The same voters live under two rulebooks. Voters weigh whether their vote can change a seat. Parties found, merge, reposition and fold. Cabinets bargain and break. Under single-member plurality and a fixed-term president, the system pulls itself to two parties and stays there. Under proportional representation with a cabinet that needs parliament's confidence, many parties survive and coalitions keep forming and falling.
Founding parties and seed restart both worlds. Polarization and strategic voting apply from the next election.
Each country's rules, simplified, run on the same simulated electorate with your current settings: two runs of 60 years each. Real figures are the effective number of parties in the legislature, from Gallagher's election indices. Use the A and B buttons to watch a country in either world above.
The model reproduces where each country sits among plurality systems well, and the gap between plurality and proportional systems. It fills every niche proportional rules allow, so it overstates fragmentation where society offers fewer cleavages (Germany, New Zealand) and caps out below highly fragmented systems (Brazil, recent Netherlands). That pattern is what the literature on rules and social heterogeneity predicts; see the review below.
The effective number of parties weights each party by its seat share (Laakso and Taagepera): two equal parties score 2.0; a 40/40/20 split scores 2.8. Below it, every government each world has had, split by the seats of the parties in it.
Three games run inside every world, played by identical agents. Only the payoffs change with the constitution. The readouts below update from the live simulation.
A vote matters only if it can change who gets a seat. Strategic voters weigh each party by that chance, read off the polls. In a single-member district only the top two can win, so a vote for a party polling 15% is worth almost nothing next to a vote for a front-runner. With 150 seats filled from one list, every party above the threshold turns votes into seats at the same rate.
Before each election a would-be founder scouts the gaps between parties, simulates the campaign that would follow (strategic voters included), and founds a party only if the seats it can expect beat the cost of organizing: 5% of the chamber. Neighbouring parties run the same sums in reverse. Under plurality, two parties splitting one camp hand seats to the other camp, so merging pays.
A parliamentary cabinet lives as long as its majority. Each month a crisis may strike. Every partner compares staying (its share of office minus the policy compromise, over the rest of the term) with its best way out: opposition, a new majority, or early elections if it would gain seats. A president faces no such vote, because the term is fixed.
Each experiment runs fresh worlds in the background with your current settings: founding parties, polarization and strategic voting.
A 150-seat parliament elected in districts of 1 to 150 seats, no legal threshold, 40 years per run, two runs per size. Cox's M + 1 rule says at most M + 1 parties stay viable in a district of M seats.
Swap the electoral rule and the executive independently. Three runs of 60 years each per cell.
435 House districts in 50 states plus DC, with real electoral-vote counts. Each district's voters are spread along a left–right scale and districts lean left or right by different amounts. Both worlds use exactly these voters: the proportional world pools all 435 districts into one national electorate.
Every voter prefers parties closer to them, plus each party's current popularity. Sincere voters back their favourite. Strategic voters (half, by default) weigh each party by the chance that one more vote for it changes a seat, computed from district polls that are republished four times per campaign (Myerson and Weber; Fey). In a single-member district that chance is concentrated on the top two. In a 150-seat list it is spread over every party above the threshold, and parties near the threshold attract or lose votes as their odds of clearing it move.
Before each election, neighbouring parties merge if together they expect clearly more seats, plus a better shot at the presidency where one exists. Each party steps left or right toward more seats, held back by its own voters, while House candidates lean toward their district. A would-be founder scouts open niches and founds a party only if it expects at least 5% of seats after strategic voting. A new party starts handicapped (no ballot lines, donors or loyal voters), and the handicap fades over about 12 years. A party with under 1% of seats in two straight elections folds.
A president serves a fixed four-year term; the House is elected every two years. A parliamentary cabinet needs a majority. The three largest parties take turns as formateur, proposing the minimal winning coalition they like best among those every partner should accept. Office is shared by seats (Gamson's law). Each month there is a 4% chance of a crisis, and a partner walks out when the crisis makes leaving worth more than staying, after paying a reputational cost for breaking the agreement (Lupia and Strøm; Diermeier and Stevenson).
The country presets add a two-round runoff (France's assembly and president, Brazil's president), ranked-choice ballots (Australia), a semi-presidential president who appoints the prime minister and gets a honeymoon assembly election (France), the constructive vote of no confidence (Germany), and a collegial council elected for a fixed term (Switzerland). Chambers larger than 435 seats are scaled down, and mixed-member systems are modelled as list proportional representation.
