Full analysis of Kramnik's allegations of Hikaru Nakamura cheating with calculations

After I saw the last video where Kramnik asks Chess.com to investigate whether a streak of 55 consecutive wins is really that suspicious. Initially, I thought that it was, but I decided to take a critical look at the data and actually check numbers without making a conclusion first. It turns out, Hikaru's streak is indeed very probable. If you notice any mistakes or inconsistencies in my calculations, please let me know!

I summarized the key findings on Reddit, and here is the full post for reference:

https://kirillbobyrev.com/blog/analyzing-long-win-streaks/

Without going into any depth at all. A 300 FIDE rating gap leaves the stronger player with a 98.3% chance of not losing. It's easy to see long unbeaten streaks can emerge from that. eg. 98.3% ^ 55 = 38.9%.

That is not correct. 300 FIDE rating gap leaves the stronger player of .85 expected score (not losing) (source) without considering White's first move advantage.

88% ^ 55 = 0.01%

In classical, Hikaru (2788.3) would win 2488.3 player (again, on average, without White's first move advantage) with a probability of 62%.

62% ^ 55 = 10^(-10)%

And even over the course of 35,000 games, the probability of scoring 55 consecutive win streak (with a win rate of 62%) is 0.

White's first move advantage plays no role when the rating difference is several hundred.
You cannot calculate with an average rating of opponents, you have to take all individual ratings of the winning streak because the function win_probability (rating difference) is nonlinear.
Probability of streak of N = (probability of win against #1) * (probability of win against #2) * ... * (probability of win against #N)
You have to retrieve the ratings for that time control of all N opponents.
Classical rating is not Rapid Rating is not Blitz Rating is not Bullet Rating.

Right. This is exactly what I address in both my post and Reddit.

"Coin flips", i.e. constant probabilities is something I didn't do yet but will soon.

Excellent analysis, thank you!

I'm a researcher (cognitive scientist) interested in games, decision-making, risk-taking, and gambling. I've taught stats, data wrangling and visualization before, and "winning streaks" is an interesting and important topic, not only for chess but more generally.

Getting all the details correct (e.g. individual opponents' ratings are dynamic, classical ratings different from rapid and blitz and so on) is very difficult and cumbersome, but also besides the point in my view. What it boils down to is a misconception (cognitive bias) of how likely winning streaks actually are, on a general level, all else being equal. An "unusually long" winning streak often seems intuitively improbable. To show this is not the case we can resort back to simplified simulations of variability in "binary outcomes" (such as win / loss). I don't think we need to consider all the fine details of chess to drive the point home -- although what you've done is certainly excellent work in my view.

I ran a simple simulation for winning streak lengths when the outcome is either a “win” with 85% probability or a “loss” with 15% probability. I’m used a streak length of 44 as a “length of interest”. If 50 000 games are played given these probabilities, then the likelihood of not having a winning streak of at least 44 games at least once, is 0.6% (0.006). Over a period of 50 000 games given the above probabilities for wins and losses, one should on average expect to have 4.961 winning streaks (SD = 2.232143) that are at least 44 games long. I also visualized the distribution of the longest winning streaks for each 1000 simulated set of 50 thousand games. This showed that even winning streaks of 80+ games are certainly not unheard of, given these parameters, see https://herrasmies.github.io/winning_streak_simulation.html .

edit: I'm sure the above type simulations have already been done, in which case I'll just mention that I fully agree with everyone that long winning streaks are, in fact, unintuitively common with large enough data

edit2: This is a tangential point, but I should also add that even if we knew for a fact that a specific winning streak had a very small chance of occurring (say, 1%), we could not conclude that the complement (1-0.01 = 99%) was the probability of cheating. If the a priori probability of cheating is low, then we are faced with two improbable events, and they should be contrasted, else we will commit base rate fallacy.In other words, instead of saying: "This winning streak is improbable, therefore he's cheating.", we could just as well say: "Him being a cheater is so unlikely, therefore this winning streak has to be legit."

Hikaru, as a top 5 player in the entire world, should be expected to win a lot of games in a row.

Just at a lowest 5 player in the world should be expected to lose a lot of games in a row.

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something something, winning streak, something, he's good at it, some more things, he didn't cheat, something something

1.8% chance of a 300 point lower player claiming victory? that has not been remotely close to my experience