Fischer vs Karpov in 1975: Who would have won?

by Matthew Wilson
11/3/2018 – In the USA, chess fans are eagerly awaiting the World Championship match to see how Caruana fares as the first American to challenge the undisputed World Champion since 1972. But what about that other eagerly awaited contest that never was: Fischer-Karpov, 1975. It was a match that the whole chess world would have loved to see. Who can say what beautiful ideas the players would have shown us in their games? And who would have won? With statistical modelling, we can at least talk about the likely outcome of the match. Dr. MATTHEW WILSON, economics professor at Truman State University, takes a crack at it. | Artwork: Unknown (via Quora)

Endgames of the World Champions from Fischer to Carlsen Endgames of the World Champions from Fischer to Carlsen

Let endgame expert Dr Karsten Müller show and explain the finesses of the world champions. Although they had different styles each and every one of them played the endgame exceptionally well, so take the opportunity to enjoy and learn from some of the best endgames in the history of chess.

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Who would have won? Probably Fischer...

The match didn’t take place because there was no agreement on the format. Fischer wanted the “10 win system with draw odds.” The first player to score 10 wins would be the victor. But there was a catch: if the score reached 9-9, then Fischer would retain his crown – that is the “draw odds” part of the system. This proposal was rejected. Afterwards, FIDE adopted what I will call the “6 wins system.” The first player to win 6 games would be the champion. There were no draw odds; if the score reached 5-5, they would keep playing. Previously, FIDE had used the “24 game system.” The first player to score 12.5 points or more would be the winner. Fischer would continue his reign if the match were tied at 12-12. Under all three systems, Fischer would have been the heavy favourite — even if I rig a few assumptions in Karpov’s favour.

Why is the model so confident in Fischer’s chances? It’s because of their ratings. Fischer was 2780, just a few points shy of his peak, while Karpov stood at 2705. A 75 point rating gap is not to be lightly dismissed. It’s about the same as the gap between Anand and Carlsen in their matches. Of course, Anand had his chances and the games were not one-sided. But to call Anand vs. Carlsen a 50-50 tossup is just not accurate.

Even worse for Karpov, the match would probably be much longer than either of the Anand-Carlsen duels. And in a longer match, an upset is less likely. That’s because it’s harder for the underdog to consistently beat the odds over many games than it is to beat the odds just a few times. For example, in a large open tournament, a 2200 might score an upset against a GM. But the GM would almost certainly prevail in a 24 game match. Anand’s chances in his 12 game matches with Carlsen were slim. His prospects in a traditional 24 game match would be even worse. That would be like drawing one of the 12 game matches and then winning the other — a pair of unlikely upsets.

Here is how the model works. The ratings give us the expected score. Fischer’s expected score is about 0.6 points per game (see the FIDE handbook for info on the rating system). So if they played a bunch of 10 game matches, we would expect Fischer to win by 6-4 on average. 

𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑠𝑐𝑜𝑟𝑒 = (1)(𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑤𝑖𝑛) + (0.5)(𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎 𝑑𝑟𝑎𝑤) = 0.6

For now, I will set the probability of a draw at 50%. We will modify that assumption later. Plug that into the equation above, and here’s what you get. In each game, Fischer has a 35% chance of winning, 50% chance of drawing, and 15% chance of losing.

There are three possible formats: the traditional 24 game match, the 10 wins system with draw odds, and the 6 wins system. The best case scenario for Karpov is the 6 wins system. In that format, he is fighting on a level playing field – no draw odds for Fischer. So I ran 10,000 simulations of a match with the 6 wins system. Here is what happened:

Fischer’s chances in 1975 were similar to Carlsen’s chances in 2013.

Player Rating Wins the match %
Fischer 2780 93.49%
Karpov 2705 6.51%

In the above analysis, I simply assumed that the draw rate was 50%. Even if we change that, Fischer is still the heavy favourite in the simulations: 

Fischer wins the match

Karpov’s prospects improve when the draw rate is lower — even though draws are not counted in the six wins system. That is not a mistake. If the draw rate falls by 10%, then in each game, Karpov’s chances of winning and Fischer’s chances of winning both rise by 5%. But that causes the ratio of Fischer’s winning chances to Karpov’s winning chances to fall.

Draw rate effect

Here’s a simple example. We already saw that when the draw rate was 50%, then Fischer had a 35% chance of winning and 15% chance of losing in each game.

𝑅𝑎𝑡𝑖𝑜 𝑜𝑓 𝐹𝑖𝑠𝑐h𝑒𝑟′𝑠 𝑤𝑖𝑛𝑛𝑖𝑛𝑔 𝑐h𝑎𝑛𝑐𝑒𝑠 𝑡𝑜 𝐾𝑎𝑟𝑝𝑜𝑣′𝑠 = 0.35/0.15 = 2.33 𝑤h𝑒𝑛 𝑑𝑟𝑎𝑤 𝑟𝑎𝑡𝑒 𝑖𝑠 0.5

What if the draw rate were 0%? Then since Fischer’s expected score is 0.6, he would have a 60% chance of winning each game and a 40% chance of losing.

