Quantifying Luck in Head-to-Head Fantasy Leagues

If you've played fantasy football you know that success is somewhat dependent on luck.

Teams face off in head-to-head matchups and the higher scoring team wins (i.e., a team wins if it happens to be matched up against a lower scoring team during that particular week). If you post the second-highest score in your league during a particular week, you still lose if you are matched up against the highest scoring team. That's bad luck. Meanwhile, a low-scoring owner can luck out by facing an even worse owner.

Sports league scheduling is a dissertation unto itself (and a multi-billion dollar business when it comes to scheduling games for professional and college leagues) but in its simplest form, a fantasy league's schedule is a round-robin tournament among the owners. (Fleaflicker's scheduling algorithm is a bit more sophisticated, balancing inter- and intra-divisional matchups according to user specifications.)

During any given season, the ordering of the matchups is random. The schedule introduces an element of luck to an owner's win-loss record.

There are several ways to quantify this luck.

The brute-force approach is to enumerate all conceivable schedules for an owner and tally the results of each to yield a probability distribution for wins. For example, in a 12-team league in which each owner faces every other owner exactly once, there are 11! possible schedules for each team (this isn't entirely accurate--there are actually far fewer ways to schedule the league correctly for all teams--but from a single owner's perspective there are 11! possible schedules).

A less computationally-intensive approach is to compare each matchup's outcome to the probability of beating any owner in the league. For example, an owner who posts the league's lowest score has a 100% chance of losing that week--no luck is involved. An owner who posts the 2nd-lowest score of the week has a 1/11 chance of winning and a 10/11 chance of losing. If the owner lost, the owner was the victim of a tiny bit of bad luck. But a victory means the owner was the beneficiary of a lot of good luck.

(We added a luck column to your team schedule page so you can see this effect on your fantasy teams. Here is mine for a particularly lucky season.)

Can scheduling luck affect a fantasy owner's fortunes?

Across all 2008 Fleaflicker leagues, teams were involved in 3 luck games on average (games involving 50%+ luck as defined above).

Call a team lucky if it finished at least 2 games over .500 in luck matchups and unlucky if it finished at least 2 games under .500 in luck matchups. 20% of all teams last season were lucky and 20% were unlucky (so about 5 teams in a 12-team league were affected by luck).

Call a team good if it finished in the top-third of the league and bad if it finished in the bottom-third. 22% of good teams were lucky and 16% were unlucky while 18% of bad teams were lucky and 21% were unlucky.

Taking it one step further, call a team very lucky if it finished at least 3 games over .500 in luck matchups and very unlucky if it finished at least 3 games under .500 in luck matchups. 8% of all teams last season were very lucky and 8% were very unlucky.

10% of good teams were very lucky and 5% were very unlucky; 7% of the bad teams were very lucky and 9% were very unlucky.