**Range Voting with Small Electorates; comparison of several Tie-breaking Methods **

by Thomas C. Smith and Warren D. Smith

**WARNING:**
This entire page is almost entirely irrelevant egghead-ness, because in large elections
TIES WILL ALMOST NEVER HAPPEN!!!
However, if you are the sort of person who has to examine
every single issue, no matter how incredibly minor,
to try to wring the last drop of juice from the topic... read on.

Range voting is probably the voting method least likely to lead to a tie, especially if a large score-range is employed (for example, an 0-99 range is less likely to lead to a tie than an 0-9 range). But the chance of a tie increases as the size of the electorate decreases. In voting by small committees, panels, or juries, ties could have a non-negligible chance of occurring, even with range voting.

Range voting has major advantages over other methods of social choice. It is:

- independent of clones,

- independent
of irrelevant alternatives (deleting a candidate from all votes and the
election does not alter the relative standings of two non-deleted
candidates),

- monotonic (raising a candidate's score in your vote cannot decrease his winning chances),

- participatory (casting a honest vote cannot hurt you versus not voting at all),

- consistent (if a candidate wins in two districts, then he also wins in the combined region),

- allows the voter (without strategic sacrifice) to never betray her favorite candidate.

We shall discuss several possible ways to lessen the opportunity for ties and do tie-breaking in range (or other kinds of) voting.

**Setting a resolution for preference intensity**

Those administering an election could start with defined notions of support levels, such as: Unacceptable, indifference, mild support, moderate support, strong support. In its basic form, the ratings for these levels of support could be respectively, 0, 1, 2, 3, 4. One could proceed to an additional step of resolution between these defined steps to obtain a range of 0, 1, 2, 3, 4, 5, 6, 7, 8, where 0 still represents unacceptable and 8 still represents strong support. Where much greater resolution is desired, one could use a range of 0-100, where 0 is still unacceptable and 100 is strong support.

The greater the range resolution, the less chance a tie will occur in an election. That is one factor to consider regarding the acceptability of resolution settings.

Drawback of the **"flip a coin" tie-breaking method**

If N candidates are tied (most commonly, of course, it would be N=2) then one can flip a coin (or if N>2 then flip an "N-sided coin") to choose the winner.

This method has the advantage of extreme simplicity. But a tiny disadvantage is its vulnerability to candidate cloning. That is, if A's party "clones" A, then if A and B (and A's clone) tie, then A and its clone have a 2/3 chance of winning instead of 1/2. I.e, cloning helped the A-team. That is a non-optimum result. It's better if cloning candidates neither help nor hurt the clone-team. However, we know that coin-flip tie-breaking cannot be too bad because range voting with coin-flip tie-breaking is equivalent to range voting with infinitesimally randomly-perturbed scores.

**An improvement with randomness: "Random ballot" tie-breaking method**

Choose a ballot at random, and use those ratings
to break the tie. (I.e. if the tied candidates are A and B, and
the randomly chosen ballot scores A higher than B, then A wins.) In the
unlikely event this ballot *still*
indicates that some or all of the tied candidates are tied, then one
chooses at random again, and continues until the number of tied
candidates is reduced to a unique winner.

This method is immune to cloning. It also encourages sincere
voting. If your vote is picked as the tie-breaker, then you risk
breaking the tie the wrong way if you rated the candidates on your
ballot in a dishonest order. With the random ballot method,
dishonesty is strategically self-defeating.

Another alternative: Selecting a median rating

This procedure is to inspect the ratings on every ballot for each
tied-winner, then find the median rating for each tied winner.
With an odd number of ballots, this will be the middle rated ballot(s)
for each tied winner, and for an even number of ballots, this will be
the average of the two middle rated ballots. In both cases, the median
will have the same number of ballots rated above them as below them.
The unique winner has the highest median score. We shall call
this method the Median Ballot Rating or MBR.

MBR disobeys the participation and consistency criterions.
With a small electorate, where tallying is done in one location only for
tie-breaking purposes, this disadvantage arguably is irrelevant.
MBR does not violate any other of the usual
voting method criteria that Range Voting complies with.

**A more sophisticated method: Tie-breaking utilizing the method itself**

We
advocate tie-breaking by use of the range voting method itself (actually its
more primitive form called approval voting), thereby retaining
compliance with these same voting method criteria. Here's one way to do
that:

Each tied candidate receives one point for each voter who rates him above her arithmetic mean score for all the tied candidates. Ratings at or below the arithmetic mean receive zero points. The candidate with the most points wins. We shall call this method Votes Exceeding Arithmetic Mean or VEAM.

VEAM
has the advantage over random ballot in that it will make more voters
happy. In fact, if a "happy" voter is one who feels that
candidate X was better than a random choice among the tied-candidates,
then this method to choose X maximizes the number of happy voters.

Conclusions

If we employ range voting with small electorates, we can design it to accurately express strength of support and lessen the chance of a tie. Even if a tie does occur, there are methods to resolve such a situation from cast ballots. Voting method criteria compliances need not be sacrificed in the case of ties, nor (usually) need ties be decided by chance.

3 March 2007