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On an 8x8 board, according to chess theroticians a chess king has a fighting value of 4 pawns (though noting it cannot be exchanged). Since a mann moves exactly like a king, even though (or especially as) it is a non-royal piece, I see no reason not to put a mann at 4 pawns in value as well.

The problem is that Knights beat Men more often than not, and that a pair of Men is totally crushed by the Bishop pair (in the presence of sufficiently many Pawns). So it appears the value 4 is total hogwash.

It may depend on how one values minor pieces, and perhaps more importantly if a single mann can handle a lone minor piece, with or without an extra pawn (with other pieces and pawns on the board, to some extent). A minor piece has been valued as much as worth 3.5 pawns (which I tend to agree with), or as little as 3 pawns. Also, two bishops are commonly thought worth a bit more than two knights on average, even if one assumes a single bishop = knight exactly on average. Perhaps Lasker and Evans (and other chess authorities) had more in mind the situation of a king vs. enemy piece [+ extra pawn] just on one side of the board, and didn't think about pitting two king type pieces against two minors.

Piece combinations pitted against each other might lead to different conclusions about the numerical value of a piece than one piece (or more) vs. one piece [+ extra pawn] battles (e.g. 7 knights beat 3 queens [probably no pawns involved, though] I seem to recall being posted, yet numerically in value the 3 queens would normally on paper be evaluated to be very much superior). Having said that, H.G.'s examples are for two pieces vs. two men battles (with a supporting cast of pawns for both sides), which seems close to a one on one battle, so I'm not at all confident any more that a king or mann is (ever, or at least on average) worth 4 pawns (on an 8x8 board).

[edit: Being foggy from daily medication aside, it should have been obvious to me that a mann quite possibly has a {significantly?} higher value than a king's fighting value, whatever that ought to be. That's since a mann doesn't have to stay out of check like a king, and a mann can be traded for something if necessary.]

P.S.: Note that in chess king and pawn endgames, a king can at times restrain or even eventually overcome (through zugzwang) up to just 3 connected passed pawns, but in other cases might gobble up many pawns that are not defended by the opposing king, which Lasker & Evans et al may have weighed too. Pieces (or combinations of them) can be somewhat different in value in an endgame as opposed to in the middlegame or opening phase (e.g. Rook + Pawn may be >= Bishop + Knight often in an endgame, but not usually sooner). IMHO its hard to feel completely sure of an exact average value for a piece, since for one thing the endgame phase, if any, comes last. An exact position can also naturally affect values: queen vs. pieces (with or without extra pawns) positions can often be sensitive to something as seemingly slight as whether most/all of the pawns and pieces of the side without the queen are comfortably protecting each other as they attempt to continue to operate.

I've edited my last comment somewhat substantially.

I've added a possibly vital edit (as indicated) to my 2nd last post.

Waterloo -- better than all-empty-space Grand Chess anyway.

Then this one mentioned as also using Commoner/Man is even better and meeting CVPage standards: Medieval_(europe).

The world expert at values, H.G. Muller joined this discussion, and I still hold out for value of nearer 6.0 for Falcon than Rook 5.0. I just played three Falcon games on Game Courier and early Falcon uses and forks were decisive in all of them whilst the Rooks mostly sat in place. It's matter of teaching programs maybe to open with Falcon; then the gap is filled in value between Rook and Queen, the way Man/Commoner/Guard can fill in between Bishop and Rook lower scale for beginners. (But Muller is right that anything towards 7.0 for Falcon among the four fundamental chess pieces was irresponsibly unsupported.)

Vickalan's point is good one, that better CVs have clear value in exchange. That was recurrent theme for years in our thread on Game Design: Value_In_Exchange. When most though not all pieces have different distinguishable values, there is intriguing Flight and Fight.

Anyway Man is just a Queen, and that's how to get its value. If full Queen is 9.0, then develop method for Queen limited to 6 steps not 7. Then get limited Queen up to 5 spaces, then Q4, Q3, Q2, Man. Since Q6 is near Q7 but others progressively detrimental, it might go: Q6 8.6, Q5 8.0, Q4 7.2, Q3 6.2, Q2 5.0, Man 3.6. No values in isolation since it depends on other pieces as well as array and rules.

Principle of value in exchange is why I dislike all yes all 50 Carrera variants, that three pieces are boringly near-same-valued.

