Lasker Piece Values

Pawn, 6; knight, 17; bishop, 17; rook, 24; queen, 47, king 20.

I need help to round down numbers into standard values.

http://www.chesshistory.com/winter/extra/value.html

I assume I prompted this? Ya, they were trying to figure out all the piece values for a while. Reuben just simplified it rendering the whole ordeal worthless.

Not sure if this method of conversation works 

P 1 / 6 = ¹⁄₆ 

N 3 / 17 = 0,176

B 3 / 17 = 0,176

R 5 / 24 = 0,208

Q 9 / 47 = 0,191

K 3.5 / 20 = 0,175

Total = 1,095 

Average Factor = 0,1825 x listed values

Q: Is the C2 pawn more valuable then the A2 pawn by reason of potential. C captures both ways while A is attack just one. I'm thinking a piece can be more valuable or less depending on the placement. iono. ..

Chess isn't purely numerical. And it isn't purely logical. There are irrational positions where material is objectively meaningless.

I don't get valuing the King. The lack of both reach and ability to be exchanged should render a figure below that of a bishop or a knight, shouldn't it?

The King is known for it's close range striking force. However it can not be exchanged or traded for other pieces, so the generall usage of gauging his fighting power have little practical points. Some experts say the king, majority of the times is the strongest minor piece in the endgame. 

The ultimate on piece values are the Kaufman values, based on statistical analysis of muillions of GM games. They are P=100, N=B=325, R=500, Q=950, B-pair bonus=50. Everything else should be considered guess work, no matter how smart and able the person doing the guessing might have been.

Note that a non-royal King (aka Commoner or Man, a piece moving as King but not handicapped by restrictions for moving in check) for most of the game is less valuable than Bishop (and two of them dramatically less than a Bishop pair). Only in the very late end-game the Commoner starts to be superior to a minor, more because of its mating potential (which makes drawing tactics of sacrificing the opposing minor for the last Pawn impossible) than for its tactical ability.

The Commoner has the advantage of 'concentration' of its moves, of which the mating potential comes as a side effect of minor importance. The Knight, which also covers 8 targets, compensates for this by having a higher 'speed'. It is interesting to note that a piece that moves (non-capturing) as Knight, but captures as King is worth about 0.5 Pawn more than either of them.

20/17 x 3.00 = 3.53 which is close to 32/9 or 3.5555>

HGMuller@ may ask for your opinion on the assessed value of 3.5?

Well, it depends on what scale. If N=B=3 then K=3.5 is way too high. If N=B=3.25, then K=3.5 in the late end-game might be correct. But in the opening it would then be more like K=3. Another strength of Commoner is that it is a 'tough defender': when protected by King it can hold out against K+Q, which even a Rook cannot.

HGMuller@ do you concede with the idea of "reverence" or "veneration" the king is worth slightly more than an commoner? | Ceteris Paribus

Just for fun, here's what Lasker wrote:

The argument underlying the thought of Euler and Janisch amounts to what follows. The pieces in chess are only of value in so far as they are able to effect captures, to defend other men, to cut off flight squares from the hostile king, to administer checks, to interpose against checks, to attack several pieces simultaneously or to defend several pieces simultaneously, and finally to support or fight against the advance of pawns. Every piece possesses these various capacities, and the greater the mobility and the number of squares dominated by a man, the greater is the probability that such a man will be used to advantage by a fine player. All this applies equally to any game constructed after the model of chess. If a game should be so constructed that each piece would have the same mobility all the time then it is clear that we might assume the value of a piece having mobility 4 to be double that of a piece having only mobility 2, because two pieces of the latter kind could do no more than one piece of the first kind, if the player used all pieces to best advantage. If the mobilities of the pieces are not always the same, then we might fancy a piece to possess always mobility 1, and compare the other pieces with this ideal unit man. In each position the "value" of a man will be, according to the principle above, equal to its momentary usefulness, and its average value will therefore be, generally speaking, equal to its "average mobility." In other words, going through all possible chess positions and counting the mobility of a man in each, we obtain the average measure of the "value" of that man by taking the averages of all these mobilities.

The "value" of a man is thus a mathematical conception and can be calculated by mathematical methods. At a later stage of this course I shall invite those interested in scientific matters to dive much more deeply into the theory of value as dimly outlined above, when, I believe, it will be seen that this chess conception of '' value'' is the germ of a truth of stupendous importance, whose capacity for application not only in games but in many questions of serious import is on a par with that of the fundamental iaw of d'AIembert in natural philosophy. At the present stage of our investigation we shall restrict ourselves to chess. In the royal game the "values" whose definitions have been outlined above, have the following significance. In any position where there are few if any possibilities for capture or for saying check, where both kings are safe and where no passed pawn is near queening, where, to be brief, no "complications" are existent, it is only necessary to sum up the "values" on each side in order to find out which of the two players, if either, has an advantage. If the totals on each side are the same, then between players of first class strength the game should end in a draw. And if the total "values" on the two sides are different then the advantage will be on the side possessing superior "values." That side will thus be entitled to make exertions to win the game, whereas the other side will have to act on the defensive.

