Computer predictions, simulations and scenarios

3/22/2016 – One can never have too much data as any Wall Street analyst will tell you. In this latest report on the Candidates based on simulations with weighted statistics, we not only see what the winning chances are for the various players, but also what games will have the biggest impact in the next rounds, and even what games have the biggest impact on Anand's winning chances.

Computer predictions, simulations and scenarios

By James Jorasch and Chris Capobianco

Sometimes threats and opportunities remain hidden in plain view because something much more captivating is holding a viewer’s attention. The thought of tiebreaks, for example, isn’t nearly as thrilling as watching the unfolding Candidates’ Tournament games. Let’s take a look at why that’s problematic.

With his loss against Anand, Aronian now needs to make up half a point to catch up with the two leaders. But our simulations show Aronian dropping from a 36.2% chance to win the tournament down to 16.8%. Why such a large drop? The answer lies buried in the tiebreak rules. These rules can have a significant effect on the outcome in the tournament.

The tiebreaks are applied in this order: head-to-head results, number of wins, Sonneborn-Berger score, rapid game match, blitz game match, and finally an Armageddon game. Our simulation software has all of the rules built in, so we can track exactly how many ties are broken by each tiebreak rule in a given simulation run.

Let’s take the example of Anand and Aronian. Aronian is only a half point behind Anand, but catching up isn’t going to help him if the tournament ends in a tie between Anand and Aronian. In that case, Anand would have a 1.5 to 0.5 record against Aronian and would thus eliminate Aronian in the head-to-head tiebreaker. The net effect is that Aronian probably needs to make up a full point between himself and Anand. This is one of the reasons that Aronian drops from 36.2% to to 16.8% in the latest simulation. In effect, Anand’s win over Aronian was  worth not 1 point, but 1.25 points, under these tiebreak conditions.

For this reason, we call games between the tournament leaders with this tiebreak structure “Power Games”. You can see the upcoming schedule of Power Games in GM Jon Speelman’s update.

As a demonstration of the importance of Power Games, see below for simulation results of the round ten game between Caruana and Anand.

Win probabilities after round nine


After Rd 5 After Rd 6 After Rd 7 After Rd 8 After Rd 9 (vs Rd 8) Point Standings
Anand 5.5% 9.4% 11.0% 11.4% 26.6% (+15.3%) 5.5
Aronian 26.8% 40.1% 42.2% 36.2% 16.8% (-19.4%) 5.0
Caruana 16.9% 10.8% 10.5% 20.2% 20.8% (+0.6%) 5.0
Giri 10.4% 8.7% 6.8% 5.7% 4.1% (-1.6%) 4.5
Karjakin 28.0% 27.2% 25.1% 25.3% 31.0% (+5.7%) 5.5
Nakamura 5.3% 1.3% 2.9% 0.5% 0.1% (-0.4%) 3.5
Svidler 5.5% 1.7% 1.4% 0.7% 0.6% (-0.1%) 4.0
Topalov 1.7% 0.9% 0.2% 0.1% 0.0% (-0.1%) 3.0

Anand moves up to 26.6%, but still trails Karjakin, who is in the lead with 31.0% in our simulations.

Three players now combine for a less than one percent chance of winning the tournament: Nakamura, Svidler, and Topalov. While they only have a small chance of winning the tournament, they could still have a huge impact in any game by beating one of the leaders.

Giri’s tense and complicated battle with Caruana ended with his ninth draw in a row as his chances in our simulation fell from 5.7% to 4.1%.

The top four players have now formed a close pack with just 15% separating the probability of their winning the tournament.

Simulations of Research Scenarios for Anand

With Anand joining the lead, these simulations look at a few scenarios for him over the remaining rounds.


Sim 1 Sim 2 Sim 3 Sim 4 Sim 5
Rd 10 Caruana-Anand Draw Draw Draw Win Draw
Rd 11 Anand-Karjakin Draw Draw Win Draw Win
Rd 12 Nakamura-Anand Draw Win Draw Win Draw
Rd 13 Anand-Giri Draw Draw Draw Draw Draw
Rd 14 Svidler-Anand Draw Draw Draw Draw Win
Simulated Winning Chances 18.7% 60.8% 85.4% 89.7% 97.7%

Sim 1 illustrates Anand’s chances when he draws all the rest of his games. He ends up winning 18.7% of those tournaments. With just a single additional win, Anand increases his chances considerably, but note the difference between Sim 2 and Sim 3.

