The values are dynamic. In a closed position knights fare better than bishops and in open it's the other. Rooks are pretty useless in early and midgame but become extremely valuable in endgame. Especially for the side winning.
Indeed and numerous individuals have developed systems which try to account for this. Additionally, the values of the pawns aren't all identical either (a connected pair is better than two in the same file and so on).
In my view, the standard and chess engine valuations of a piece generally estimate its value in the endgame. In the middle game, position and coordination of pieces matter more for which player is able to deliver a checkmate or decisive attack. If you are checkmated in the middle game, the value of your pieces does not matter, as the Opera game demonstrates - but even without a checkmate, a material advantage can be temporary in the face of a decisive attack. A chess game is won either by a checkmate in the middle game or by reaching an endgame with a material advantage. Hence for individual piece values, it is better to consider them in the endgame.
Wrong. The equation is non linear. So the coefficients of a piece are dependent on high order terms. However there is an objective exact average value for each piece. But we aren't anywhere near proving that. But the estimates we use are likely +/-1 of the true value given engine with heavy training are using these values
@@theanimatedchemisthaving watched GothamChess, he mentions the fluidity of the points system depending on the position of individual pieces. For example, a rook wasn't able to move more than a couple of squares, due to it's position, so became, for all intents and purposes, a 3 point rook.
N + N + N = # and B + N = # thus N + N + N = B + N therefore N + N = B also B + N = # and B + B = # this B + N = B + B therefore N = B so N ≤ B ≤ 2N And my calculation puts B (3.0) between N (1.8) and 2N (3.6) 😱 Before anyone thinks I'm being truly serious here I'm just having a bit of fun :)
Mating potential is a very small component of material value. A piece which moves like a king but cannot be checkmated can force checkmate against a bare king with the help of almost any conceivable royal piece, which is more than can be said for a Rook. But clearly no sane person would consider a non-royal king to be worth more than a rook, and in many situations it wouldn't even be preferable to a knight or bishop.
You're definitely correct. I did some new calculations giving additional weight to whether the piece can capture or defend and the Knight and Bishop values are more aligned with an ever-so-slight advantage to the Bishop. Probably worth a video update. I've got a chemistry one in the queue then two more to make before I go back to chess.
@@theanimatedchemist We're also only talking about relative average value. The values of pieces change based upon how they are positioned. A Bishop that is hemmed in by its pawns has little value. If you watch a lot of lower-level games you will see people that will leave the a, b & c pawns on their starting positions and never move their queen's side Rook or Bishop. In this situation that Rook was nearly useless because all it did was sit on A1 and do nothing but guard the unmoved Bishop and a-pawn, and it is doubtful that the a-pawn even needed a defender.
@@kingrex1931 That's also true. The other calculation I've done also considers it on an open board for simplicity. I guess AlphaZero's calculation of the values may be the most sensible given that they were derived from actual games. I wonder what the deviation would be if the analysis was done with human games instead.
@@deepalnanayakkara1887 That is true because a Bishop pair controls both the dark and light squares. But it is still relative. A well-placed Knight + a good Bishop is better than a Bishiop pair where one of the Bishops is completely hemmed in by its pawns.
I think perhaps the knight was so overvalued over the bishop because you should have halved the Bishop’s average influence value. You averaged its squares influenced value over what it could influence on the light squares, but technically it has 0 influence on the dark squares (and the dark square bishop has 0 influence on the light squares). Had you halved the bishops average, it would have been comparable to the knight, though both would have been undervalued.
You're not wrong. I have another calculation which brings the two pieces more in line without simply doubling one or halving one because it seems sensible.
I agree. The difficulty with this approach is defining "influence" as "average number of squares a piece can move to in principle". It overvalues pawns, for example, because they cannot take forwards and they cannot move diagonally without taking. It also undervalues knights because they cannot be blocked by intervening pieces.
Although bishops are great and fast weapons (I favor them) they are quite predictable pieces, especially compared to a knight giving you fits if it has spots in your house it can jump around in. A knight is more feared, unlike the other pieces. There is something in that the fork power is devastating, but how do you factor it? Can't really, it lives on a pipe dream, that has vanished by the time it arrives on the scene. I think they're only good for some immediate tactic or just defense, not great for planning.
This is not correct. The reason is because in the centre of the board, the bishop controls corner and edge squares, which are less important, but he valued them the same as controlling of the central squares. You can't just take half of the squares away just because it's a different color complex. It's also controlling MORE of the color it does control. If you take half of its influence, you're not considering that aspect.
That was an interesting analysis. Prior to neural networks, tables would provide values for each piece depending on where it was on the board, thus a white pawn on the 7th rank is worth a lot more than on it starting 2nd rank, and a knight on the rim is dim etc. Furthermore these values would be adjusted for the end-game. Ultimately, assigning static values to pieces is not going to work well in practice.
I think if you watch analysis by top players, they are usually thinking of piece value in terms of the endgame: thus for example, doubled pawns or isolated pawns are worth less, but advanced, protected and/or passed pawns are worth more. In the middle game individual piece values are not enough, even taking position into account, because the coordination of the pieces is also vital to determine which side has the stronger attack and is more likely to deliver checkmate or reach the endgame with a material advantage.
Knight vs bishop: the Knight's influence can never be blocked, it always influences that number of squares. The Bishop's influence (like Rook and Queen) stops at the next piece.
