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A Tiebreak Win and the Problem of Draws

December 2, 2018

Carlsen impressed in fast chess, but what of classical?

Cropped from AFP/Getty source (Irish Times)

Magnus Carlsen retained his title of World Chess Champion on Wednesday. He thumped the American challenger Fabiano 3-0 in a best-of-four tiebreak series of games played at Rapid time controls. Despite his having only one-fourth the standard thinking time, Carlsen’s quality in the tiebreakers was plausibly higher than in the twelve regulation games.

Today Dick and I congratulate Carlsen on his victory and discuss implications for future top-level competitions in chess.

Chess ratings via the Elo system are based on results of games and the ratings of the opponents. My Intrinsic Performance Ratings (IPRs) are on the same scale but use only the quality of the player’s moves as judged by strong chess programs. The mapping from quality measures to Elo ratings comes from training data of millions of moves by players of all ratings. Per remarks in my previous post, I don’t claim the mapping captures all aspects of chess skill—but it does provide firm ground for judging players’ performances relative to their peers. My training sets go from beginner clear up to the Elo 2800+ level of Carlsen and Caruana.

One use of IPR is to measure how thinking time affects quality. The title match tiebreaker gave 25 minutes plus an extra 10 seconds per move, the same as in the famous Melody Amber tournaments, whereas the World Rapid championships give 10 fewer minutes. My preliminary results show an average dropoff of 200–210 Elo in the Ambers and 280–290 Elo in World Rapids. Caruana’s quality of 2575 (with huge error bars from under 100 relevant moves in the three tiebreak games) was consistent with this, but here is what I measured for Carlsen:

2945 +- 190.

Since the error-bars are two-sigma, this was more than one standard deviation higher than Carlsen’s rating at standard time controls, and higher than his IPR for the twelve regulation games of the match. Clearly Caruana ran into a buzzsaw.

Carlsen’s match two years ago also went to tiebreakers and I measured no dropoff there either: 2835 +- 250. Carlsen won the World Rapid Championship in 2014 and 2015 and is the reigning World Blitz champion to boot. His prowess upheld his reasons for taking a draw rather than press his advantage in the twelfth regulation game, which I discussed before. Here we will consider what it says about standards and expectations for the chess title in general.

Two Issues

The first thing to note is that although many chess luminaries have voiced dissatisfaction with the 12 draws and tiebreaker and match rules that produced this outcome, neither of the two players is among them. This contrasts with great fights over rules by Bobby Fischer and Garry Kasparov in particular. Caruana has simply offered his congratulations and thanks. Carlsen in a post-match interview considered using Rapid and Blitz formats more not less.

That the tiebreak games were so decisive makes the outcome seem juste. My near-namesake Kenneth Rogoff opined after Wednesday’s conclusion that nothing is amiss. He compared the twelve draws to a 0-0 soccer match that has tantalizing near-misses before its penalty-kick shootout. Most of the draws were hard fights where one or both sides had chances. All six with Caruana as White were Sicilian defenses, considered a fighting choice by Black. The more-drawish Berlin defense never happened, and of the two Petroffs chosen by Caruana as Black, one produced the fascinating game-6 endgame which Caruana could have won. Two games with the Queen’s Gambit Declined (QGD) were fairly tame, but in two others, Carlsen’s 1.c4 English Opening led to sharp play.

There are at least two separate issues:

  1. Is it right to mix formats? Suppose a 1500 meter championship footrace ended in a dead heat—as did the 1500 meter speed skating final of the 1960 Winter Olympics. Would we have the runners do a 400-meter tiebreaker? Or blitz out a series of 100-meter dashes? Consensus is no: the format is so different as to make many mile-pace skills irrelevant, let alone how it changes the race strategy. Is the situation in chess similar?

  2. How terrible was having all 12 regulation games drawn?—for the general public and prospective sponsors, not just chess cognoscenti. What if all games had been Berlins and Petroffs and QGDs?

Thoughtful considerations and proposals have been given here by my fellow International Master Greg Shahade (whose father Michael I knew in the 1970s and whose sister Jennifer was a US commentator on the match), here by master teacher Dennis Monokroussos on his blog The Chess Mind, and here by the noted economist and blogger Tyler Cowen (who played on teams with me in the 1970s). See also this from 2016. Dick and I will not pretend we can solve all the issues in one post. But we can offer a few particular observations.

Single Format

To find an opinion against the inclusion of games under fast time controls, we need look no further than Shahade two years ago, when the regular part of the match had 10 draws and one win apiece:

Just as in the 20th century, the Champion should retain the title on a drawn match. There should be no rapid tiebreak. … I don’t agree with giving anyone the chance to become World Chess Championship by tying a Classical match and then winning some rapid games. At every moment in the match, someone will be behind on the scoreboard [and will] fight harder every game.

