Quorum in the Debian Voting System

The Debian voting system differs from plain Condorcet voting in several points. For example it introduces the concept of a quorum. Traditionally a quorum should assert that no decision is made without sufficient support among the electors. The Oxford Advanced Learner's Dictionary of Current English defines quorum as the

In the Debian voting system the main purpose of quorum seems to be different, namely to weed out options without enough support before they enter the voting machinery (cited from Buddha Buck's summary of the rationale for the general resolution):

The question is, which method is better for our votes? A first observation: whenever the whole vote would be dropped because of global quorum, with the per-option quorum method the default option would win; both outcomes are effectively the same. So the methods can only differ when there are at least R votes cast.

In this text I will give some examples, which show the different deficiencies of the two approaches. I will assume here, that you have read my explanation of the voting system at my Debian Voting System page. If you didn't, go straight there and read it!

When reading the remaining part of this text, please keep in mind that ties between options are very rare and (to the best of my knowledge) in none of our previous votes an option did actually fail to meet the quorum requirement. So the discussion below is mainly of theoretical interest. But then we want to get things right and we should think about these seldom-to-occur cases before they actually do occur in one of our future votes.

In all examples on this page, option D denotes the default option. Grey parts of the vote graphs are dropped because of quorum. Edges in the vote graph, which are removed by the tie resolution mechanism, are marked red.

Why is Per-Option Quorum Strange?

Dropping the Ideal Democratic Winner

Image of the vote graph	Detailed votes	Result
	ABD 15x 213 10x 132	25 votes cast Quorum: 20 (per-option) Option B dropped because of quorum Winner: A

Table 1. A situation where the ideal democratic winner is dropped.

Here the option B is preferred to every other option by a majority of voters. (Such an option is sometimes called an ideal democratic winner.) Nevertheless option B is dropped from consideration, because it does not meet the quorum requirement of 20 (only 15 voters prefer B over the default option D).

In my opinion, if one option is preferred over every other option, none of the other options should win the election. Thus I consider this behaviour as an defect of the per-option quorum mechanism.

Dropping Uninteresting Options Makes a Difference

The following (more complex) example shows another problem with per-option quorum. In short: if we drop (because of quorum) an option, which is part of a tie but has no chance to win, then the winner of the vote can change in a hard-to-predict way. This makes it more difficult for the voter to understand the impact of his vote.

Initial vote count

Image of the vote graph	Detailed votes	Result
	ABCDE 13x 241-3 9x 131-2 9x 3124- 9x 323-1 4x 312-4 3x 123-1 2x 12543 2x 312-3 1x 232-1 1x 412-3	53 votes cast Quorum: 45 (per-option) Option E dropped because of quorum Winner: option B

Table 2. Example vote count (continued in table 3).

Here we have five options, A, B, C, E, and the default option D. All non-default options defeat the default option, but option E is dropped because it does not meet the quorum requirement of 45. (It has only 44 votes which prefer it over D.) The Schwartz set consists of the three remaining options A, B, and C.

We break the tie between the options in the Schwartz set, using Schwartz sequential dropping algorithm, i.e. we omit the edges from the graph one-by-one, always choosing weakest defeat. From the image we can find the weakest defeat by finding the edge, which has the smallest number before the colon. This is the edge from A to B, it has strength 28. After dropping AB the cycle is already broken up and option B wins.

One more vote makes a difference

Now someone, who likes option B, enters the scene. He knows that B is best and A is almost as bad as E. So he votes 312-4, indicating that his preferences are B, then C, then A, and only then E. This gives E another vote over the default option and thus option E passes the quorum requirement now. The new situation is as follows:

Image of the vote graph	Detailed votes	Result
	ABCDE 13x 241-3 9x 131-2 9x 3124- 9x 323-1 5x 312-4 3x 123-1 2x 12543 2x 312-3 1x 232-1 1x 412-3	54 votes cast Quorum: 45 (per-option) Option E passes quorum Winner: option A

Table 3. Another vote is added to the situation from table 2. The additional vote is in favour of B, but causes B to loose.