Your comment is why I specifically did this with respect to an open board. It gets hilariously complicated to even think about otherwise. But your statement is not fully correct: a knight cannot land on a square occupied by a piece of the same color. But even my counter-statement is not fully correct: a knight can *protect* that piece, of course…but that piece does not necessarily always require protection! I am nuts enough that I might consider doing another version of this while attempting to take your observation into account. It would be more complicated like I said but that should make it all the more more fun to work on.
@theanimatedchemist Good point about pieces of the same colour. The same goes for bishops. This sums up the notion that on open boards, bishops are preferred. I did want to take issue with two points about the worth of a pawn - one an ignorably pedantic view of your model : on an open board, it can't take anything for there isn't anything to take; and secondly if on the seventh rank, its "influence" is to move forward or take sideways, but the effect is promotion. I wonder if that should be added in as a "pawn-on-eighth-rank" case, or if its worth should be upgraded on the 7th rank for the potential it has to promote?
@@jeremywilliams5107So you mean the value of a pawn increases depending on what rank its on? Thats getting silly. The whole point is to try to assign pieces objective values outside of any given position. Not "a pawn is worth 1 point... unless you are in a game where you have a pawn on the seventh rank and the opponent must suffer material loss or checkmate in order to stop the pawn from promoting thereby increasing its value". 😂😂😂
Your assumption is incorrect. The number of squares a knight can move to is not always the same. Knight attacks 16 squares of 8 attacks 128 16 squares of 6 attacks 96 20 squares of 4 attacks 80 8 squares of 3 attacks 24 4 squares of 2 attacks 4 242 attacks from 64 squares. 3.78125 average Bishop attacks 2 squares of 13 attacks 26 6 squares of 11 attacks 66 6 squarrs of 9 attacks . 54 4 squares of 8 attacks. 32 14 squares of 7 attacks. 98 276 attacks from 32 squares 8.625 average Yes, knights can't be blocked. However, they can not pin opponent pieces either.
Great video, but as a mathematician, I think the main challenge is define what you mean by "influence" and "value". As others have noted here, there are difficulties in comparing the influence of pawns versus knights versus bishops, rooks and queens. Pawns cannot take forwards, and cannot move diagonally with taking; knights can jump over pieces and fork, but bishops, rooks and queens can pin and skewer. In my view the material value of individual pieces and their positions is most critical in the endgame. In the middle game, coordination of pieces is also very important, whereas in the endgame, it is only the local coordination of pawns that is significant (doubled, isolated, protected, passed), so estimating material value here is crucial. The goal in the middle game is to develop a stronger attack than the opponent in order either to deliver checkmate during the middle game, or reach the endgame with a material advantage. Since engines rarely fall for checkmate in the middle game, they are estimating a combination of activity (attacking potential) and endgame material value.
The pawn's influence should be adjusted to account for the fact that it is the only piece which cannot capture on 1 (and occasionally 2) of its influenced squares. This would explain why normalizing to the pawn's influence left the queen, rook, and knight undervalued compared to the standard model, as well as all four (non-pawn) pieces relative to alpha-zero. Basically, I believe that the 2.916 value is too high considering that the pawn's influence is not the measuring the same thing.
You raise a good point. Here is my version of a 'back of the envelope' calculation. Capture = 1, Move = 0.75 results in an average for the pawn of 2.64. P = 1, N = 1.99, B = 3.3, R = 5.3, Q = 8.6, K = 2.5 Capture = 1, Move = 0.5 results in an average for the pawn of 2.33. P = 1, N = 2.3, B = 3.8, R = 6.0, Q = 9.8, K = 2.8 Given the Q crosses the 9pt value between the two calculations, perhaps a more appropriate "influence" of the pawn is between 2.64 and 2.33. Certainly taking into account @acebharath's point below about the knight's fork would up the knight's value in these calculations too. However, what would a promotion move's value be? Then what about a capture that results in a promotion?
@@theanimatedchemist I don't think this back-of-the-envelope calculation is the right adjustment, because a pawn cannot move diagonally unless it can capture. Thus all its moves are limited by context, not just the forward ones. However also I don't think you can do a useful a priori calculation without being clear what is meant by "influence".
A good next step might be to compare the pieces occupancy rates on particular squares during / after common openings instead of treating all squares as having an equal chance. I also feel certain pieces lose their value in the late game especially the knights, they’re more useful for capturing rooks and queens early game. Definitely a very interesting topic!
As others have rightly pointed out, the values are certainly dynamic. What you suggest first might be within my coding capability. I can definitely go through a database and get board positions for games of a certain length. What exactly to do from there would take a bit of thought.
Good analys!! But i have another varient, if we will count squares, where pieces cant go (pawns and bishops) we get this result: Pawn - 140 (1) Knight - 336 (2.4) Bishop - 280 (2) Rook - 896 (6.4) Queen - 1456 (10.4) King - 420 (3) Less similar to standard, but also interesting
The knight has a super power. It can attack all other pieces on the board without being in danger from the piece it's attacking ─ which makes it the only piece that can attack the queen.
There’s a fundamental flaw in your calculations: the pawns can’t go backwards once they’ve moved, but they can turn into different pieces, and a pieces value is also determined by its compatibility with other pieces.