In Sim 2, he defeats Nakamura to rise to 60.8%. In Sim 3, he again has only one win, but it is in a Power Game against Karjakin and thus results in a bigger leap to 85.4% winning chances! Some of this gain is from taking down a leader, and some of it is the tiebreak benefits as mentioned above.

Sim 4 and Sim 5 illustrate two wins for Anand, with the Sim 5 win against Karjakin and Svidler virtually locking Anand into first place.

Quantifying a Power Game

Power Games are going to be a focal point in the remainder of the rounds. Anand won the Power Game of round nine, pushing Aronian’s chances of winning the tournament down to 16.8% in our simulations.

Round 10’s Power Game will be Caruana-Anand. The table below shows the results of three simulations, examining all three possible outcomes of the game and the effect on the winning chances of all tournament participants.


After Round 9 Sim 1

Caruana Wins
Sim 2

Draw
Sim 3

Anand Wins
Anand 26.6% 6.3% 31.1% 56.2%
Aronian 16.8% 19.6% 17.0% 9.4%
Caruana 20.8% 42.5% 13.3% 2.1%
Giri 4.1% 3.2% 4.6% 3.9%
Karjakin 31.0% 27.6% 33.3% 28.0%
Nakamura 0.1% 0.1% 0.1% 0.1%
Svidler 0.6% 0.7% 0.6% 0.3%
Topalov 0.0% 0.0% 0.0% 0.0%

A win by Caruana would put him in the driver’s seat with a tournament leading 42.5% winning chance, with Anand pushed down to 6.3%.

A draw benefits Anand (who moves up to 31.1%) since he has half a point more than Caruana in the current standings. Every round that he is not losing ground gets him closer to a win (though his head-to-head score against Karjakin could hurt him later). Conversely, the draw moves Caruana down to 13.3%.

A win by Anand moves his chances up to 56.2% in our simulation and brings Caruana down to 2.1%. This is clearly a Power Game that Caruana cannot afford to lose. Even if Caruana follows this loss with a win against Topalov in Round 11, his chances only rise from 2.1% up to 6.3%.

Tournament Analytics

All games of the Americans (Caruana and Nakamura) and Russians (Svidler and Karjakin) were drawn in Round 9, though Anand’s win improved Karjakin’s chances, so the Russians once again move upward relative to the Americans with a gap of more than 10%.

The expected winning score for the tournament moved lower this round to 8.56. The trend in the chart below shows the biggest gain in 8.5 (+3) as a winning score - happening 459,000 times (45.9%) in our simulation of a million tournaments.

The probability of a tie grows as Anand joins Karjakin in the lead. The number of tournaments expected to be decided in a tiebreaker is now 31.9%, up several percent over the 28.6% of round eight. Of those extra ties, the majority are being decided in the first head-to-head tiebreaker because of the Power Game results between Anand and Aronian and between Karjakin and Anand.

Money

The simulation tracks prize money earned by each participant at the conclusion of the event, so we show in the chart below what the players average over the million simulations. Note that this does not attempt to incorporate the extra value of winning first place in the form of the minimum prize money earned from the World Championship.

Anand adds a robust €13,400 to his expected prize money with his win over Aronian. Aronian drops €11,600 in simulated winnings.

Methodology Note

This report was created by running millions of simulations of the Candidates Tournament using our tournament software. We introduced this software the day before the tournament began. All actual game results to date are included in the simulation, and the remaining unplayed rounds are simulated based on a number of factors. This report discusses angles of interest on simulated tournament results and each player’s expected probability of winning.

About the authors
James Jorasch is the founder of Science House and an inventor named on more than 700 patents. He plays tournament chess, backgammon, Scrabble, and poker. He lives in Manhattan and is a member of the Marshall Chess Club.
Chris Capobianco is a software engineer at Google. He is a two-time finalist at the USA Memory Championships and has consulted on memory for finance, media, advertising and Fortune 500 companies such as Xerox and GE.

If you have questions about tournament statistics that have not been covered, please get in touch. We’ll try to answer it in a future round report. Feel free to forward this along to friends. Interested in getting a copy of each round report? Send an email with the subject line “subscribe candidates tournament.”

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Bertman Bertman 3/22/2016 11:39
@ MrT

Statistics aren't for everyone, and can seem like a lot of gibberish if you cannot read them.
firestorm firestorm 3/22/2016 11:44
How about respecting chessbase as a family friendly site and not swearing in the discussion section?

