Preference Profiles

The preference profile controls do the following:

Motions

Set the number of motions (alternatives, candidates, choices, etc.) for the next profile.

Original cycle

Reset to the Original cycle; a preference profile in which no matter what choice is picked, \(60%\) of the electorate will agree a different choice would be better.

(no term)

New profile ;; Randomly generate a New profile, showing all possible orderings and how many voters share any given ordering of the choices.

Preference order lists

The count of voters for each specific order can be changed at any time. Any changes are immediately reflected in the pairwise results, Copeland scores, and the tournament digraph.

This electorate model is based on the assumption that every voter can produce a list of all the motions in a strictly linear and fully transitive preference order. A collection of these voter orderings and their multiplicities is called a preference profile. Ties within a preference orders are sometimes allowed, but it seems more usual to avoid them. Things to notice about preference profiles:

1. Every preference profile can be used to generate a tournament by going through the preference orders and adding their multiplicities to the vote count of the more preferred motion in each pair.

2. Every tournament can be generated by many different preference profiles.

   You can play a little game by randomly generating a tournament over \(3\) motions in the tournaments explorable, and trying to reproduce that tournament in Figure 1 by changing the preference profile. As there are \(3\) possible states for each pairing, and \(3\) possible pairings, there are \(3^{3}\) or \(27\) possible combinations for this very small tournament. (If you need help or want a general solution, you can see McGarvey, David C. (1953) for the full procedure, it's only three pages including introduction, theorem, proof by construction and two illustrated examples.)

3. Preference profiles record only the relative preference order of the motions listed. Thus each voter v's actual preference for any motion x is represented only by its ordinal position in v's preference order, such as first choice, second choice, last choice, etc. A preference profile does not provide any information about whether any of the voters actually like or dislike the motions.

   A voter's first choice could be only the least evil of a bad set of possibilities, and a voter's last choice could be perfectly acceptable because the motions all seem pretty good. This means they are useful for studying ordinal (rank-based) voting systems but are inadequate for studying cardinal (score-based) voting system. (Briefly, rank-based voting systems have ballots requesting a full or partial ordering of the motion, while score-based voting systems have ballots that grade or score the motions.)

4. While a simulated tournament with \(N\) motions has to generate a result for each pair of motions, a simulated preference profile needs to generate a vote count for every ordering of the motions. This mean while a tournament over \(N\) motions has \(N(N-1)/2\) control buttons in the first simulation, a preference profile over \(N\) motions has \(N!\) numeric inputs.¹ Put another way, the biggest tournament in the first simulation has \(66\) buttons for \(12\) motions while this preference profile simulation has \(120\) numeric inputs for \(5\) motions. (I felt \(720\) controls were ridiculous for a browser simulation so do not allow \(6\) motions here. If I allowed \(12\) motions so the preference profiles and tournaments covered the same range of motion, the display of \(479,001,600\) numeric inputs would probably break something.)

I suggest playing with this simulation long enough to get a feel for what is possible with preference profiles, and how likely it is for there to be a clear winner when more than two motions are in the mix.

A preference profile is a collection of preference orders of the proposed motions along with a count of the number of voters in the electorate holding those preference. This representation is fairly easy to work with manually, particularly with only a few motions, and so is often used to present an electorate's preferences for examples of voting systems and various impossibility results.

This is perhaps the most common representation of the electorate, at least in academic journals. It was probably used by Condorcet and Borda when they presented their methods in the 1700s, and was used by Dodgson, C. L. (1873), Black, Duncan (1948), Arrow, Kenneth Joseph (2012), Gibbard, Allan (1973), and Satterthwaite, Mark Allen (1975) (among many others), and is often used on websites and discussion groups to this day.

But there are some problems with computer simulations of preference profiles.

While preference profiles are vastly more informative about voters than tournaments are, they are still not very informative. All we know about the voters are their individual preference orders over a fixed set of motions.

As with tournaments, even though any randomly generated preference profile is possible, it is not at all clear whether the distribution of any given method of generating random preference profiles is at all similar to real electoral preferences.

The orders, as mentioned earlier, carry no information about whether any of the motions are subjectively good or bad. The order only tells us better or worse.

This last point is quite significant to me because I'm also interested in cardinal (score-based) voting systems.