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> PS: I've never seen a player with a high bullet rating but a low blitz/rapid rating, so I'm not sure how to interpret a negative result either.

I don't have a "high rating" in bullet, but it's higher than my blitz/rapid ratings.

Why ? I'm impatient. When I wait for my opponent to play their move, I lose focus. When they take 30s to 1min to think about a move that seems obvious to me, it just makes me lose my train of thought.

Because of this, I play mostly only bullet games and puzzle rush. I'm at ~2400 in tactics on lichess, and ~1900 in bullet.

> Isn't this pretty much true by definition? Fast thinking is the only thing you can do in quick games, so it has to be more prominent.

I think you might be implicitly defining "quick games" as ones where you only have time to think fast, so it comes across as a tautology to you. Meanwhile, the author writes in terms of continuums quicker and slower. I think the emphasis is on time pressure crowding out slow stuff that might make a difference. I.e. if the slower stuff made a difference, people that beat you at quick games might not beat you at slow ones ceteris paribus.

> And now I'm just confused. How do we know that fast thinkers aren't using "slow thinking" or anything else when they play at slower time controls? Maybe good fast thinking is correlated with good slow thinking, but this doesn't mean it is the dominant factor!

I believe that the author is just restating the NH, which is the hypothesis that slowing down time controls will make the dominant player more dominant (or at least maintain the advantage). The alternate hypothesis is stated afterwards

> AH: NHC is false. On average, given some rating difference \(x\), if \(A\) beats \(B\) at the quick time control with a probability \(p\), then \(A\) beats \(B\) at the slower time control with probability \(p' < p\).

Makes sense, but I don't think this hypothesis is helpful.

Right at the beginning of the article the author says the main goal is to:

> investigate the extent to which fast thinking is the dominant factor affecting game outcomes at any time control.

To achieve this goal, I think we'd need an hypothesis that isolates the effects of fast thinking, and not one that could confound correlated factors with the effects of fast thinking itself.

By just comparing probabilities of winning at various time controls, we're evaluating players as a whole and not isolating any aspect of their thinking, so this hypothesis doesn't seem helpful.

Perhaps if the study looked at individual moves and how much time they took to make that could help?

It’s an observational study and the author volunteers a bunch of caveats at the end of the article including with regards to construct validity which I think is the crux of your challenge.

If you think of chess as two people with different capacity CPUs calculating against each other (many people do) then time should only ever make the outcome more certain. The author basically says that if that was true, people that beat you at fast chess should be even more likely to beat you slower chess, but that doesn’t seem to be the case.

Thank you for this reframing. The CPU speed analogy finally made me understand what the author meant with "fast" and "slow" thinking.