5:36 for the bishop’s value, I disagree, it should be 1/2 of 8.75 = 4,375. I think we should considerate the full 64 squares and say that he has a 0 influence value on half of them.
The value of pawns is way more fluid than anything. You must check if the pawn is passed, if it is connected and protected, if it is on the 7th rank (only 1 square away from promotion) instead of say 3rd rank, if it is on the edge of the board and the opposing team has only the wrong bishop left, how solid is the base of the pawn island, etc etc. I think these considerations must skew a bit the value of the pawn and therefore the relative value of all other pieces
One thing I find interesting is comparing the value of pawns. Obviously the value is 1 by default, but by looking at the other values, we can see that influence values pawns more than standard because the other numbers are lower and vice versa for AlphaZero.
The reason why your bishop and knight values are so off, is because you're valuing influence on every square equal. In the centre of the board, the knight attacks 8 squares, and the bishop attacks 13. However, 4 of those squares the bishop attacks are on the edge/corner of the board. You could weight influence, giving greater weighting to squares controlled in the centre.
That would be an interesting approach. I think it's generally agreed upon that the center is more valuable to control. I could keep the influences as I defined them and adjust what it means to be a particular square. The only issue I see is with the pawn. A pawn on a7 isn't really influencing anything but it's also eyeing promotion. I do love the idea though. It's making me think a lot.
@@theanimatedchemist Threat of promotion is the threat of having the value increase to that of a queen on the enemy's home row. I'd imagine you'd just have to know the probability of that happening to assign a value to that possibility. Looking at AI games and dividing the number of promotions by the number of pawns that make it to the 7th row should give a pretty accurate value for that probability. But then, the value of a pawn on the 6th row is influenced by the fact that it's eyeing becoming a 7th row pawn... [edit:] also, the pawn still has to take up one move for that promotion, so the act of expending that has to be assigned some value and it has to be subtracted from the points increase due to the promotion (before their result is then multiplied by the probability of it happening).
@@theanimatedchemist maybe weight the # of squares influenced by the % of the time the piece stays on that square by extracting the info from a chess database?
I get what you're saying, and other's have said similar before, but it's not that simple. You are correct a bishop can be blocked by either color but a knight's movements are restricted by like-color pieces occupying the "landing" square. For example, all knight's have 3 potential squares to move from the starting position but 1 of those 3 is blocked by a pawn.
I think that really the only thing that needs to be added to the “classical” picture is that having the bishop pair is worth about 0.5-1 pawns. It explains why trading one bishop for a knight is bad (you lose the bishop pair), and why trading your second bishop for a second night is almost always dubious (you lose the ability to force a trade between your “twin-less” bishop and one of your opponent’s pair).
I would question the approach of only dividing the bishop's value by the total number of squares it can influence (32) rather than the total number of squares on the board (64), given that it is the only piece on the board that is limited to half of the squares (and given that this means that its value is -- I would argue -- more than halved by not acting as a pair).
That type of deduction is flawed, as the number of squares it can influence inherently already determined what squares it could attack/defend. There is no need to divide it's value in half. One could argue that the knight has a similar limitation. Although the knight can access all squares, what can it access per move? While a knight is on a light square, it can ONLY access dark squares and vise versa. Same limitation. If you want the knight to be able to attack dark squares while it is on a dark square, it costs you an extra move.
@@exisfohdr3904 _"If you want the knight to be able to attack dark squares while it is on a dark square, it costs you an extra move"_ And if you want the bishop to attack the other color of square, you're SOL. A true analysis of piece value of course should include that bishops only control half the squares and therefore half this influence calculation. And, when you do this, you can also establish why chess players don't actually favor the bishop per se, but the bishop _pair_ . That is to say: two bishops of opposite color squares are better than a bishop and a knight, or two knights. The reason being that these two bishops cannot hinder each other's movement, they are exactly twice as good as one bishop. And the same is not true of the other combinations.
Here is the intrinsic (note 1) piece value system I use and have taught: P = 3, N = 10, B = 10 or 11 (note 2), R = 15, Q = 29, K = 13. As you can see, if you exchange a N & B for 1 R and 1 P, you give up 21 to get 18. Not good! But if you get 1 R and 2 Ps (21), now we're talking! Also we can see that 2 R = 30, a bit better than 1 Q. Note 2: 2 Bs = 21. Which B is worth 11? The first one to leave the board. This may seem cumbersome, but with a bit of practice it's very easy. Primarily use this when making exchange decisions. Note 1: the actual value of any piece changes with its role or position. A white N placed on an out[post on e5 for example, is usually considered to be as strong as a R. The K is a fighting piece in the endgame so use it as such!
That's an interesting system. How did you develop it? It's not exactly the same but it seems like it has the same feel as Sarratt's (1828) where it's just not as simple as a piece having a constant value. It's on page 2 of his book and I think this link will bring you directly there (books.google.com/books?id=_0wCAAAAYAAJ&pg=PA2#v=onepage&q&f=false).
@@theanimatedchemist It's not my system but one that I heard about a long time ago. I'm sorry, but I can't find the originator of it anymore. In any case, it's the best one I have ever seen! I say it's the intrinsic value by which I mean their basic value. But once the game progresses, the actual value will change due to all the positional aspects of their locations. For example, a white N on e5 is quite a dangerous piece so its actual value is greater than 10. Put it on d6, then it's worth a R.
