‘The winners take it all, the losers standing small’: Party centralization and Duverger's law

Do electoral rules shape party centralization? Duverger posed the question seven decades ago; confounding factors have kept it open ever since. We argue that centralization also responds strategically to institutional and ecological incentives, not only to inheritance, imitation, or top-down imposition. Brazil's bifurcated mayoral rules offer a quasi-experimental test: municipalities under 200,000 registered voters elect mayors by single-member plurality (SMP); larger ones use a two-round system (TRS). A regression discontinuity design at this threshold identifies the causal effect of electoral rules on party organization, using elected (decentralized) versus appointed (centralized) local branches as the outcome. TRS municipalities sustain about two to three more elected branches than comparable SMP municipalities, and under SMP the probability of gaining a new branch drops sharply once about two parties hold one — consistent with Cox's M+1 rule. A cross-national analysis of 365 parties in 56 democracies with directly elected presidents is consistent with the pattern. Restrictive rules centralize parties from the bottom up.