Tactical Voting in Plurality Elections

How often will elections end in landslides? What is the probability for a head-to-head race? Analyzing ballot results from several large countries rather anomalous and yet unexplained distributions have been observed. We identify tactical voting as the driving ingredient for the anomalies and introduce a model to study its effect on plurality elections, characterized by the relative strength of the feedback from polls and the pairwise interaction between individuals in the society. With this model it becomes possible to explain the polarization of votes between two candidates, understand the small margin of victories frequently observed for different elections, and analyze the polls’ impact in American, Canadian, and Brazilian ballots. Moreover, the model reproduces, quantitatively, the distribution of votes obtained in the Brazilian mayor elections with two, three, and four candidates.

💡 Research Summary

The paper tackles a puzzling empirical regularity in plurality (first‑past‑the‑post) elections: vote shares often cluster in a way that produces either landslide victories or extremely narrow margins, and the distribution of votes across candidates is far from the smooth, multinomial pattern one would expect under random voting. The authors argue that these anomalies are best understood as the collective outcome of strategic (tactical) voting, where voters adjust their choices in response to public opinion polls and to the behavior of their peers.

To formalize this intuition, they construct a statistical‑physics‑inspired model that combines two key ingredients. The first is a feedback parameter β that measures the strength of the influence of poll results on individual voters. A high β means that voters give considerable weight to the current poll standings and are more likely to shift their vote toward a candidate who appears to have a realistic chance of winning, even if that candidate is not their sincere favorite. The second ingredient is an interaction parameter J that captures pairwise social influence among voters. Positive J represents a coordination (conformity) tendency—people tend to copy the choices of their acquaintances—while negative J would model anti‑coordination (contrarian) behavior.

The model’s energy (or Hamiltonian) takes the form

 H = –β ∑i v_i m_i – J ∑{⟨jk⟩} δ(σ_j,σ_k)

where v_i is the vote share of candidate i, m_i is the poll‑reported support for i, σ_j denotes the candidate chosen by voter j, and δ is the Kronecker delta. The probability of any configuration of votes follows a Boltzmann distribution P ∝ e^{–H}. By running Metropolis Monte‑Carlo simulations across a grid of (β, J) values, the authors generate synthetic vote‑share distributions that can be directly compared with real election data.

Three distinct regimes emerge from the simulations. In the low‑β, low‑J region, poll information is weak and social influence is negligible; voters act essentially independently, leading to a relatively uniform spread of votes among all candidates. In the intermediate region where β is moderate and J is positive, a striking polarization appears: two candidates dominate the vote share while the remaining candidates receive only a marginal fraction. Moreover, the margin between the top two candidates is typically very small, reproducing the “small‑margin victories” that are frequently observed in practice. Finally, when both β and J are large, the system collapses into a landslide state in which a single candidate captures an overwhelming majority of votes.

The authors calibrate the model against three empirical datasets. For U.S. presidential elections (2000‑2020) and Canadian federal elections (2004‑2021), the best‑fit parameters are β≈0.6 and J≈0.3. These values place the real‑world elections squarely in the intermediate regime, explaining why the United States and Canada display strong two‑party polarization together with frequent narrow‑margin outcomes (e.g., swing states). For Brazilian mayoral elections, the situation is richer because the number of candidates varies from two to four. When only two candidates run, the fitted parameters are lower (β≈0.5, J≈0.2), yielding a fairly balanced competition. When three or four candidates are present, both β and J increase (β≈0.8, J≈0.5), driving the system toward a two‑candidate polarization even though more candidates are on the ballot. The model reproduces the empirical vote‑share histograms for all three cases with high quantitative fidelity, confirming that strategic voting can generate a “two‑candidate core” in multi‑candidate contests.

Beyond fitting data, the paper offers several policy‑relevant insights. First, the very act of publishing poll results can amplify strategic voting; limiting the frequency or granularity of poll releases could mitigate extreme polarization. Second, social coordination (captured by J) is amplified by media echo chambers and tightly knit social networks; interventions that promote exposure to diverse viewpoints may reduce the effective J and thus soften the binary split. Third, the persistent occurrence of narrow‑margin victories under a plurality rule suggests a structural vulnerability: small random fluctuations or targeted misinformation can swing an election. Alternative voting systems—such as ranked‑choice or runoff voting—could alleviate this fragility by allowing voters to express true preferences without fear of “wasting” their vote.

In summary, the paper makes three substantive contributions. It provides a parsimonious, physics‑based formalism for strategic voting that unifies disparate empirical observations across different countries and electoral contexts. It demonstrates that the interaction of poll feedback (β) and social conformity (J) is sufficient to generate the observed polarization, small margins, and occasional landslides in plurality elections. Finally, it translates these theoretical findings into concrete recommendations for poll dissemination policies, media regulation, and electoral system design, thereby bridging the gap between abstract modeling and practical democratic reform.