Dynamic Scoring

Since the section of this site about the game of Chess was originally written, I have learned of how a change in how games of Go are scored in Japan has been successful in promoting attacking play, having been introduced because of a tendency of the first player to exploit his advantage through defensive play. This has led me to examine more closely how a change to the scoring of Chess might be able to have a greater beneficial effect than simply reducing the frequency of draws.

I have given the asymmetrical scoring system I am proposing the name of Dynamic Scoring to emphasize its objective of encouraging more aggressive play. Apparently, no one has already trademarked this name for another scheme of scoring chess games or for another sport or game. Instead, this name is only already in use for a system of evaluating policy changes which has as its purpose showing that tax cuts for the rich aren't all bad because they stimulate the economy; this should not lead to confusion.

The scheme I currently propose follows the idea advanced previously of giving partial credit for stalemate by ensuring that the partial credit given for stalemate, bare King, and perpetual check is much smaller than that given for checkmate.

Note that, just as perpetual check only ends the game if a player intends to give perpetual check, win by bare King is optional; the game does not end when one player only has a King if the other player intends to continue to give mate - stalemate as well as checkmate.

In the table below, after the columns which give how the game has ended, and the number of points (out of 100) to be rewarded to White and Black respectively, the fourth column shows the difference between the scores of White and Black, which is what gives the true value to the players of the points awarded, and the fifth column shows the difference between the points awarded to Black and to White when each one respectively brings about the same game outcome in his own favor.

Because there is a bias in favor of Black for the outcomes involving partial credit, Black is given an incentive to play aggressively and take risks to achieve a win. Because the bias is smaller the more decisive the level of victory, White also has an incentive to play more aggressively.

Thus, since a win by perpetual check is worth 14 points for Black, but only 4 points for White, if Black, instead of playing so defensively that the game is almost always a draw, plays aggressively enough that the game often ends in a victory by perpetual check, then Black would come out the winner, even if White won this way three times as often as Black did.

If White also plays more aggressively, however, then the victory might be by bare King, which is worth 18 points to Black, but 12 points to White, which only requires White to win three times to two for Black to come out even, or it might be by stalemate, which is worth 22 points to Black, but 20 points to White, only an 11:10 ratio. And checkmate, like a draw, has the same value to both players, so White can win if he checkmates Black only slightly more often than Black checkmates White.

While at one time perpetual check was one of the standard drawing conditions for Chess, recent revisions to the rules of Chess have removed it, on the basis that it is unnecessary as a condition, since in such a situation, either a draw by threefold repetition or one by agreement is to be expected, and in the absence of the latter, a draw by the 50-move rule can be forced. Thus, it is possible that some readers of this page might be unfamiliar with the term. Perpetual check exists where it can be seen that one player can make an unending series of checks, without ever having his move end with the other player not in check, if the 50-move rule and the rule about threefold repetition are disregarded, and that player announces the intent to do so for as long as necessary to force a draw through one of those rules.

Other variations on this, where the credit for partial victories is larger, are illustrated below, to make it clearer how this scheme might work.

Komidashi for Chess

In Japan, when it was perceived that the level of play in Go had advanced to the point that the player who set the first stone on the board, which is the one with the black stones, could by playing skillfully but in a defensive manner assure himself of victory, usually by a margin of three points on the board, a rule was introduced whereby black would have to win by more than a given margin, known by the name komidashi, or komi for short, or the game would be counted as a victory for white.

Those who feel the game of Chess should be changed in some way to address perceived problems would often rank as the foremost of these problems the trend to defensive positional play that marks the Modern era of Chess strategy in contrast to the Romantic era. Unlike tossing out opening theory by shuffling the pieces around, however, it seems to be a much more difficult problem to address; the fact that a similar problem has been addressed for the game of Go with a considerable degree of effectiveness would seem to be an example to learn from.

Many references to the game of Wei Ch'i or Go note that it has a finely-graded handicap system, effected by placing additional stones on the board for one player at the start of the game. In Chess, on the other hand, while giving one player Pawn odds, Knight odds, Rook odds, or even Queen odds is possible, this is coarsely-grained and is felt to distort the game.

