Interpolated Consensus Vote Aggregation

Here is how and why the 2007-2008 UBC VFM Committee proposed the contest voting structure (now called “Interpolated Consensus” or IC) the way they did:

The previous VFM contest, in January 2007, used a very simple approval voting structure.  The ballot had just one check-box next to each media contestant’s name, and voters checked the boxes of (any number of) contestants they thought deserved a prize.  The contestant with the most votes got first prize, etc – see results here.  The $8000 award pool was sliced in advance by the 2006-2007 VFM Committee – 1st prize $1500 down to 8th prize $500.

The Interpolated Consensus structure, however, gives voters more choice and more power.  Next to each contestant’s name are five checkboxes: No Money (i.e. $0), $500, $1000, $1500 and $2000.  Voters choose the award they think each contestant deserves.  The method for aggregating votes into awards will be discussed below, but first here are some general comments:

Interpolated Consensus should encourage voters to look at more of the media contestants.  Last year, many voters may have just thought about one or two media they liked, and voted for them.  Interpolated Consensus emphasizes that each voter will directly affect the amount of award (or non-award) for each contestant.  The ballots will say that non-votes are counted as votes for No Money, and the electronic ballot will have No Money preselected for all contestants.  Having intermediate amounts in the voting choices ($500, $1000, $1500) lets voters express degrees of approval instead of just yes or no.  We hope many voters will take advantage of this, encouraging media to broaden their appeal to as wide a range of students as possible.

Interpolated Consensus doesn’t need the VFM Committee to decide in advance how the award pool should be sliced up.  Voters will do that after seeing how the media contestants perform.  It may make sense to have fewer but larger prizes, or it may make sense to have more but smaller prizes.

After each voter has selected No Money or $500 or $1000 or $1500 or $2000 for each contestant, how should we aggregate the votes to determine the actual awards to give out?  There are many possible ways of doing this.  The VFM Committees this year and last year debated a wide range of methods and variations.

Averaging = Voucher Systems

One simple way is to average the voted amounts.  The $8000 budget could be matched by proportional scaling before or after averaging, or by programming the internet voting interface to require each voter to allocate exactly $8000.

Notice that averaging is equivalent to a voucher system that divides the prize pool by the number of voters.  For example, suppose there are 4000 voters.  Then each voter gets to allocate $8000/4000 = $2 worth of media vouchers.  You allocate your $2 among the media contestants as you see fit.  The $500 step size for voting amounts translates to a $500/4000 = 12.5 cent smallest unit for vouchers.  So you would get 16 vouchers worth 12.5 cents each, and distribute those among the contestants.

With vouchers, we would divide by the number of voters and then allocate, and add up each contestant’s allocations.  Or with averaging, we would add up voted amounts and then divide by the number of voters.  The math comes out the same either way.

The VFM Committees this year and last year unanimously rejected voucher or averaging systems, because that would create incentives for media to favour narrow interest groups.  Each voter could allocate vouchers to a contestant regardless of how many other voters support that contestant.  This would tend toward an equilibrium where each contestant preaches to its own choir.  For a full discussion of this issue, see section 2 of the paper “Voter-Funded Media” at www.votermedia.org/publications.  The $2000 (= 4 vouchers) limit on how much you can vote to each contestant alleviates the problem somewhat, but the pervasive narrow incentives would remain.

Consensus System: One-Media-Org Example

Instead, Interpolated Consensus (IC) is one of a class of aggregating methods that reward contestants appealing to a broader consensus of voters.  To see where the IC design came from, consider the following simplified voting situation:

Suppose we are voting on the funding for just one media organization, and we use a ballot where each voter types in any amount between $0 and $5000.  Suppose most people think a reasonable amount of funding for the organization is around $1000 to $2000, but the high limit of $5000 is used to let voters express a wide range of opinions.  The funding source is not an external sponsor, but comes from a large pool (over $100,000) of taxes or fees already paid by the voters, that is used for their general benefit.  So voters know they are trading off funding this media organization versus other spending that benefits them.

Now let’s compare two methods of aggregating votes: the mean (average) and the median (50th percentile or midpoint voted amount).  The mean encourages strategic voting, while the median does not.  To see this, suppose you think the ideal amount of funding is $1000 but most other voters would choose higher amounts, and you expect the aggregate result to be $1500.  Since you are one of many voters, your $1000 vote will have a small downward effect on the final result.  If the mean is used, you can magnify (triple) your effect by voting $0 instead of $1000.  If the median is used, voting $0 would have the same effect as voting $1000, so you might as well be honest and vote $1000 – after all, what if you’re wrong in your expectation of how others will vote?  Likewise, voters who want higher funding could increase their impact if the mean is used, by voting for the maximum possible amount.  Politics becomes a tug-of-war between two extremes.

The median can be seen as the amount of funding that would be chosen by majority vote.  Under reasonable assumptions about the regularity of voter preferences, the median would win a two-way vote against any other (higher or lower) proposed funding amount – see http://en.wikipedia.org/wiki/Median_voter_theory.

