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The Future, Future_2005, has those fifteen comments locked in before upgrade CVP comments. Mentioned as computer-difficult by Lavieri there in original ''Future'' eight years ago is Arimaa.
Hi Kenneth, you are right. It is more important the horizontal of action (the exponent) than the branching factor (the base) for that exponentiation formula. But I believe that the formula num_turns^branch_fact is not a reliable measure for the depth of a game. I have two arguments: 1. We should count the number of important decisions per turn, not the general number of move per turn. Take this stupid game played on a goban: in every turn the player drop a single stone, the winning player is the last dropping player. This game is as deep as Go with that formula, actually it has zero depth. 2. If we have an opaque and/or a highly tactical game (with zero strategy) that has an average of 300 turn, those turns don't add extra depth to the game. About the first point, we usually can ignore it. But I believe the second point is important for designing a game. If we want true deep games, difficult for AI, we have to focus on big branching factor and extremely stratigical games. In my view a strategical game is a game in which is "easy" to look ahead and that this looking ahead is the central skill for winning the game. I’m trying to follow this way designing my cv Kingdrops: http://www.chessvariants.org/index/displaycomment.php?commentid=29503
Forced to choose, Mats Winther's two future chesses would probably be first Future_Chess, not quite preserving 64 squares but going to 68, and second, what has with indulgence other informal name ''pawn-Knight-drive,'' Valiant_Chess.
Gustavian board of 68 would seem to be minimal increment to 1000-year western practice and is itself over 200 years, Board_68, providing a slot for a single paired piece addition without awkward unaesthetic drop mechanism, as Winther's cvs do for many a new Bifurcator piece-type on that 68 squares. Individual subvariants of Future Chess, otherwise preserving the mobilization phase, could fill the vacant corner squares with one piece-type two a side at a time, bifurcator or otherwise.
The reason why the Capablanca big-board variants haven't caught on is because the board alters the relation between pieces. There are now ten pawn, a fact which affects the pawn value. The center of the board isn't sharply defined anymore. The knight is worth a pawn less than a bishop. The rook is now worth equally much as a bishop + knight. There is now plenty of space on the board, so one is unlikely to achieve spatial advantage. All this means that important strategical themes are lost, especially the important relation between knight and bishop, and it's hardly possible to sacrifice a rook for a knight anymore. However, the Gustavian board retains all the relations of Fide chess. There are only eight pawns, and the relation between knight and bishop is retained. Possibly, the knight value increases slightly because of the extra corner square, which makes it easier to maneuver with the knight. The queen is probably somewhat more valuable since it can now invade the enemy position via the extra corner square. I hold that the 68-squared Gustavian board is much superior to the 90-squared board, although it also depends on the rules of the game. However, if we simply add the chancellor and the archbishop to the Gustavian board, we get a much better game than Capablanca Chess or Gothic Chess. This is because all the fine nuances of Fide-chess are retained. http://hem.passagen.se/melki9/gustaviii.htm /M. Winther
"The reason why the Capablanca big-board variants haven't caught on is because the board alters the relation between pieces. There are now ten pawn, a fact which affects the pawn value. The center of the board isn't sharply defined anymore. The knight is worth a pawn less than a bishop. The rook is now worth equally much as a bishop + knight. There is now plenty of space on the board, so one is unlikely to achieve spatial advantage. All this means that important strategical themes are lost, especially the important relation between knight and bishop, and it's hardly possible to sacrifice a rook for a knight anymore."
I understood that this was Capablanca's staled aim. He felt that advanced knowledge of strategy was making Chess stale, and that a game for which that knowledge base had yet to be built up would make for more interesting play for players and audience alike.
H. G. Muller wrote on Wed, Jan 9, 2013 03:51 AM EST:> The knight is worth a pawn less than a bishop. The rook is now worth > equally much as a bishop + knight. There is now plenty of space on > the board, so one is unlikely to achieve spatial advantage. If you are talking about 10x8 board, note that the intrinsic value of the Knight is only 0.5 Pawn below Bishop, and that the other 0.5 Pawn comes from the Bishop-pair bonus. So a Knight is only one Pawn below the FIRST Bishop you capture. Furthermore, the opening value of the Rook seems to be only equivalent to (first) Bishop + Pawn, and a Pawn is worth significantly less than Knight. Once on open files the Rook value could be upgraded by 0.25 Pawn.
Regardless of the exact values, it seems like many very fine strategical qualities are lost on the 9x10 board. The space factor, so important in Fide-chess, has taken on a completely different meaning. Suddenly there are large areas of space always available. Of course, there are new aspects of the game, such as the enormous tactical capacities of the super-knights. This is great fun, but many important factors are lost, too. However, with the Gustavian board they are retained, while new tactical qualities can be added. I don't repudiate the Capablanca variants; they are an interesting complement. But I don't think they will ever become popular. Seirawan Chess implements the super-knights on the 8x8 board, however, they are introduced in an erratic manner. To place them on the extra corner squares of the Gustavian board is much more to the taste of the modern player, I think. Gustav III's Chess, with Amazons on the extra squares, is actually a very good game. A preset is here: http://www.chessvariants.org/index/msdisplay.php?itemid=MPgustaviiisches /M. Winther
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The horizon of action, also called the depth d, has more influence than the branching factor b. The overall complexity is b^d (b raised to the power of d), which can be re-written as 2^{d*log(b)}. A doubling of the branching factor adds only 1 to log(b), and hence adds d to the exponent. Whereas, a doubling of d adds d*log(b) to the exponent. OK, for b as low as 2 this makes no difference, but for a healthy b like 30 it does. Anyway, my intuition for why Go is so much more difficult points to the long horizon, rather than the up-to-361 branching factor. Also in Shogi, storming with pawns and the weaker pieces is much of the strategy, and takes a long time---especially using paratroops. That horizon, rather than the branching of paratroops, explains the greater difficulty for computers, IMHO. More simply put, I believe in a common version of Moore's Law for games, phrased in terms of Laszlo Mero's concept of "depth of a game." Namely, say two players are a "class unit" apart if the stronger one takes 75% of the points in head-to-head play. Under the Elo system this corresponds to roughly a 200-point rating difference (was exactly so before a re-basing on logistic curves). The depth is the number of class units from "adult beginner" to world champion. When I first saw reference to Mero's work 20 years ago, this was reckoned as 11 for chess (600 thru 2800), 14 for Shogi, and "25-40" (sic!) for Go. If the CCRL 3200+ ratings for top programs are accurate, then that corresponds well to computers knocking on the door of the best Shogi players but being still tangibly behind. That is, based on Moore's Law, top programs like Houdini on 8-core hardware are at "Depth 13". Use of Monte-Carlo techniques may have computers further along in Go than this idea would predict, however.