It is true that in actual play this law often appears to be faulty, inasmuch as frequently games are won by the side which according to the law above ought to be under a disadvantage. The law is only correct if the very best play on both sides is presupposed. Observation has however shown that even with very weak players if only they are of about equal skill, the side having a theoretical advantage in a position will win oftener than the other side, no matter how great the mistakes committed by the two players may be. Indeed, this happens so often, that the weaker players are the staunchest believers in the law of values, entirely forgetting its limitations.

The values of the pieces calculated with sufficient accuracy for over the board play are as follows:

Pawn, 6; knight, 17; bishop, 17; rook, 24; queen, 47.

These values have not been found by executing the indicated calculaton, but are those which come nearest to express my experience in hard fought games. It would be comparatively easy to find the values of bishop, rook and queen, by a mathematical calculation, but owing to the absence of obstruction for the knight, the change of power on the eighth row for the pawn, also the difference in the regions of mobility and of capture for the pawn, and finally the peculiar position of the king, the determination of values for these pieces by theory would be very hard and experience as a method of determination is preferable. The king, of course, is always on the board and hence for the above law it is unnecessary to find its value. But if we want to compare its capacity for supporting defence and attack with that of the other pieces, we may again use the above values and add that of the king—20.

I'm just glad we don't use the metric system for piece values because it would be even more confusing.

This seems pretty non-sensical to me. A Commoner can do anything a King can, you are not forced to expose it to check. But it can do more, and when that more is bad, you won't do it, but when it gives you an extra advantage you would. So logical a Commoner should be worth more than a King.

In this case I'm refeering to the royal-piece aka the King, for standard chess. Is the monarch somehow worth more due to reverence, as it always remains on the board? 

Actually this is wrong. Empirical value determination through playing materially imbalanced games between computers have shown that the value of a piece that can directly (i.e. with unblockable moves) leap to N destinations not further than two squares away (such as Knight and Commoner, with N=8) is

(30 + 5/8*N)*N

This contains a quadratic term, so that a piece with 16 targets is worth more than double the value of one with 8 targets. This expression calculates a relative value of 280 for the Knight, which could be multiplied by a scaling factor 1.1 to bring it to a scale of centi-Pawns (N ~ 308). A piece with 12 targets would then be worth 495, about a Rook, and with 16 targets 704 centi-Pawn.

These values in fact describe the average value of all symmetric pieces with that number of moves. In practice there are differences for pieces with the same number of targets, based on global properties of the particular combination of targets. Some pieces with 8 targets are color-bound, which will surpress their value (unless you have a complimentary pair of them). This cannot be blamed on any particular  single board step it can make. The Commoner (non-royal King) is slow (bad) but has concentrated attacking power (good). A Knight is actually one of the best 'short-range' pieces with 8 targets.

For asymmetric pieces it turns out that forward moves contributes about twice as much to the total value as sideway or backward moves. 'Forwardness' is a desirable quality. But it should not be overdone; being 'irreversible', (meaning the targets ly all in the same general direction so that you cannot always return to where you came from), is quite bad. For 'divergent' pieces that have different capture and non-capture moves, capture targets contribute about twice as much as non-capture targets.

Blockable moves are worth much less than direct leaps. The Horse from Chinese Chess moves as a Knight, but does not jump, and can be blocked on orthogonally adjacent squares; this makes it worth only half as much as a Knight, i.e. just 1.6 Pawns. (But this is an extreme example, as one blocker can block two moves, on a square that is not attacked by the Horse itself.) For sliding moves it is more complex; sliding pieces like Rook and Bishop have many targets, but the distant ones can be blocked on many squares, and on one nearby square you can block almost all distant moves. And although they potentially have many moves, in each direction they have at most one capture. So they are not nearly worth as much as their high mobility would suggest. It turns out all the moves more distant than 2 together on the Bishop are worth only as much as the power to jump to the second square instead of needing a clear path to it. For Queens the power to jump to the second square is worth as much as all moves more distant than 3 together.

Note that the idea described by Lasker that the average value of a position with a certain amount of material can be obtained by simply adding the value of the pieces is also just an approximation. In real life there are cooperational and adversarial effects. We all know about the Bishop pair, which is worth more than twice the Bishop value. But it could also never be explained by individual piece values why 3 Queens lose so badly against 7 Knights (in the presence of only Kings and Pawns).

If you are not satisfied with my previous answer, I don't understand the question. More than what (if not a Commoner)???

More than bishop value essentially.

And yes, without taking the commoner into consideration this time 

Take it up with Dr. Lasker.