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 in Western Europe averaging a little over two years with a third to a half ending early, and talks lasting two to three months. At most eight parties can exist at once, one per colour. Stability here means two things: the party system returns to the same shape after shocks, and governments last their term. It does not measure democratic survival. Linz argued that fixed terms make presidential regimes more brittle when president and legislature clash, and divided government, the presidential version of deadlock, is reported in the scoreboard.
Sources and the debates behind these choices are in the literature review below.
The simulator stitches together results from three research programmes: the effect of electoral rules on the number of parties, the bargaining theory of coalition governments, and the comparative study of presidential and parliamentary democracy. This review summarizes where each stands, what the model takes from it, and where the evidence is contested.
Duverger (1954) proposed that simple-majority, single-ballot elections favour a two-party system, while proportional representation and two-ballot majority systems favour multipartism. He named two mechanisms. The mechanical effect: plurality turns votes into seats so unevenly that third parties are under-represented. The psychological effect: voters and politicians anticipate this and abandon parties that cannot win. Riker (1982) traced the idea back to Henry Droop and other nineteenth-century writers, and presented its revision and testing as a rare case of cumulative knowledge in political science. He also refined it into a probabilistic, district-level claim, since plurality countries such as Canada and India sustain more than two national parties.
Game theory turned the psychological effect into equilibrium analysis. Myerson and Weber (1993) showed that when voters weigh candidates by the probability of being pivotal, plurality elections have Duvergerian equilibria in which only two candidates get votes. Non-Duvergerian equilibria exist too, but Fey (1997) showed that they are unstable once voters update on pre-election polls, so iterated polling drives the race to two serious contenders. Palfrey (1989) gave a formal proof of Duverger's law for large electorates. Cox (1997) generalized the logic to districts of magnitude M: strategic voting leaves at most M + 1 viable lists, and strategic entry and withdrawal by elites reinforce the result. The simulator's voter game is a smoothed version of this pivot calculus, and its Duverger's-curve experiment reproduces the M + 1 pattern for small districts.
Electoral rules are permissive, not determining. Ordeshook and Shvetsova (1994) and Amorim Neto and Cox (1997) found that the number of parties depends on the interaction of district magnitude with social heterogeneity: diverse societies produce many parties only when the rules allow it, and permissive rules produce many parties only when society supplies cleavages. Clark and Golder (2006) confirmed the interaction on postwar elections, though later reanalyses dispute how robust it is. The cleavage tradition behind this view goes back to Lipset and Rokkan (1967). Taagepera and Shugart's seat product model (1989; Shugart and Taagepera 2017) predicts the effective number of parties from district magnitude and assembly size alone and fits large samples well on average. The simulator has a single left–right cleavage and fills every niche the rules allow, which is why it overstates fragmentation for Germany and New Zealand and cannot reach Brazil's.
Duverger's logic operates within districts. Why the same two parties compete nationally is a separate question. Cox (1997) called it linkage, and Chhibber and Kollman (2004) traced it to the centralization of power: national parties dominate when the national government holds authority, and regional parties gain when power devolves. They also documented substantial minor-party voting in the United States before the 1930s. A powerful national presidency is a strong force for linkage because it is a single prize that only a broad coalition can win (Shugart and Carey 1992), and concurrent presidential elections reduce legislative fragmentation (Golder 2006). In the model this appears as the value parties place on the presidency, which makes neighbouring parties merge.
Two-round systems lower the cost of a first-round vote for a small party, because the decisive contest is reaching the runoff. Duverger expected them to sustain multiple parties grouped into alliances, and France's Fifth Republic has averaged about 3.3 effective parties in the assembly. Ranked-choice voting, as in the Australian House, lets voters rank a minor party first without wasting the vote, so minor parties do much better in votes than in seats: Australia's 2022 election had over five effective parties in votes and about three in seats. The model captures the first effect by making the race for the top two places the pivotal one, and the second by letting ranked-ballot voters vote sincerely.
