Polkadot elects its validators with a voting method designed in 1890s Sweden

A parliamentary fix, repurposed for blockchain

Most proof-of-stake networks are straightforward: the more tokens a validator stakes, the more influence it carries. @Polkadot takes a different approach. Every elected validator on the network holds equal voting power, which turns the selection process into a balancing problem rather than a simple ranking exercise.

The method behind it is sequential Phragmén, named after Lars Edvard Phragmén, a Swedish mathematician who developed it in the 1890s. Phragmén proposed the voting rules to address the needs of Swedish elections, which sought a more representative and proportional process for selecting parliament members. The core idea was to stop the most popular candidates from sweeping every available seat, ensuring minorities retained a fair share of representation.

In the late 19th century, Phragmén proposed a load-balancing approach for selecting committees based on approval ballots. Polkadot's adaptation of that logic applies the same principle to its validator set: stake must be spread as evenly as possible across all elected nodes.

How Polkadot applies the algorithm

In practice, $DOT holders nominate up to 16 validator candidates. At the start of each era, through the use of the sequential Phragmén method, each nominator's stake is allocated to elected validators in their selected set. Each era, roughly 24 hours, Polkadot's sequential Phragmén algorithm selects the active validator set, currently 297 validators, from all candidates.

The election optimises for three things simultaneously: maximising the total stake in play, maximising the stake backing the weakest validator, and minimising variance across the set. Since validators are paid equally in Polkadot, it is important that the stake behind each validator is spread out. That equal footing is also the network's main security consideration, because the chain is only as strong as its least-backed active node.

Given a large set of validators and an even larger set of nominators, the stake distribution algorithm becomes a computationally intensive task. That is why Polkadot computes the result off-chain and then proposes a transaction with a set of validators that should be elected. Off-chain workers submit the winning set, sparing the chain from processing lengthy calculations at the close of each era.

The elected committee satisfies the property of Proportional Justified Representation, a popular axiom in election theory indicating that an election is fair. That formal guarantee, borrowed from political science and adapted by Web3 Foundation researchers, is what gives the mechanism its theoretical grounding alongside its practical function.

Sources:
W3F Research: Sequential Phragmén Method
Wikipedia: Lars Edvard Phragmén
Staking Rewards: Polkadot Nominator Behaviour Research