A new measure of tournament toughness: Harmonic Field Strength

mikepage

mikepage

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How Strong Was the Field?
Harmonic Field Strength (HFS) as a New Way to Measure Tournament Toughness​



Pool pundits battle about whether the European Open or the World 10-Ball Championship had the stronger field—or which event truly ran deepest in 2025. Or which tournaments should be considered majors.

These debates almost always come down to some version of the same tradeoff:
star power vs. depth — comparing the top few entrants to the overall quality and density of the field.

Panozzo might argue the star-power side this time, pointing out that Filler, Gorst, SVB, and FSR were all in the field of some tournament. Or he might count how many of the World Nineball Tour top 10 were there. Or how many players were rated over 830.

The other Mike might counter that while Panozzo’s event had a few more marquee names, this other field, with more players traveling from Asia, ran deeper and included more of the top 100 or more players over 800 or 780.

Both sides have a point. It’s a good instinct to care about both star power and depth. How hard it is to win — or just to go deep — really does depend on both. It is more likely for a top contender to get snake-bit early when the field runs deep.

Until now, though, we haven’t had a unified way to balance that tension in a single number.

Introducing: Harmonic Field Strength (HFS)

You’ve probably come across some version of the 80/20 rule, also known as the Pareto Principle -- 20% of the people do 80% of the work. Or 20% of the seals do 80% of the barking.

That idea reflects a deeper pattern called a power law, where impact isn’t spread evenly but instead tapers off from the biggest contributors to the many smaller ones.

The same pattern appears in tournament field strength.

  • We care a lot about the presence or absence of the top five or ten players.
  • We care somewhat about the next 30 or 40.
  • And we care very little about whether the bottom half of a 256-player field is rated 680 or 610 — or even whether they’re there at all.
To account for that uneven importance we borrow a concept from economics and physics: A harmonic-style average that naturally emphasizes the top-ranked players while still incorporating the strength of the rest in a balanced way.

What is HFS?

Harmonic Field Strength (HFS) is a single-number summary, a rating-style average that weights each player’s contribution by the reciprocal of their rank, where rank here comes from a top-to-bottom ordering of the players in the field by Fargo Rating.

  • The 2nd-highest rating counts half as much as the top rating.
  • The 3rd counts one-third
  • The tenth counts one-tenth.
  • And so on — through the top 64 players.
This produces a number that:

  • ✔️ Looks like a player rating
  • ⚖️ Reflects both star power and depth
  • 📊 Allows clean comparison between tournaments
📊 An Illustration: Bali vs Hanoi

Two overlapping events happened in 2025:

  • 🌴 World 8-Ball Championship in Bali (Filler, SVB, …)
  • 🏙️ Hanoi Open (Gorst, FSR, …)
Which field was tougher?

By average rating of the top 16, Bali is a smidge ahead, by less than a point. But when we reach further to consider the top 32 or top 64, the tide changes and Hanoi looks stronger. The average rating of the top 64 players is 801.3 for Hanoi and 792.2 for Bali. HFS conbines the influence of the top 64 players and reveals Hanoi as the stronger field overall.

Event HFS Score
Hanoi Open 825.2
World 8-Ball 824.6

What About Other Events?

Here’s a taste of what HFS reveals:

  • Eurotour events, Derby City 9-Ball and Super Billiards Expo come out comparably tough.
  • The Battle of the Bull (Roanoke VA) ranks right alongside Bali and Hanoi, surprising perhaps, until you note its proximity to the US Open
1760539282307.png


🏆 Which tournament had the toughest field in 2025?

What about these big ones?

  • China Open
  • Las Vegas Open
  • US Open 9-Ball
  • World 10-Ball
  • UK Open
  • Florida Open
  • World Pool Championship
  • Peri Open
  • International Open
Which one do you think had the strongest field?
What are the top three picks?

We’ll compile the full numbers and share the results soon.
 
I'm a big fan of the prospect of ranking tournament toughness. I'm a bit skeptical of the presented method. The value of top players seems to drop off rather sharply.

More importantly, completely throwing away the bottom players, 65+, seems rather extreme. Just for example, in the Hanoi there were a number of upsets of tip-top players: Gorst, Raga, Roda, and FSR. The more rounds there are, the more players like Antonakis, Kiet, and Sevastyanov the top 64 have to fight through. So more players--and therefore more rounds--makes the tournament tougher, even before considering that more rounds just makes the marathon that much longer. Tyler Styer lost round one to an unknown in Hanoi--how much does having 256 entrants vs 96 matter when you have to fight more rounds to get to the money?

Again, I think it's a very interesting idea, but the method presented sounds predicated on faulty/poor assumptions.
 
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I like the idea here, Mike, but what matters certainly varies from fan to fan. You can put me with the crowd that cares much more about star power than depth. Field depth matters little to me, as I rarely watch stage one at the WNT majors in the first place, in part because of the lower quality of play, but also because there is no shot clock.

That is why I'm so upset that the bickering between WPA/Predator and Matchroom has made it very rare to have all, or nearly all, the superstars in a single field.

Right now, our game, arguably, features three of the ten greatest 9ball players in history in Filler, Gorst and SVB. It used to be quite common that we saw all three in the same event, but pool politics have made it rare, and the fan is the loser.

Depth of field is nice, but like most other sports fans, pool fans want to see the superstars more than anything and field depth can never be enough to overcome their absence.

PS My guess is that the World 9ball will come out on top in your study.
 
Rocket354 said:
matter when you have to fight more rounds to get to the money?

