SAVE Simulated Preference Profiles

Preference profiles are based on the assumption that every voter can produce a list of the alternatives 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 alternative 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 alternatives 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 3³ 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, its only 3 pages including introduction, theorem, proof by construction and two illustrated examples.)

3. Preference profiles record only the relative preference order of the alternatives listed. Thus each voter v's actual preference for any alternative 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 alternatives.

   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 alternatives 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 alternative, while score-based voting systems have ballots that grade or score the alternatives.)

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

A preference profile is a collection of preference orders of the proposed alternatives 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 alternatives, 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 alternatives.

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 alternatives 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.