𝑅𝑎𝑡𝑖𝑜 𝑜𝑓 𝐹𝑖𝑠𝑐h𝑒𝑟′𝑠 𝑤𝑖𝑛𝑛𝑖𝑛𝑔 𝑐h𝑎𝑛𝑐𝑒𝑠 𝑡𝑜 𝐾𝑎𝑟𝑝𝑜𝑣′𝑠 = 0.6/0.4 = 1.5 𝑤h𝑒𝑛 𝑑𝑟𝑎𝑤 𝑟𝑎𝑡𝑒 𝑖𝑠 0

So when the probability of a draw falls, Karpov’s chances get relatively better in the decisive games.

But was Karpov underrated?

Possibly. You can see that Karpov was rising rapidly during the early 70s: 

Karpov rating curve

Karpov's rating progression in the 1970s

However, even if we account for this, Fischer remains the clear favourite. And Karpov’s improvement had clearly tapered off by the time he was supposed to face Fischer. I calculated Karpov’s performance rating in 1974 and 1975. He actually underperformed in 1975 and his rating fell to 2695. So I’ll bias the model in Karpov’s favour by focusing his best year. His performance rating was 2726 in 1974.

Let’s say Karpov’s true strength was 2726. He would still be well below Fischer’s 2780. The rating gap would be comparable to that between Carlsen and Giri; they were 60 points apart as of October 2018. Of course you would say that Carlsen would be the favourite in a match with Giri. Only two world championship matches in nearly half a century featured 50+ point upsets. And both of those matches (Kasparov-Kramnik in 2000, Kramnik-Topalov in 2006) were relatively short, meaning that upsets were more likely. Karpov’s chances do improve; Fischer’s expected score shrinks to 57.7%. However, Karpov is definitely still the underdog.

Chance the Fischer wins against draw rate

What about the model’s other assumptions?

The model implicitly assumes that the games are independent of each other. I tested this assumption many times and never rejected it.

Test #1: If one game is decisive, does that affect the probability of a win or loss in the next game? Perhaps the winner gets a confidence boost and the chance that he would win again would rise. But this is not true, as I demonstrated in an earlier article. Of course decisive games affect the players psychologically, but it does not have any significant effect on the winning and drawing rates.

Test #2: Here is another way to detect a confidence effect. In super tournaments, if you win your previous game, do you outperform your rating in the next game? The answer is no.

Test #3: Suppose that winning your previous game gives you a confidence boost that helps you in the rest of the tournament. If that is true, then results in actual tournaments will be more spread out than my simulations would suggest. My model would have too many players bunched up at 50% and not enough at the extremes. However, a statistical test rejected this argument.

Test #4: Once a player gains a significant lead in the match, he will start playing for a draw. Thus, the draw rate should not be constant in the simulations.

This argument doesn’t apply to the six wins system that I have investigated here. However, it would be plausible if FIDE had insisted on the traditional 24 game match. However, there is no evidence that the draw rate changes once a player takes a 2 point lead. Naturally, the leader is inclined to play safely, but this is balanced by his opponent’s extra motivation to fight for a win.

At the top level, the games really do seem to be independent of each other.

Conclusion

It was a tragedy for chess that the match never took place. However, we can be quite confident that Fischer would have prevailed. It is true that the model does not directly account for factors such as Fischer’s mental health. However, his 2780 rating that was used in the simulations was his actual rating — the rating he attained in spite of his issues. This is not an imaginary problem-free version of Fischer.

Are nerves a factor in a world championship match? Of course they are. But the ratings used in the simulations come from real tournaments and matches in which nerves were also a major factor. Thus, the results in the simulations cannot be easily dismissed.

Karpov’s chances are low even if we twist the model’s assumptions in his favour. Even if we pick a system in which Fischer does not have draw odds and even if Karpov were underrated, Fischer is still the heavy favourite. Fischer’s chances are — at the bare minimum — 70%. I do not know how long Fischer’s reign would have lasted; perhaps Karpov would learn important lessons from their match and come back stronger in 1978. But we can be quite confident about the outcome in 1975.