Under this system, Carlsen would have retained his title on two straight drawn matches, as did Mikhail Botvinnik in his 1951 match against David Bronstein and 1954 match against Vasily Smyslov. Kasparov retained his title once that way against Anatoly Karpov by winning the last game to make a tie. Those matches, however, were 24 games each, and even at that length, Fischer led a chorus of many who felt the “draw odds” were too steep on grounds that the defending champion could play for draws.

Fischer’s solution was to have draws not count. It was used for the 1978 match between Karpov and Viktor Korchnoi, which had a dramatic ending but took 32 games over three months. The format cratered in 1984–85 when Kasparov, after losing 4 of the first 9 games, shifted to grinding for draws. The match was annulled after 48 games with Karpov still shy of the six wins needed.

Multi-month matches are out, not only because of the expense and time commitment for sponsors, but for the positive reasons that today’s champions have been attracted to play in a more diverse array of chess events than their predecessors. Cowen advocates the old 24-game format (with fewer rest days) but with prescribed openings away from Berlins and Petroffs and QGDs and lines that players already know 30 moves deep.

Otherwise, the only way to mitigate the champion’s draw odds is to alter chess so that draws are less frequent. This can be done by branching the opening phase so greatly as to reduce the expected span of computer-aided opening preparation and by changing rules to provide more symmetry-breaking.

Here, too, Fischer was a visionary. His Fischer Random format, also called Chess960 for its number of starting positions, continues to be used in some high-level events, but has issues in that these positions retain symmetry that in numerous cases gives White more advantage than in the classical setup. I stand by my proposal to combine an older non-random, non-symmetrical form by Bronstein with Fischer’s rules and a few tweaks.

If such rule changes are too drastic, and the classic 24-game format falls on grounds of length and inequity, then we must vary time allotments. Here there is much creative leeway.

Mixed Formats

Dick suggests one simple idea which has also been remarked in comments to the above-linked proposals and elsewhere: Play a day of fast games first. The winner would effectively have draw odds for the regular match. That removes the inequity of giving those odds to the defending champion and creates Shahade’s situation where someone always needs to win.

A variant on this that if a regular game finishes drawn within a certain short time, the players follow with one of the Rapid games. Then the Rapid standings would loom over the match. This prefigures ways to integrate the faster-paced games into the scoring system. This month’s London Classic has become one of several elite tournaments and series to blend Classical, Rapid, and Blitz into one system.

Shahade’s new ideas begin by estimating that a 25–30% time reduction would bring only a 30-odd point reduction in quality. Since I measured 200 points from a 75% time reduction, that seems about right. The time saved from today’s standard playing session of up to 7 hours allows the following schedule:

  • You play one game at 90 minutes plus a 20 second increment. The winner gets 10 points the loser gets 0.

  • If that game is drawn you reverse colors and play one game at 20 minutes plus a 10 second increment. The winner gets 7 points and the loser gets 3.

  • If that game is drawn you keep the same colors as the rapid game and play one game at 5 minutes plus a 3 second increment. The winner gets 6 points and the loser gets 4. If this game is drawn, both players get 5 points.

The points division makes a classical win 2.5 times more valuable than a win at Rapid pace, and the latter worth twice a win at Blitz. But he notes that all formats would count and the overall match score would be less often tied. That a Blitz game is a relative crapshoot would reduce playing to preserve a tie over the whole session, though the Black player in the classical would have incentive to stodge and enjoy White in the faster game or games.

Shahade’s idea still gives every playing day the same structure. If we are willing to vary the structure in return for lowering the prescribed use of Rapid, here is my suggestion:

  • Play 2 games at the classic time, so each player gets one turn as White, one as Black.

  • If the score is tied, then play 2 games with 25–30% reduced time.

  • If the score is still tied, then play 2 games per day at 50–60% reduced time.

  • Whenever a 2-game set produced at least one win, the next set uses the longest time format.

  • A tie after the last 2-game set in regulation still brings a one-day fast-chess playoff.

  • The match follows prescribed playing days except when the last 2-game set needs what is otherwise designated the rest day before tiebreaks.

This might set up a subtle feedback reward system for “not playing to draw”—insofar as I can attest that having more time feels nicer when playing. Except for the possible playoff, the Rapid pace would not intrude. The match length expands in the absence of decisive games.

Open Problems

Update 12/4: A review of the match by Carlsen; incisive analysis by former world champion V. Anand. And an extended discussion of both the games and the match format by Steve Gardner in 3 Quarks Daily.

Does the format of championship chess need fixing? Should we just have a 16 or 18-game classical match with fewer rest days where the champion retains the title in a tie? Note this was the rule in last night’s drawn heavyweight boxing fight.

How should the two highlighted issues be weighed—also for possibly reducing draws in other kinds of tournaments?

Shahade’s scoring scheme would also apply to round-robin tournaments, for which soccer-style scoring of 3 points for a win has sometimes been employed to encourage combat. It would also apply to “Swiss System” tournaments since it conserves the total of 10 points between the players—though demands on the tournament officials to oversee so many more rapid and blitz games could be prohibitive.

