Physics Law That Turns Every US Election Into a Coin Flip

The numbers keep coming up the same. In 2000, the popular vote margin was half a percentage point; in 2016, just over two; in 2024, 1.5. Three elections across a quarter-century, spanning wildly different candidates, media climates, and geopolitical moments, all landing within statistical spitting distance of a draw. Political scientists have been arguing about why for decades. A team of physicists in Vienna think they may have found the structural answer, and it has almost nothing to do with the candidates themselves.

The explanation, published this spring in Physical Review Letters, borrows its mathematics from the physics of magnets and phase transitions. Which is, admittedly, an unexpected place to look for insight into American democracy.

Jan Korbel, Remah Dahdoul, and Stefan Thurner at the Complexity Science Hub in Vienna started from a simple observation about how voters actually behave. Two forces compete for any voter’s allegiance: the social environment around them (family, colleagues, the neighbours they actually talk to) and the political campaign messaging they follow. In the language of statistical physics, that’s homophily on one hand and an external field on the other. The researchers modelled this as a random field Ising model, the same mathematical framework used to understand how ferromagnets snap between ordered states. Each voter is a spin, pulled by social interactions and pushed by campaign spending. The model has a temperature parameter, too, roughly representing how susceptible the electorate is to changing its mind.

The critical question was whether you could calibrate this against reality. So they did. They fed in data from 6,357 US House races across 435 congressional districts and 21 election cycles, 1980 to 2020.

What came out was a threshold. Below about 1.8 million dollars per campaign (in 2020 dollars), social dynamics do the heavy lifting. “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,” says Korbel. In this regime, elections can be decisive. Incumbents hold structural advantages. Who wins depends on local social texture as much as on money.

Cross that line on both sides and something abrupt happens. The social fabric, in the model’s terms, stops mattering. The election trends toward a draw.

The Point of No Return

“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 unexpectedly, not a decisive win,” says Thurner. The data backs this up rather starkly: in closely-matched high-spending races, more than 70 percent of outcomes cluster near 50:50. Below the threshold, only around 39 percent do.

This is, in the physics sense, a phase transition. The same kind of relatively sudden, system-wide shift that turns water into steam. There’s a critical point, and either side of it the system behaves differently. What the Vienna team did, and what makes this study unusual, is extract that critical point directly from historical electoral data, fitting a temperature parameter of roughly 0.92 (on a scale where values below 1 indicate the presence of hysteresis) and a critical spending threshold of just over $1.83 million. The thermodynamics, in other words, were inferred from the ballot box rather than assumed in advance.

What Incumbency Is Actually Worth

The model also sheds new light on something political scientists have documented but struggled to quantify: the incumbency advantage. When both campaigns are spending in a middling range, the model predicts a hysteresis zone, a region where the outcome depends not on current spending but on who held the seat last cycle. Officeholders carry the memory of the system into the next election. The team put a number on this: even if an incumbent spends nothing at all, a challenger must invest roughly $140,000 just to neutralise the baseline structural advantage. When the incumbent is spending around $900,000, the challenger still faces a disadvantage equivalent to something like 20 percent of total campaign cost, purely from phase structure rather than individual candidate quality.

There are limits to what the model can say. It was calibrated on bipartisan House races, where spending data is clean and district populations are roughly comparable. Senate campaigns and presidential races involve far higher absolute spending, and the smaller number of such elections makes precise calibration difficult. European multi-party systems are a natural extension in principle, though most lack the per-candidate transparency available in US federal data.

“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 note. The incentive structure is perverse: no rational campaign wants to be the only one spending below $1.8 million, so both sides push past the threshold, which collectively makes elections closer while making voters more estranged from each other. An arms race that leaves everyone worse off, socially, than when it started. The sharp rise in races where both campaigns exceeded the threshold in 2018 and 2020 suggests the system has been tilting deeper into the polarized regime over recent election cycles. Whether that trajectory continues, and whether any regulatory intervention could reverse it, is now, the researchers argue, a question with a quantitative answer for the first time.

The physics, it turns out, was always there in the data. It just needed someone willing to look for a phase diagram in a ballot box.

DOI: 10.1103/9gjj-1df6

Frequently Asked Questions

Does this mean spending more money on a campaign is pointless?

Not exactly. It depends on where both campaigns are starting from. When one side crosses the $1.8 million threshold and the other hasn’t, the better-funded campaign gains a significant advantage. The problem emerges when both sides spend above it, at which point additional spending stops changing who wins while still deepening political division. The research suggests there’s a rational incentive to overspend that collectively harms everyone.

Why does heavier campaign spending lead to a 50:50 result rather than helping the bigger spender win?

Above the spending threshold, campaign messaging drowns out the social networks that would otherwise differentiate voter behaviour across communities. When both campaigns are broadcasting at full intensity, voters essentially sort uniformly by partisan affiliation rather than being nudged by local social dynamics. The result is that outcomes mirror the underlying partisan split in the district (which in competitive districts is close to even), regardless of how much more one side spends.

Could campaign spending caps actually reduce political polarization?

That’s essentially what the model implies. If regulatory interventions kept both campaigns below the critical threshold, social networks would reassert their influence on voter decisions, potentially producing more decisive outcomes and less affective polarization. The researchers are careful to note the model still needs broader empirical validation, and multi-party systems introduce complexities the current framework doesn’t yet address.

How does the incumbency advantage fit into the physics model?

In statistical physics terms, incumbency shows up as hysteresis, the system’s tendency to remain in its current state based on history rather than present conditions alone. The model predicts a spending zone where the prior election result, not current spending, determines the outcome. This lets the researchers put a dollar figure on structural incumbency advantage for the first time, quantifying something political scientists have long described qualitatively.