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# Comments by Robert Shimmin

In all honesty, I don't think humans will lose interest in chess just because computers can beat us at it. It's already the case for the vast majority of chess-players, and yet they still play. Or to make another comparison, we didn't stop holding foot-races because we had built machines that can go faster than us.

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Having looked at USP 5,690,334, and noting that it claims 'A method of playing an expanded chess-like game ... comprising the steps of ...,' I have to ask, what is it that the inventor thinks he has actually patented? It seems that, the 'method of playing ... a chess-like game' having been patented, the activity which infringes the patent is the playing of a chess-like game according to that method: i.e., that anyone who plays Falcon Chess infringes the patent! While it is true that many chess-variants are invented to be admired more than played, rarely is this design goal backed up with legal force.

Sometimes changing the board size can actually change to outcome. In shogi on a board of 10x10 or smaller, king and gold vs. king is usually a win. On boards of 11x11 or larger, it is almost never a win. This phenomenon usually rears its head where most of the mating power available is in short-range pieces.

Michael -- Here's the data on the augmented knights, obtained from Zillions vs. Zillions using 5-ply fixed-depth searches. The augmented knights are placed in the rook positions and played against the orthodox army. All the augmented knights give an advantage vs. rooks, probably in part due to their ease of development and the rook's lack thereof. The following values are the various pieces' advantages over the rook, in centipawns. ND, NA = 27 NW = 38 NF = 82 (!) The standard deviation for these measurements is about 10 cP.

On another note, can anyone think of any reason why NF ought to be noticeably stronger than the rook, or any of the other knight+(one atom) pieces? Zillions wins rather more often with it than the others, and I'm trying to puzzle out if this is a quirk of Zillions', or if it could be duplicated in theory or human playtesting.

<i>'Archangel is Gryphon plus Bishop'. If your numbers do not show it as
supeirior to Q, mustn't that be an eror in the numbers?</i>
<p>
The crowded-board mobility calculation for the Gryphon predicts that is very similar in value to a Cardinal. Because the Gryphon can move like F anyway, adding Bishop to its move is really only adding the longer Bishop-moves, and the Archangel is only one half-knight stronger than the Gryphon, and a piece one half-knight stronger than a Cardinal-class piece is a Queen-class piece, or so the numbers say.
<p>
Still, my gut instinct would think, like you, that an archangel would be noticeably superior to the queen. This is something that can only be resolved through playtesting. If the numbers are right, well, this somewhat bolsters our faith in using the mobility calculation to say two pieces are roughly the same in value. If the numbers are wrong, it is very interesting, because both the queen and the archangel are well-balanced pieces, and the first step to improving a theory is trying to identify those cases where it fails.
<p>
After calculating the 'forking power' for a number of these pieces, these are my second thoughts...
<ul>
<li>What I call 'forking power' is really only the average number of two-square combinations attacked by a piece, and therefore treats pins and forks as the same phenomenon. Perhaps this is wrong.
<li>If two pieces have similar mobilities, they will almost certainly have similar FP's. Therefore, the theory can't make any predictions that couldn't also be made by invoking a multi-move mobility component to value, or simlper yet, a power-law dependence of value on mobility. I originally thought up the archangel to think of a piece with similar mobility as the queen, but vastly greater forking power. But this 'vastly greater' wasn't nearly as great as I hoped. :(
<li>Not all two-square combinations of attack are created equal. Maybe two squares in the same direction are more or less valuable than two squares in different directions.
<li>In short, I originally invoked this definition of forking power because it was something that would be calculated without any undue difficulty or assumption, and would be much larger for strong pieces than weak pieces, so with the right scaling factor, could be made to give the queen the right value. It may be an improvement to the theory, but I can't think of any test for it that would distinguish it from a few alternate improvements.
</ul>

In response to Ralph's comment, I've done the forking power calculation for a few more pieces. The magic number is 0.67 Piece Mobility Forking Total % Fork ------------------------------------------------------- Nightrider 7.82 29.53 9.09 14.0 Rook 7.72 29.23 8.97 14.0 One thing I've noticed (and should have expected) is that the 'forking power' value is very close to being proportional to mobility squared. These pieces illustrate about the most variation I can create in FP for 'normal' pieces of about the same mobility. Archangel is gryphon + bishop. Piece Mobility Forking Total % Fork -------------------------------------------------------- Archangel 13.10 98.07 17.32 24.4 Queen 13.44 91.32 17.37 22.6 FAND 13.56 95.38 17.66 23.2 Clearly, these differences are too small to test. So while we know there is some superlinear dependence of value on mobility, we can't yet say whether that is most related to forking power, multi-move mobility, or what.

