# Surplus Handling

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In a sequential Multi-Member System which uses a Quota Method to ensure the Hare Quota Criterion is satisfied, there is an ambiguity about which voters should be in the quota when there are more than needed. This amount is referred to as a surplus and the various methods to deal with this situation are referred to as surplus handling.

## Surplus allocation

In allocation-based systems, the surplus can be transferred to other candidates with some form of surplus allocation. The number of surplus votes is known, but none of the various allocation methods is universally preferred. Alternatives exist for deciding which votes to transfer, how to weight the transfers, who receives the votes and the order in which surpluses from two or more winners are transferred.

### Random subset

Some surplus allocation methods select a random vote sample. Sometimes, ballots of one elected candidate are manually mixed. In Cambridge, Massachusetts, votes are counted one precinct at a time, imposing a spurious ordering on the votes. To prevent all transferred ballots coming from the same precinct, every $n$ th ballot is selected, where $\begin{matrix} \frac {1} {n} \end{matrix}$ is the fraction to be selected.

### Hare

Reallocation ballots are drawn at random from those transferred. In a manual count of paper ballots, this is the easiest method to implement; it is close to Thomas Hare's original 1857 proposal.

### Cincinnati

Reallocation ballots are drawn at random from all of the candidate's votes. This method is more likely than Hare to be representative, and less likely to suffer from exhausted ballots. The starting point for counting is arbitrary. Under a recount, the same sample and starting point is used in the recount (i.e., the recount must only be to check for mistakes in the original count, and not a second selection of votes).

Hare and Cincinnati have the same effect for first-count winners, since all the winners' votes are in the "last batch received" from which the Hare surplus is drawn.

### Wright

The Wright system is a reiterative linear counting process where on each candidate's exclusion the quota is reset and the votes recounted, distributing votes according to the voters' nominated order of preference, excluding candidates removed from the count as if they had not nominated.

For each successful candidate that exceeds the quota threshold, calculate the ratio of that candidate's surplus votes (i.e., the excess over the quota) divided by the total number of votes for that candidate, including the value of previous transfers. Transfer that candidate's votes to each voter's next preferred hopeful. Increase the recipient's vote tally by the product of the ratio and the ballot's value as the previous transfer (1 for the initial count.)

The UK's Electoral Reform Society recommends essentially this method. Every preference continues to count until the choices on that ballot have been exhausted or the election is complete. Its main disadvantage is that given large numbers of votes, candidates and/or seats, counting is administratively burdensome for a manual count due to the number of interactions. This is not the case with the use of computerised distribution of preference votes.

### Hare-Clark

This is a variation on the original Hare method that used random choices. It allows votes to the same ballots to be repeatedly transferred. The surplus value is calculated based on the allocation of preference of the last bundle transfer. The last bundle transfer method has been criticized as being inherently flawed in that only one segment of votes is used to transfer the value of surplus votes denying voters who contributed to a candidate's surplus a say in the surplus distribution. In the following explanation, Q is the quota required for election.

1. Separate all ballots according to their first preferences.
2. Count the votes.
3. Declare as winners those hopefuls whose total is at least Q.
4. For each winner, compute surplus as total minus Q.
5. For each winner, in order of descending surplus:
1. Assign that candidate's ballots to hopefuls according to each ballot's preference, setting aside exhausted ballots.
2. Calculate the ratio of surplus to the number of reassigned ballots or 1 if the number of such ballots is less than surplus.
3. For each hopeful, multiply ratio * the number of that hopeful's reassigned votes and add the result (rounded down) to the hopeful's tally.
6. Repeat 3–5 until winners fill all seats, or all ballots are exhausted.
7. If more winners are needed, declare a loser the hopeful with the fewest votes, recompute Q and repeat from 1, ignoring all preferences for the loser.

Example: If Q is 200 and a winner has 272 first-choice votes, of which 92 have no other hopeful listed, surplus is 72, ratio is 72/(272−92) or 0.4. If 75 of the reassigned 180 ballots have hopeful X as their second-choice, and if X has 190 votes, then X becomes a winner, with a surplus of 20 for the next round, if needed.

The Australian variant of step 7 treats the loser's votes as though they were surplus votes. But redoing the whole method prevents what is perhaps the only significant way of gaming this system – some voters put first a candidate they are sure will be eliminated early, hoping that their later preferences will then have more influence on the outcome.

### Gregory

Another method, known as Senatorial rules, or the Gregory method (after its inventor in 1880, J.B. Gregory of Melbourne) eliminates all randomness. Instead of transferring a fraction of votes at full value, transfer all votes at a fractional value.

In the above example, the relevant fraction is $\textstyle\frac{72}{272 - 92} = \frac{4}{10}$ . Note that part of the 272 vote result may be from earlier transfers; e.g., perhaps Y had been elected with 250 votes, 150 with X as next preference, so that the previous transfer of 30 votes was actually 150 ballots at a value of $\textstyle \frac15$ . In this case, these 150 ballots would now be retransferred with a compounded fractional value of $\textstyle \frac15 \times \frac{4}{10} = \frac{4}{50}$ .

An alternative means of expressing Gregory in calculating the Surplus Transfer Value applied to each vote is

$\text{Surplus Transfer Value} = \left( {{\text{Total value of Candidate's votes} - \text{Quota}} \over \text{Total value of Candidate's votes}} \right)\times \text{Value of each vote}$ The Unweighted Inclusive Gregory Method is used for the Australian Senate.

## Fractional Surplus Handling

In Cardinal voting systems the problem of surplus handing is simplified because the vote aggregation is arithmetic. This means that the surplus voters do not need to be allocated to other candidates. Instead, they can have their ballot weight reduced proportionally to the surplus and the tabulation process can continue unaffected. This method is better than allocation because it is completely deterministic and unbiased. This down-weighting of ballots can be applied to all ballots or to a subset depending on the desired effects and the tabulation system.