Where Swimming & Strategy Meet: An Analysis of the ISL’s Scoring System

Courtesy: Steve Gambino

The sport of swimming is taking a big step in a new direction this fall with the inaugural season of the International Swim League (ISL).  While swimming professionally seems to have been on a slow and steady rise, particularly since the Phelps era, the introduction of the ISL aims to truly expand the sport into a profession.  This is a big change for swimming in its own right, but furthermore, until this point, professional swimming has existed almost entirely as an individual sport; the ISL format switches this to a team focus.

While you could argue there certainly appears (at least from an outsider’s perspective) to be an element of a team dynamic amongst the members of the USA National Team (and perhaps other countries – I’m admittedly not as familiar) when it comes to actual competitions like World Champs and the Olympic Games, this isn’t reflected in the inherent structure; rather, the only real team-oriented elements are that of national pride and moral support of teammates.  To be clear, I don’t mean to be dismissive of these.  I’m not saying they aren’t important or valuable, they’re just not built into the events’ structure.  This changes with the ISL.

In the ISL, swimmers compete to score points for their team.  Here, an individual’s victory or defeat is only really relevant in the context of how it contributes to or detracts from the larger whole.  As you may expect, this completely changes the dynamic, as the goal has transitioned from having a best possible single (or collection of) swim(s) to contributing the most possible points for the team.  This adds an element of strategy, and dare I say, mathematics and statistics, that hadn’t previously existed in the sport, while also begging a host of new questions.

For example, do you, as a manager, put your best swimmers in more relays or more individual races?  Should you prioritize paying distance swimmers or sprinters?  Are freestylers more valuable than IMers?  How might we rearrange our meet lineup to best stack up against a specific opponent?  Does this vary significantly depending on the specific strengths of the opponent?

These are all examples of questions I’d like to explore in theory here, but regardless of the theory, it will be truly exciting to watch how their answers unfold in practice throughout the season.  That being said, I must admit, part of this makes me a little uneasy.

Swim Meets vs. Swim…Games?

Perhaps I’m a bit of a purest, but part of me loves the simplicity and elegance of 8 finalists lining up to compete for a given event with the first person to touch being crowned the victor, period.

Sports like swimming, track and field, rowing, etc., exist in their own class because of their uniquely objective determination of outcome.  The athlete that completes 100m in the least amount of time wins.  The athlete who can throw the javelin the furthest distance wins.  The athlete who can jump the highest wins.  These sports are all based around fairly clear-cut metrics (i.e., time, distance, etc.), which can then be compared and ranked easily amongst competitors.  This differs from sports like gymnastics or diving, which feature some level of subjectivity via judging, or more relevantly, football, where for example, Team A could dominate Team B, Team B could dominate Team C, and Team C could dominate Team A in a “rock-paper-scissors” like scenario.  In other words, you can directly compare the results of a 100m time of Sarah Sjostrom swum in Europe to a time by Simone Manuel swum in the U.S, but you can’t directly compare an Energy Standard victory to an LA Current victory unless they compete directly.  The ISL straddles the lines that previously clearly separated the class containing sports like swimming and the class containing sports like football.  This isn’t necessarily a bad thing, especially since these “game” sports tend to be the most commercially popular, but I do wonder if it yields a negative effect on athletes who achieve a level of dominance within a specific context.

For example, Katie Ledecky, because of her complete dominance in distance events, often with margins of victory measured by body lengths, is one of the greatest swimmers of all time.  Will that translate to her being a dominate ISL competitor?  Adam Peaty has similarly become one of the best swimmers in history via his dominance in the 100m Breaststroke, but in the ISL there may be more to the story; because again, an individual victory is now only relevant in context of its contribution toward the team.  This is where the structure of the rules and scoring system will now play a big part in who succeeds and who fails.  Does the ISL structure value the dominance of athletes like Ledecky or Peaty?

I guess my concern is that success will become too much about “gaming the system,” rather than the actual swims itself.  Now, am I overthinking this, being too conservative, or overly cautious?  Admittedly, probably (that is half the fun though, right!?).  But regardless, I am curious to see how the season unfolds and equally curious to see if, from the rules / scoring system the ISL has constructed, we can derive any conclusions in support of or contrary to this idea; and perhaps even discuss some optimal strategies teams might employ.