Regarding the simulation and winning probabilities, one thing is apparent- if you win games you have a greater chance of winning the tournament- yes, sorry for the statement of the obvious. I'm not really not sure what these millions of simulations and probabilities are achieving (unless you're trying to set odds on a gambling site), since you want an a priori accurate prediction, not an update based on what the results actually are- of course Anand's chances are better than Aronian's (tie break rule) and worse than Karjakin's- same time break rule with Karjakin having beaten Anand. Its an analysis without any chess in it ...
amosburn amosburn 3/23/2016 12:10
They can still seem like a lot of gibberish if you do understand them
tom_70 tom_70 3/23/2016 01:13
I like Anand, but I'd kinda like to see one of the young guns win this. We need some fresh blood in the World Championship match.
karavamudan karavamudan 3/23/2016 02:33
Well I hope all these so called statistical predictions match with reality at least at the end of the candidates matches.

In one of the graphs, same color (red) has been used for both Aronian and Nakumara.

vixen vixen 3/23/2016 03:09
to me now the one and most important game now is Anand vs Karjakin
thlai80 thlai80 3/23/2016 03:22
One scenario not mention is if Karjakin, Anand and Aronian finished tied on points, with Aronian beating Karjakin in their personal encounter. In this case, They need to look at tiebreak 2 as well since Karjakin beat Anand who beat Aronian who beat Karjakin?? Where's the probability?
Bertman Bertman 3/23/2016 03:51
@thlai80

Then Anand would win as he has the largest number of wins.
thlai80 thlai80 3/23/2016 06:27
@Bertman, I know, but the probability calculation (even with 1mil sims?) does not seem to account for such scenario. What I trying to say is that the initial premise to calculate the probability of Anand winning is incomplete/wrong, so the resultant calculated % of him winning is wrong.
Beanie Beanie 3/23/2016 09:20
To test the accuracy of the program it was tasked with retrodicting the result of the 2000 World Championship Match. It's conclusion was that Kramnik had a 31% chance of winning and Kasparov 69%. Kasparov felt vindicated and said, "I told you so!" Kramnik was puzzled and asked for a recount. So the actual result was fed in and the new conclusion was Kramnik 100% and Kasparov 0%. The commentators were mightily impressed at the accuracy of this and also at the computer's ability to "learn". Then it was tasked with retrodicting the result of the 2006 WCM, but this time it hit a snag and produced no output at all. The problem was it had no system for processing the command, "Toilet Breaks."
victorhoogland victorhoogland 3/23/2016 11:22
@firestorm: I agree. It is difficult to interpret a 26.6% winning probability. Moreover, it seems that a simple predictor of the tournament winner is just whoever is at first place in the current standings (with some tie-breaking rule).

It would be interesting if the authors would evaluate their predictions on other tournaments in history. How often would their pre-tournament predicted winner be correct?
idratherplay960 idratherplay960 3/23/2016 12:12
anand and karjakin are probably groaning after reading this given that the model originally predicted caruana and naka to have the best chances...curse of the supposed predictive model
Steven Gerrard Steven Gerrard 3/23/2016 02:47
What a wasteful way to burn a lot of CPU to get colourful graphs
Stupido Stupido 3/23/2016 05:14
Even now that Caruana won and the wind is clearly in his sails, I fail to see any reason why he would be "in the driver seat". Karjakin is in great shape and will play with White in their last round game.
billbrock billbrock 3/23/2016 06:19
If I'm a player and I'm reading this, how would my strategy change?

Seems clear that Anand has to beat Karjakin tomorrow. And Karjakin has to beat Caruana in round 14.

But Caruana might be a more interesting case. He could go +2 in rounds 11-13 and still lose if he can't hold off Karjakin. By the same token, he can't afford to be trailing Karjakin after round 13. So how much risk can Caruana afford against an out-of-form but always dangerous Topalov? (Cf. Ivanchuk's influence on the final results in the London Candidates.)
Chessspawnvt Chessspawnvt 3/23/2016 08:22
About the only thing this makes clear is that your statistical predictions, while fun, are of suspect value in the real world of competition with human beings who have variables that you can't statistically account for. Still, keep at it and we'll see what statistics has wrought at the end.
Kurt Utzinger Kurt Utzinger 3/27/2016 10:20
The first prediction - with the exception of Caruana - had Karjakin on the second last place. I give nothing on such statistics when it goes to predict the winner of a single tournament.
Chessspawnvt Chessspawnvt 3/29/2016 12:31
Karjakin..........So much for the statistical analysis. Fun to read through, none the less.
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