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I don´t agree that potential advantage in the exchange comes ever from significant differences in piece values, and good examples comes from positional games like Xian-Qi or Hexetera/Etcetera. In Hexetera, my subjective estimation of values are, fixing Pawn in 1: Man 1.5, Flyer-Elephant 2.5, Guardian 4, Rook 5.5; but in this game the usual exchanges for advantage are strictly positional, and many times (really many times)this kind of exchange is performed exchanging a major piece for the capture a piece of less value, i.e., conceeding material. In this game there is not permissed to change pieces of the same type, making this game almost estrictly positional, and sacrifices are not only usual, but many times necessary for a definition, finishing a game in around 40 moves. In Xiang-Qi, material advantage is not as important as positional advantage, and other of my games, Deneb, is clearly a very positional game, being that all the major pieces have approximately the same medium value, around a little less than a FIDE-Rook, but the extinction rules induce games that lasts in average 25-35 moves. It is difficult establish good measures for positional games in which material advantages are not determinant. I´ll be back with other games in which good measures are not easy to stablish properly.
Excellent analysis, Antoine. I have to add some comments to your lines, and some other comments about George´s interesting ideas. I think that measures are good for a first view in abstract, but the measures needed are not ever easy to standarize, and I have a lot of examples. I´ll coming back to this in the next days, when I have a bit of time to write something about it.
Antoine Fourriere mis-reads Larry Smith's idea, which I agree with, that potential for advantage in the exchange comes from significant differences in piece values, regardless whether many an exchange may appear equal. I incorporate these piece-value disparities numerically in what is called Exchange Gradient. In Antoine's words, 'a useful variable' of 'over-all strength by piecetype variance' is exactly what EG is.
I don't believe piece-type density is so relevant. Pocket Mutation Chess is an excellent game with a lot of piece types. To me, the acid test is that the pieces aren't difficult to memorize. (But of course, Pocket Mutation Chess can't be simply defined by its armies. There must be a different standard for PMC or Anti-King Chess than there is for games which simply pit two armies, like Chess, Xiangqi, Shogi or Ultima. (TakeOver Chess and Alice, which are blending classic pieces with new rules that make them formally equivalent to the introduction of new pieces, must lie somewhere in-between.) While Tamerspiel and all Shogi variants look overbloated, Chess on a Longer Board with a few pieces added, which features only two unusual pieces, passes that test. There is also a sense of legitimacy. Rooks, Knights and Bishops appear in several historic variants, while many Japanese types, and perhaps even the Gold and the Silver Generals, seem to have originated out of the blue from the brain of a drunk goblin. Conversely, the lack of some pieces may be disturbing. I tend to decree that, on a square board, a piece other than a Pawn should have its 'hippogonally symmetric' equivalent (that is, a piece with its orthogonal moves turned diagonal and vice versa, such as the Rook for the Bishop or the Queen for itself) on the board. Although Chinese Chess features an interesting opposition between (mainly) orthogonal attackers and diagonal defenders, Shako feels strange with its orthogonal Cannons and diagonal (Firz+Alfil)s known as Elephants but not the corresponding Vaos and (Wazir+Dabbabah)s. (Eurasian Chess, or my Can(n)on-featuring games offer that symmetry, but one can't help wonder why pieces which hop one piece to capture are legitimate, but pieces which hop two or more pieces to capture are absent. Absent too are pieces which are always hopping, like the Korean Cannon, or pieces which hop neutrally, but capture as riders. Why? Legitimacy is in the eye of the beholder, might comment Peter Aronson, but the feeling remains that if two closely-related pieces look as legitimate as each other, say Pao and Vao, or Camel and Zebra, and one doesn't stand on the board, maybe the other also doesn't deserve to stand there. Fusing them into a somewhat downgraded brand, like a Can(n)on which is most of the time a Cannon and the rest of the time a Canon or a Falcon which is a lame Camel + Zebra, seems the best answer.) Thus, although Heroes Hexagonal Chess is interesting, I would prefer three colorbound, clearly-defined Bishops to pieces which can move two squares in this situation or three squares in that situation. (Bishops differ enough from Rooks that, though they remain legitimate on hexagons, the Glinski Queen becomes as contrived as a Marshall or a Cardinal.) Which hints as another presentation of the same idea: if you don't remember the exact rules one month after having read and reread them, the