Evaluating Scoring Systems:

Now, I’ve written a bit about scoring systems before, so some context as to where I’m coming from may be useful here:  About 2 years ago I wrote “A Rio Retrospective: Swim Stories told by Statistics,” where I discussed a number of ways we might hypothetically “score” the Olympic games as a competition amongst countries and the consequences of each scoring system in terms of a country’s ranking.

To summarize that piece, the distribution of points across a heat does have an effect on the outcome of the countries’ rankings.  Sometimes this is more substantial than others, but for example, a system where we highly value gold medals like (8-4-2-0-0-0-0-0) helped improve the ranks of a country like Hungary because of Katinka Hoszzu’s impressive performances, whereas a system that scores the top 8 as (9-7-6-5-4-3-2-1) helped boost Australia’s rank, as they performed better through depth.  The scoring structure itself tells a story and choosing a certain scoring system is very much a matter of deciding what to value.

Further, since the ISL is not employing a dual meet structure, rather, opting for a quad meet setup, it is mathematically impossible to create a “perfect” scoring system that accounts for some basic discrepancies (this is an actual proven mathematical theorem known as Arrow’s Impossibility Theorem, which I described in a bit more detail in the previous article).

Suffice to say, when the ISL decides to score individual events (9-7-6-5-4-3-2-1), relays (18-14-12-10-8-6-4-2) and shootouts (27-21-12-10-4-3-2-1), it is both a reflection of their values and relevant to the outcome; and thus, worth breaking down and investigating further.

A Breakdown: What is worth what?

First and foremost, the women and men both earn the same number of points.  So, if optimizing for a win is the highest goal of a team manager, there is no difference in value between genders.  I believe this is appropriate.

Secondly, with the case of a tie being an exception, there are 1,640 points available in a given meet, 1,110 for individual events, 370 for relays, and 160 for “skins” races, a.k.a. shoot-outs.   As percentages, that breaks down as follows:

Note: In case you are looking closely here, yes, I rounded to the nearest tenth, so yes, I realize it doesn’t add perfectly due to the rounding!  

I don’t think this really tells us all that much.  While it is true that individual events in total will make up the largest percentage of the overall score, there are also more individual events (15 in total) than relays (3 in total, excluding the tiebreaker) or skins (1 in total), so it seems natural that this would be the case.  Instead, breaking it down by a single event gives us as follows:

Analyzing Table 2 shows that relays are worth twice an individual event, and a skins race is worth 2.16 times an individual event (in total) and 1.08 times a relay.  This gives us ever-so-slightly more of an insight into what the ISL values.  Skins are obviously a really exciting event, so they’ve set those events to have the greatest value on a single basis.  However, there is even more to this story.  Understanding the breakdown by event type gives us a tiny bit of info, but breaking it down by distance might be even more useful.

Table 3 clearly shows that the ISL values sprinting.  While I expected this to be true, it makes a compelling statistical case that sprints are a clear priority over distance swimming for the ISL.  This is further evident when you consider that a pure distance free specialist can essentially max out at 9 points, or about 0.5%.  Even if that swimmer has a fantastic 400 IM too, it is still only about 1% of the total value.  Conversely, Sarah Sjostrom, a talented sprinter who earned 63 points in this season’s first meet in Indianapolis, accounted for 3.8% of the meet’s available points (without even considering her relay contributions).  That makes her about 7 times more valuable than a 400-freestyle specialist.

So, sprinting has more value than distance swimming overall.  Given the nature of a competitive swim league in contrast to events like the Olympics or World Champs, this is arguably appropriate as well – it certainly has made for exciting meets so far and overall, I’m happy with the ISL’s decision to value these races.

We can also further break this down to understand how different stroke specialists can add value, though we have to be careful with how we count this.  Specifically, I’ll count a medley relay’s worth as ¼ toward each stroke, but separate individual medleys as its own category, since a true flyer, for example, wouldn’t add a similar ¼ value to an I.M.

It’s tough to learn a lot from this without a deeper analysis of event overlap (Swim Swam has done a similar analysis for the NCAA), but I guess we still can generally conclude, that being a great freestyle sprinter is the most valuable role to an ISL team.  Again, not particularly surprising here and also arguably appropriate.

All of these ideas give us a vague breakdown of what the ISL values, and thus, some general strategies for which teams can employ in order to be successful.  Pay your sprinters, pay your freestylers, train to crush the skins races.

