The D'Hondt Divisor Sequence: How Proportional Representation Systems Allocate Seats Based on Highest Averages

In the 2024 European Parliament elections, the distribution of 720 seats across 27 member states relied on a mathematical algorithm first published in 1878.[2]

Developed by Belgian mathematician Victor D'Hondt, the formula is a mechanism designed to translate raw vote totals into whole legislative seats without fracturing them into unworkable fractions.[3]

"The d’Hondt method is a highest averages method for allocating seats in party-list proportional representation," according to the research group UK in a Changing Europe.[4]

The mechanism operates through a sequence of divisors. After all votes are tallied, each party's total is divided by 1, then by 2, then by 3, and so on, creating a grid of quotients.[2][4]

Seats are then allocated one by one to the highest numbers in that grid until the district's entire delegation is filled.[4]

For example, if Party A receives 100,000 votes and Party B receives 40,000 votes in a three-seat district, Party A's quotients are 100,000, 50,000, and 33,333. Party B's quotients are 40,000, 20,000, and 13,333.[2]

The first seat goes to Party A, which holds the highest quotient on the board at 100,000. The second seat also goes to Party A at 50,000, because its second-round quotient remains higher than Party B's first-round quotient of 40,000.[2]

The third and final seat goes to Party B, which claims it with its 40,000 quotient, beating Party A's third-round figure of 33,333.[2][4]

While the system is broadly proportional, the specific sequence of integers—1, 2, 3, 4—creates a structural advantage for larger political formations.[1]

Because the divisors grow slowly, a large party's divided totals remain relatively high in subsequent rounds, allowing it to capture multiple seats before a smaller party's undivided total clears the threshold.[1]

This large-party premium is a deliberate feature. It is often favored by constitutional designers seeking to manufacture stable governing majorities and prevent extreme legislative fragmentation.[1]

The Canadian Parliamentary Review observes that "Common Seat Allocation Methods in Proportional Representation Systems" often contrast D'Hondt with the Sainte-Laguë method to demonstrate how the choice of formula alters the balance of power.

The Sainte-Laguë method uses odd-number divisors: 1, 3, 5, 7. By dividing vote totals more aggressively in subsequent rounds, it rapidly diminishes the value of a large party's surplus votes, making it easier for smaller factions to win seats.

The D'Hondt method's proportionality is also heavily dependent on district magnitude—the number of seats available in a given electoral boundary.[1]

In a massive 50-seat district, the D'Hondt formula produces highly proportional results that closely mirror the popular vote.[1]

However, in a district with only three or four seats, the mathematical premium for large parties becomes overwhelming, often shutting out parties that secure 10 to 15 percent of the vote.[1]

Today, the D'Hondt sequence is the most widely used seat allocation formula in the world, governing elections in Spain, Poland, Israel, and dozens of other democracies.[3]

In the United Kingdom, it is utilized to allocate regional list seats in the Scottish Parliament and the Welsh Senedd, operating as a corrective mechanism alongside first-past-the-post constituencies.[4]

The choice of divisor sequence demonstrates that proportional representation is not a single, objective standard, but a spectrum of mathematical choices. By selecting D'Hondt over Sainte-Laguë, a state mathematically prioritizes legislative stability over exact proportional reflection.[1]