game may be somewhat objectionable. Regarding exchanges, it is certainly important to have pieces of comparable values. I prefer Chess to Grand Chess, but Grand Chess offers much more assymmetric endgames, say Queen against Marshall. In Chess, you usually trade a Queen for a Queen. Period. (CLB is even better in that respect.) Etcetera/Hexetera, which forbids the capture of the major pieces by their opposite numbers, is also efficient in leading quickly to assymetric armies. Chess has to content itself with assymetric positions. Another important criterium in my view is to have piece types which exert comparable influences. (That criterium is a bit of the other side of having assymetric exchange opportunities.) Chess is very good in that 2 Rooks are slightly superior to 1 Queen, which is slightly superior to 8 Pawns, which are slightly superior to 2 Bishops, which are slightly superior to 2 Knights. Conversely, I wouldn't have objected if Rococo had given two Withdrawers to each side and would indeed suggest to find a way to add one Withdrawer to Maxima (and to Ultima as long as you do not replace the second Long Leaper and the second Chameleon by an Advancer and a Swapper) but two Long Leapers unbalance an otherwise fascinating game. (Cavalier Chess, which I don't like anyway, also suffers from the presence of two Marshalls as opposed to only one Queen. I would suggest to add another Queen on a 9x8 Board.) To translate this into numbers, a useful variable would be overall strength by piecetype variance. But there is more to comparable influence than simply comparable strength. An Immobilizer is much stronger than a Coordinator, but one Coordinator still looks enough in Ultima/Maxima because it affects many decisions, such as 'can I have my Immobilizer immobilized?', as would one Shield. The overall strength is certainly important. In that respect, Chess and Shogi are both balanced. Chess pieces, which are stronger than Shogi pieces, don't switch owner when they are captured. Hostage Chess and Mortal Chessgi are in my view much better than Chessgi, because they implement offsetting mechanisms which keep reasonable armies on the Board. So, the overall strength factor should be doubled by prisoner recruitment, but only multiplied by a smaller parameter for Hostage Chess and Mortal Chessgi, leading to a mildly pathological result only for Chessgi. (True, Takeover Chess is even more shaky than Chessgi - the pieces there are very powerful: a piece can be captured, or converted - and remains enjoyable, but then again, there must be a different standard for games which come up with new rules and for games which simply pit new armies. Besides, not all the pieces in TOC remain on the Board.) There is also the problem of White's initial advantage. A number of games, including PMC or Pocket Polypiece Chess (quickly-evolving armies, both topologically and functionally) and TOC (very strong armies) or Viking Chess (quick, well-protected Pawns) may have an automatic win at Grand Master level. Finally, the fact that Zillions plays a game badly (AKC, in particular) is also a good sign.
Wildebeest Chess design analysis: # squares: 110 # piece types: 8 Piece-type density: 7.27% Est. piece values: P1, N3, B3, R5, Q10, K3, C4, W8 Initial piece density: 40% Power density: 1.27 Exchange Gradient: 0.499; (1-G) = 0.501 Ave. Game Length Projected: #Moves=((3.5)(110)(0.0727))/((1.2727)(0.499)) = 44 Moves Features: Unbalanced initial positioning suggests a hundred more variations on the same board with the same pieces. Comments: Despite large Z board size,low PTD suggests average-length games.
Wildebeest Chess design analysis: # squares: 110 # piece types: 8 Piece-type density: 7.27% Est. piece values: P1, N3, B3, R5, Q10, K3, C4, W8 Initial piece density: 40% Power density: 1.27 Exchange Gradient: 0.499; (1-G) = 0.501 Ave. Game Length Projected: #Moves=((3.5)(110)(0.0727))/((1.2727)(0.499)) = Moves Features: Unbalanced initial positioning suggests a hundred more variations on the same board with the same pieces. Comments: As Z increases, mostly this board size determines #M, but the other factors remain important adjustments
Regarding Jacks and Witches, I believe a)it is R=7, C=5 (a Rook is worth two Cannons in Chinese Chess, and although my Can(n)ons are obviously stronger than Cannons, the diagonal moves suffer from the shape of the board) b)all three games ended with the help of quick blunders which lost the King once and the Witch twice.
Predictions for the length of games (#M) is not the main goal for looking at CVs analytically. Yet results from Courier completed games interesting: -predicted ave.#M- -Game Courier- Jacks&Witches 37 11-03-04 23 = 24 Moves, (anticipating checkmate) 07-10-03 14 = 16 Moves 28-10-03 26 = 36 Moves, checkmate maybe 10 moves ahead Rococo 60 15-12-03 44 Moves 16-01-04 55 = 60 Moves, (checkmate five moves ahead) 23-12-03 53 = about 58 Moves played out The trend is apparent that, with Z Board size more or less constant, Exchange Gradient especially has high predictive value for length (#M).
