Lasker Piece Values

Well, if he reads here, he is free to argue with the facts whenever he feels like it.

Somehow I don;t think he will...

I don't get why you can't comment my question 

I simply asked if the King in normal chess is slightly stronger than bishop due to reverence, as it always remains on the board. 

C'mon it can't be so hard to get the logic here 

Weird... I don;t see the word 'Bishop' anywhere in your question. Only the word 'Commoner'.

If you now want to know about the difference between royal King and Bishop... Well, if a non-royal Commoner already is weaker than the Bishop during most of the game, logic dictates that the King must be far weaker, as it is handicapped by royalty. The King, for instance, cannot be used to block the Bishop from guarding the promotion square of your last Pawn, for instance, something a minor could easily do.

To know exactly how much royalty weakens a piece, you could define the concept of 'semi-royal' pieces: a semi-royal piece is a piece that is not allowed to step into check, and is obliged to evade a check when it can, but if it cannot, it would be allowed to stay put (and so possibly can be captured then, without consequences).

This avoids the complication of being checkmated, making it possible to test pieces against their semi-royal partner, to see how much the disadvantage of semi-royalty is. You can be sure that true royalty is even more of a handicap, because when a true royal cannot escape capture the game is over, even where a semi-royal woul djust be traded.

So how much is the commoner worth compared to bishop?

Please indicate the number in points.

In my opinion 

Knight: 3.05

Bishop: 3.25

Commoner: ?

It depends on the game phase. In the opening about 2.90. In the late end-game perhaps 3.25. But I would rate a lone Bishop a bit lower.

Like with Bishop vs. Knight it depends on a lot of factors. With Pawns on both wings Knight and Commoner have a clear disadvantage. WIth very few remaining Pawns Commoner would have the advantage of mating potential, which gets irrelevant when there are many Pawns. For Bishops color-binding is a major issue (good Bishop vs. bad Bishop, like vs unlike Bishops).

These circumstances affect the value by a much larger amount than the difference between the 'average' value.

So seemingly both the royal-king and commoner appear to excel in the endgame.

Do you belive the pair of commoners suffers from "Principle of Redudency"

Kaufman's article also discusses penalties for rook and knight pairs, citing a "principle of redundancy"

The bishop pair can never interfere each other. Pair of commoners can in theory. 

Waiting for proof that 3 queens gets beaten by 7 knights...

I believe redundancy is a red herring. The 7 Knights vs 3 Queens result points to exactly the opposite.

Another thing that might be interesting to know: a second King is worth about nothing. Playing games where one side lacks the Queen, and the others side has the Queen replaced by a King, is well balanced. Of course when the side with two Kings can survive to a Pawn ending, he wins very easily.

Wild 9 variation on the Internet Chess Club is interesting.

9: There are two kings. You win if you checkmate the king that is closest to the a-file. If both kings are on the same file, you win if you checkmate the one that is closer to rank 1. For example, if your opponent has kings on a7 and c5, you win if you checkmate the king on a7. The other king is just an ordinary piece. It can be checked and captured with no consequences. The king you must mate can change during the course of the game.

According the rules, the second behave like commoner.

Indeed, the second King is effectively a Commoner. There is one catch, however: it can change during the game which one. According to the rules the King closest to a1 is the royal one. So if your royal one gets apparently checkmated, you can still save the game by moving the Commoner, passing the royal King towards a1, and thus take over the royalty, so that the opponent only captures the Commoner.

Something similar occurs in Spartan Chess. There you also have two Kings, which are both royal, but 'extinction royalty' instead of 'absolute royalty'. That means you don'tlose when you lose your first King, but only when you lose all your Kings. And the Spartans start with 2 Kings:

So here royalty automatically transfers to the remaining King, although there is a rule that you cannot leave both in check at the same time. You can leave one in check, and let it consequently be captured. But it turns out the fact that the one you treat as a Commoner can take over the crown if the other one gets captured makes it worth 4 to 4.5 Pawn.

So you'd better not allow one of your Kings to be traded for a minor. But if Rook or Queen attack it, you can simply protect that King, as capturing it will then be a bad deal for the opponent. (Depending of course on how safe the remaining one is tucked away; if that is on e4 in the middle game you won't survive long when it is the only remaining King.) But if you lost one, you can always get it back through promoting a Hoplite!

As they say, the proof of the pudding is in the eating. You will only be able to appreciate this when watching it done. Just set up a position like

for your favorite top Chess engine (Stockfish, Houdini, Komodo...), and let the engine play itself to see the drama unfold. It is very entertaining to see Houdini evaluate itself +8.5 Pawn ahead initially, and see the score steadily drop when the Knights push forward, and the Queens have to be traded for a Knight one by one. Or better yet, let the top engine play with the Queens, and play it against the approximately 1000-Elo weaker engine QueeNy that you can download from my website at http://hgm.nubati.net/QN.exe , and watch a patzer bury Stockfish. (QueeNy is biased against trading 2 Knights for a Queen, an error the top engines easily commit, when they see the opportunity. QueeNy will just wait for a better deal, which sooner or later will present itself.)