@@belaji Dynamic values are probably the way to go. Finding one that is sensible and teachable may be difficult though it seems like you were lucky enough to find one!
How can a king be assigned a value? Anyway regarding rooks, my experience says, practically queen is actually worth more than two rooks in most circumstances. Only if your opponent has some major permanent weakness like a shelterless king or destroyed pawn structure, may be in those cases two rooks may offer you some hope. Even the great Carlsen couldn't win with two rooks versus queen in an otherwise equal position.
Plain Q vs 2R is difficult for either side to win. But if you have an extra pawn the Rooks will find it easier to push. Certainly 3 minor pieces are a bit better than a Q, but they have to be coordinated, so how do you score that?
Appreciate the kind words! I do mostly chemistry and science here but there will be a couple for chess videos in the future…if my laptop can manage all the PGN analysis that is!
While the knight lacks in board influence, it gains a powerful ability that when a knight puts your king in a check, you either have to take that knight or move your king because you cant block the knight's attack with other peices. This jumping mechanic makes one of the most unique checkmate pattern in my opinion, the smothered mate.
I think the fact that the knights are unobstructed by your own pieces, or your opponents pieces, needs to be considered. The board is rarely fully clear. I would have liked to see this but considering the number of pieces you would typically see on the board and the influence of the pieces in those positions. I.e., knight is more powerful at the bigging of the game and less so at the end, assuming that the board is open.
Woah, the influence section gave me a huge realization: Knights are known as being the powerhouses in a closed position (a position with many pawns that aren’t able to move on the board. The knight cant be obstructed except by same color peices, so it can still make of it’s moves and captures. However the bishops lose their advantage worse than the knight. My hypothesis is that if you had an application running that calculates turn-by-turn influence of each peice, the knight’s avg influence would be higher for closed games, or games where you put it on a “good square.”
I also believe that the number of pawns on the board, and their ability to move, devaluese the bishop when the pawns are on it's color. This also increases the value of the knight. Closed positions favor knights, open positions favor bishops.
The main issue with this analysis is that it assumes the chess board is empty, negating the main advantage the knight has over all the other pieces. If we counted the average available squares in positions from real games, you'd get a much higher number for knights as, unlike bishops, rooks and queens, they don't get stuck between other pieces.
everything is slightly under which can be explained by the fact the pawn is off. it considers all pawn moves as equal but it usually doesnt have 3 moves since its rare for a pawn to have both captures and forward be available. a pawn wont always have a capture and can easily be blocked meaning it can even have 0 possible moves sometimes and the knight has special powers such as attacking a queen while not being attacked itselt and jumping over pieces which is not considered
I think the most crucial aspect in your underestimate of Knights (or overestimate of long range pieces) is the fact the long range pieces can be blocked in by enemy or even your own pieces. As a crude example, at the starting position, none of your long range pieces actually have any available moves, while Knights have two possible moves.
Maybe split the evaluation into max mobility count and max attack count. E.g. the bishop can attack max 4 pieces, while the knight can attack max 8 pieces. This would bring them closer together. Also, the pawn is currently overvalued as it should be much worse when ignoring its promotion ability.
This is an interesting idea and "max attack count" can be adjusted for position. On a8, for example, a knight attacks 2 squares and the bishop only 1. In terms of percentage, the knight can attack 100% of it's total influence while a bishop only 14.3%.
Only going off of legal moves on any given square is not the best way to evaluate piece value. You need more factors than that. For example, the number of squares until promotion is a significant factor in determining the value of a pawn. A couple of connected pawns on the 7th is usually more valuable than a rook.
A few things: 1. The pawn's influence is extremely overestimated. When we say that a pawn can influence 3 squares, it cannot actually capture or move to all of them. Pieces that move and capture differently are difficult to analyze. 2. Leaping pieces are far less likely to be blocked than sliding ones. In reality, a rook on one side of the board on the middlegame has basically no chance to make those cross-board 7 square moves. This can be shown by the fact that a jumping rook is pretty much a fair alternative to the queen. 3. Movement direction and the ability to jump over pieces or defended squares have quite a bit of influence on value. One reason that a knight is more useful in the early game than a non-royal king is that a knight has 4 ways to make forward progress and two of them completely bypass a row of pieces. One reason a pawn is useful at all is that all of its moves point toward the enemy and the promotion squares. A pawn with no forward bias in its movement pattern would be absolute hot garbage. 4. Pieces are more likely to be on good squares than bad squares even after moving randomly. For example, if you jump a knight around the board at random, it's much more likely to end up in the middle of the board than in the corner, because there are many more squares that a knight can move from to reach the middle of the board than the corner, and this applies recursively too. Low mobility due to board edge effect is way less important than placing pieces on random squares suggests, as in reality everything but a rook is more likely to be in the center even if moved by a monkey, and human or engine play tends to compound this effect, not negate it, since random play assumes you're equally likely to open with Nf3 or Nh3, while grandmaster level classical and rapid games played Nf3 as their first move literally one thousand times more often than Nh3. Real humans do not like moving pieces to low mobility squares even when the option is available, and even a random move agent will avoid those squares most of the time because low mobility means for pieces that move symmetrically they are also difficult to get to. The "correct" average influence of a knight is much higher than 5.25. Likely over 6. With a random move agent you could argue that knights are more likely to end up sitting still and not moving on the rim than in the middle, but that's a rather moot point unless you actually plan on using a random move agent. 