I tend to agree with Dutch world chess champion Euwe, who put a queen at worth 10 pawns (Q=R+B+P, or Q = 5.5 + 3.5 + 1 = 10). In that case, oddly enough, if George's value for a mann (3.6, if a queen =9) is correct, if a queen is supposed to be 10 then by ratio a mann would be put at 4 pawns. However, I'd have to check whether Lasker or Evans put a queen at 10 pawns to guess if they saw things this way.

George's is the sort of estimating method I've used for other pieces on occasion. One problem might be that it doesn't take into account that the mann is not a long-range piece (a reason why a colour-bound bishop is a match for a non-colour-bound knight; the latter also has a leaping ability to compensate for moving to less squares on average than a bishop). The disadvantage of being only short-range would show up even more on a larger board than 8x8.

Larry Kaufman has shown by statistical analysis of a huge number of GM games that the game outcomes are optimally predicted by the following values: (P-N-B-R-Q):

100 - 325 - 325 - 500 - 975

with the caveat that a pair of Bishops is worth 700 (rather than 2 x 325 = 650). Of course the value of a Pawn is not defined very well; we know there are many kinds of Pawns, with very different values, from backaward / doubled / edge Pawns to centralized protected passers. The opening value of the Commoner in my tests would be about 310 on this scale.

And yes, those claiming it would be 4 are very much off.

It is true that piece values in themselves are already a highly simplified approximation to reality, and that the power of an army cannot be written as a plain sum over its individual pieces. The Bishop pair already shows that. The 3Q vs 7N (which was in the presence of Pawns, BTW) is another example of that.

Conclusion: Grandmasters can win even more games if they knew a king's value is 3.1, not 4.0

I don't think this is true.  The King can't be traded - each player has exactly one at all times, so any consideration of it's value is purely theoretical (at least in orthodox chess.)

There's a book called King Power that I haven't read in a long time. Sometimes a king assists in attacking the other king, even in the middlegame once in a while, and a king is often able to defend itself (or defend squares/pieces/pawns around it) from attack. The king regularly comes into its own in the endgame phase. In Grandmaster Secrets: Endings, American GM Andy Soltis mentions the concept of creating a mismatch in one area of the board, where a superior force overpowers a weaker one locally. Often a king is part of such a superior force. To a strong player, it's often not too hard to tell whether a king is going to help to make progress of this sort, and he really doesn't need to tally up the theoretical value of the attacking force (with a king among it) vs. the defending force to know which side will win out in a local battle, or in other words, in a mismatch massacre in the making.

Thanks Kevin for the information.

I recently started using Fairy-Max, and did a test to try to confirm HGMuller's information. Using guards on a 10 x 8 board, each test starts with at least 2 knights and 2 bishops for each side, and guards on only one side.

asymmetry: [2 guards vs. 2 bishops]
guards win (score) = 40/80 = 50.0%

asymmetry: [2 guards vs. 2 knights]
guards win (score) = 46/80 = 57.5%

asymmetry: [2 guards vs. 1 bishop and 1 knight]
guards win (score) = 101/200 = 50.5%

I think these tests confirm HGM's conclusions. A guard seems to be very nearly equal to a bishop, and slightly superior to a knight. When the game is [3 knights, 3 bishops] vs. [2 knights, 2 bishops, 2 guards] the guards are almost exactly equal to the average of knights and bishops.

note: this test started with an unknown value for the guard. By fine-tuning their value, they might be able to play slightly better, so the guards value might be even a little more than summarized here.

Btw, here's an updated image of a guard (what I see more often in variant chess games).

Just a guess, but the value of having even just a single guard in the endgame could be lower than I imagined on even a relatively small 8x8 board. This is because there could be less chances of creating a mismatch somewhere, if both sides have a king plus guard, and material is pretty equal in other ways too. That's since a potential attacking king can be opposed by a king (or sometimes even a guard?) in one area (or side) of the board, while elsewhere a potential attacking guard can be opposed by a guard (or sometimes even a king?) as well. Then it would seem harder to sufficiently distract a defensive side's king or guard in either area (or side) of the board for an offensive king/guard to break through somewhere, via a mismatch. The guards might thus be considered to have at least some good defensive value, but I suppose if attempting to measure the value of a lone guard vs. a lone minor piece (with a supporting cast of equal armies for both sides), a high number of draws tends to devalue it, if it scores just a few more wins than would be par (for instance).

