Kramnik's massive blunder

Kramnik's Unfounded Accusations Against Hikaru: A Case of Misconceptions and Lack of Evidence
The chess world has witnessed a recent surge in cheating allegations, with Grandmaster Vladimir Kramnik emerging as a vocal figure in the anti-cheating movement. However, his recent accusations against fellow Grandmaster Hikaru Nakamura have sparked controversy and raised concerns about the legitimacy of his claims.
Kramnik's Baseless Suspicions
Kramnik's primary basis for his accusations stems from Nakamura's impressive online performance, particularly in blitz and bullet chess formats. He has suggested that Nakamura's high win rates and seemingly superhuman moves are indicative of engine assistance.
However, these claims lack substantial evidence and fail to consider other factors that could contribute to Nakamura's success. Nakamura is a renowned chess prodigy with exceptional talent and experience, and his online achievements align with his established prowess.
Countering Kramnik's Narrative
Several points counter Kramnik's narrative and support Nakamura's innocence:

1. Lack of Concrete Evidence: Kramnik has yet to present concrete evidence, such as statistical anomalies or engine-like patterns, to substantiate his claims. His accusations rely primarily on subjective assessments and personal opinions.
2. Inconsistent Accusations: Kramnik's accusations have been inconsistent and selective. He has refrained from accusing other top players with similar online performance, raising questions about the objectivity of his approach.
3. Nakamura's Willingness for Scrutiny: Nakamura has consistently demonstrated his willingness to undergo scrutiny, including playing under strict anti-cheating measures and inviting third-party investigations. His openness and cooperation contrast with Kramnik's reluctance to provide concrete evidence.

