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Has any extensive research been done into how values change when the board is expanded to 10x10? 1) How much weaker do the stepper atoms become? 2) Does the ratio of power between rooks, bishops and knightriders change? If so, which pieces become weaker and which become stronger, and by how much? 3) How much is a 3,1 rider or 3,2 rider worth?

<i> Remember, the principle is that Pawn and move may be 2:1 odds if the
stronger player is rated 1800 USCF and the weaker is 1600; but if the
stronger player is 2600, he can only give P+move to a 2200 (numbers are
made-up examples for rhetorical effect).
</i>
<p>
If chess is a theoretical draw, then this principle won't always apply, or rather it applies in a weaker form to small advantages than to big ones.
<p>
Let's assume chess is a theoretical draw. Two equally matched and very strong chess players (stronger than any grandmaster, but not perfect -- they usually lose when they play against God, but draw often enough to make things interesting) play at odds. The odds are small enough that the game is still a theoretical draw, but large enough that any larger advantage would be a theoretical win. Half the time, the side giving odds will make that infinitessimal slip-up that allows the other side to win, but the other half of the time, the side given odds will make that infinitessimal slip-up that gives the other side enough breathing space to ensure a draw. So the value of these odds to these inhumanly strong players is 3:1 money, or 190 ratings points.
<p>
When God plays Himself at these same odds, the game is always a draw, since He plays perfectly and the game is a theoretical draw. When weaker chess players play at these odds, the side giving odds may occasionally win, and the odds are worth somewhat less than 190 points.
<p>
The mathematics that inspired this thought experiment yields the following results:
<p>
(1) If chess is a theoretical draw, then no odds small enough to keep the game a theoretical draw are worth more than 190 points at any level of play.
<p>
(2) All such small odds have some level of play at which they are worth a maximum. At weaker levels of play, the side given odds is too weak to fully exploit them, and at stronger levels of play, the side giving odds is strong enough to overcome its disadvantage.
<p>
(3) As the odds increase, this critical level of play above which stronger players actually notice the odds <i>less</i> also increases.
<p>
(4) For even the smallest odds, however, this critical level of play is stronger (not too much stronger, but stronger nonetheless) than any human being plays.

> became convinced that with good statistics (thousands of compiled games), Ja, ja, I said that more or less. My vision was to have rated opponents play odds games, and it did not matter whether the opponents were humans or computers (except that there are now no computers that play as weakly as this 2330-rated human FM!). Remember, the principle is that Pawn and move may be 2:1 odds if the stronger player is rated 1800 USCF and the weaker is 1600; but if the stronger player is 2600, he can only give P+move to a 2200 (numbers are made-up examples for rhetorical effect). I depend on you to complete this work. I took it as far as I could, and now that I am older I find that I cannot wrap my mind around it as I once could. Expect no more goodies in this field from me.

>>So I wrote some scripts to play Zillions against itself and compile the results whenever I'm not using my computer<< Could I get a copy of those scripts? I've been doing this manually, and didn't realize it could be scripted. How did you script it?

On a re-read of parts 2-4 of About the Values of Chesspieces, I finally became convinced that with good statistics (thousands of compiled games), two things should be possible. (1) A workable handicap system for chess players of different rank that could tie the 19th-century handicaps of two moves, pawn and move, knight odds, etc. in with the modern rating system. (2) A theory of piece values that has better predictive power than what we have now. So I wrote some scripts to play Zillions against itself and compile the results whenever I'm not using my computer, and if, Zillions' quirks notwithstanding, these numbers have any relation to games played by human beings, some of my initial results are intriguing. Among them are (1) Ralph Betza's intuition in designing Chigorin Chess seems to be correct: averaged over the course of the entire game, knights may be more valuable than bishops. Conventional wisdom holds the opposite because by the time the game gets around to trading knights for bishops, things have often opened up enough to close the gap between them. (2) The 19th-century source was nearly dead-on in calling pawn-and-move at 2:1 money odds -- if we can assume Zillions' strength is on par with that of the average 19th-century club player, then my statistics so far indicate pawn-and-move is worth about 130 USCF ratings points. Knight or bishop odds seems to be around 400 points so far, and rook odds (with nowhere near enough games to have good statistics) seems to be worth a little over 500 points. Of course, since advantages become bigger with the increasing skill of the players, it very much matters _which_ 500 points those are... Anyway, as I've alluded above, the chief barrier in proceeding with this work, or even in determining whether the numbers have any value at all, is getting enough games. Figuring that I actually have to use my machine, I can only crank out a few hundred games a day. So if anyone has interest in donating their computer's downtime to the cause, please email me at [email protected] with the particulars of the machine (processor, memory, and operating system) you'd like to run it on, and I'll send you my scripts for automating Zillions. The first step will be seeing how much Zillions' strength varies from system to system, but after that, we may actually be able to answer some of these questions.

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10x10 BOARDS:See Opulent Lemurian Shatranj for my opinion on the values of some of Joe Joyce's favorite pieces, including the General, which moves like a nonroyal King. See Unicorn Great Chess for Chancellor and Queen and Unicorn.At the end of my

Notesto Rose Chess XII are some brief comments on my theory of relative endgame values of pieces on a 12x12 board. I like to hold the Knight constant at 300 points on all boards. Rooks and Bishops increase in value on the larger boards, while one-step pieces like the General, Silver General, Gold General decrease in value. Pawns have constantly shifting values during the course of a game - it is simplest to just assign 100 points value. For what its worth, I have also considered: Rook = 700 points on a 16x16 board, Knight = 300 points, General (also called Commoner) around 275 points. These values are obtained by multiplying the 8x8 values by 1.4 for pieces with limitless range and 0.7 for one-step pieces.