Early theories predicted which coalitions form from seats or policy alone: Riker's (1962) size principle favours minimal winning coalitions, Axelrod (1970) minimal connected ones, and Gamson (1961) proposed that portfolios are shared in proportion to seats, a regularity that has held up well empirically. Baron and Ferejohn (1989) modelled formation as sequential bargaining in which a recognized formateur proposes and others accept or reject. Laver and Shepsle (1996) argued that ministries give parties control of policy areas, so cabinets form at stable allocations of portfolios. Strøm (1990) showed that minority governments are common and rational where opposition parties hold policy influence without office. Formation can take months, and bargaining delays grow with fragmentation and uncertainty (Diermeier and van Roozendaal 1998). The simulator uses formateur bargaining over minimal winning coalitions with portfolios split by seats.
Research on cabinet duration moved from structural attributes to events. Browne, Frendreis and Gleiber (1984) treated terminations as the result of random critical events. King, Alt, Burns and Laver (1990) unified this with attribute-based accounts, showing that majority status, fragmentation and polarization shift the hazard of collapse. Warwick (1994) found that ideological diversity within the cabinet is among the strongest predictors of early termination. Lupia and Strøm (1995) modelled termination as a strategic choice: a partner brings a cabinet down when new information, such as favourable polls, makes an election or a new coalition worth more than staying. Diermeier and Stevenson (1999) separated replacements from early elections as competing risks. The simulator's cabinet game follows this design: random crises, stay-or-leave comparisons, credible alternative majorities, and early elections when a majority expects to gain.
Germany's Basic Law (Article 67) allows parliament to remove a chancellor only by electing a successor. This constructive vote of no confidence has succeeded once, in 1982. Switzerland's Federal Council is elected by parliament for a four-year term, cannot be dismissed by a no-confidence vote, and has been shared among the largest parties under the "magic formula" since 1959. Both show that the confidence relationship, not proportional representation as such, drives cabinet instability. In the model, Germany's partners can walk out without toppling the chancellor, and Switzerland's council cannot fall.
Linz (1990) argued that presidentialism is the riskier design: fixed terms make conflict between president and legislature hard to resolve, and winner-take-all elections raise the stakes. Mainwaring (1993) found the danger concentrated in "the difficult combination" of presidentialism and multipartism, and Shugart and Carey (1992) and Mainwaring and Shugart (1997) showed that outcomes depend heavily on presidential powers and party discipline. Cheibub (2007) challenged the causal claim: across all democracies from 1946 to 2002, presidential regimes broke down more often, but he attributes the difference to their frequent emergence from military dictatorships, not to the institution itself. Cheibub, Przeworski and Saiegh (2004) showed that coalition governments form under presidentialism about as readily as under parliamentarism, as in Brazil's coalitional presidentialism (Abranches 1988). For semi-presidential systems, which Duverger (1980) first named, cohabitation is the analogue of divided government (Elgie 2011).
Fixed terms buy government durability at the price of possible deadlock. Mayhew (1991) found that divided and unified governments passed similar numbers of major laws from 1946 to 1990. Binder (2003) found more gridlock under divided government once the size of the policy agenda is taken into account. The simulator reports divided government but does not model legislation, so it says nothing about whether durability translates into effective policy.
Electoral systems are not handed down from outside. Boix (1999) showed that ruling parties adopted proportional representation in the early twentieth century when new socialist parties threatened them, and Colomer (2005) argued that party systems often choose electoral rules rather than the reverse. Rare reforms provide cleaner tests. New Zealand switched from plurality to mixed-member proportional representation in 1996, and its effective number of parliamentary parties rose from an average of 1.96 (1946–1993) to 3.11 (1996–2020). The simulator treats rules as fixed, which isolates their effect but cannot capture why countries adopt them.
The effective number of parties (Laakso and Taagepera 1979) weights parties by their squared shares. Disproportionality is usually measured with Gallagher's (1991) least-squares index. Country figures on this page come from Gallagher's election indices (edition of 18 January 2023). Comparable data on cabinets are collected in ParlGov (Döring and Manow), and on electoral systems in the Democratic Electoral Systems dataset (Bormann and Golder 2022).
The model is a teaching device that chains established mechanisms: pivot-based strategic voting, strategic entry and merger, formateur bargaining and event-driven termination. Its main claims match the core findings. Plurality with a national presidency produces a two-party equilibrium that absorbs shocks; proportional rules sustain many parties; confidence-dependent coalition cabinets are shorter-lived than fixed-term executives. It does not settle the contested questions: whether presidentialism endangers democracy, how much society rather than rules drives fragmentation, or whether stable governments govern better.