Defending Nakamura's Integrity
Nakamura has faced these accusations with composure and maintained his focus on the game. He has expressed disappointment in Kramnik's actions, emphasizing the importance of fair play and the need for concrete evidence before making such serious allegations.
Moving Forward: A Call for Evidence-Based Approach
The chess community should prioritize evidence-based approaches to address cheating concerns. While vigilance is crucial, unfounded accusations and unsubstantiated claims can tarnish reputations and damage the integrity of the game.
In conclusion, Kramnik's accusations against Nakamura lack concrete evidence and fail to consider alternative explanations for Nakamura's online performance. Nakamura's willingness to undergo scrutiny further strengthens his position and highlights the need for a more objective and evidence-based approach to addressing cheating concerns in the chess world.Kramnik has also ruined his popularity and respect with such accusations. Even a mathematician has confirmed it, that the odds of Hikaru pulling such feat is not impossible and fairly common and occurs multiple many times in each year. Hikaru has spent his entire career playing blitz and Kramnik thinks he will cheat in a non money event( I am not saying he will cheat in money event).Here is what the post by Piotr Bugaj( Mathematician):"With all my great respect to You and with all my admiration to Your incredible chess career, I believe it is clear You don't fully understand how probability works. "Below 1% according to mathematicians". Yes, but that's over ONE EVENT. Hikaru plays thousands of games a year, so yes, You roll such chances of way below 1% but You roll them over thousands of possible sets of 46 games, so overall he is VERY LIKELY to have at least a few streaks of 40+ wins. (And Yes, I am a mathematician myself, greetings from Adam Mickiewicz University in Poznań, Poland)
Consider this: You flip a coin and want to get at least 5 heads in a row. If You toss a coin 5 times You get the probability of 3,125% (1 winning scenario out of total 32). But if You flip it 6 times You get 4,6875% chance of success (3 winning scenarios, 011111, 111110 and 111111, out of total 26=64). If You flip it 7 times it's already 6,25% (8 in 128). For 8 it's 7,8125% (20 out of 256 possible outcomes, I hope I didn't make any mistake here). If You only looked at the 5 tosses on which the event occurred and made a claim: "This was a 3% chance event!" You would be wrong, because it was a nearly 8% chance one, because it also COULD have occurred in different 5 in-a-row within these 8 tosses. That's the first mistake You are making here.
And to be clear, You're not the only one who fell for this, I see this mistake made pretty much always (out of the instances I witnessed of course), among respectable chess statistics commentators, sometimes even it the academic setting, with best addressing of this issue being "Of course 1 tournament like that can happen, but when it happens multiple times it's suspicious". People often say something like: "This suspicious GM had a 1 in 10000 performance this tournament, this must mean he cheated!" - and I'm like, ok, but we have, let's say a 1000 active GMs, let's say on average they play 10 tournaments a year each (Idk if these numbers are right, but the point stands), so actually, on average every year someone should have such a stellar performance. It's only if, let's say in a few consecutive years there is a dozen such performances, then it becomes highly unlikely that all of them are fair, but again, a few of them SHOULD be legit with high probability.
Now Hikaru's case is of course not as simple as the coin flipping, because You need to factor in specific chances of victory in any given game, but according to the Elo system You're expected to score this well against opponents with such rating disparity:
400 pts. - 91/100 350 pts. - 88/100 300 pts. - 85/100 250 pts. - 81/100 200 pts. - 76/100
So let's say Hikaru normally has a 70% chance to win a game against a guy, with some percentage points left for draws. If he's on a really good day (because that's also a factor when we deal with humans and not coins) it may be like 90%. No let's say over the year he has 20 days where he plays 100 games in a row with an average win chance of 90% (I encourage You to mess around with these numbers, I'm picking nice, round ones here, but You can insert whatever You think reality is close to). Then the chances that he gets at least one 40+ winning streak on one of those 20 days are... about 90% as I'm estimating. And If You only looked at 90% win chance out of specific 40 games You'd get ~1,48%. And if on top of that You didn't factor in "good day" and just assumed that every game ever played with this rating disparity is 70% chance of a win You'd get ~0,0000637%. You see how enormous these disparities are? And we didn't even consider the whole year and all games which Hikaru plays with different wining chances based on opponents ratings and external factors. He can also get for example 15 wins at the end of a "60% win chance day) and the following 25 the next day, when he farms some weaker players he has 85% chance to beat, so we need to treat games in one continuous bunch, rather than separate days, or whatever.
Long story short, I believe it's almost certain that Hikaru gets multiple 40+ winning streaks a year. To get the exact answer You'd have to model individual win chances for every game and then put a chain of events from an entire year (or career) and then calculate the probability of an n+ streak occurring k times. Impossible to do, but of course we can start with a simplified model and then add complexity to it as we go.
For example, if we take Hikaru's last year blitz stats we get 77,7% wins over 4094 games (as of right now). Assuming all the games to be equal (maximum simplification) we get a chance of at least 1 40+ winning streak being about 3,5%. Seems low, but now let's add complexity. Let's assume he plays 750 games with 62,5% win chance, then 750 with 70%, then 750 with 77,5 and so on, so we have 4000 games with an average win chance of 77,5% (Still highly simplified, but at least we recognize not all games are equal). Now we get ~9,5% chance of such win streak. Of course it's still far from the reality of his games, the frequency and pattern of his "farming", but the point is, the closer we get to the reality of life, when good and bad conditions for wins are not distributed evenly the higher the likelihood of an abnormal event occurring somewhere down the line. I mean just modeling 20 days of 100 games with 90% odds gave us ~90% chance that at some point during these 20 days there would be a 40+ win streak. That's why I said that I believe Hikaru is mathematically expected to have a few such streaks a year. We just need to build an accurate enough model, with still manageable level of complexity, to estimate his real win streak chances over a desired period of time.
If You have any questions about my calculations and reasoning or if You would like me to perform calculations on a model with numbers given by You as You believe they are in real life, feel free to contact me, I see it as my duty as a mathematician to do such informative work whenever required. Once again I bow my head to You and Your incredible chess career.
Piotr Bugaj"
Its quite weird to see 14th world champion do this type of things but lets see what would happen in the future. I will keep you all updated and post my views too.