5. A piece that doesn't develop anywhere can't really do much other than sit there and defend. This is part of the reason why the Ferz (1,1 leaper) is stronger than the Wazir (1,0 leaper) despite being more influenced by the board edges and also colorbound. The reality is that if you have a Wazir, marching it to the middle of the board is almost never gonna be a good use of 5 moves in a row. But marching a Ferz to the middle of the board in 3 is a different story, and bringing out a knight attacks the middle immediately. Being able to quickly and efficiently bring a piece to attack or defend an important square is very much needed for attacking. It is one reason that the King does not feel stronger than the Knight in practice outside of the endgame even if you make it not royal. The king moves significantly slower. 6. Colorbound pieces usually suffer somewhat if there are some parities of them which aren't available. 7. Move synergies often have important effects on a piece's strength. A queen has a power that neither the rook nor the bishop has. That is, the ability to move like one and attack like the other. A king can reach a square a knight's hop away in 2 moves, while a Wazir takes 3, and a Ferz can't do so at all. Another advantage that the Knight has over the king is that it has substantially more synergistic movement than the king. That is, a king can attack 25 different squares including its initial square should a knight can attack 41 (of which 8 require it to remain in place, a knight cannot "triangulate," unlike a king). Meanwhile, a pawn can only attack 4 squares unless it captures, and 2 of them require it to stay in place. It cannot attack its own square either. This is significant because making a non-capturing move to threaten a square is probably the vast majority of moves made in chess, and even if they are not made, they force your opponent to consider them. The fact that a rook or queen can instantly decide to attack any square on the board in a sufficiently late endgame is quite valuable. And the fact that the queen can potentially fork or pin most of the combinations of squares on the board is hugely valuable as well.
Is there a way to take those those tables and that influence value and create the Piece Value Tables the chess engines use to determine where pieces are best positioned?
I took a very naive approach to finding piece values but I think at best some "principles" of chess are revealed by them. For example, centralize the queen, put the bishop on the longest diagonal, "a knight on the rim is grim", and place rooks on open files/ranks. As for engines, my knowledge of how they work is about as deep as my opening theory so here are a couple of papers regarding AlphaZero which may give you a sense of how they work (arxiv.org/pdf/1712.01815.pdf and arxiv.org/pdf/2009.04374.pdf).
I have somewhat of an understanding as to how the engines work. They use Piece Value Tables as a way of determining if a piece is on a good square or not. That then gets included in the calculation of the position to determine what the next best course of action is. I was just wondering if there was a way to take the tables you showed with how many spaces each piece can move from certain squares and derive those Piece Value Tables. But thanks for the info I'll check it out.
@@Michael-sb1tt ok, now I get where you're coming from. The second of the two links describes how AlphaZero's piece values were derived (Section 3.8 - Material).
I like the general idea, but I feel there are a few flaws with the concept. First of all, I guess it's obvious that pieces can only influence all these squares if the board is empty. Which isn't the case. The fact that the knight can jump over pieces and therefore ALWAYS influences the numbers written on screen whereas a bishop's will nearly always be reduced is something that should be accounted for. Secondly, by only averaging over the squares the piece can land on, you are overvaluating certain pieces, i.e. the pawns and bishops. Overall, they can influence less squares, but since we are only averaging over the ones that they can access, this disadvantage is not taken into account. It seems a bit weird to use a system like that and then don't account for these things at all. Which gives the feeling that you searched for a way to get close enough to the standard values and then worked your way backwards from there. Because if you included the squares the pieces can not control, the bishop would already be weaker than the knight. And we haven't even taken into account that the knight can jump at this point. Which I feel already shows that the system isn't really telling you much about the pieces' values.
Idk if u r gonna update this video ever with the feedback. But the value of the queen is bigger because it can cause "draws" from losing positions via perpetual checks. And its the only piece that can do this by itself
I was always taught that the Knight was sligtly higher value then a Bishop (3.1 to 3) due to its superior attacking and area influencing potential. The Knight retains its threat on all squares it can reach irregardless of the pressense of pieces which would otherwise block every other piece type, while the bishop can only each reach half the board. I think you would find that statistically knights are moved more then Bishops are and are only the pieces most likely to put the king in Check.
Why knight’s ability to fork the king and the queen is overlooked? I mean , its the only piece that can fork the two most important piece on the board and eat up the most valuable piece (queen) without being in the danger of being captured. Sure , the bishop , the rook and the pawn can fork the king and the queen but they cant do it on their own. They need a supporting piece to do the royal fork. The knight OTOH dont need any piece. The royal fork by the knight is considered as game over for 99.9% of the players. Its also really really tough to understand what the knight’s intention are when he is on F5. A protected knight on F5 is definitely worth a queen as it has great chances to fork A, The Rook and the Queen B , The rook and the king C, The queen and the king D, The Queen and an unprotectable Bishop These forking adds lots of value to the knight.
I purposely "overlooked" plenty and I state as much. You are certainly correct: a knight does have a devastating move that it can pull of on its own that no other piece can. This absolutely would increase its value over the simplistic calculation I did but the question would be by how much? You say "lots of value" but it's not as if every move by the knight results in a fork let alone a royal fork. So how on earth would I go about quantifying that? If I ever do a follow up, and I might, I would absolutely take it into consideration. Probably still wouldn't satisfy everyone though!