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Matthew Wilson is an economics professor at Truman State University. He is a 2100 player and runs a chess and statistics blog.
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twamers twamers 11/3/2018 01:21
It's always hard looking back on these things. However my own view is Fischer would have won had the match taken place and whether that was a traditional 24 game match (still the best format in my opinion) or first to 6 or 10 wins I still think Fischer wins. Karpov is clearly one of the games greatest players but he only just squeaked through against Korchnoi in the 1974 candidates final (3-2 on wins) and that suggests 1975 would have been too soon for him versus Fischer. Karpov clearly got stronger as he got a bit older but he also had a very close result against Korchnoi in 1978 match (6-5) and it wasn't until 1981 that he was clearly much stronger than Korchnoi in match play (6-2). Fischer for me in this assessment.
Jack Nayer Jack Nayer 11/3/2018 12:36
I have always thought that Karpov would have won against Fischer if the match would have taken place. I have no statistics to back this up, but there are some arguments.
It’s not valid to use Fischer’s 2780 rating after three years of complete inactivity. Even if Fischer was obsessively studying openings and the like, he lacked tournament practice and there is no substitute for that.
Karpov, working hard, would have had the assistance of several grandmasters and he would have listened to them, in contradistinction to Spassky, who, in the words of a furious Geller, read too many novels and played card games and tennis instead of checking and memorising analysis.
Third, while it is true that Fischer reached his peak suffering from mental health problems, the question is where the threshold was that prevented him from function properly, not only as a human being but as a top grandmaster. Isn’t the fact that the match never took place an indication that this threshold had been reached? Fischer, rightly or wrongly, was terrified of playing. The new generation had arrived. This was not Tal, Botvinnik, Petrosian, Spassky, Kortsnoj, Geller or Stein – this guy was even better. Here you go: the match didn’t take place because Fischer couldn’t do it.
woodshifter2016 woodshifter2016 11/3/2018 12:28
If the predictions are based solely on the expected results between 2 players with given ratings: what if there is distortion in the way one of the ratings was calculated?

I was naively surprised at just how much fixing of results there was between Soviet players: draws agreed to give rest days either as a mutual assistance between Soviet players or to favour the State's new favourite, or players throwing games/matches.

Their achievements are obvious and the games they played (at least when not between 2 Soviets) speak for themselves (if you are good enough to understand them). This isn't intended as a moral judgement on the players - only someone who lived in those countries at that time could really understand the situation.

A digression: Fischer's insight was usually very reliable on any Chess related matter (outside of Chess his opinions can be seen for what they were and cannot be defended): Fischer Random, Fischer Clocks etc. He said that the Karpov Kasparov matches were fixed - I don't think any strong player has ever seriously investigated. this. Maybe Fischer's non-Chess mental abnormalities intruded into his judgement here and distorted his analysis of the games. I don't think he ever published anything to back up his accusation. He wouldn't have been objective where the World Championship was concerned.
ebit ebit 11/3/2018 12:06
It was an impressive effort and success, that Fischer became champion on his own, in fact without any help. But he had to pay his price, was getting mental ill and was probably aware of it, so he hadn't the mental strength for a match and let's not forget the fact, that Fischer had something to lose: his title, a situation, he obviously couldn't handle.
By the way, the author is guessing about Karpov's real strength, but doesn't really question Fischer's 2780 rating. Why not?
lajosarpad lajosarpad 11/3/2018 11:11
I agree that Fischer was afraid of Karpov and even if he was much stronger, his psyche might have made it difficult for him to play his best against Karpov. Also, all the grandmasters of the Soviet Union would have helped Karpov. I would not be so confident in stating that Fischer was the heavy favorite.
Nezhmetdinov1919 Nezhmetdinov1919 11/3/2018 10:34
"...in the end Fischer, having full information, choose not to play Karpov. And that says everything."
Well put, Bill
Nezhmetdinov1919 Nezhmetdinov1919 11/3/2018 10:32
Thanks for the article, Mr. Wilson. Good reading with nicely done calculations.

There are other factors, however, which could also play a role in the match. But first, let's remember that Fischer reached his incredible rating of 2780 (said rating being one of the main pillars of your argument) under certain conditions which I do not think would be present in a match against Karpov:

- Fischer had been hiding in his cave for years and if in one hand that could compromise his shape (which I do not think was the case), on the other it also allowed him to build up a formidable array of surprise weapons such as openings and middlegame ideas. I can not think Fischer would be able to rebuild such impressive repertoire in time to meet Karpov, particularly because being the stuborn person he was, he was bound to do all the work alone;
- Fischer was able to surprise Spassky in some critical games (by playing 1.c2-c4, for instance as well as defences he had never employed in his life). Karpov would be hardly surprised by 1.d2-d4, 1.c2-c4, or any 1.b2-b3 anymore. The aforementioned defences wouldn't cause much trouble either (check Karpov's record against the Alehine, or Benoni, for instance. One could hardly draw Karpov emplyoing the Alekhine!) and if it's true Fischer could well be able to come up with new ones, keep in mind that there is only so many good defences one can apply in elite chess, particularly at World Chess Championiships, so the well was already drying;
- Spassky was a rebel during those years and didn't pay much attention to what assistants, trainers and even high-ranking members of russian politburo would suggest him to do. That certainly played in Fischer's hands. Karpov, on the other hand, was just the opposite. In case Fischer did accept to play, he would have a real hard time getting things his own way;
- Finally, let´s not forget Fischer's instability. A nice -- and curious -- additional calculation to your well-written article would be the odds of Karpov withdrawing the match against the odds of Fischer doing the same (!). With that in mind Karpov would be the clear favorite!