[added link for a version of playing the fast-game series first, link for 2016 discussion, and a note about match length in my suggestion at the end, plus some word tweaks; added 12/4 update]

12 Comments leave one →
  1. Bill Gasarch permalink
    December 2, 2018 9:28 pm

    I like the idea that was posed of:
    Play two normal games
    If both are draws then
    Play two faster games
    If both are draws then
    Play two faster still games

    Also- 12 games may not be enough

    Not sure how many iterations, maybe just one.
    Reminds me of overtime in football.
    Only drawback (pardon the pun) is that someone who is very good at speed chess may play for a draw.

    Don’t like the idea of forbidding certain openings. Restricts the game too much.

    As for `if its really a tie in the end have the champion retain the title’ – if you
    also do the above (say 16 games, and the 2-2-2 format) that could be okay.

    Were the 12 draws a bad thing?

    For the chess world no since the games were interesting.

    For the general public yes since they tend to think that draws are boring.

  2. Agris Šostaks permalink
    December 3, 2018 1:55 am

    No problems with the rapid tiebrakes. If you run a marathon and come to the finish shoulder by shoulder with your opponent, then you should sprint.

  3. December 3, 2018 4:30 am

    Among tied finals, there was also the 2004 Kramnik-Leko, where the challenger Leko was up by a point before the final game, which Kramnik won to keep his crown.

  4. December 3, 2018 9:56 am

    “and here by the noted economist and blogger Tyler Cowen”

    The same one who’s quite happy to repeat the nonsense about 600 million chess players?

    • December 4, 2018 10:39 pm

      Fans, not players. I don’t play football. The best quantification of “fans” I can think of is to sum up all newspapers that have had chess stories on their sports or arts pages, then use the expected % viewing such a page multiplied by the subscriber base. Russia surely nets more than half its population, say 100M, its neighbors at least 50M. Can we get to 600M?

      Added: The source indeed said “players”. See 2013 review with source links. I stand by my thought above. One might need to postulate and estimate a further deflation factor for the range of readership appeal in which a paper would include a chess story prominently on the page.

  5. none permalink
    December 3, 2018 8:05 pm

    Is there any decent model of drawing probabilities based on ratings of the players and the incentives of the moment? I.e. if someone is playing for a draw as white (or black) because of the match or tournament standings at that moment, how likely are they to get it? Similarly if they are playing for a win?

    I think we need a model like the above to study the anti-draw proposals in a game theoretic framework. Chess fans might have preferred if game 12 was played out but taking the draw was a Nash equilibrium from the players’ perspectives. So it’s unsurprising that they took it.

    Also what’s wrong with a drawn match ending up with co-champions, other than the question of who the next challenger gets to play against? Maybe both co-champions could get seeded into the next Candidates’ cycle with 0.25 point lead, the equivalent of draw odds against the rest of the players.

    • December 4, 2018 10:27 pm

      Not that I know. I’ve been interested to see whether “drawishness” can be quantified from the tables of engine values, which in my case include the changes in values across depthsof search. Note the update with Carlsen’s statements on the match including game 12.

  6. December 6, 2018 3:53 am

    f^{k}(n)= f^{l+m}(n)= \frac{n\times 3^{l} + 3^{l-1}\times 2_{a1}^{x} + 3^{l-2}\times 2_{a2}^{x+y} +…+ 3^{l-l}\times 2_{al}^{x+y+…+z}}{2_{a1}^{x}\times 2_{a2}^{y}\times …\times 2_{al}^{z}\times 2_{al+1}^{m-x-y-…-z}}

  7. Alberto Ibañez permalink
    December 6, 2018 3:57 am


    f^{k}(n)= f^{l+m}(n)= \frac{n\times 3^{l} + 3^{l-1}\times 2_{a1}^{x} + 3^{l-2}\times 2_{a2}^{x+y} +...+ 3^{l-l}\times 2_{al}^{x+y+...+z}}{2_{a1}^{x}\times 2_{a2}^{y}\times ...\times 2_{al}^{z}\times 2_{al+1}^{m-x-y-...-z}}

  8. anonymous permalink
    December 12, 2018 12:01 am

    If of any interest, a team of 4 human players headed by a GM managed to get a draw against Alpha Zero in an exhibition game. The played 1 e4 hoping A0 would play its usual Berlin (which it did), then simplified as fast as they could, unabashedly going for a draw from the very beginning and eventually getting it.

  9. December 18, 2018 12:18 pm

    A question to Ken. Could it be that the higher quality you measured for Magnus in the tie-break is due to his opponent having less time?

    • December 19, 2018 11:55 am

      Jan Arne, you have a point insofar as I believe from observing a few cases that when one player is trounced and seems to put up little resistance, then the statistics for the victorious opponent are often inflated. I do not have large data on this, and it is among many investigations awaiting the resolution of issues with my expanded model and “challenge created” metric—which should help tell such cases. Here, however, Caruana’s play being measured at 2575 was not only typical for Rapid—here I have a lot of data—it was reasonably competitive in general. In chess-judgment terms, Carlsen’s play was simply awesome.

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