Rococo design analysis: # squares: 82 [counting rim squares as 1/2] # piece types: 8 Piece-type density: 0.098 Est. piece values: P2,W3,K3,C4,S5,L7,A8,I10 Initial piece density: 32/82 = 39% Power density: 126/82 =1.54 Exchange gradient: 0.69; (1-G) = 0.31 Ave. Game Length: #M = (3.5(82)(0.098))/(1.54(0.31)) = 60 moves Other features: Reasonable to count as 1/2 border squares, reachable only by capture. The high exchange gradient (low exchange potential) reflects steady continuum of piece values. Comments: Long games, high # moves predicted, and Rococo is game that player can recover from being down in material.
Jack & Witches design analysis: # squares: 84 # piece types: 9 Piece-type density: 0.101 Est. piece values: P1,L2,N3,B2,R5, J1(in hand), K2,C7,W12 [Probably Pawns are less than 1 and Witch greater than 12, but convenient to stay at these limits] Initial piece density: 48% Power density: 122/84 = 1.45 Exchange gradient: 0.444; (1-G) = 0.556 #M = (3.5(84)(0.101))/(1.45(0.556)) = 37 moves [Still fine-tuning constant now 3.5 instead of 4] Other features: Transporter cells do not disproportionately affect piece values. Comments: Power density is high substantially from number of pieces paired, five(5).
Are you sure this is right? In an extreme case the pieces all had the same values G would be 1, and based on your comments that would be very poor exchange possibilities...
I think that some might be leaping to premature conclusions. These formulae are only to assist in any evaluation, they cannot be the final word. Although game_x might score 7.5 and game_y is 8.5, this does not say that one is better than the other. Only that they score differently in the formulation. After the evaluation of many other games, these can be charted and compared with known quantities. For instance, where do some of the most favorite games fall within this pattern? When a large enough sampling has been accumulated, one can then state that if a game falls within certain parameters it might either be bad or good. And still this will not be an absolute statement.
I wonder if Piece Type Density needs to be considered in conjunction with Move Type Density. FIDE Chess has six piece types in 64 sqaures and also has 7.5 move types (King, Rook, Bishop, Knight, normal pawn move, normal pawn capture counted at full value; Castling, Pawn double step, and e. p. counted at half value.) No move type for the Queen as it combines the Rook and Bishop. Capablanca's Chess has 8 piece types on 80 squares, but has type same 7.5 move types. Does this mean that Capa's game is clearer than the 8/80 ratio and its Power Denisty would indicate? Perhaps PTD and MTD need to be averaged in some way? My own Pocket Mutation Chess scores poorly on clarity by its PTD of 12/64 (the six starting piece types counted at full value and the 12 promotion/mutation types counted at half value). But its MTD is only 8.5 (FIDE moves plus Nightrider). My own playing experience is that Pocket Mutation isn't as clear as FIDE, but that the disparity seems less than PTD would indicate.
Moises Sole asks about G Exchange Gradient in move equation. See my comment here 'To go with Depth-Clarity....' Heuristically, G is average of all the possible ratio-pairings of piece values, King included. Informally: to avoid 'infinities,' put smaller value always on top, normalizing. In specific case of Isis with piece values 1,2,3,4,8, it becomes: (1/2 + 1/3 + 1/4 + 1/8 + 2/3 + 2/4 + 2/8 + 3/4 + 3/8 + 4/8)/(10) = 0.425. Then (1-G) for right directionality with the other factors in #M equation is 0.575. The first use of G, or (1-G), is to predict average number of moves in a game-concept. This predicts closely game length for those tested so far: M = 4(Z)(T)/(P)(1-G), where M #Moves, Z board size, T piece-type density, P Power density, G Gradient as above.
When I designed Heroes Hexagonal Chess, I first started with the idea to design a game on a hex board that was unencumbered by Glinski's adaptation. I then developed the thematic pieces based on a liking for the ancient variants, Shatranj, Makruk, for example. The idea of the Hero as a source of power for his army was inspired by the role of the Hero in ancient folklore. To determine the power of the pieces, I made a rather simple estimate of power density and I tried to come close to that of FIDE chess. I did this because FIDE seems to have achieved a nice level of power density, probably through countless attempts. With some play-testing, some good feedback, and some calculation, I then refined the specific characteristics of the pieces. For me, Chess has to engage the imagination as well as the intellect to be interesting. Here, predilection play a big role. The game must also be playable. Here, some calculation helps.