Again, thanks for the article.
Cheers
Bill Alg Bill Alg 11/3/2018 10:06
Quite funny article. Of course, Fischer knew, better than anyone else, his own condition/playing form after 3 years' inactivity and he had Karpov's games at hand to analyse so he knew his opponent as well. There is no need to make such statistical analysis and other assumptions: in the end Fischer, having full information, choose not to play Karpov. And that says everything.
RayLopez RayLopez 11/3/2018 09:47
Good article, except I don't understand why you would model a draw rate that's not symmetric. E.g., if a player is rated 300 points higher, the expected win ratio is about 15%, which means on each side of the 15% line there should be a draw probability that's symmetric giving exactly 15% expected wins. One player should not be expected to draw any more than the other? Other than that, the model is very understandable and I've coded something similar myself.
goeland goeland 11/3/2018 08:35
Lets have a teacher/researcher/professor from Moscow do the same analysis.
Bartali Bartali 11/3/2018 08:17
Replay the Karpov-Spassky 1974 match!
Keshava Keshava 11/3/2018 07:05
"But we can be quite confident about the outcome in 1975." We can be quite confident that YOU are quite confident. That is all. The rating has something to do with style - like Fischer trying 100% to win games (perhaps even showing preparation) when a match or tournament is already decided. Or the Soviet idea of stiving only to win with White and accepting a draw with Black (unless White seriously errs). Yes, even Karpov felt that Fischer would have been the favorite had he found the stones (my words) to defend his title in 1975. However to say that "Fischer’s chances are — at the bare minimum — 70%" is an insult to a player of Karpov's stature. Let us not forget that chess is a sport and if Spaasky had not (surprisingly) defied the Soviet authorities by refusing to accept a win of the match by default then Fischer probably would have never even became champion in the first place. I think Karpov would not have been so accommodating and so one or two forfeit wins against Karpov would be a huge challenge to overcome. Fischer was so nervous about playing Spaasky (who he felt that he could beat) that the match almost never happened. He must have known that Karpov was superior to Spaasky so who knows how much that affected his psyche. It is Karpov's mental advantages that would have made the match more competitive than the author thinks imo.
stevef0531 stevef0531 11/3/2018 06:16
Fischer in 1975 could easily discard any perceived rust. He would not be ignorant
of all the new theoretical developments. As long as Fischer did not lose his mental state
[no guarantee that he wouldn't; he practically did already in 1972] he would win
w/ his undeniable mental and chess-playing pressure against Karpov.
Interesting would have been the Fischer of 1972 at the 1962 Curacao Candidates.
ChatNoir ChatNoir 11/3/2018 05:16
I was expecting a serious article here - unfortunately it is not the case. One of the two basic assumption for all calculations, which should at least be questioned, is Fischer's level. His last published Elo rating was indeed 2780, but how to account for the effect on his level of his three years (1972-1975) inactive, away from official competition? Does Mt Wilson really think that chess level (as shown by your FIDE rating) can be maintained without play? Already in 1974, after two years "retired", Fischer was aware that the probabilities were to be around 50-50 in such a match - and he deprived the chess world of it.
dysanfel dysanfel 11/3/2018 04:43
Fischer would have remained champion until he lost it to Kasparov in either the first or second match. Time and age would have taken Fischer's title. It is important to remember that Kasparov score against Karpov was only +2 after 144 or so world championship games. Hardly a dominant world chess champion match player.
SoMuchNonsense SoMuchNonsense 11/3/2018 03:35
Of all the "predictions"we have seen over the years on this topic this is probably the more nonsensical one: Maybe we have to remind the author of when the ratings where implemented and how developed were they by the time.

Anyways, this is just another futile exercise to "prove" the author's wishes. We will never know

By the way, Karpov accepted ALL the ludicrous Fischer's conditions, including the Draw Odds: What he did not bow to was the request for the contest to be officially named as the "Professional World Chess Championship", but anyway Fischer, being so existentially scared, would certainly have kept coming with more and more outrageous requests, as for the match never taking place. It was not to be, to Karpov's disgrace.
syuanjiang syuanjiang 11/3/2018 02:57
This article does not count that Fischer had not played for 3 years; he was rusty and could cost him a couple of games. Also we don't know how hard Fischer trained during those 3 absence years.

Anyway, I am a Fischer fan and I do believe he would win in 1975 (I am not sure about 1978; very likely Karpov would win in 1981).