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A Logical Flaw

On the previous page, I note that a points schedule for Chess games such as the following:

Checkmate              1000      0
Stalemate               600    400
Bare King               525    475
Perpetual Check         505    495
Material Superiority    501    499

might be enough to make Chess exciting and suspenseful once more.

As an example, I noted that if the main element of suspense was whether White would win by bare king or perpetual check, being able to win by the greater victory more often is still something for the players to struggle over.

But it is also possible that such a measure might not be enough to achieve the desired goal of making Chess more intensely competitive.

Players can choose whether to play defensively or aggressively. This can be reflected in their choice of openings, or in other aspects of play.

Thus, it might be that Black does not have to settle for being defeated by perpetual check most of the time. Instead, by able defensive play, it could be that most games between masters or between grandmasters under this system were decided by material superiority.

While that would still avoid draws, I suspect that this would still leave Chess in an unsatisfying state. Since Chess is played to inflict checkmate, and not to grab material for its own sake, the amount of material left at the end of the game might not correlate well to how well each player was playing during the game.

Of course, the players would end up adjusting their behavior to play the way they need to in order to win under the new setup. But if material superiority ends up being the goal, I fear that the type of Chess that would result would not be an improvement.

It seems to me that the following assessment of what has been considered so far is at least plausible:

The number of levels of victory is not large enough that choosing a point between two of them as the threshhold for a win is likely to be able to make the two sides equal, and having minor victories count is not, by itself, enough to encourage players to take risks and play for at least a win by perpetual check.

But I think that the system of multiple partial victories with graduated scoring that we have is a basis on which we can take the next step that does have a reasonable chance of encouraging players to compete more intensely.

While there are still only a limited number of victory conditions, the points awarded for a win can vary from 501 points to 1000 points.

So one could perhaps think of granting Black more points than White for winning a game, varying this based on how much more often the player with the White pieces wins. While one could equalize the situation of the two colors in this way, it's not clear that this, in itself, would necessarily make a difference to how players choose to play.

Instead, I have another idea. And this is the idea I call Dynamic Scoring.

Before plunging into the heart of that idea, I will first note an adjustment I make to the points schedule we've seen so far.

Because the final level of victory, Material Superiority, although it serves the goal of almost eliminating draws, and making near-draws favor Black, so that White is denied the option of eliminating his disadvantage in scoring by playing more defensively instead of more aggressively, somewhat mitigates against the style of play it is desired to promote, however, I decided to adjust this points schedule to more strongly encourage players to achieve one of the higher levels of victory:

Checkmate              1000 -    0   1000 points value
Stalemate               600 -  400    200 points value
Bare King               550 -  450    100 points value
Perpetual Check         510 -  490     20 points value
Material Superiority    501 -  499      2 points value

Now there is a large jump from Checkmate to Stalemate, so that the game of Chess is not changed too much by this innovation, but the jump from Stalemate to Bare King is considerably smaller, since both were considered victories in some historic forms of Chess, while that from Bare King to Perpetual Check is large again, since perpetual check is particularly easy to achieve in a game of Chess, and so making it a minor victory has to be approached with caution. Narrowing the gap from Stalemate to Bare King produces extra room so that the largest gap can be the one from Perpetual Check to Material Superiority.

Before introducing the victory condition of Material Superiority, I had noted in my description of Dynamic Scoring that adding Perpetual Check as a victory condition could be viewed as "scraping the bottom of the barrel", which is an argument for limiting oneself to the victory conditions originally proposed by Emmanuel Lasker, even though I believe that his point schedule for them was too generous.