This has already paid off big time for Energy Standard, as coach James Gibson led his team to 14 individual (10 of which were races 100m or less), 4 relay, and 2 skins victories as their strategy rocketed them to a big win over the field at the Indy meet.  The Condors, Trident, and Centurions respectively had ten, four, and two individual wins and one, zero, and zero relay wins.  So, this strategic approach does seem to be working in practice so far.

Relays vs. Individuals: Are relays opportunity-cost efficient?

One thing that is noticeably absent from the ISL rules is a limit on the number of events an athlete can swim.  For example, in an NCAA championship meet, a swimmer can compete in a max number of 4 events, where up to 3 may be individual.  This is relevant because it forces a manager to determine where a given swimmer can add the most value; in other words: to strategize.

Now, this doesn’t mean this type of strategy is completely non-existent in the ISL.  Though there could be cases in which a particularly resilient swimmer is able to squeeze out a number of points from a number of different events, I suspect this element of strategy will still exist to at least some degree since swimmers will naturally need to limit themselves due to fatigue, especially with the fast pacing of these meets.  To say how much exactly, we’ll need to wait and see how things evolve over time, but for purposes of this discussion, let’s assume that the pacing will be reasonably limiting (perhaps, a swimmer can compete in a max of 3 events per day).  Now, do you prioritize placing your best swimmers in relays?

In the ISL, winning an individual event is worth 9 points.  Winning a relay event is worth 18.  This seems like more, but since 4 members make up a relay team, their theoretical contribution to the team would actually only be 4.5 points.  Does that mean then, that foregoing “stacking” your relays to spread out the individual point winners might be a useful strategy?

Let’s consider a hypothetical example.  Going into the second day of a meet, the Cali Condors might determine that Caleb Dressel has a very high chance of winning the 100 FR, the 50 FL, and the 50 FR shootout.  They might also determine that they have a good shot of winning with him in the 200 mixed medley relay.  It could potentially be a common situation that the coaches would have to choose.  So, what might those options look like?

1. He could swim 2 individuals and the skins. If he wins all 3, he earns 9 + 9 + 27 = 45 points.
2. He could swim one individual, one skins, and a relay, which would earn the team a total of 18 + 9 + 27 = 56 points.
3. He could skip the shootout and his team would earn a total of 9+9+18 = 36 points.

Both of the first two options are clearly better than skipping out of the shootout (that is where ISL places the most value after all), but picking between option one and two isn’t as clear.  It really all comes down to how well he would do relative to his replacement.

Looking through the Condors Roster, there doesn’t appear to be another athlete who could rival Dressel in any of those 3 individual events.  (I’ve only skimmed through their roster and don’t have every athletes’ performance metrics memorized, so it is possible that there is some wiggle room here, but let’s assume not, for purposes of this hypothetical).  So, choosing for him to opt out of an individual race for someone else, could cost them up to 8 points.  However, they might have one or two swimmers who could throw down a decent enough 50 in a relay to help them still get 2nd or even 3rd.  Let’s revisit our table and two options keeping in mind the points of his replacement.

In this case, entering Dressel into 2 individual events and the skins could actually prove more valuable than entering him in 1 individual, the skins and a relay.

Of course, in order to get to this conclusion, I made a lot of hypothetical assumptions, and even then, the difference is still pretty small, so I wouldn’t expect this to be very common.  However, Energy Standard did elect to sit Sjorstrom and Heemskerk out from relays in order to be as fresh as possible for the skins event at the meet in Naples this past weekend.  Their strategy seemed to have paid off as the two women were able to pull off a one-two finish in the skins race with Heemskerk just barely out-touching Olivia Smoliga.  This secured them a minimum of an 18-point swing toward Energy Standard.

Imagine, on the other hand, they had both been entered into the mixed medley relay a few events earlier.  In the best case, this would have improved their B relays’ finish from 5th place to 2nd place (they also had the winning relay here).  In total, this would have bumped them up by 6 points, and lowered the Cali Condors score by 2, yielding an 8-point swing.  However, this is not nearly as much as the 18-point difference that losing to Smoliga would have cost them if Heemskerk’s slight additional fatigue added even six hundredths of a second!  Considering they ultimately won the meet by only 3 points; this strategy clearly placed a crucial role in Energy Standard’s success.

Determining exactly how much this strategy may be relevant is tough to say, especially without having seen many meets play out, but it is evident that it will play a role.  Even if this only affects meet lineups in rare cases, I do think it detracts slightly from the overall excitement of the relay events if the best swimmers are strategically opting out of them on occasion.

