stay.

6. The last column is not deadlock. It counts months in which the president’s party lacks a majority in the House. Under US rules, with two parties, that is close to divided government in the usual sense. With a proportional congress, no single party ever wins a majority, so 100% is almost automatic. It does not mean an opposition majority, and it does not mean the president can’t pass legislation. Presidents in that position usually build coalitions: Cheibub, Przeworski and Saiegh (2004) found that coalitions form in more than half the cases where the president’s party lacks a majority, and that minority governments were not less successful legislatively than majority coalitions. The model doesn’t simulate coalitions behind a president or legislation, so it can’t speak to deadlock.

How it works

Three strategic games run in every electoral cycle.

1. Voters: is my vote wasted? (Myerson and Weber 1993; Fey 1997). A strategic voter values a vote for party k by the probability that one more vote changes a seat, multiplied by how much they prefer k to the party that would otherwise win it:

value(k) = P(one more vote for k moves a seat) × [u(k) − u(seat it would displace)]

These probabilities come from district polls published four times per campaign. In a single-member district the calculation reduces to the race between the top two. In a 150-seat list nearly every party above the threshold is about equally pivotal, and parties near the 2% line gain or lose support as their chances of clearing it change. Half of the voters vote strategically by default; the rest vote for their favorite.

2. Parties: found, merge, reposition, fold (Cox 1997; Palfrey 1984). Before each election, neighboring parties merge if together they expect clearly more seats (and, under a presidency, a better chance at the office). Each party moves toward the position that wins more seats, constrained by its own voters. House candidates lean toward their district’s median, a stand-in for primaries. A prospective founder looks for an unoccupied position, simulates the campaign that would follow, and enters only if it expects at least 5% of seats after strategic voting. New parties start with a handicap that fades over about twelve years. A party below 1% of seats in two consecutive elections dissolves.

3. Cabinets: form and break (Baron and Ferejohn 1989; Lupia and Strøm 1995; Diermeier and Stevenson 1999). The three largest parties take turns proposing their preferred minimal winning coalition among those every partner should accept, with ministries shared in proportion to seats. Each month a coalition or minority cabinet may face a crisis. A partner leaves when leaving is worth more than staying, net of a reputational cost for breaking the agreement. Leaving can mean opposition, a new majority, or an early election if the party has gained in the polls. A president simply serves a fixed term.

Limitations

• Calibration is not validation. The defaults were tuned to stylized facts: about 14 points of spread in district vote shares, under 3% of House seats for third parties, coalition cabinets averaging a little over two years with a third to a half ending early, and coalition talks of two to three months. Results that match those targets, including the US third-party share and cabinet durations, are not independent confirmation. The model does not yet hold any observations back from tuning.
• Politics in the model has a single left–right dimension. Real party systems also divide along regional, religious and linguistic lines, which is precisely how regional parties survive under plurality rules.
• The number of parties under proportional rules depends on the party ceiling, the 5% entry hurdle, the 1% exit rule and the newcomer handicap, which jointly limit how many parties can persist.
• “Stable” has two meanings here: the party system returns to the same shape after a shock, and governments serve their full term. It does not measure the survival of democracy itself. Juan Linz (“The Perils of Presidentialism”, 1990) argued that fixed terms make presidential systems brittle when president and legislature are in conflict; the model doesn’t test that either.

What would make it a stronger test

• Model coalitions behind a president and legislative support, so the presidential worlds can measure something closer to deadlock.
• Let single-party majority governments face crises too (internal splits, scandals), so durability is not decided by assumption.
• Report the number of viable competitors district by district, to test the M + 1 bound where it actually applies.
• Separate calibration from validation: tune on some countries and outcomes, and check the rest.
• Report full distributions across many more seeds, recovery times after shocks, and how often party identities change while the count stays the same.

What the simulator is good for today is making the mechanisms visible: change the rules in either world, trigger a third-party surge or split the largest party, and watch how each system responds.

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