Unfought draws – mathematical, logical and practical considerations

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Grandmaster Draws: mathematical, logical and practical considerations

By Kung-Ming Tiong, Kota Kinabalu, Malaysia

In this article, the "grandmaster draw" is dissected and analyzed. Firstly, a more accurate and proper working definition of the problem is given. Secondly, the effects of "grandmaster draws" are outlined and thirdly, some probable motivations for achieving "grandmaster draws" are critiqued by aligning them with the working definition of "grandmaster draws". Fourthly, the merits of proposed solutions to the problem so far are discussed, with some rejected while some requiring improvements. Finally, a mathematical, logical and practical effort to solve the problem is offered (I stop short of claiming a successful solution as the proposal should first be tested in practice, which have not been done).

1. Definition of "grandmaster draws"

Firstly, let us revisit the problem of "grandmaster draws" which have been the source of much debate within the chess community. The problem of "grandmaster draws" warrants a proper working definition and drawing from various comments in Chessbase, the problem can be defined more precisely as:

It is also important to realize that a draw is not a problem when the game is hard-fought. Ron Dorfman pinpoints this exactly when he wrote:

Alexander Aguilar (Chessbase, December 29, 2007) went further to clarify precisely that "playing for a draw is the right of a player". To support this, I would say that playing for a draw actually produces quality, competitive games as well. If only one player plays for a draw and the other is not, the game could still be competitive.

A "grandmaster draw" occurs only when both players "agree" during the course of the game to the make the game uncompetitive.

Eric Duker (Chessbase, December 29, 2007) meanwhile correctly points out the fact that the length of draws is meaningless (this view is supported by Julio Kaplan, Chessbase, December 29, 2007) and does not reflect on the competitiveness of the drawn game. Another important point which Mr. Duker mentioned was chess positions are not easily evaluated by an average chess spectator.

2. The effects of "grandmaster draws"

Secondly, the effects of the "grandmaster draw" problem can be categorized into:

1. the integrity of the game – chess is a fighting, strategy game and as such should be played in accordance to its intended characteristic, where the outcome of a game based on agreement and not on performance is not in the spirit of chess (Mihael Ankerst, Chessbase, December 29, 2007)

2. fairness and reliability of results – results should reflect the true performance of a player (Mihael Ankerst, Chessbase, December 29, 2007) in a tournament/match and thus, any collusion or "team" playing is highly frowned upon and unacceptable

3. spectators' satisfaction – spectators pay to watch players play in tournaments or matches and naturally, expect to see interesting, well-fought games

4. sponsors' satisfaction – sponsors provide the prize fund and bears the organizing costs, therefore, expect well-played games which attracts and pleases spectators, thereby ensuring the commercial value of present as well as future tournaments/matches

3. Motivation for "grandmaster draws"

Thirdly, the motivation behind "grandmaster draws" cited involves one or more of the following:

1. achieving or maintaining a certain ELO rating
   - higher rated players sometimes would rather not risk losing rating points to a (much) lower rated player
   - a player playing to achieve a norm would be content to a achieve draw

2. placing in a tournament, and in effect, financial rewards
   - e.g. when a player is in a position to win a tournament or be within the top positions, the player is more inclined to get a draw to maintain the advantage (say, in a 9-game tournament, a player has achieved 7.5 points with the nearest rival at 7 points, the player with the higher points will be more than willing to achieve a draw)

3. strategical purposes in a tournament or match
   - a player may play for a draw in some games so that he could recuperate and have a better overall performance in a tournament or match (Alexander Aguilar, Chessbase, December 29, 2007)
   - a player may also choose to draw to revive confidence say e.g. where the player have been losing games in succession in a match

4. personal reasons
   - a player who may be feeling sick or exhausted in the middle of a game may choose to offer or play for a draw (Alexander Aguilar (Chessbase, December 29, 2007)
   - perhaps a player already had "nothing" to play for e.g. the title have been decided and even a win would not suffice to tie for first

Granted, the above are all probable motivations for participating in a "grandmaster draw". However, they do not accurately reflect the problem of "grandmaster draws". Going back to the definition of the problem of "grandmaster draws", it should be reiterated that "grandmaster draws" can only happen with the participation of both players. A short, unfought draw is only possible when both players veer to that conclusion. If one player is aiming or playing for a draw and the other player is playing to win or at the very least aiming for a compelling competitive game, the "grandmaster draw" would not occur.

Thus, all the points mentioned above require the extra element of both players having one or more of the above at the same time for a "grandmaster draw" to occur.

Josu Fernandez (Chessbase, November 15, 2007) aptly points out the Minimax Strategy in game theory where:

Except of course:

Enough said.