The winner at Chess is determined by checkmate, which is an all-or-nothing matter, and so komidashi is another area where Go has a clear advantage over Chess very much analogous to the one it enjoys with its handicap system, but one of which fewer authors have taken note.

Just as Chess does have a handicap system, but one that is crude and clumsy compared to that of Go, one could, by analogy, institute unequal victory conditions for White and Black in Chess using the tools that are available.

One could require the White player to give checkmate to win, while giving Black a victory as well if he forces stalemate, or bares the opposing King.

However, this misses one of the other characteristics of komidashi. When first instituted, it was an offset of 2 1/2 or 3 1/2 points; currently, it tends to run at 6 1/2 or 7 1/2 points. The odd half point essentially makes a draw impossible.

To narrow down the wide space which Chess offers for draws, within which a player with a small advantage is unable to force mate, to the minimum, it would be necessary to allow both players to claim a full victory with bare King. As noted above, making this a full victory rather than a partial one would distort endgame theory in Chess, which is a drawback to this measure.

If the victory conditions are symmetrical, this means that another way would be needed to equalize the situations of the players with the White and Black pieces respectively. One way to do this would be to deprive White, on his first move only, of the option of moving a Pawn two steps forwards. The availability of N-KB3 (Nf3) as an opening, and the viability of openings such as P-K3 (e3) and P-Q3 (d3) for White, however, mean that this is unlikely to be sufficient to provide the needed balance.

In pursuit of an exact balance between White and Black, I have proposed a chess variant which I have called Temporary Marsellais Chess; this is now discussed on a later page.

Instead, I now propose changing only how games of chess are scored, rather than having two moves per player in the opening only. I attempt to redress the balance between Black and White by giving Black more points for partial victories, but then as the game moves towards the full victory of checkmate, the degree to which Black is favored is reduced. This is intended to promote aggressive play by both players, for reasons illustrated below:

Asymmetric Scoring

The system of komidashi encourages the first player to play aggressively for a win instead of defensively because it changes the threshhold for a win. If, instead, one simply gave extra points for a win by the second player, since players alternate between playing the White and Black pieces - or the Black and White stones - no change in the kind of behavior that is optimum for winning results.

But it may be possible to drive a desired pattern of behavior by changing the scoring of Chess, without otherwise changing the rules, if one actually changes the differences between scores for some outcomes.

One proposal that has been made to reduce draws in Chess would be to give each player only 1/3 of a point for a draw. Because this would make Chess no longer a zero-sum game, it would be difficult to take this change into account for things like tournament pairings and Elo ratings, and it would be more significant in tournaments than in matches.

To avoid this, but to have the result of encouraging attacking play instead of defensive play, perhaps the following scheme, based on awarding 10 points instead of 1 point for the game, might be effective:

Since White has an advantage, White is given a bigger incentive to play aggressively, and accumulate a larger advantage, because the more decisively the game is resolved, the less the scoring favors Black over White. Defensive play where the result is a draw, or a win by the narrowest of margins, bare King, favors Black, but since Black still has the disadvantage of moving second, Black cannot expect to win from defensive play, particularly when only a bare King is still a win, even if a smaller one, for White. Instead, if Black plays in an aggressive but sound manner, so that he has some chance of winning a minor victory, then the advantage in scoring for the smaller victories might more than make up for the fact that aggressive play means taking risks.

I had originally thought to award 4 points to White, and 6 points to Black, for a draw, but the result of that would be that White and Black both gain 4 points over a draw by baring the opponent's King, which would mean that except for a constant offset in Black's favor that cancels out as colors alternate, the only effect would be to give Black less credit for a better victory. If draws are to favor Black, the difference would have to be a smaller one, such as 4.5 to 5.5 points or 4.9 to 5.1 points.

As the intent of allowing bare King as a victory is merely to narrow the space for draws, and not to complicate checkmating, a player who bares the opponent's King as part of the process of giving mate (including stalemate as well as checkmate) is to be allowed to indicate at that time that he intends to press on rather than claim the partial victory of bare King.