Understanding the expected value of points earned for a given event lineup, along with measuring points relative to replacement could prove useful towards a winning strategy, as exemplified by Energy Standard thus far.  This means there does exist at least some element of “gaming the system.” Personally, I’m not big on this.  That being said, though it won’t be perfect, I think the ISL system is still set up reasonably well in general and this minor flaw (if you will) won’t detract from the overall spectacle substantially.

Examining Possible Outcomes

Disclaimer for those who are a bit (or a lot) math-phobic: We’ve now reached the point where the mathematics gets a little heavy and computational.  Proceed carefully.

With 8 athletes racing in a heat, there are 8 options for who touches first (we’ll exclude ties again for simplicity here).  Then, once the winner is determined, there are 7 remaining options for who earns 2nd, then 6 options for 3rd and so on, hence 8(7)(6)(5)(4)(3)(2)(1) = 8! = 40,320 permutations of a given heat.

That seems like a lot of possibilities, however, most of them are not relevant.  For instance, two members of the same team swapping positions doesn’t affect the outcome at all.  Rather than considering them as 8 unique athletes, we can better understand the possible outcomes for a given event by considering them as 4 teams, each earning two of eight positions in a heat’s ranking.  We can compute this by dividing out all of the repetitions for swaps from our original calculation.  In other words, since swapping two Cali Condors athletes makes for a repeated ranking, we’ll divide our total by 2 for them, as well as each of the other three teams.

So, each event has 2,520 unique outcomes?  Well, not exactly.  While there are 2,520 unique permutations of a heat’s rankings, this doesn’t necessarily mean there are 2,520 ways the ISL’s scoring system distributes the points.  For example, consider the following two permutations:

Notice that if we compare the total points for each team from each possibility, both cases yield the same outcome even though the rankings look very different.

So how can the points actually be distributed?  There are actually 1,448 distinct outcomes.  This is a pretty tricky thing to count, but with a little python code comparing all 2,520 permutations and checking for the ones that yield unique values, a computer gets the job done pretty nicely.

Of these 1,448 outcomes, each team can earn a minimum of 3 points (for a 7th / 8th finish) up to a maximum of 16 points (for a 1st / 2nd finish) and any whole number in between.  If ranks were determined randomly, scoring between 7 and 11 points would be most common, each with 3 possible ways to occur.  The following histogram represents the number of ways a team can earn each amount of points.

In particular, note that scores of 14, 15 and 16 points can only be earned in one way: 1st / 4th, 1st / 3rd, 1st/ 2nd respectively.  This allows us to make a reasonably obvious conclusion:  If your team wins every event and takes at least 4th place, you’re guaranteed a victory overall.  Again, perhaps this is obvious, but I think we can even take this one step further to gain some valuable insight.  Since a 1st / 5th or 1st / 6th finishes yield 12 and 13 points respectively, and each have only two possible associated outcomes, these performances nearly guarantee overall victory as well.  The only way this would not occur is if another team also placed 2nd / 4th (earning 12 points) or 2nd / 3rd (earning 13 points).  Considering how competitive a given heat is, I would suspect with reasonable likelihood that these particular rankings won’t occur all that often, and even if they do, you’ve at worst tied with another team.

This shows that winning an event goes a really long way within this scoring system, especially if you have a second swimmer that will be competitive with the rest of the field.  Even a 1st / 6th place finish still puts you at a pretty high likelihood (80% chance) of earning or tying for the most points in that event.  In fact, breaking down the theoretical probabilities of each outcome, we have:

Note: These are theoretical probabilities.  In other words, given placing for one team, we are computing the number of outcomes where another team will score more, and dividing that by the total number of ways the remaining teams can be placed.  This does not exactly represent the chance of your team winning, given its placing (other than the instances of 100% or 0%), because talent may be distributed amongst teams in such a way that some outcomes will be more or less likely than others.  In other words, it certainly may be the case that the experimental probability here differs noticeably from the theoretical probability. 