4. Proposed solutions to "grandmaster draws"

The various proposed solutions (and I only take those which are worth considering) to the problem of "grandmaster draws" so far can be classified into:

1. rules-based
   - the Sofia Rules (proposed by Silvio Danailov), where a player is not allowed to offer a draw directly to the opponent but instead are only allowed through the arbiter in these cases: perpetual check, triple repetitions of the position and theoretically drawn positions. The results of statistical analysis by Josu Fernandez (Chessbase) for a list of top tournaments in 2006 and 2007 show some fascinating results where it shows that draws in 30 moves or less were reduced significantly when the Sofia Rules were applied (3% in 2006 and 7% in 2007; compared to more than 23% in other tournaments in the study). Mr. Fernandez, however, pointed out correctly that the Sofia Rules is only partially successful as it only ensures that draws cannot be arranged or agreed by both players but it "does not ensure that players play to draw, play without taking risks, or that both players play with the intention of reaching a draw, therefore arriving at a balanced situation that will obviously lead to draws."
   This situation regrettably occurs because the role of arbiters in Sofia Rules is limited. Suggestions on how the roles of arbiters can be enhanced for more effective results will be discussed in the next section.

   - The Christmas Rule (mentioned by Brian Karen, Chessbase, December 29, 2007; touted as a promising rule by Chessbase) where when a player proposes a draw, the draw offer would stand for the rest of the game. According to Mr. Karen, this would mean that only a player who is certain the position is "dead draw" would make the draw offer and once offered, it allows the opponent to play sharper lines while keeping the draw in hand. Agreeably, this would result in interesting and enterprising lines of play after the draw offer (it does not, however, ensure that a game would be interesting and enterprising up to the point before the draw offer) but this in effect creates an artificial situation as the player offered the draw now has, literally, nothing to lose and has the incentive to win "to prove to the player offering the draw that the draw offer was a mistake". Thus, out of fear (perhaps), the first player would not offer the draw. This artificial situation clearly benefits the second player. If the first player had misevaluated the position when offering the draw (and this definitely could happen), he is at a disadvantage as there is the possibility that he could now end up losing the game. It is true that the draw offer is discouraged but would the win by the second player be justified when it was achieved under a "nothing to lose" situation where in the normal course of the game (if the draw was not offered), the second player may not have played the interesting and enterprising lines. Certainly (in my opinion), the win can be considered as botched and unworthy.

2. mathematical (i.e. changing the scoring system)
   - Mark Galeck (Chessbase, November 08, 2007) proposed a situation where the value of a draw is not split evenly between white and black to encourage players to play more aggressively but his argument on determining the "imbalance" through negotiation is seriously flawed and unnecessarily complex. A similar proposal, which is simpler and more practical, were offered by Joseph Ellis (Chessbase, November 06, 2007) where the point values of a game (total = 1.0) were arbitrarily determined as: draw with white (0.45), draw with black (0.55), win with white (1.0), win with black (1.1), loss with white (-0.1), loss with black (0.0).
   Both these suggestions were perhaps motivated by the agreed fact in chess that playing white is slightly advantageous (this advantage disappears if white does not maintain his advantage; and in top-flight chess, it is not uncommon to hear these comments emphasized in chess commentaries: GM B wins with the black pieces against GM A!; GM A, playing black, would have a difficult time in the next game against GM B; GM B scores an important win with the black pieces). Therefore, if a black player can win or manages a draw, he should be rewarded more for his efforts. This suggestion is definitely interesting, and worth serious consideration by the chess community (the merits which will be explained in the next section), but there were little enthusiastic response to this suggestion, except for Iman Khandaker (Chessbase, November 8, 2007). Mr. Khandaker went further by saying that this scoring system would sort out tie breaks and make the rating system more accurate.

   - The Bilbao Rule, where a win is given 3 points, a draw 1 point, and no point for a loss. The purpose and effect of this rule is that it increases the incentives for playing for a win. On its own, however, it fails to address the problem of collusion/cheating and may even encourage that problem (Frederic Friedel, Chessbase, November 06, 2007) where there were various negative comments on these by various readers of Chessbase, citing the Bilbao Draw possibility (which was propounded by Ron Dorfman, Chessbase, October 28, 2007). Having said that, the proposal actually has merit in that it recognizes the importance of wins in chess by giving a higher point incentive to wins.

3. practical
   - Rob Eisler (Chessbase, January 04, 2008) proposed the advantage of having arbiters to decide on a dead draw but pointed out incorrectly that this is a negative aspect as it involved a "third party". The arbiters are definitely part of the competition in the game of chess (to jest, I was going to say competitive game of chess, but then realized, as many have, that some games are definitely not competitive). That's why they are there in the first place, to be the referee for games in a tournament or match (but their roles should be expanded).