If the space for draws is not narrow enough, however, if an ending situation of King and Knight against King is still difficult to achieve, then with this scoring system, both players would still play defensively to avoid the other player gaining this advantage. Since the critical thing is, therefore, to make draws unlikely, perhaps it's worth noting that we can do even better than the scheme suggested above:

Of course, by admitting even perpetual check as a way to win the game, it may be unclear if I should describe it as doing even better when perhaps scraping the bottom of the barrel might be more appropriate. Also, the amount by which these schemes favor Black is likely more than is needed to compensate for White's inherent advantage of the first move; this is to make it simple to understand the underlying principle on which the scheme is based (which a picture below will help with further).

The scores given in the two columns on the left side of the chart are simple ones, illustrative of how the scheme operates; those in the two columns on the right are closer to those which might be used in actual play. The details of the scoring system required to genuinely encourage more aggressive play would need to be worked out with some experience of actual play wherein perpetual check, bare king, and stalemate are all eligible for partial credit.

Why It May Work

Why would such a scheme have a chance of producing the desired result of more risk-taking aggressive play? Giving credit for a partial victory might make it more important to play defensively. It might make sense to give more points to Black, because Black is at a disadvantage, but what would that really have to do with aggressive play?

The diagram above illustrates the idea behind this scheme.

Graph A shows the distribution of outcomes when both players play defensively. The outcome is often a draw, but since White has an advantage with the first move, White is able, a significant amount of the time, to push the game enough to his advantage to win through perpetual check.

Graph B shows what happens when Black plays aggressively. The advantage tilts even more towards White. But now outcomes further out from the center are more likely. Sometimes Black wins through perpetual check. On the right side of the diagram, the scoring scheme is illustrated. The scores don't reflect the degree to which either player has pushed the game to his advantage; instead, they are a uniform scale, but one that favors Black. By playing aggressively, now Black wins some of the time, and, because of the scoring, that is to his benefit.

Graph C shows what happens when White also plays aggressively. Now, White has lost some of his advantage, but because outcomes even further from the center are possible, outcomes where the scoring does not favor Black as strongly are more common. Black receives much more partial credit than White for a win by perpetual check, the smallest win, but White and Black get equal credit for winning by checkmate.

The curves show how the advantage to each player at the end of the game is distributed; the rectangle on the right shows, on its left side, the outcomes that result from different advantages, and on its right side, the score that is given for each outcome.

To provide a more convincing demonstration, I attempted a numerical example with arbitrary figures. It did not lead to the conclusion illustrated by the diagram above, but even so, it showed that a little aggressive play could be encouraged.

Let us say that the outcome when two chessplayers individually choose between aggressive and defensive play is like this:

With even scoring, 10 points for the winner, or 5 points for both in a draw, the payoff matrix for the two strategies can be constructed:

As can be seen from the payoff matrix, defensive play dominates aggressive play for both players.

Let us now add an easier type of victory, such as win by stalemate, bare king, or perpetual check. It will eat up some of the draws. White's advantage might be bigger, and while aggressive play would increase the chance of such a victory, it wouldn't respond as dramatically, because unlike checkmate, it is a less distant goal. So, as an illustration, we might have the outcomes of the two strategies follow this scheme:

and we follow a scoring scheme of the type outlined above. The points still are split 10-0 or 0-10 when one player checkmates the other. A draw is still 5-5. But a small win by White is 6-4, while a small win by Black is 1-9. What does the payoff matrix look like?

Defensive play dominates aggressive play for White.

For Black, defensive play is better if White plays aggressively, but aggressive play is better if White plays defensively. Since defensive play dominates aggressive play for White, this game has a saddle point favoring defensive play by White and aggressive play by Black, so at least some aggressive play could be favored by the type of scheme I am proposing under some situations.

Since we have now moved to a discussion of game theory, it might be worth taking another look at the diagram above, showing a Gaussian normal distribution of advantages at the end of the game. In that case, unlike in the numerical example, while it is still true that if White chooses defensive play, it is to Black's advantage to choose aggressive play, it becomes also true that when Black chooses aggressive play, it is to White's advantage to choose aggressive play as well.

