# Highest averages method

The **highest averages method** is one way of allocating seats proportionally for representative assemblies with party list voting systems.

The *highest averages method* requires the number of votes for each party to be divided successively by a series of divisors, and seats are allocated to parties that secure the highest resulting quotient, up to the total number of seats available. The most widely used is the d'Hondt formula, using the divisors 1,2,3,4... The Sainte-Laguë method divides the votes with odd numbers (1,3,5,7 etc). The Sainte-Laguë method can also be modified, for instance by the replacement of the first divisor by 1.4, which in small constituencies has the effect of prioritizing proportionality for larger parties over smaller ones at the allocation of the first few seats.

In addition to the procedure above, highest averages methods can be conceived of in a different way. In this manner, what was called the **divisor** above will now be the **quotient**, and what was called the quotient will now be the divisor. For an election, a **divisor** is calculated, usually the total number of votes cast divided by the number of seats to be allocated. Then, each parties' quotient is calculated by dividing their vote total by the divisor. Parties are then allocated seats by rounding the quotient to a whole number.
Rounding down is equivalent to using the d'Hondt method, while rounding to the nearest whole number is equivalent to the Sainte-Laguë method. However, because of the rounding, this will not necessarily result in the desired number of seats being filled. In that case, the divisor may be adjusted up or down until the number of seats after rounding is equal to the desired number.

The tables used in the d'Hondt method can then be viewed as calculating the lowest divisor necessary to round off to a given number of seats. For example, the quotient which wins the first seat in a d'Hondt calculation is the lowest divisor necessary to have one party's vote, when rounded down, be greater than 1. The quotient for the second round is the lowest divisor necessary to have a total of 2 seats allocated, and so on.

An alternative to the *highest averages method* is the largest remainder method, which use a minimum quota which can be calculated in a number of ways.

## Comparison between the *d'Hondt* and *Sainte-Laguë* methods

### The unmodified *Sainte-Laguë* method shows differences for the first mandates

d'Hondt method | unmodified Sainte-Laguë method | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

parties | Yellows | Whites | Reds | Greens | Blues | Pinks | Yellows | Whites | Reds | Greens | Blues | Pinks | |

votes | 47,000 | 16,000 | 15,900 | 12,000 | 6,000 | 3,100 | 47,000 | 16,000 | 15,900 | 12,000 | 6,000 | 3,100 | |

mandate | quotient | ||||||||||||

1 | 47,000 | 16,000 | 15,900 | 12,000 | 6,000 | 3,100 | 47,000 | 16,000 | 15,900 | 12,000 | 6,000 | 3,100 | |

2 | 23,500 | 8,000 | 7,950 | 6,000 | 3,000 | 1,550 | 15,667 | 5,333 | 5,300 | 4,000 | 2,000 | 1,033 | |

3 | 15,667 | 5,333 | 5,300 | 4,000 | 2,000 | 1,033 | 9,400 | 3,200 | 3,180 | 2,400 | 1,200 | 620 | |

4 | 11,750 | 4,000 | 3,975 | 3,000 | 1,500 | 775 | 6,714 | 2,857 | 2,271 | 1,714 | 875 | 443 | |

5 | 9,400 | 3,200 | 3,180 | 2,400 | 1,200 | 620 | 5,222 | 1,778 | 1,767 | 1.333 | 667 | 333 | |

6 | 7,833 | 2,667 | 2,650 | 2,000 | 1,000 | 517 | 4,273 | 1,454 | 1,445 | 1,091 | 545 | 282 | |

seat | seat allocation | ||||||||||||

1 | 47,000 | 47,000 | |||||||||||

2 | 23,500 | 16,000 | |||||||||||

3 | 16,000 | 15,900 | |||||||||||

4 | 15,900 | 15,667 | |||||||||||

5 | 15,667 | 12,000 | |||||||||||

6 | 12,000 | 9,400 | |||||||||||

7 | 11,750 | 6,714 | |||||||||||

8 | 9,400 | 6,000 | |||||||||||

9 | 8,000 | 5,333 | |||||||||||

10 | 7,950 | 5,300 |

### With the modification, the methods are initially more similar

d'Hondt method | modified Sainte-Laguë method | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

parties | Yellows | Whites | Reds | Greens | Blues | Pinks | Yellows | Whites | Reds | Greens | Blues | Pinks | |

votes | 47,000 | 16,000 | 15,900 | 12,000 | 6,000 | 3,100 | 47,000 | 16,000 | 15,900 | 12,000 | 6,000 | 3,100 | |

mandate | quotient | ||||||||||||

1 | 47,000 | 16,000 | 15,900 | 12,000 | 6,000 | 3,100 | 33,571 | 11,429 | 11,357 | 8,571 | 4,286 | 2,214 | |

2 | 23,500 | 8,000 | 7,950 | 6,000 | 3,000 | 1,550 | 15,667 | 5,333 | 5,300 | 4,000 | 2,000 | 1,033 | |

3 | 15,667 | 5,333 | 5,300 | 4,000 | 2,000 | 1,033 | 9,400 | 3,200 | 3,180 | 2,400 | 1,200 | 620 | |

4 | 11,750 | 4,000 | 3,975 | 3,000 | 1,500 | 775 | 6,714 | 2,857 | 2,271 | 1,714 | 875 | 443 | |

5 | 9,400 | 3,200 | 3,180 | 2,400 | 1,200 | 620 | 5,222 | 1,778 | 1,767 | 1.333 | 667 | 333 | |

6 | 7,833 | 2,667 | 2,650 | 2,000 | 1,000 | 517 | 4,273 | 1,454 | 1,445 | 1,091 | 545 | 282 | |

seat | seat allocation | ||||||||||||

1 | 47,000 | 33,571 | |||||||||||

2 | 23,500 | 15,667 | |||||||||||

3 | 16,000 | 11,429 | |||||||||||

4 | 15,900 | 11,357 | |||||||||||

5 | 15,667 | 9,400 | |||||||||||

6 | 12,000 | 8,571 | |||||||||||

7 | 11,750 | 6,714 | |||||||||||

8 | 9,400 | 5,333 | |||||||||||

9 | 8,000 | 5,300 | |||||||||||

10 | 7,950 | 5,222 |

## Notes

Highest-averages methods in some sense simulate vote management, whereas highest-remainder methods do not. Example: If there are two seats, with Party A getting 50 votes, and Parties B through Z each getting 10 votes, then most highest-averages methods give Party A both seats, because they can put 25 votes into both seats, whereas any other party can at most put 10 votes into even one seat. Highest-remainder methods would generally give Party A one seat, but then give one of Parties B through Z the second seat; this is because the Hare quota (and even Droop quota) far exceed Party A's 50 votes, therefore all of Party A's votes would be "spent", leaving only Parties B through Z with any votes (in a tie, actually) to take the second seat.

A common misconception is that only largest remainder methods pass any kind of quota-related criteria. In reality, while all highest averages methods fail the quota rule (i.e. a party can theoretically get more seats than would seem fair), many do guarantee a minimum number of seats a party will win based on its number of quotas of votes. For example, D'Hondt guarantees that in the party list case, a party will win at least as many seats as it has Hagenbach-Bischoff quotas.