SAVE Simulated Tournaments

There are a few electoral situations that show up nicely in tournaments and it is useful to know what they look like and what they are called.

1. Condorcet winner: named after a well-know French intellect from the 1700s is an alternative with only out-links (all arrows with an end at that alternative point away from it), A strong Condorcet winner has no ties and beats all other alternatives. A weak Condorcet winner is undefeated, but may be tied. Voting systems that select such an alternative when one exists are said to meet the Condorcet criteria.
2. Condorcet loser: the opposite of a Condorcet winner, this is an alternative with only in-links (all arrows with an end at this alternative are pointing at it). A voting system that does not select a Condorcet loser when one exists is said to meet the Condorcet loser criteria.
3. Majority cycles: are a set of alternatives in which one can start at an alternative and follow arrows along a path of domination that leads back to the starting alternative.
4. Fully transitive orders: If you set the Win (or Lose) probabilities to 100%, there will be no ties and in every pair the alphabetically earlier (later) alternative will win. If you look at such an order, you will see that for any x, y, and z, if x≻y and y≻z then x≻z. That is the definition of a fully transitive order.

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

This tournament explorable can show every possible tournament outcome for 2 to 12 alternatives. In real tournaments ties are unlikely and sometimes impossible. This model starts with the assumptions that the probability of a tie is 10%, and when there is no tie, the chances of winning or losing are equal. The Copeland score simply gives 1 point to the winning alternative in a pair, and 0.5 points to each pair when there is a tie.

The tournaments shown in the simulation are mathematical objects called graphs, specifically directed graphs or digraphs. The data generating the digraph is complete in the sense that there is a known relationship between every pair. An arrow points from a winner to a loser. A missing arrow indicates a tie. As long as you remember the arrows originate from a winning alternative to a losing alternative, it is fairly easy to see what is going on.

One good thing about tournaments is they are based on pairwise majority decisions. A majority decision between two alternatives is pretty much the only voting method and outcome on which everyone agrees. Since there are only two possible outcomes (or three if a tie is allowed), a voter's decision is simple: vote for their preferred outcome. Therefore, a tournament, in theory, can be assumed to reflect the actual pairwise preferences of the electorate.

The challenge of a tournament is to pick the best alternative overall given the true pairwise collective preferences. If you have looked at a bunch of these tournaments, you have probably come to realize this problem is not easy.

But that is only one of the problems with computer simulations of tournaments.

Since this section is about representing electorates in computer simulations, it is a fair question to ask, "where is the electorate in a tournament?" The answer is the electorate has been abstracted away. The idea is that the only completely valid information we can get from an electorate is the pairwise election results, so once we have those, the electorate is no longer necessary.

With the electorate reduced to only (randomly generated) majority decisions, there are no vote counts, and so no margins of victory, close races, or landslide victories. Therefore, none of those measures are available to help decide which alternative is better

It is worth remembering each and every one of the pair-wise outcomes represents a full election by majority decision between the specific pair of alternatives. In a real collective choice situation, determining the actual relationship between each pair comes at a cost, and there is an awful lot that has to happen before anything like an accurate tournament digraph can be drawn.

Even though every possible tournament can in fact be generated by the preferences of a group of voters, the probability of any given tournament occurring in a real life election is not known. When tournaments are randomly generated, we can learn a lot about randomly generated tournaments, but it is very unclear whether we are learning anything at all about elections.

This point was driven home by Maurer Maurer, Stephen B. (1980) when he showed (among other things) that random graphs tend to be very far from linear orders, while the pecking order of real chickens tends to be very close to linear. Which basically means studying random tournaments is unlikely to be productive in a quest to improve voting systems.

On the other hand, tournaments can still be useful as an output form if the underlying results are generated by some other method.

Maurer, Stephen B. (1980). The King Chicken Theorems.