All non-default options defeat the default option, the Schwartz set now consists of all the four options A, B, C, and E. Again, we use the Schwartz sequential dropped algorithm to break up the tie. We have to drop AB first (strength 28), and then CA (strength 30). Now the cycle is broken up, but this time option A wins. Thus, in this example, adding another vote in favour of B causes option B to loose!

Summary

The example shows that the introduction of the per-option quorum requirement can change the winner of an election in a hard-to-predict way. Even dropping of an uninteresting option (one which had no chance of winning) can change which of the remaining options wins.

Why is Global Quorum Strange?

Strategic Voting for Low-Participation Votes

Image of the vote graph	Detailed votes	Result
	ABD 19x 12-	19 votes cast Quorum: 20 (global) The whole vote is dropped because of quorum Winner: none

Table 4. The whole vote is thrown out, because there are not enough voters.

In the situation depicted above, the whole vote is dropped, because one vote was missing for the quorum requirement. Adding any vote to this ballot will make option A win. Of course this causes the following problem: adding a vote in favour of B (like 213) will change the outcome from no decision to A wins. This can be a problem under the following circumstances:

In this (unlikely) situation it may be better to not vote for B even if you want it to win.

Note that (to the best of my knowledge) in our previous elections the number of participants was always much larger then the quorum requirement.

Winning Options Do Not Need To Defeat The Default Option

Image of the vote graph	Detailed votes	Result
	ABD 15x 123 13x 231 11x 312	39 votes cast Quorum: 20 (global) Winner: A

Table 5. An option which does not directly defeat the default option wins.

When the default option D is contained in the Schwartz set, then any option may win, which only transitively but not directly defeats the default option. This may or may not seem strange to you, depending on how strong you value transitive defeats.

With a per-option quorum of 10 option A would still win here, but with a per-option quorum of 20 the option A would be dropped (because only 15 people directly prefer A over D) and thus option B would win instead. Note that with our current voting mechanism option A will be dropped independently of quorum, because it fails the implicit super-majority requirement of 1:1 in section A.6(3).

The decision here is to either ignoring the 24 voters who preferred, let's say, none of the above over option A (by making A the winner) or to ignore the 28 voters who preferred option A over option B (by making B the winner). Personally I prefer to drop as less votes as possible, so I think that this is no argument against global quorum.

Summary

We have seen that with global quorum in known-very-low-participation votes, voters of unpopular options may be tricked into not voting at all. I doubt that this kind of problem would really occur in our votes. For reference: in the 2003 DPL elections we had 488 votes cast and a quorum of 14.4. It is hard to imagine a vote where only 14 or 15 votes are cast.

Then, global quorum allows winners which are not directly preferred by many voters over the default option. Here I see no real problem. Weighting direct defeats stronger then transitive defeats here, does not buy us much for the price of ignoring more of the votes.

For per-option quorum we have seen that it can drop the ideal democratic winner without dropping all the other options, thus causing one of the remaining options to win. I think that this a problem, but again I doubt that this kind of result will occur in any of our votes. When the number of voters is much larger than quorum, this situation is not very likely to occur.

Finally, dropping individual options from the Schwartz set disturbs the carefully constructed tie resolution mechanism. Dropping an option here, even one which would not win anyway, can change the outcome of the vote. Whenever the default option is part of a tie between options, this effect can easily occur, even for a large number of voters. So this effect is more likely to occur, and it makes it harder for the voter to understand the impact of his vote. I think there is no point in dropping single votes here. We should just resolve the tie, if the option in question finally would win, we should let it be the winner.

Considering all this I prefer the global quorum method. It is the simpler method, it will successfully prevent stealth decisions by only a few developers, and at the same time it will cause less problems then the local-quorum approach.