The knight is a great piece ofcourse, with all the abilities you have described, but its negated to some extent by some of its shortcomings. All the other pieces can continue attacking from different positions, but once a knight is dislodged it needs two or more moves to attack the same point. Also if you are playing on one side of the board, and your opponent switches the attack to the other side, your knight will need 1 or two more moves to reach that side, while rooks and bishops can travel instantly.
check mate is usually worked by surprise! In my opinion, the Knight has the great advantage of stealth, valued in the same way as the `assist` to a goal scorer. The value of a football `striker` is huge in comparison to a great winger of midfielder, BUT without that pass made to the striker, the striker would be pretty useless! This is why I personally value the knight highly because he can creep up all of a sudden from out of nowhere, to afford the glorious queen her checkmate move.
What's missing from your assessment method is the fact that the knight, as you said, moves uniquely, any piece it attacks cannot take it. For undefended pieces, the Queen can take any of them except for the knight, to give an example. Also a king can never capture a cheking knight, altough a king might be able to capture any other checking piece. This is one of the reasons your knight was ranked to low relatively to the gameplay and experience based assessment that's 'default' today.
I came across this issue when trying to create my own heuristic for a chess bot. We often hear that a rook is valued at pieces in a open board but what about a closed position? I tried providing two heuristics: A control "Frontier" that explores the squares it directly controls, and a hard value. The intuition is that the hard value is a long term evaluation of how desirable it is to preserve this piece and acts as a value "floor" when computing trades. This way, a trapped queen doesn't immediately get traded for a knight. Unsurprisingly enough, the rook had the largest hard value, reflecting the discrepancy between the early game and the late game.
A couple rational steps to improve the influence rating could be to factor in how likely it is in a real game for a piece to be on that square and be able to move to the squares it can see with rational play. Another step could be to factor in how many moves it would take for a piece to get to each square on the chess board
Assuming that reach (or influence, as you call it) is a good measure for the value of a piece… the number of surviving pieces on the board limits reach.; Your own pieces even more so than opponent pieces, since you cannot take your own pieces. So if the pieces were randomly distributed (which they aren‘t), you‘d need to adjust each piece‘s value by how many white and black pieces are left at any given moment. The knights would be the least negatively affected by this analysis, which explains why a) your analysis undervalued them and b) why they are relatively more powerful on crowded boards (which correlates with closed positions). There are additional complexities stemming from the caveat that pieces indeed aren’t randomly distributed, particularly at the beginning of the game. However, the conclusion that the value of pieces changes over the course of a game as pieces get eliminated - even before considering their position on the board - will be familiar to every chess player.
The main flaw with just looking at number of squares is that being unable to switch colour seriously hurts its point value. In an endgame you can just move every pawn and your king off of that colour. This obviously applies to the bishop, but this principle holds true for a lot of alternate and historical chess pieces too. An alternate version of the bishop that is just given the ability to step one to the left or right to switch colours goes up almost to the value of a rook (like 4.5 points or something). The "Camel" which is a variant on the knight that can jump 3-1 (instead of the Knight's 2-1 jumps) ends up stuck to one colour and therefore ends up with a value of around 2 instead of the value of 3 for the knight. The same way the queen is a hybrid bishop/knight, there are historical hybrid knight/rooks and bishop/rooks, and they actually end up with very similar piece values to each other because once again the bishop gets a lot stronger when it can switch colours--which knight movement allows. (These compound pieces are worth like 8 and 8.5 respectively). With just standard pieces, there is often a point bonus that is given for owning the bishop pair, because both bishops together can cover every square (the bonus is roughly half a pawn).
All pieces except the pawn can attack on all counted squares; the pawn can either capture or occupy an empty space without fighting. It seems to me that the pawn was overrated based on the squares. But a pawn can become a piece, this is not taken into account at all, there is a slight underestimation here. The knight is also noticeably undervalued, since its moves, unlike all other pieces, cannot be blocked by enemy pieces. Your own or someone else's piece on the bishop diagonal blocks all the squares behind it. Same with other figures. This does not happen with a knight; blocking each of its cells only blocks that one.
Your "averages" are calculated based on the total number of squares any piece *could* move to *IF* there are no obstacles (such as its allies in the way; enemies in the way; or tempo issues such as checks, pins, and opponent's attacks on higher-value pieces). But the true value of a piece is intrinsic to what can do in the current position, extending to its realistic potential. I would submit that the value of a piece is a very dynamic number. We've all seen games where a single pawn becomes more valuable than everything else on the board, simply because its next move results in checkmate, and the opponent cannot prevent it.
Even if the true valuations are wrong due to a variety of factors, just the "influence" grid is interesting to see, and probably useful for all beginning chess players to try to maximize their coverage/utility of pieces.
bishops are usually considered having more value than knights but a piece has a different value depending on the scenario of the game. At times, a knight pair can be stronger than a bishop pair but it is rare. Knights are not long range pieces,which is one of the reasons they get weaker in end games or open games. but some times a bishop can be stronger than a rook, depending on the game. I have been playing chess for a long time and I am pretty good at it so I sort of think about this practically.