Now, this gives us some really useful information.  Earlier, I had questioned whether or not the athletes who are particularly dominant in their respective events would be as valued within a scoring system focused on team.  This may not be true in terms of their ability to post faster times relative to the rest of the field, but to secure points and wins, I think this evidence would suggest dominant athletes are quite valuable.  In other words, if I was a manager examining these probabilities, I would focus more on signing as many top stars to my team as possible (particularly top sprinters if we factor what we learned earlier).  For example, if I’m the London Roar manager, I’m essentially going into the 100m Breaststroke with a minimum of a 25% chance of winning or tying the event, even if my number two takes last.  If my number two can beat one or two guys, that percentage jumps up pretty high, pretty quickly.  That makes a dominant athlete like Adam Peaty pretty valuable to the team.

The importance of winning is amplified even further in the Skins races since reaching the second-round doubles the individual events’ score and reaching the third-round triples it.

As you can see from Table 11, winning a Skins race gives a 60% chance of winning the event regardless of the performance of your team’s second swimmer. This is huge in its own right without even considering how much of a point differential they will win by.

If you’ve read any of Malcolm Gladwell’s work, you may be familiar with his perspective of “strong-link” vs. “weak-link” sports. A strong-link sport is one where a single talented individual can make a dramatic difference in your team’s success. Consider basketball for example. If Michael Jordan joins any team, they could instantly go from worst in the league to a championship contender. A weak-link sport is the opposite; no single superstar can make a dramatic positive difference, but upgrading “weaker” players can, so valuing depth really matters to a team’s success. It appears that the ISL will lean towards the strong-link side. Considering the success of the NBA as a professional league, I think making the ISL star-focused in this way will help expand the sport’s and league’s popularity.

Now, on the other hand, you’ll similarly be guaranteed to earn the least amount of points if you finish 7^th / 8^th or 6^th / 8^th and guaranteed to lose or tie for last for finishes of 4^th / 8^th and 5^th / 8^th. For all of these options, finishing last gives a 0% chance of winning an event unless your team also takes first.

Thus, to summarize in terms of strategy, we might say: win every event that you can and recruit swimmers who have as near to guaranteed wins as possible.  Lose as few events as possible.  Then, secondarily, worry about the specifics of what happens in the middle.

Is the ISL’s system…good?

This is of course somewhat subjective, but I’ll try to be as clear and objective as possible in how I’m making this determination.  What makes a system good?  My personal philosophy dictates we should consider these axioms (in no particular order):

• If both genders are competing together, they should be valued equally.  The ISL does this.
• The system should value the most exciting / intense races the sport has to offer.  What makes an event exciting is something to be determined in its own right, but I think the ISL’s decision to place a high value on skins races and sprinters facilitates this.
• The system should promote entering top swimmers in relays as the optimal outcome.  The ISL probably does this most of the time, though there are small set of cases where teams may decide to be tricky.  In my opinion, this does detract from the event a bit.
• There needs to be enough variety in outcome that the results don’t become repetitive and predictable.  I suspect 1,448 outcomes across all the events will probably be enough to get the job done here, though we’ll need to see a bit more to know for sure.
• The system should value the purity of the swimming competition itself.  Achieving this to the fullest extent is impossible for the ISL, or any type of league for that matter.  The best we can hope for is that “gaming the system” is minimized and all of the managers become students of the system, educating themselves on what strategies can help them “game the system” most effective.  This would ensure every team has a similar advantage when recruiting, constructing rosters, and deciding how to pay athletes, as well as where to place them within a meet lineup.  This will make it as much about the actual swim competition, rather than the numbers game, as possible.

All in all, I think the ISL’s system is mostly good.  The system is broad enough that it shouldn’t be overbearing. For the most part, at least for now, the focus still feels to be very much on what happens between the lanes, rather than between the numbers.  Where the numbers do become relevant, they tend to highlight the most commercially valuable and exciting aspects of the sport, which is definitely a good thing.  And, most importantly, it’s been incredibly fun to watch the first two meets so far.  With the lights, music, and graphics, the swimmer walk-outs before each race, the intense matchups like Sjorstrom/Heemskerk vs. Smoliga and Manadou vs. Dressel, it is certainly a spectacle.  I’m excited to see what lies ahead and am glad the ISL is here to push our sport forward.

About Steve Gambino

Steve grew up swimming in Middletown, CT.  He has been coaching age group swimming for the past 7 years, first as an assistant with Charter Oak Aquatics in Hartford and currently in his fourth year with Crimson Aquatics – Rhode Island.  Steve has an M.S. in Mathematics from University of Rhode Island and currently works as an Assistant Professor of Mathematics at the Community College of Rhode Island.