Since the details of the system I'm proposing are given on this page, I will recap the details of the minor victories once more:

These minor victories would be awarded points according to a schedule such as the following:

Game Outcome Points for White Points for Black Difference Offset favoring Black Ratio of Differences
White forces checkmate. 1000 0 1000 0 1:1
White forces stalemate. 600 400 200 10 10:11
White bares Black's King. 550 450 100 40 5:9
White gives perpetual check. 510 490 20 40 1:5
White has material superiority. 501 499 2 0 1:1
Draw 500 500 0 0 -
Black has material superiority. 499 501 -2 0 1:1
Black gives perpetual check. 450 550 -100 40 5:1
Black bares White's King. 410 590 -180 40 9:5
Black forces stalemate. 390 610 -220 10 11:10
Black forces checkmate. 0 1000 -1000 0 1:1

How is this points schedule, or one similar to it, expected to benefit the game?

Let's start with a simpler example.

Suppose we take the points schedule with which we began this section:

Checkmate   1     - 0
Stalemate     3/5 -   2/5
Draw          1/2 -   1/2

and modify it to become the following:

White checkmates Black  1     - 0
White stalemates Black    3/5 -   2/5
Draw                      1/2 -   1/2
Black stalemates White    1/3 -   2/3
Black checkmates White  0     - 1

With this point schedule, stalemate is worth only 1/5 of a point to White, but it is worth 1/3 of a point to Black. The value of draws and checkmates is not changed.

Incidentally, note that in addition to resignations and draw offers, it would be necesary under such a system to allow a player who expects to be stalemated but not checkmated to be able to make a semi-resignation offer to his opponent; like a draw offer, it would be subject to acceptance, but the result of acceptance would be splitting the points as if a stalemate had taken place.

What might we hope for as the outcome of such a system?

In the original system, allowing stalemate as a partial victory encourages players to keep fighting during the course of a game if they see hope to win in this fashion, even if they no longer hope to inflict checkmate.

But before the game starts, players may decide which openings they will use, and what general approach they will take to the game. Will it be an aggressive one or a defensive one? Will they take risks to win, or will they devote all their efforts to avoiding a loss?

Allowing stalemate as a partial victory doesn't necessarily, in itself, change how that decision will be made.

If a stalemate is worth more to Black than it is to White, then, what I might hope is that Black will reason as follows: while Black is at a disadvantage, and thus playing a more open game in an effort to win would benefit White more than oneself, playing only a slightly more open game, in an effort to win by stalemate is a reasonable thing to do; even if that creates the risk of White winning by stalemate, as long as Black wins almost as often as White, Black comes out ahead.

And then, I might hope that White would react based on the following reasoning: if I open up the game still more, and the game is won or lost by checkmate, then Black no longer has the benefit of a handicap; checkmate is 1 point for either player.

I might hope both players would reason this way. But what's to stop White from deciding that he is not at a disadvantage when the game is a draw, scored at 1/2-1/2?

Initially, instead of putting a bias in favor of Black in the score for a draw, once I learned of how material was counted up by its point score in some Korean chess tournaments as a way to avoid ties, I decided to add the win condition of Material Superiority to Dynamic Scoring in addition to Stalemate, Bare King, and Perpetual Check. As I also had previously included a 1/2 point bias in favor of Black in counting piece values, partially following the example of Korean Chess, which used a 1 1/2 point bias, it seemed like it would be almost impossible for onver-the-board play using this variant to result in a draw - and, thus, giving a bias in favor of Black to the score for this victory would eliminate any incentive to White to play more defensively instead of more aggressively as a reaction to, for example, an attempt by Black to win by perpetual check.

However, as can be seen from the table above, I have now avoided giving any advantage for Black in the scoring of victories by material superiority. This is because I realized a logical fallacy in doing so.

Initially, one can think of three scenarios existing, representing a more major victory, a minor victory, and a draw:

A Lesser advantage to Black
B Advantage to Black
C No advantage to Black

Black is motivated to play more aggressively to move the situation from C to B, and White is motivated to play more aggressively to move the situation from B to A.

Since White could instead move the situation from B to C, this is a problem. But if I attempt to fix the problem by changing the choices to:

A Lesser advantage to Black
B Advantage to Black
C Greater advantage to Black

then Black has no motive to start the cycle by playing more aggressively to move the situation from C to B.

Given that the flaw in the first table can't be remedied, then, on what basis can I then hope that Dynamic Scoring will work at all?

