A physics-inspired model calibrated on 40 years of US congressional data pinpoints a spending threshold of roughly 1.8 million USD at which campaigns stop influencing who wins and start fueling polarization instead.
American presidential elections have landed near 50-50 with regularity. In 2000, the margin in the popular vote was 0.5 percentage points; in 2016, it was 2.1; in 2024, just 1.5. Conventional wisdom points to candidate quality, media dynamics, or a polarized electorate. But a new study from the Complexity Science Hub (CSH) suggests a more structural explanation for these near-dead-heat outcomes – one rooted in the physics of phase transitions.
The research, published in Physical Review Letters by CSH researchers Jan Korbel, Remah Dahdoul, and Stefan Thurner, identifies a spending threshold in US House races: roughly 1.8 million USD per campaign (in 2020 dollars). Below it, social dynamics shape outcomes. Above it – on both sides – elections systematically trend toward a draw, no matter how much either party ultimately spends, while driving polarization higher.
“When both parties spend below the critical threshold of around 1.8 million USD, social networks dominate: how voters interact with neighbors, friends and family, who they talk to at work – all this shapes the outcome. The bigger spender has an advantage, but community dynamics still matter," explains Korbel. When one party crosses the threshold and the other doesn't, the better-funded campaign gains a large, systematic edge. Its messaging drowns out the social fabric. "But when both parties spend over 1.8 million USD, social influence becomes negligible and the election very often ends in a close race. Even if one party in a swing district spends 10 million USD and the other ten times that amount, the outcome barely changes but people's opinions drift further and further apart. What you get is polarization but – completely unexpected – not a decisive win," says Thurner.
OVER 6,000 ELECTIONS
For the study, the researchers applied a model from statistical physics to bipartisan elections. They tested it against 6,357 US House races with only two relevant candidates spanning 435 congressional districts and 21 election cycles from 1980 to 2020.
What they found is that political polarization behaves like a phase transition – the same kind of relatively sudden, system-wide shift that turns water into steam. There is a critical spending threshold at which more campaign money only deepens polarization without moving the needle on who wins.
THE INCUMBENCY ADVANTAGE, QUANTIFIED
The model also provides a fresh perspective on a well-documented phenomenon: the incumbency advantage. In the intermediate spending range, the model predicts a hysteresis zone – a region where the outcome depends not on who spends more today, but on who held the seat before. Officeholders carry the "memory" of the system into the next cycle.
The researchers put a number on this structural advantage. Even if the incumbent spends nothing, a challenger must invest roughly 140,000 USD just to neutralize the baseline incumbency effect. When the incumbent spends around 900,000 USD, the challenger still faces a disadvantage equivalent to about 20% of total campaign cost, purely as a consequence of the system's phase structure, not the incumbent's individual qualities.
EUROPE NEXT? WHAT THE MODEL DOESN'T YET CAPTURE
The current study is calibrated on bipartisan US House races, given the relatively high data quality and straightforward comparability in this context. Extending the framework to multi-party systems is a natural next step, though most European democracies lack the per-candidate transparency available in the US – aggregate data and differing electoral structures make direct comparisons difficult.
The critical thresholds estimated here – around 1.8 million USD for House races – are also district-level figures. Senate campaigns and presidential races operate at much higher absolute spending, though the comparatively small number of those elections makes precise calibration harder. What the model does establish, robustly, is the existence and shape of the transition itself.
POLICY IMPLICATIONS
"Rising campaign spending may be one of the mechanisms driving the global increase in polarization. Small increases in spending can have large systemic effects – and these findings could be directly relevant to campaign finance regulation," the authors say. From a rational-actor standpoint, there is an incentive to spend more (you don't want to be the only one below the line of 1.8 million USD), but the collective result is a kind of arms-race equilibrium that leaves everyone worse off socially because the society ends up more polarized.
The same model, the authors note, extends well beyond US politics: These phase-transition dynamics appear wherever two campaigns compete for allegiance.
Service
The study “Empirical Validation of the Polarization Transition in a Double-Random Field Model of Elections” by Jan Korbel, Remah Dahdoul, and Stefan Thurner was published in the journal Physical Review Letters (doi: 10.1103/9gjj-1df6).
ABOUT THE COMPLEXITY SCIENCE HUB (CSH)
The Complexity Science Hub (CSH) is Europe's research center for the study of complex systems. Drawing on large-scale data across economics, medicine, ecology, and the social sciences, CSH develops quantitative methods to understand the interconnected networks that underlie society – from financial markets and supply chains to public health and urban development. The goal is to provide a rigorous basis for navigating the challenges of an increasingly complex world.