Consensus System: Multi-Media-Org Adaptation

So if we think the median is a good way to aggregate funding votes, how should we adapt this to our situation of allocating a fixed budget ($8000) among many media contestants?  If we try to use medians here, several problems arise.  How do we make the awards add up to equal the budget?  Even if every voter allocates a total of $8000 across the contestants, the medians might not sum to $8000.  We could rescale them proportionately…

But there’s a more serious problem: faced with a large number of contestants (13 last year), most voters are unlikely to read all the media.  So it may be hard for any of the contestants to get positive votes from at least half the voters.  The VFM Committee debated how to treat “non-votes” – cases where a student votes on some of the media but leaves some others blank.  Should we treat them as votes for No Money ($0), or omit them and look at percentages of actively cast votes?  Omitting them might tend to favour media that most voters are unaware of, but are zealously supported by a small group of friends.  So the Committee preferred to count non-votes as votes for No Money, since this encourages all media to reach a broad range of students.

All these problems can be solved by adjusting the award cutoff from the 50th percentile (median) to whatever percentile level allocates the exact total $8000 award pool.  This is what the Committee recommends.  Most likely the percentile will be higher than the 50th.  In my example spreadsheet, using the 74th percentile allocates exactly $8000, so each contestant gets the award where 74% voted for that amount or less, and 26% voted for that amount or more.  But whatever the percentile, media will be competing to appeal to as many voters as possible.  And like using the median, it minimizes the incentive and impact of strategic voting.

Multiple-Choice & Interpolation

Instead of asking voters to type in a dollar amount for each contestant, the Committee proposes a multiple-choice format, with options No Money, $500, $1000, $1500 and $2000.  This simplifies the decision process for busy voters.  It’s well known in marketing that giving consumers too many choices may cause them to throw up their hands and make no choice at all.  Students prefer multiple-choice exams, right?

However, this would make the awards jump by $500 increments, so that two contestants with quite different degrees of voter appeal could get the same award, while two others with similar appeal could get awards $500 apart.  Hence the Committee recommends the interpolation adjustment described in the detailed rules below and spreadsheet example.  This will allocate awards in $100 increments, even though students vote in $500 increments.

It is neither necessary nor desirable to require each voter to allocate a total of $8000.  This is because consensus aggregation works quite differently from taking averages.  With a consensus method, voting for more funds does not give your vote a greater impact.  It only matters whether your vote is above or below the consensus, not how far above or below.  Notice that in the very simple consensus method used last year (approval voting), a student who voted to approve all 13 contestants would have zero effect on the outcome.  Likewise with Interpolated Consensus, changing from voting No Money for all contestants to voting $2000 for all contestants would merely cause the consensus percentile P% to be lowered by one voter, with awards unchanged.

Detailed rules of Interpolated Consensus system:

(a) Interpolation –

(i) Each vote for $500 shall be interpreted as 1/5 of a vote for $300, plus 1/5 of a vote for $400, plus 1/5 of a vote for $500, plus 1/5 of a vote for $600, plus 1/5 of a vote for $700.

(ii) Each vote for $1000 shall be interpreted as 1/5 of a vote for $800, plus 1/5 of a vote for $900, plus 1/5 of a vote for $1000, plus 1/5 of a vote for $1100, plus 1/5 of a vote for $1200.

(iii) Each vote for $1500 shall be interpreted as 1/5 of a vote for $1300, plus 1/5 of a vote for $1400, plus 1/5 of a vote for $1500, plus 1/5 of a vote for $1600, plus 1/5 of a vote for $1700.

(iv) Each vote for $2000 shall be  interpreted as 1/5 of a vote for $1800, plus 1/5 of a vote for $1900, plus 1/5 of a vote for $2000, plus 1/5 of a vote for $2100, plus 1/5 of a vote for $2200.

(v) Each vote for No Money (including blank votes) shall be interpreted as one full vote for $0.

The term “vote” in the rest of this sequence of steps refers to these interpolated interpretations.

(b) [Comment: A percentile is the value of a variable below which a certain percent of observations fall.] After the votes are tallied, the Elections Committee shall, if possible, select a percentage, P, of votes such that the sum across contestants of the Pth percentile of each contestant’s award votes is $8000. Then each contestant shall be awarded the Pth percentile of its award votes.

(c) If there is no percentage P% as described in step (b) above, then:

(i) [Comment: This is the unlikely case where everyone votes low for all media.] Where the sum across contestants of the highest amount voted for that contestant is less than $8000, each contestant shall be awarded its highest voted amount. In this case the awards given out would total less than $8000.

(ii) [Comment: This is the unlikely case where everyone votes high for all media.] Where the sum across contestants of the lowest amount voted for that contestant is greater than $8000, each contestant shall be awarded its lowest voted amount scaled down by the proportion that makes the awards add up to $8000.

(iii) [Comment: This is the case of discontinuity.] If the total of awards as a function of consensus percentage P% described in paragraph (b) above jumps discontinuously from below $8000 to above $8000 when P is continuously raised, then the vector of awards given out shall be the weighted average of the award vector just before the jump and the award vector just after the jump, with weights chosen to make the awards add up to $8000.

(iv) If the voting results do not correspond to any of the above cases, the Elections Committee shall decide the award allocations in whatever way it deems appropriate.

See spreadsheet example of Interpolated Consensus award calculation.