First, I need to address the situation of what happens when players opt for more defensive or more aggressive strategies.

I'm assuming that with conventional Chess scoring, where both players receive equal credit for any type of victory, even if we modify the normal scoring by admitting minor victories, it will still be true that a player's best outcome comes from careful defensive play, and when players are evenly matched, a player who attempts to open up the position and try to win, while he may succeed some of the time, is placing himself at a disadvantage, even if only a slight one.

Also, while each player can somewhat influence the type of game that is played, the experience of frequent draws in Chess seems to imply that a skilled defensive player can largely impose his will on the game, making the position closed and drawish, despite the efforts of the opponent to open up the position and create attacking possibilities. If a piece sacrifice is sound, it can still also be declined, and sacrifices have been a favored method of achieving positional advantage and attacking opportunities.

But White has the advantage, and since a Chess player does need something better than an endless series of draws to win, it is assumed that White is already, to some extent, motivated to take attacking risks. What is important is to get both players to agree to do this, which is why Dynamic Scoring is still worth doing even though the possibility of White preferring a draw (or the game "temperature" of material advantage, as it were) to a "game temperature" where either player is likely to win by perpetual check or even better remains present.

Another way to think of this is to think of two Chessplayers in a match being trapped in the "Prisoner's Dilemma".

Each player can choose one of two strategies: (A) to play attacking Chess, accepting gambits, choosing wild tactical openings, aiming at flashy sacrificial combinations for himself but with the risk his opponent might achieve the same, or (B) to play sound careful defensive Chess, aiming at accumulating small positional advantages which might, given good fortune, on occasion allow checkmate.

If one player chooses (A) and the other chooses (B), the player choosing (B) is more likely to be the winner of the match. If both choose the same strategy, their chances are even.

But if both players choose (A) instead of both players choosing (B), over the long run, at the metagame level, the prizes for matches such as that one would be larger given greater public interest in the game.

The way to solve this problem is to change Chess so that (B) is a more winning strategy than (A), then players aren't caught in the vicious trap of the Prisoner's Dilemma any longer.

But I found that goal too difficult to achieve without changing Chess too much - I have no doubt that someone could change the powers of the pieces to make Chess favor the offense over the defense; allowing Pawns to move backwards, for example, would make it unreasonable to tie them up in a wall in front of one's King - so instead I aimed at the more modest goal of making the optimal strategy a tiny distance from (B) to (A), with the partial victories also making that strategy look more like (A), thus making it almost as entertaining as (A).

And the reason I have opted for this limited strategy of offering a mere ersatz, over just allowing Pawns to move backwards - the simple, obvious, solution that took me little thought to arrive at - can be found in the words of Siegbert Tarrasch: "Pawns are the soul of Chess": so, here, as with my lower score awarded for Stalemate than the one awarded by Emmanuel Lasker, while I want Chess to gain popularity and money, I refuse to propose that it do so at the cost of losing its soul.

After all, what would it profit Chess to do so? (This is, of course, a paraphrase of Mark 8:36.)

Despite the original rationale for adding the victory level of material superiority of facilitating putting a bias in favor of Black on the scoring of what are, in effect, draws, no longer being seen as valid, I left it in because it still served another purpose; that of making draws nearly impossible as the result of over-the-board play. I removed the 1/2 bias in the sum of piece values in favor of Black because there was no need to make the draw completely impossible. As noted above, to avoid players having to expend effort pointlessly in playing out games that are already decided, not only do they need to be able to offer draws, but they will also need to be able to offer partial resignations: to offer to split the points as if an outcome of victory for the other player by material advantage, perpetual check, bare King, or stalemate is already predetermined, or as if the issue of the game lies between any arbitrary two outcomes.

So, at this point, material advantage is present as a victory condition for two basic reasons: because it serves as a tiebreaker, and because it provides an excuse to award 1000 points for a chess game instead of 100, which makes it easier to set allocations of points for bare King and perpetual check that are easy to work with and follow a uniform pattern.

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