Again, I think it's a very interesting idea, but the method presented sounds predicated on faulty/poor assumptions.
Click to expand...
Given the battle of the bull has highest HFS, I think it isn't a particularly useful measurement point.
 
Nice idea. IMO this seems a little aggressive in weighting the top players and is basically a measurement of the top few players with some stray extras thrown in. Players 24 through 64 combined count the same as Player 1, which seems more like "barely caring" than "caring somewhat".

How about a harmonic exponent? 1 / i^p. p = 0.7-0.8 could be a good middle ground.
 
Good idea. Spectators are more interested in watching matches streamed on TV/YouTube that have 2 strong players that’s why there is much more interest at the single elimination stage. I could not watch any match for the Hanoi Open streamed on YouTube, albeit I was speeding through looking for matches so I could have missed a good one.

I think a HFS based on 32 might be a better indicator, but you can also give the results for 16 and 64.

Just to understand the math, if you have 2 events with 2 players each where event 1, the players Fargos are 850 and 780 and event 2 with the Fargos being 830 and 820, what will be HFS for these events?
 
sjm said:
I like the idea here, Mike, but what matters certainly varies from fan to fan. You can put me with the crowd that cares much more about star power than depth. Field depth matters little to me, as I rarely watch stage one at the WNT majors in the first place, in part because of the lower quality of play, but also because there is no shot clock.

That is why I'm so upset that the bickering between WPA/Predator and Matchroom has made it very rare to have all, or nearly all, the superstars in a single field.

Right now, our game, arguably, features three of the ten greatest 9ball players in history in Filler, Gorst and SVB. It used to be quite common that we saw all three in the same event, but pool politics have made it rare, and the fan is the loser.

Depth of field is nice, but like most other sports fans, pool fans want to see the superstars more than anything and field depth can never be enough to overcome their absence.

PS My guess is that the World 9ball will come out on top in your study.
Click to expand...
Yep. Agree, US Open has WAY too much dead-money watering down its 'HFS'. I wanna see good-to-great players at the end. Sure field depth means something but i just want to see the best at the sharp end of the draw.
 
Black-Balled said:
Given the battle of the bull has highest HFS, I think it isn't a particularly useful measurement point.
Click to expand...
I think it's pretty interesting that a "minor" event could have such a strong field. I think the schedule was a big factor in getting the top players there.

A few details that aren't obvious:

The top player contributes about 20% to the final FR average.

The top 6 together contribute about 50%,

The top 20 of 64 contribute 75%, so #21 through #64 contribute 25% to the average.

Note that if the top player withdraws, it reduces the average only by roughly the difference between him and the #2 player and usually less. The top 64 FR players have a HFS of 838.1. If Filler is replaced by #65, the HFS drops to 835.2.

I like the HFS number, but I think there are more intuitive ways to express it. One way would be to state it as a percentage of the maximum possible HFS, which is 838.1 right now. The 740 event listed above ought to be something like 50%, which would be the games won ratio for those two ratings.
 
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What if we just did the HFS only at 64 or 32?

It's a little goofy, though, TBH... Guess we need something new to discuss on occasion.
 
VVP said:
... Just to understand the math, if you have 2 events with 2 players each where event 1, the players Fargos are 850 and 780 and event 2 with the Fargos being 830 and 820, what will be HFS for these events?
Click to expand...
The first is (850/1+780/2)/(1/1+1/2) = 826.66

The second is (830/1+820/2)/(1/1+1/2) = 826.66
 
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skip100 said:
Nice idea. IMO this seems a little aggressive in weighting the top players and is basically a measurement of the top few players with some stray extras thrown in. Players 24 through 64 combined count the same as Player 1, which seems more like "barely caring" than "caring somewhat".

How about a harmonic exponent? 1 / i^p. p = 0.7-0.8 could be a good middle ground.
Click to expand...
We are working on Monte Carlo simulations to find an optimum exponent. I have a sneaking suspicion it may actually need to go in the other direction! We’ll see.
 
Bob Jewett said:
The first is (850/1+780/2)/(1/1+1/2) = 826.66

The second is (830/1+820/2)/(1/1+1/2) = 826.66
Click to expand...
I would be definitely more interested in the second event (830 vs 820) over a 850 vs 780 although the HFS is the same for both events. Maybe going to 16 or 32 will present a truer picture of the strengths of the events.
Funny that the numbers came out the same in my example though 😆
 
sjm said:
I like the idea here, Mike, but what matters certainly varies from fan to fan. You can put me with the crowd that cares much more about star power than depth. Field depth matters little to me, as I rarely watch stage one at the WNT majors in the first place, in part because of the lower quality of play, but also because there is no shot clock.

That is why I'm so upset that the bickering between WPA/Predator and Matchroom has made it very rare to have all, or nearly all, the superstars in a single field.

Right now, our game, arguably, features three of the ten greatest 9ball players in history in Filler, Gorst and SVB. It used to be quite common that we saw all three in the same event, but pool politics have made it rare, and the fan is the loser.

Depth of field is nice, but like most other sports fans, pool fans want to see the superstars more than anything and field depth can never be enough to overcome their absence.

PS My guess is that the World 9ball will come out on top in your study.
Click to expand...
Depth here kinda means 800s, not the qualifier crowd.
 
VVP said:
I would be definitely more interested in the second event (830 vs 820) over a 850 vs 780 although the HFS is the same for both events. Maybe going to 16 or 32 will present a truer picture of the strengths of the events.
Funny that the numbers came out the same in my example though 😆
Click to expand...
Bob had a typo. The one you prefer is 840 and the other is 826.
 
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