What if influence was calculated not by what they can move to, but on a given square, how many ways could a piece move to that square? What about captures? Considering that pawns can only capture in certain directions but they can move in a different directions, would it be better to evaluate the influence based off of their threat or defense to a square if they were in every board position? Idk.
Knights extra value compared to this comes of course from the fact it's the only piece that can attack any other piece without being threatened, for its movement. And bishops value gets demoted once we consider only one bishop.
He addressed this, when a party reaches the end it is no longer a pawn so it's a different piece with a new value separate
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a knight can attack other pieces without supports (like you can chase the queen with a single knight while other pieces need at least one cover). It can also move better in close game.
The influence of all pieces but the knight can't go through other pieces except via x-ray attacks and fossilizations. That's probably the biggest thing that this video didn't cover. because if you factor in that most of the pieces trajectories will be blocked in some way, the knight then regains its status as being very similar in value to the bishop
I expected that stockfish evaluates a bishop distinctly above 3, but I was most surprised when the rook had a value above 5.6. Shows that endgame really is everything, huh.
Many things not accounted for, like the value of a pawn changing into a queen, or how near it is to changing to a queen, or how a bischop en pawn can auto/mutual defend itself, these are also very valuable properties.
I definitely thought about these things but there are so many permutations, never mind trying to assign values to each, that trying to properly account for them all would just get in the way.
I think considering the number of squares covered by two moves into would be fairer on the knight, it doesn’t reach the bishops level, but it does double its influence for most squares it can land on, whereas the bishop at most adds 3/4s of its own and lowest doesn’t add anything
adding an addendum, the second move should only consider the average of each potential second move, that wasn’t very clear, so for example, a bishop in a corner would move to each square on the diagonal, covering those 7 squares + (2 * (2 + 4 + 6) + 1) / 7 squares for the average number of squares covered by each of those moves and not considering the 7 squares along the diagonal or the corner square the bishop started on, because otherwise the queen and rook cover the whole board each, and ask any chess player, a queen and rook are not equal
Chess960 drastically changes the values of the pieces. Theoretically, they can be vaued the same, but its starting position can change the outcome & strategy of a game.
He took influence only as the number of squares controlled. You talk about the dynamic value, but he talks about the static value. There is no point in calculating dynamic value since it changes with every position and only a computer would be able to evaluate all the pieces according to the position on the board for most of the positions, and still it would not be 100% correct since chess isn't solved to this day (with more than 7 pieces on the board)
If I am playing a Benoni type I am killing my bishops for their knights. Same with closed Dutch. Typically - I like more open games and with that I will throw my knights away to kill their bishops. One thing with your analysis. Take any board position on any move. Calculate the influence for that piece, on that move. Very early in the game, the pawns and knights have more spaces. Bishops, queens, and rooks are useless. So you can make an argument that early in a game - pawns are important at impeding influence of competing pieces. And since knights can jump, this helps their mobility. Meaning… The time of the game can have an influence on value. The type of game (open/closed) can create a skewed value for knights over bishops/rooks. So time of game is a big deal. An end game wide open- queens and rooks rule, with bishops the next important and knights the last.
well one thing is that most pieces are blocked by other pieces. So the bishop can only hit the first piece in every direction, and it's rarely covering behind it, whereas the knight always covers all squares it can reach and is good at covering backwards. If you counter the max number of square a knight control it's 8, but for both a bishop and a rook it's 4 if you account for being stopped by other pieces of course that's not a good way to look at it either, just a way to point out that the board coverage without other pieces on board isn't always a good metric.
Fair comment. This is actually a simple way I could have "fixed" the low value of knight after seeing I got 1.8. Of course, simply doubling it to 3.6 doesn't seem right either.
I think the flaw in the Influence scores is that they do not take into account the fact that the knights can jump. At any one given moment your opponent can block the full range of moves open to any of your pieces. Not so the knight. So this adds to its value - though not sure how one could use maths to factor this in ...
Yeah that's beyond me as well. I would attempt something like this: given the position of a knight, how likely is it that it one of its potential moves is blocked by a piece of the same color? For example, say white is down to a knight and king. The knight can move to b6 or c7 as long as the king does not occupy one of those squares. Since the king has the potential to be on 63 other squares, it has a 2 in 63 chance of blocking one of the knight's moves. Now if white has a bishop on a8 and a king, then the king has a 7 in 63 chance of blocking one of the bishop's moves. Obviously more pieces would get absurdly complicated but maybe it's worth a go for just 1 piece and 1 potential blocker. Maybe it'll give some type of "factor" which would increase the knight's value compared to the others.
Every endgame up to 7 pieces has been solved. The results of the endgame table bases confirms that in most endgames, the bishop outperforms the knight by far, especially when there are two of them. Two bishops can even outperform 3 knights. There’s an article that has all of the data called “Bishop vs Knight”. While the knight is very handy in the early game, I believe trading it for a bishop is typically a good idea.
I've heard of the endgame table bases before but haven't looked into them much. I wonder if there will be a number of pieces where the knights are superior or if it will always tend towards the bishops.
@@theanimatedchemist I forgot to address your question at the end of your comment! Bishops will always be better than rooks in higher numbers. A common trend in chess programming is to actually modify the value of pieces depending on their position and other factors. Pawns, for example, loose half a point of doubled or isolated. The double bishop is often hard-coded to be worth significantly more than 6 material because they can make an incredibly strong mating net.
