These are some problems I'd like to posit to the sports analytics community, as food for thought and future research starters.
The scheduling of 31 teams (NHL Ice Hockey)
What scheduling problems, will arise in the national hockey league by having a prime number of teams? It must restrict their options for the number of games in a season, at the very least.
Are there even any other major leagues with odd numbers of teams? Leagues without a divisional structure such as Barclay's Premier League in soccer could get away with it, but they have the complications of Champion's League promotion and of relegation to deal with.
It's not just about creating a set of pairings so that every team has the same number of home and away games as well as games against same-division and same-conference opponents. There is also scheduling to consider, in that the limited number of weekend nights are preferable for business purposes, and that teams must be able to physically be in the same location at the same time, presumably with a rest buffer. There were already issues with the 30-team, 82-schedule because of injury risk (and greater injury consequence in terms of games missed). Adding a team will compound this, and the forced asymmetry of a prime number of teams reduces the flexibility of the schedule to account for weekends and rest.
As an aside, look to Balanced Incomplete Block Designs as a basis for designing team schedules.
Ratings for Sports Officials (All Sports)
In baseball, the location of the pitch as it crosses home plate is recorded for every throw, so people can tell where each umpire considers the strike zone to be, and the level of consistency of that zone.
What ways are there to evaluate the accuracy and consistency of officials in other sports? Tracking live decisions vs after the fact decisions from recordings? Could you do it as a 'deviation from superhuman AI' metric like how professional chess players are sometimes rated? For example, we could probably existing trackers (which are 2D for players and 3D for the ball) in NBA basketball to check the level of enforcement of traveling.
What are the ethical and behaviour implications of tracking and reporting officials' performance statistics like we do with players? Will officials lose focus on the game if they are also concerned with their stats? Would there be sufficient value to sports to do this?
Playoff Match Draft (All Sports, NHL Hockey example)
Instead of having first-round playoff match-ups determined solely by seed, matches should be determined by draft. Take the national hockey league (please). The NHL is split into two conferences of 15 and 16 teams respectively. In each conference, the 1st seed team plays the 8th seed team (Ignoring wild card complications), the 2nd seed team plays the 7th, 3rd plays 6th, and 4th plays 5th. If a team's skill were one-dimensional, this setup makes sense and the team that does best in the regular season is rewarded with the best chance to advance by playing against the weakest qualifying team.
Reality is messier. This setup occasionally leads to teams being punished for playing well in the regular season by being played against a team that they do particular poor against in the first round the playoffs.
Imagine a playoff draft instead. The eight teams in each conference qualify for the playoffs as before, but instead the top seeded team CHOOSES their opponent from the other seven qualifying teams. Then, the top seeded team among the remaining 6 chooses their opponent from the remaining 5 and so on. By default, teams could always select the lowest seeded available opponent, which leads to the same 1-8, 2-7, 3-6, and 4-5 pairings as the current setup. However, doing very well (3rd or better) in the regular season earns you some discretion if there's an opponent that counters you that you would like to avoid.
There would be absolutely no reason for a team to covet a lower position over a higher one. There would be no implication of strategically losing games at the end of the season, because there would be no conceivable reward in it.
A playoff draft also adds the potential for dramatic scenarios. If a team choose anyone other than the default lowest seed opponent, that implies a lot of confidence in being able to beat that opponent specifically. A lot of bragging rights come from a draftee beating their drafter. Would a team draft a stronger but less physical team to reduce their injury risk for the 2nd round, if it happens.
Would a team deliberately develop a reputation for being physical in the hopes of being drafted later against a weaker team?
Home Team Advantage (MLB Baseball)
Is batting last in baseball really an advantage? Sure, the team who bats last has more information when they do bat, which can provide a strategic advantage. However, that information advantage is nothing like it is T20 or One-Day cricket, in which each team only bats once, and teams still opt to bat first sometimes. Also, in baseball, batting last means always pitching for 9 innings in non-tied games; that's on average 6% more pitching than the other team needs to do. Is the information in the current game worth the extra pitching fatigue from the next game?
In Major League Baseball, games are usually played in 3 or 4 game series between opponents. The home team is given the supposed advantage of batting last for all of those games. Would it be a greater advantage to bat first for the first 1 or 2 games? Even if it did, would it be 'better' to increase the home team advantage?
Designated Hitters (MLB Baseball)
What would happen if the catcher didn't have to bat, and was also replaced with a designated hitter like the American league pitcher? Would that speed up the game, or is the time to change gear minimal? Would it lead to more hits by pitch because of the reduced opportunity for retaliation?
Does there need to be 9 players in a batting lineup, or can the designated hitter simply be removed in favour of an 8-player lineup? What second-order effects to the typical roster would there be by eliminating the designated hitter?