Perhaps it has already been mentioned ==>>

Bishop pair is worth more than the sum of the individual bishops, perhaps as much as 1.

The Knight takes on more value depending on its outpost square -- e.g.  worth more at f5 than at h3.  It is generally stonger in the opponent's half of the board and on the "middle" squares.

Komodo gives away Knight for Pawn after 5 moves? Did you use it in suicide mode??? That is not normal, and certainly not forced. Of course it loses after blundering away a Knight.

I guess Komodo is no good; perhaps it suffers from overflow in such extreme positions. So I guess you should really try QueeNy to show how it is done. I admit Stockfish is a tough cookie; they must have improved it a lot since last time I tried this. In this very tactical game, black really has to search deep enough not to run into a surprise mate,with 3 enemy Queens. But with 40 moves/15 min QueeNy searches about 15 ply, and that is enough not to fall into any mate traps. (Pondering was off. I have reasons to suspect QueeNy has a ponder bug.) Although Stockfish searches 6-7 ply deeper, it doesn't stand a chance:

Indeed, this is a gross misevaluation. All engines do it (except QueeNy, where I set the Knight value to 5 (where Q=9.5). This is also not correct, so QueeNy overestimates its initial position at +4. The drop in score of the misevaluating engine during the game as reality starts to sink in is quite typical. I see now that you had to delete one white Pawn to prevent the Komodo blunder, but as you can see from the QueeNy game this really isn't needed to make it a won position. There is no reason whatsoever to move so many Knights to 6th rank that they become subject to Pawn forks.

I don't know in which context Korchnoi made his remark, but the proper theory behind this is as follows:

Strong pieces devaluate by the presence of weaker opponent pieces, because the latter interdict access to part of the board for them. You cannot allow them to be traded 1 for 1 against the weaker pieces. Now if a single Queen balances 3 Knights, putting 2 Queens on a 16-wide board against 6 Knights would also be balanced: the Queens would on average be too far apart for one Knight to bother more than one Queen. But put them all on an 8x8 board, and now each Queen is not only hindered by the Knights that were balancing it, but also by the Knights that were supposed to balance the other. Each Queen suffers twice the devaluation caused by 3 Knights, so the Queen side suffers 4 such devaluations in total.

With 3 Q vs 7 N each Queen suffers 2.33 times the devaluation caused by 3 Knights (that would make it balanced for one Queen), so in total you suffer 7 times the devaluation, rather than 3 times (which would balance you against 9 Knights). The 4 extra devaluations are apparently worth more than 2 Knights, so they make up for more than the apparent deficit in material.

P.S. It would be fairer if Black could only promote to Knight in this game. I could modify QueeNy to do that, but of course opposing normal Chess engines would expect it is going topromote to Queen, and often already sacrifice a Queen for the Pawn to prevent that. I pre-programmed this game in Fairy-Max as variant 'light-brigade'; there it does only allow black to promote to Knight. Fairy-Max is a quite weak engine, that a good Chess player should be able to beat, so if you want to try your luck with the Queens yourself it would be a good opponent.

An empirical study. Even if you don't agree with the conclusion, it's interesting, as it tries to determine the strength of nontraditional pieces.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.95.326&rep=rep1&type=pdf

There is nothing empirical about that one, is it? It is just numerology: guessing piece values by counting squares, and dreaming up methods of counting that give the same results as what you already know for the orthodox Chess pieces. Empirical means based on play testing and game statistics.

The method used, in that paper, counting 'safe checks', is quite non-sensical: it would predict zero value for the Commoner! Also it would predict the same value for pieces that differ from each other by adding King moves. Like the Dragon from Shogi, which adds King moves to a Rook when the latter promotes. Of course in reality a Dragon is worth substantially more than a Rook (about 7), these extra moves make it devastating. Also a piece that combines King and Knight moves is worth substantially more than two Knights, with its 16 unblockable move targets, while the presented theory would give it the same value as the Knight.

So not surprisingly, the results presented in that paper for some of the unorthodox pieces are totally off. In particular, it predicts the value of the Archbishop (B+N compound) as 6.5-6.9, while empirically it is more like 8.75 (when Q=9.5): In an otherwise complete FIDE army Archbishop + Pawn have the upper hand over a Queen. So the guess is off by about two full Pawns!

Another nice game where Stockfish with 3 Queens bites the dust (imagining itself > +800 initially!), this time at 40 moves/10 min (ponder off). This time promotion did not play a role. (As is actually more typical.) Just go for the enemy King with the pack of Knights, not being afraid to march the King with them, so that you won't have to leave any cavalry behind to guard it.

If I force black to promote to Knight in the first Stockfish-QueeNy game, the continuation would be:

After f1=N