A physics-inspired model calibrated on 40 years of US congressional data pinpoints a spending threshold of roughly 1.8 million USD at which races tip into a structural draw. Campaigns stop influencing who wins, and start fueling polarization instead.
THE STUDY IN A NUTSHELL
- A model from statistical physics, calibrated on 6,357 US congressional races (1980–2020), identifies a critical spending threshold of around $1.8 million per campaign.
- Below this threshold, social networks shape the outcome – whoever spends more has an advantage, but social contacts play a substantial role.
- Above this tipping point, campaign messaging crowds out social influence: the outcome barely changes, close races systematically trend toward a draw, but opinions drift further and further apart.
- The model also quantifies the structural advantage of the incumbent.
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. The question of why has been debated for decades. A new study by CSH researchers Jan Korbel, Remah Dahdoul, and Stefan Thurner may have found a structural answer – one that has little to do with the candidates themselves, but one that is rooted in the physics of phase transitions.
The team applied a model from statistical physics to partisan elections and calibrated it on 6,357 congressional races spanning 435 districts and 21 electoral cycles from 1980 to 2020. Their findings were published in Physical Review Letters.
The central result: there is a critical spending threshold of around $1.8 million (in 2020 dollars) per campaign. Below this threshold, social networks shape the outcome. Whoever spends more has an advantage – but local communities still play a role. Korbel explains: “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.”
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.
PHYSICS MEETS POLITICS
The model used is mathematically equivalent to those physicists use to describe magnets. In such models, every atom aligns either up or down depending on the forces acting on it. In the political model, every voter represents such an atom, with a binary preference: Democrat or Republican. Two competing forces determine this alignment: party campaign messaging on the one hand, and the local social environment on the other.
What matters is the ratio between these two forces. In physics, there are points at which a system fundamentally changes its character – so-called tipping points – such as when water turns to ice. This transition does not happen gradually, but abruptly, once a certain temperature threshold is crossed. The researchers observe exactly this dynamic in electoral campaigns: once both campaigns cross the critical spending threshold, the system tips. Social influence is crowded out, party affiliation dominates – and the race converges toward a 50/50 outcome.
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.
A SPENDING RACE THAT LEAVES EVERYONE WORSE OFF
Looking at the data over four decades, the share of races in which both sides crossed the critical threshold remained relatively low for most of the period. In the 2018 and 2020 election cycles, that share rose sharply. The researchers interpret this as an indication of increased political polarization – but explicitly note that this finding is based on model projections and requires further empirical validation.
From a theoretical standpoint, the findings describe a kind of collective dilemma: individual campaigns have a rational incentive to spend more – no one wants to be the only one below the $1.8 million mark. The collective result, however, is a spending arms race that leaves everyone worse off socially: cohesion within districts erodes, while electoral outcomes barely shift.
“Rising campaign spending could be one of the mechanisms driving the global increase in polarization. Modest spending increases can have large systemic effects – and these findings could be of immediate relevance for the regulation of campaign finance,” the authors write.
EUROPE NEXT? WHAT THE MODEL DOESN'T YET CAPTURE
The study draws on US House races, where data availability and comparability are particularly favorable: decades of candidate-level financial disclosures, a two-party system, and hundreds of comparable districts. Most European democracies lack these conditions. Aggregated party finances, varying electoral systems, and the multiplicity of relevant parties make direct comparisons considerably more difficult.
The researchers nonetheless consider an extension of the model to be feasible in principle – and politically relevant. The phase transition dynamics described are not specific to the American system, but arise wherever two campaigns compete for the support of the same electorate.
BEHIND THE SCENES
How did the researchers do it? And where does the complexity lie?
The model treats voters as agents in a social network, each pulled between two forces: the opinions of friends and family, and the messaging of their preferred party’s campaign. Because voters typically follow only one campaign, each side is pushing its supporters in opposite directions, while social influence flows more freely across the whole network. This is what creates the tipping point.
Mathematically, this maps onto the random field Ising model, a framework from statistical physics used to study how magnetic systems respond to competing local and external forces – the same mathematics that describes how a magnet flips. “Fitting the model to historical spending and outcome data, we identified the existence of a phase transition – which was not obvious”, Korbel says, “and the transition point which is the critical spending threshold of around 1.8 million USD”. Two independent modeling approaches returned the same result, suggesting the tipping point is a robust feature of electoral dynamics rather than a mathematical artifact. The model also significantly outperforms a simpler baseline (that the higher-spending candidate always wins) confirming that polarization is very real.
About the study
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).