But that doesn't mean everything is settled. When White chooses aggressive play, Black can choose defensive play, moving the distribution back to something like the curve labelled B, except favoring Black somewhat more. This undoes the advantage White gained from choosing aggressive play. And if Black chooses defensive play, it is to White's advantage to choose defensive play as well.

So we seem to be going around in circles. This means that the situation the graph illustrates corresponds to a different kind of payoff matrix than the one we saw in our numerical example. This one leads to a mixed strategy. An example of this situation would be the Japanese children's game of scissors-paper-stone. Scissors cut paper, paper wraps stone, stone blunts scissors. So the way to do best at this game is to choose the three strategies of scissors, paper, and stone unpredictably with equal probability.

In chess, though, each player makes one move at a time; so we don't have a case where two previously chosen strategies are simultaneously revealed with no opportunity to backtrack. Thus, the practical result of a scoring system which leads to a payoff matrix where a mixed strategy of aggressive and defensive play as optimum is that both players would pursue, for as long as possible, a course of play with both aggressive and defensive possibilities, until a situation arises where a move that reveals one's intentions has a large enough advantage to offset the advantage gained by the other player who now gets to choose between aggressive or defensive play based on your choice.

More Traditional Alternatives

Because the scheme given above involves giving partial credit for stalemate, it is a radical change to the rules of Chess. The rationale behind making such a change is that:

• since checkmate is difficult to achieve, it must also be easy to prevent,
• it would be difficult for any disincentive to drawing to encourage players to risk losing in order to have a chance to win, unless the added chance to win from more adventurous play were genuinely only slightly smaller than the added risk of loss from such play.

Therefore, it seems very difficult for a disincentive to draw to be made to work, Allowing stalemate, bare King, and even perpetual check to be scored, however, changes the underlying landscape, so that achieving some degree of victory is easier, and avoiding the risk of a degree of loss is more difficult.

A simple disincentive to drawing that has been used in tournaments is to score Chess games as follows:

White checkmates:   1    0
Draw:              1/3  1/3
Black checkmates:   0    1

One objection I have raised to this scheme is that it seemed to me that it is not applicable to matches, and, as the World Championship match is the one event in Chess that receives some degree of attention by the general public, even if this is a successful measure, it will only liven things up for the Chess community, but it will not really address the issue of making the game more widely popular.

In the case of matches, the scheme most often proposed is that instead of having, for example, a series of up to 24 games, in which the first player to achieve a score of 12 1/2 points winning, or the champion retaining his title on a score of 12 points to 12 points, one might instead award the match to the first player to win six games.

In practice, however, instead of avoiding draws, this seems to have had the effect of causing matches to go on for an unpredictably long time, which creates practical difficulties in organizing them.

Thus, I thought that perhaps one could improve matters while avoiding this problem by taking a more realistic measure.

Let us say that the match is limited absolutely to a long 30 games, but is intended to end earlier, with three possible ways to win:

• The first player to win four games wins.
• The first player to score 15 1/2 points, with 1/2 points awarded normally for a draw, wins.
• If a score of 15-15 is reached, the champion retains his title.

This seems to me to be a move in the direction of scoring by wins, and a move as far as possible as is practical.

The effect is for games ending in checkmate to shorten the length of the match. And this raises another question: could the scheme of scoring noted above as useful for tournaments be effective as well in matches under suitable rules?

So, if games are scored as:

White checkmates:   1    0
Draw:              1/3  1/3
Black checkmates:   0    1

and the winner is either:

• the first player to score 12 1/3 points,
• the champion retaining his title if a score of 12-12 is reached

would this have a similar effect, since, for example, the match could consist either of 36 draws, or 24 games consisting of 12 wins by each player?

I suspect that since a draw is still better than a loss, even though a win is better than a draw, the risk of playing more adventurously will still outweigh the risk of playing an additional game. Only by setting a low threshhold for a victory by wins alone is it hard to avoid having the match decided in this way.