4. financial
   - Several suggestions were offered (e.g. those mentioned by Frederic Friedel, Chessbase, November 06, 2007) but these financial incentives were only for number of wins; the more games won, the more the financial incentive. Although this is a good incentive and definitely encourages playing to win, it is rather restricted and should be expanded (discussed in the next section).

5. A mathematically, logically and practically (perhaps) satisfying solution

The solution to the problem of "grandmaster draws" should address the characteristics and reasons of the problem as mentioned in Section 3. At the same time, the effects of the proposed solution should fulfill what was mentioned in Section 2.

In view of these, and concluding from the comments in previous sections, the solution to "grandmaster draws" should perhaps be mathematically, logically and practically satisfying. I propose:

Now, some words on the proposed solution.

S1 Encourages fighting games as imbalanced incentives are provided for players of different colors. Since the number of games is usually even in number where a player plays white and black an even number of times (whether it is double round robin in tournaments, double mini-match, or a long match). For illustration, let us look at a simple four-game match simulation for two players, GM A and GM B. Both players played two white and two black games with GM A playing white first.

Original system = 2.0 points each

In the new scoring method, GM A would win the match by virtue of having scored two "stronger draws" with black, whereas in the original scoring system the outcome is equal. (The advantage in this scoring system compared to the one proposed by Joseph Ellis is that it uses only one decimal point which is simpler and more elegant; yes, a mathematician likes these aspects). Another advantage is that it prevents unfought games which sometimes occur at the end of tournaments/match where a player leading the tournament/match would protect his position by playing for a draw. With the imbalance in draw value (as in the example above, after 3 games, the positions are tied with 1.5 points for each player), in the last game GM A would be happy to achieve at least a draw and win the tournament/match but GM B would most likely play for the win as a draw is insufficient for him. The difference in incentive above encouraged different objectives for each player.

S2/S3 If other grandmasters can spot and comment on an uncompetitive draw (Iman Khandakher, Chessbase, November 08, 2007; Julio Kaplan, Chessbase, December 29, 2007; as well as GM commentaries that usually accompany a tournament/match) then they must be able to do the job as competent arbiters of a particular tournament or match. Players must be professionally refereed; as in soccer, as in basketball, as in tennis, as in kumite in karate. There should be substantial disincentive for participation in playing uncompetitive chess games.

S2/S3 On the concern of collusion/cheating, it could be overcome with the small team of independent, qualified grandmaster arbiters who would referee the tournament or match. With professional refereeing of a tournament or match, the effects of probable collusion/cheating is minimized. There should be substantial disincentive for participation in collusion/cheating (based on evidence of course). The remark "I cannot define it but when I see it I know" certainly applies.

S4 Players would be more inclined to win, and more importantly, win successively. In top tournaments or match, it is very much harder to win successively and it is not uncommon to hear that when a player scores 3 or more successive wins, the player is on form and performing extremely well. (Extra bonus points, perhaps 0.2 points, should be awarded as well for achieving the successive wins as a further incentive to play to win.) And there is incentive to produce the best game with white and the best game with black. Best games do not happen because one player plays well; it takes both players to produce an excellent game. And an excellent game need not be a win; it could be a draw as well.

The proposed solution above, in my opinion, fully fulfills the situations described by Jerry Olsen (Chessbase, December 29, 2007) who precisely pointed out that the solution to the problem of "grandmaster draws" should be effective, with minimal side effects, simple and realistic.

The proposed solution, S1-S4, I believe, fulfills these admirably. It is, however, open to debate.

Endnotes

Chess may once be a gentleman's game but the present situation of pervasive "grandmaster draws" clearly shows that more stringent external pressures (peer-review by arbiters and/or computer programs), internal incentives (mathematically better scoring and rating calculation) and financial incentives (for winning streaks and excellent games) must be employed.

For chess to be fully competitively entertaining, the chess world must be prudent, shrewd and decisive in maintaining the quality of play. After all, chess is about quality and not lame games or professionally unsatisfying players.

Kung-Ming Tiong is a mathematician and logician at School of Science and Technology, Universiti Malaysia Sabah, Borneo, Malaysia. He is interested in the application of logical and mathematical-statistical reasoning in aspects of daily life, and the factor of incentives which are related to personal reasoning decisions. Kung-Ming is also an avid chess player who has participated in state and university level competitions. He can be reached at victortiong(at)yahoo.com.

Links – ChessBase articles on unfought draws