I certainly enjoy this Pages, and some of the designs are interesting and nice to me, this is a sane entertainment seeing others ideas and show my own ideas to others too, Chess is not a unique concept, in certain way it is a meta-concept, and explorations around it is a cool matter. Of corse, there are ever some predilections, and it is natural, as the natural resistance to changes, but time to time the things change, if not, we would be playing Chaturanga or Shatranj now. Changes come after exploration of new ideas, rejecting old ones and making sustitutions that colective feels good for the purpose of the game. I like the things we are doing, if all of us dislike our work, it is better close this nice site and migrate to any of the multiple Pages in which we can play FIDE-Chess and write opinions about it. It is good too, but I think that many of us are happy with the things we can see in The Chess Variants Pages, Not all the things are superb, but this is the way the things are: Some are good, some are bad, and it depends on the eye that is watching a particular thing in a certain moment, not everybody has the same opinion about a topic everytime, this is one of the biggest characteristics of human beings, and this characteristic is great, it is one of the paradigms of freedom.
If would-be designers had curbed their addiction to designing CV's, these pages wouldn't exist and we wouldn't be having this (genuinely fascinating) discussion.
How's G calculated?
Of course Larry Smith and Michael Nelson are right that predilections rank high in importance. No one yet addresses multiplicity of chess game-rules sets, more than anyone can absorb at the level of play. Maybe would-be designers could curb or arrest addiction to design. Or, a change in rules of a long-established game like Ultima, for ex., should be a very cautious act, as a recent Comment under Ultima advises. David Pritchard from Introduction to Encyclopedia of Chess Variants: 'Anyone can invent a new CV within ten seconds and unfortunately some people do' and 'Probably most CVs are best consigned to oblivion.'
One CV by way example, Isis posted week of 25 March, design analysis: # squares: 48 # piece types: 5 Piece-type density: 10.4% Est. piece values: P1, B3, K2, Q4, M8 Initial piece density: 50% Power density: 68/48 = 1.42 [Orthodox Fide's is about 1.25 or 1.30] Exchange Gradient: G = 0.425, using range of values here 1,2,3,4,8 [Orthodox Fide is about 0.50, and Isis shows better exchange potential with lower G] Ave. Game Length projected: #Moves = (4(Z)(ptD)/(PD)(1-G)) = (4)(48)(0.104)/(1.42)(0.575) = 24 Moves So, Isis games should not be very long because small Z (board size) and high potential advantage in exchange (low G). Other features: River reduces value of Q. Comments: Obviously, some values are estimates not completely amenable to analysis. From description only, comparing different games shows trends in useful, compact numerical information, able to complement clearly-written game rules.
George, Men and women are about than 2% genetically different--but it's a really important 2%! Similarly, some people love apples and hate oranges and vice versa. I believe that you are making a real contribution to the 'Science of Chess Variant Design' while denigrating the 'Art of Chess Variant Design'. I think we need both. Preferences and not the be all and end all of design, but neither are they irrelevant--what is the point of designing a 'mathematically perfect' CV that no one wants to play? And aren't clarity/depth and drama/decisiveness important precisely because they speak to game players' preferences?
Although Game Theory can be used to quantify real-world events into a Game Design, a Game Design is not subject exclusively to Game Theory. Particular aspects of games cannot be quantified as they exist purely on the emotional level of the players. For example, how do you evaluate the potential for frustration or joy? Each player will react subjectively, some enjoy frustrating games. But objective values can be assigned so that a potential developer can make decisions while designing a game. But this will not cause a developer to create a good game. Their own prejudices will often effect their design. Some might never develop a large game while others will not develop small ones. And some do not appreciate game with themes, while others will not try the pure abstract.
A Comment says that comparing Games is like apples and oranges. The analogy speaks for itself: we know that biochemically, Apples and Oranges (trees) are mostly alike sharing 95%+ of their 30,000 (60,000?) genes, partly-sequenced basis to compare. So, Chess Variants compare strict equality or not in board size, pieces, and Power Density, Piece-type density. piece Gradient, Event Frequency, if one cares to try other than entirely subjective approach, and also not to dwell on the extreme values where theory less effective. Clarity and Depth alone seem too general unless something measures Clarity-Depth, besides opinion poll. After all topic of interest is Game Design not Preferences.
It's also possible that some of these numbers have non-linear relationships. For example Hectacomb with Amazons instead of Queens might not be that much different in playablity in spite of the high PD difference (aout 40%)--the PD is huge in either case. Simiarly, assuming an 8x8 board, a game with 100 piece types might be scarely less clear than a game with 50 (clarity approaching zero in both cases), while 10 piece types vs. 5 makes an easily perceptible difference. It is also very possible that numerical criteria are best at comparing games of somewhat similar types, and become more and more 'apples and oranges' as the game types diverge. The latter is why I objected to George comparing PTD in Fugue to PTD in Chess. Compare it to Ultima and Rococo and it doesn't look so bad by this criteria. It is by this measure less clear than Ultima or Roccoco but the difference in not as extreme as the the difference with Chess.
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