I wonder whether a player could drop enough anti-tiles to force a draw when each side could not bring enough force to bear on the other.

For anyone who was curious about my previous prediction that an amazon may be a full rook more powerful than the queen, I ran the following experiment. Whether it means anything is up to you to decide. I ran scripted Zillions to play against itself for 500 games where black's queen was promoted to amazon, but black was missing its queenside rook. At strength 4, results were 249-62-189, or 85 ratings points in white's favor. At strength 5, results were 265-57-178, or about 110 ratings points. For comparison, samples of 1000 games each found pawn-and-move to be a 135-point advantage at strength 4 and a 260-point advantage at strength 5, while giving white two opening tempi instead of one is a 50 point advantage at strength 4 and a 140-point advantage a strength 5. Based on this, I would guess that the amazon falls short of being a full rook stronger than the queen by perhaps half a pawn, but that still leaves the amazon a pawn stronger than a queen and a knight.

I've had this thought (2nd-move mobility etc.) before, and I think the correct way to express it is this:
<p>
Averaged over the possible locations on the board, let M1 be the average number of squares that can be attacked in one move (crowded-board mobility), M2 the average number of squares that require two moves to attack, etc. Then the practical value might be some weighted sum of these quantities:
<pre>
PV = k1 M1 + k2 M2 + k3 M3 + ...
</pre>
Of course we don't know these weighting values. But it is reasonable to believe the value of being able to attack a square diminishes by the same factor for each tempo required to do so, and if so, there's really only one adjustable parameter:
<pre>
PV = M1 + k M2 + k^2 M3 + k^3 M4 + ...
</pre>
This is at first sight a very promising approach, since it lets us lump a number of 'weakening' factors such as colorblindness, short range, etc. into one root cause: not being able to get there from here. Also, it provides an alternative explanation for the anomalous extra strength of queen-caliber pieces. Moreover, it would for the first time give a basis for calculating the practical values of pieces that move and capture differently.
<p>
However, there's one problem I've run into when I've pursued thoughts along these lines. The probability of being able to rest on a square is different from the probability of being able to pass through a square, so we need a second 'magic number' to calcuate the various M-values. Also, because the number of squares strong pieces can safely stop on is smaller, it may be necessary to make this value smaller from strong pieces than for weak pieces to account for the levelling effect. (Although I've <i>almost</i> convinced myself the levelling effect may cancel itself out for M1, I'm far less certain that it does for M2, etc.) Anyway, I've rambled about this enough. I think it's a very promising path to go down, but there are at least two arbitrary constants we need to know to go down it.

**Pocket knight**.

**Each player has a knight that he can drop during the game.**(Recognized!)[All Comments] [Add Comment or Rating]

I forget whose idea it was or where I saw it, but the idea was you could pocket any piece (as a move) and place a pocket piece (as a move), and there was no restriction about how many pieces you could have pocketed, but immediately following your drop, your opponent got a double move. My reaction at the time was that the rule changes were only of tactical value because in most situations the doublemove response should be able to easily answer the dropped piece (not to mention that the teleportation of the dropped piece required two tempi to complete, and in the meantime, the piece did had only second-order usefulness. It defended nothing and attacked nothing, but could only threaten to defend or attack things. Granted, it threatened to attack and defend EVERYTHING, but I still think the doublemove response is overkill.)

If the goal is to avoid confusion, then you should be aware that Chess Plus is the name of an existing commercial four-player variant. The author sells it at http://www3.sympatico.ca/thejohnston/chess_plus.htm If you're not concerned about avoiding confusion, then why bother with calling dibs or anysuch? Just look at how many superchesses there are.

Ahem. The contest says the rules will be selected by poll, but I've been unable to find instructions as to how the poll will be conducted. Vote for one, vote for two, rank all in order of your preference, what? Thanks for any clarification.

