The Marginal Vote Table

Votes in the Competition are tallied by the Condorcet Ranked-Pairs system. For each completed competition, you'll see a mysterious vote table.

Here's a quick explanation of that table. Say there are three games in the competition (with code names AAAA, BBBB, and CCCC). Four ballots are received.

(Each ballot lists the candidates, ranked from best to worst. Note that a ballot does not have to list every candidate.)

This produces a marginal vote table: 

VOTES   4
Margins
        AAAA  BBBB  CCCC
AAAA       X     1     2
BBBB      -1     X     1
CCCC      -2    -1     X

This simply means that, for example, two more people ranked AAAA over CCCC than vice versa. (Two people ranked AAAA over CCCC, and none lower.) The margin of BBBB over CCCC is one. (Two people ranked BBBB higher than CCCC, one the reverse; the margin is two minus one.) And the margin of AAAA over BBBB is also one (two minus one).

The lower half of the table is just the inverse of the upper half. (If the margin of AAAA over CCCC is 2, the margin of CCCC over AAAA is -2.) The diagonal entries are marked "X", because an entry can't be ranked higher or lower than itself.

Determining the Outcome

The Ranked-Pairs algorithm is simply to find the largest margin (AAAA over CCCC, in this case) and "lock it in". This does not mean that AAAA will win the competition, but it does mean that AAAA will beat CCCC in the final rankings. We then look at the second-largest margins, add those in, and so on.

In this example, there is a tie for the second-largest margin: AAAA beats BBBB by one, and BBBB beats CCCC by one. Happily, these are consistent with the previous determination, and with each other. We get a unique outcome: AAAA, BBBB, CCCC in that order.

If two vote margins contradict each other, we go with the larger margin, and ignore smaller margins that contradict it. If margins of equal size contradict, we have a tie at that point.

But Does That Really Work?

All of the contests so far (the first four, as I write this) have had ties at some level -- often at the top. A tie can indicate two games which have a margin of zero (equal number of votes for each beating the other). Or it can mean a preference loop, where equal numbers of players prefer AAAA over BBBB, BBBB over CCCC, and CCCC over AAAA.

These problems can occur in any voting system which involves more than two candidates. A purist would say that I should leave the ties as the official results -- there genuinely is no voter preferences between the choices.

I'm willing to post ties, but this system would lead to a few too many ties for my taste. So I apply two tie-breaker rules first. Among games which are Condorcet ties, I select the one which definitely beat the most other games. If there are still ties, I select the one which definitely lost to the fewest other games. If there are still ties, I post them as ties.

Last updated June 6, 2005.

Ice Game Design Competition