Somehow I cannot shake the feeling that on an 8x8 board, in many endgames at least, a single [modern] elephant (alfil+ferz compound piece) would not be worth quite as much as a single general (aka mann), with numerous pawns on the board. Such an elephant leaps, but is colour-bound while lacking a bishop's mobility. At the least, there could be many games featuring a 'bad elephant', sort of like there are cases of 'bad bishops' in chess - that is, just in regard to having many of the elephant's pawns fixed on the same colour as it moves on. Has anyone tried to evaluate the value of an elephant, at least in the opening phase? I estimate it at being worth just a fraction less than a knight is worth. Modern Shatranj is an 8x8 variant that includes elephants and generals, but no queen-like pieces, making it somewhat endgame-like from the start of a game (noting also that the board is not checkered like for a game of chess, done for that game so as to aid calculation, according to world chess champion Lasker's Manual of Chess).

Man/Commoner Sovereign Value is 1: http://www.chessvariants.com/index/listcomments.php?subjectid=SOVEREIGN_P_Ts. That means King plus Man versus King should win, already mentioned here. Bishop S.V. is 2 regardless exact piece value; one Bishop is insufficent material. Knight S.V. is 3, as Knappen clinches from table reference; two Knights are insufficient material. That in itself on smaller 8x8 makes Man equal or greater than Bishop or Rook.

The development, Queen_One, for Q7, Q6, Q5, Q4, Q3, Q2, and Man as Q1 I am only concerned about being off a bit for the last step Man because the next step would be Q0 or some Null piece. So the last Q1 or Man may be too high at 3.6, instead dampened to 3.4 or 3.3, but ahead of Bishop and Knight on the standard little board.

Here's a 2001 list I missed on CVP, with 200 8x8 board piece values suggested by Zog (many of the values are clearly too low, e.g. for certain chess pieces, when using chess pawn = approx. 1888); included are the commoner (aka mann), and a [modern] elephant by another name (if I'm reading the notation right), the mann given a value of about 3.4 pawns when I worked out the ratio. The [modern] elephant (aka ferfil!?) relates in value to a knight here by having a slightly higher value than it, which may be slightly surprising. Note that a knight is given a value of about 2.7 pawns (that is, about 0.055 pawns less than a ferfil), i.e. it's thought worth under 3 pawns(!), while a bishop is given a value of about 3.14 pawns:

http://www.chessvariants.com/piececlopedia.dir/whos-who-on-8x8.html

I've added the checkmating potential note to this page.

I wanted to call out that the "vocabulary" section here seems to disagree with our current definition of adjacent in the glossary, when referring to hexagonal variants.  I think such sections in the Piececlopedia should generally be replaced to a link to the glossary, but we should consider whether we like the glossary definition of "adjacent" for hexagonal boards.

By the glossary, do you mean the old one or the new version I've been working on? The definition here, which I wrote, is in line with the first two definitions I wrote for the new version. The new one differs by also covering what adjacent means for 3D spaces.

https://www.chessvariants.com/terms/#adjacent

I've added the checkmating potential note to this page.

Nice! But perhaps you should also mention that the man loses this mating potential on boards of size 15x15 or larger. Also note that the symbol used for Man in the WinBoard/XBoard GUI is

@Fergus, sorry, I looked up the recent thread but went to the old glossary anyway.  The newer one is more general and matches the use here, but I'd still suggest to remove the Vocabulary sections of the Piececlopedia in favor of links to the glossary.

@H.G., thanks, I'll add the board size comment later today.  (I just noticed you already brought up this complication almost a year and a half ago!)  Can the checkmating applet extend to accept a board size (as a URI query parameter)?  Is 14x14 or 15x15 too large to generate the endgame table in an online setting?

Is 14x14 or 15x15 too large to generate the endgame table in an online setting?

I am not sure how far this can be pushed. It probably depends on the computers people use. But some people might want to try it on a phone... I suppose we would like it to work for almost everyone.

The problem is that the required amount of memory increases very aggressively with board size. With 3 pieces on an NxN board you have N^6 possible positions. So going from N=8 to N=16 requires 64 times as much memory. The time it takes would be proportional to size. For 3 pieces even 16x16 only requires 16MB, which by today's standards is small. So I suppose that can be done, and based on how long it takes now, it would then take some 10 sec to generate the table. I will have a look at it.