I don't think the 2-vs-1 scenario is all that big of a problem in 3-player games, since if one player begins to pull ahead, it only becomes natural for the other two to ally against the leader. One problem in 3-player strategy games I've seldom seen solved though is the kingmaker problem. Suppose one player is losing hopelessly, but is either able to hurt one of the other players enough on the way out to give the other one the game, or is able to determine the winner more directly (by deciding whose mating trap to walk his king into...). Then the winner ends up being decided not by who played better, but by whom the _loser_ was feeling better disposed towards.

Offhand, I suggest that since we're voting in the rules one at a time, we might just agree to the convention that later rules supercede earlier ones. At least this makes the most sense to me. Any other thoughts?

When I'd read it the first time, my interpretation was that a player could deliberately place the king in check and force the opponent to capture it, but that if the opponent checked the king, that check had to be lifted or the game was lost. (ie, that placing the king in check was legal as a deliberate sacrifice, but that if the the opponenet started the check, it had to be responded to normally.) This made sense to me because it kept the king sacrifice (with mandatory capture) open as a tactical option, but a multi-move mating combination found by the opponent still worked. But I can definitely see Glenn's interpretation, too. Would it be rude to ask the inventor for one final clarification on the issue?

I can't imagine this game was ever played much, or if played much, ever played well. If the second player adopts a symmetrical defense, then the first player is forced to sacrifice material, possibly a lot of material, in order to break symmetry. The only way to break symmetry is to capture one ship (rook) with another down a file, or to check the enemy king. But either of these require opening a gap in the pawn structure, and the second player can always come out ahead materially in the process of opening such a gap.

And there's another detail that all of us forgot! When the crooked bishop is on the edge of the board, one of its paths in the (0,2) direction is blocked, even though the (0,2) square might be on the board. When I included this in my evaluation, I got the result that the crooked bishop has about 1.2 times the mobility of a rook for a broad range of reasonable values of the magic number.

<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.

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.

I'm going to weigh in on Peter's side. The probability of getting to (0,4) should be 0.51793. There are two ways to get there. One requires (1,1) and (1,3) open. The other requires (-1,1) and (-1,3) open. Both require (0,2) open. The probability of (1,1) and (1,3) open is 0.49. (Same for the other route.) So the probability of this part of each route being closed is 0.51. So the probability of both being closed is 0.51^2 = .2601, and the probability of at least one of them being open is 1-.2601 = .7399. Of course, in either case (0,2) must be open, so the final probability of getting to (0,4) is .51793. This generalizes to the probability of getting to (0,2n) being (1-(1-p^n)^2)*p^(n-1), where p is the magic number. This is equivalent to Peter's original formula, although a more compact writing of the formula is (2-p^n)*p^(2n-1) or 2*p^(2n-1) - p^(3n-1). If you need help convincing yourself your original statement > Would 0.91 times 0.7 times 0.7 be correct? Yes, this is the answer > to 'it can move there if either d2 or f2 is empty AND e3 is empty AND > the corresponding square (d4 if d2, or f4 if f2) is empty'. is wrong, try thinking about it this way: suppose both d2 and f2 are empty. Then the crooked bishop may pass through either d4 or f4 on its way to e5. And the probability of being blocked on BOTH squares is 0.09, not 0.3. So the difference between your calculation and the answer Peter and I get lies entirely in the cases where d2 and f2 are both empty. In these cases, it doesn't matter which square (d2 or f2) it passed through to get to e3, so no binding decision about which path was taken has been made yet. In fact, no such decision has to be made until the first d-file or f-file blockade is encountered. Your method of calculation forces the decision to be made at the very first step, even when both paths are open. Make any sense?

Note: in the graphical diagram of the opening setup, the black viceroy and squire on files i and j have been reversed from the other two descriptions on the page. By symmetry arguments, I assume the textual descriptions are correct?

Interesting side note: the crab, although able to visit every square on the board, must change 'color' in a much more complicated way than the fully-powered knight. You can divide the squares of the board into six sets in such a way that the crab must cycle through these sets as it moves. Just as the knight must change color with each move, the crab's moves must go from squares of type 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 1... This makes using the crab a rather tactical experience, since once moved, it takes a total of six moves to get back to the square it started on. In practical terms, this means that if it ever relinquishes an attack on a particular square, it is unlikely to ever be able to return to it.

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