In the previous post I gave an example of students deciding who best guessed some lecturers’ ages.

I chose the numbers carefully so that under three reasonable methods of measuring:

Method  First place  Second place  Third place
Method a Adam Beth Charlie
Method b Beth Charlie Adam
Method c Charlie Adam Beth

This is actually almost identical to Condorcet’s voting paradox:

Voter  First preference  Second preference  Third preference
Voter 1  A  B  C
Voter 2  B  C  A
Voter 3  C  A  B

If three people in an election vote for candidates A, B, and C this way, then even using a method that takes account of all the preferences in one of the Condorcet voting system leads to a deadlock. The Condorcet methods are fair systems of voting, which generally have better properties than the Alternative Vote (AV) system, though is more complicated to count (but equally simple for the voters who just list their preferences in order).

In each case two out of three people each prefer A to B, B to C, and in turn C to A, which can seem a bit bizarre when you first come across it, hence why it is referred to as a paradox. You can find examples on Wikipedia based around this principle supporting AV (also known as IRV, Instant Runoff Voting) over the Condorcet methods, and visa-versa, where multiple voters are now in each category instead of a single one. Again, this goes to show why you should choose your evaluation method (here voting systems) from first principles rather than from the outcomes (the parties they will elect) or solely from considering rare cases where they throw up unexpected results.

Of course, this cyclical behaviour shouldn’t seem at all strange: most children have happily played Rock, Paper, Scissors. Rock beats Scissors which cuts Paper which envelops Rock.

People are instinctively happier when you can order things nicely like the natural numbers or the real numbers: if $a < b$ and $b < c$ then we must have $a < c$. This is a nice property known as transitivity, that the three examples above lack.

Intransitive dice image from Wikipedia. Numbers on the unseen opposite sides, unlike on normal dice, are the same.

Another nice example is the nontransitive dice, which are so designed such that: Die A will on average roll a higher score than Die B; which in turn usually beats Die C; which also wins with a greater than 50% chance against the original Die A. Apparently, Warren Buffet failed to trick a suspicious Bill Gates into gambling with them: Gates was to pick first.