The most unintuitive thing to me as a chess player is that Stockfish values rook+pawn higher over a knight and bishop or two knights and approxilately as much as two bishops.
I'm a little confuser. What is the point of the video? I came up with my own value system for the pieces, but everyone says it is wrong, and actually so do I? The reason the horse is more valuable is because it can hit every square, not just half of them like a bishop. It can also smother mate. It is the best piece for attacking a queen. It cannot be blocked. It can get in and out of high traffic areas by jumping over them, where rooks, bishops, and queens need higher clearance to pass through. If it is harder to anticipate, so many people get tricked by simple forks that they would easily see from another piece. It is also the only piece, othe than a pawn, that can start a game, giving it high stretegic importance. It is harder to master, but it can be extremely useful if honed carefully.
The point? I think I made it clear around 9:45 when I said "So will this analysis, flaws and all, add much to the discussion or end up in the hallowed digital halls of Wikipedia? Probably not - *but I enjoyed giving it a go anyway*."
Nuh uh. you're clearly a beginner. I assumed this because first of all you said the knight is a "horse" lol. Knight is decent in the middlegame and gets almost completely dominated by the bishop in the endgame. Smother mate is very rare you'll very likely not get it in most of your games. It can't be blocked that's true but it's easier to trap it. knights get trapped more than bishops. Those jams are mostly temporary after trading few pieces it clears up. It's not hard to anticipate knights movement in fact most high rated people keep an eye on knights movement so they usually know what's going with knight. Bishops > knights in most cases Knights > bishops in closed position Bishop's long range makes it effective. even tho it can only control half of the board it's a very good attacking piece Bishop pair is nightmare against 2 knights
This was an extremely interesting video! I'd like to echo what some other comments have said in that I believe the reason bishops and knights are rated as equal in the standard system is their value compared to each other is determined largely by the position as a whole. They're equal by standard, but the player elevates them or hinders them with their handling of the position. Did you put all your pawns on light squares? Well your dark squared bishop DEFINITELY is now worth more than your knight, but your light squared bishop may be worth less than one. Excellent video though, seeing piece value expressed as influence is something I need to consider more in my own games.
I'm glad you found it interesting. You and the other similar comments aren't wrong, of course. I haven't included any consideration to actual positions which is an obvious flaw but it did keep things simple. We have all seen or played games where a bishop is reduced to a pawn with a pointy hat and the knight is getting all the glory so there is certainly more to the story.
Knights have an ability to jump so it can't be restricted. Its influence always stays the same. If you put other pieces on the board, knight still can move the same however bishop movement can be restricted highly. So, your measurement has a big flaw.
Knights can still be restricted by other pieces of the same color but a knight would need 8 pieces on 8 particular squares (or whatever number in other situations) to lose access to all of them while a bishop can lose access to 7 squares (say from a8) with a single piece on b7.
The authors of the A0 paper do note "...there is usually an additive adjustment while in possession of a bishop pair" but I could not find a specific bonus. Perhaps I missed it though.
It should be a weighted average; the influence of a piece in a position of the board should be weighted by the probability that it will be there in the first place - which naturally changes across time, and the probability that it's influence will be neutralized by surrounding pieces. This is a calculation only the sophisticated neural net of an AlphaZero is able to do.
My guy, the reason the knight's valuation comes out as lower is because this system doesn't consider that other pieces are usually locked in or prevented from actually having access to all of their squares they could theoretically reach because of the pieces on their respective side.
Props for addressing me as "My guy" 🤣 Yes I did not make such a consideration. All pieces can be restricted to some degree and trying to account for all possibilities was beyond what I wanted to do. I may take the idea of move restriction into account in a follow up as many people have posted ideas or criticisms, some of which I never would have thought of on my own.
Right but it is rather relevant since Knights can't be blocked by their own team while bishops can. assuming the same number of pieces on each side (usually) for a fair evaluation, this is important to consider, MY GUY@@theanimatedchemist
Thanks for watching! I’m glad you enjoyed it. I’m a long ways off on another chess video but my computer is plugging away at millions of PGNs as I type.
@@theanimatedchemist imma be watching your other stuff too ☺️ seems like you care about the vudeos yiu produce and keep the good work going friend, best regards 🙂
@@burimsaliji23 Thank you! I will do my best but other than the 2 chemistry videos I've got in the queue it's going to be some time before I've got others ready to add. Since my comment 11 days ago, I've analyzed every Lichess PGN (rated, classical) from Jan-2013 to Dec-2017...I'll leave it a mystery as to why...and I'm downloading 2018s as I type!
I think that the reason your influence analysis was a little off was because it didn't consider some more strategical elements. Influence isn't the whole story in a Chess game, rather it's the Quality of that influence. Knights may control fewer squares than most other pieces, but their ability to jump pieces, the fact that they can see both colour complexes, and the fact that no piece other than another Knight can easily challenge their influence I think increases their value.
If you saw me play, you would understand why I didn't include any "strategical elements" :) As for the knights, definitely something is missing in what I did to result in such a low value. I am still curious if the three things you mentioned, or something you didn't, could somehow be evaluated and added on to it to some degree.
@@theanimatedchemistI'm by no means an expert I was just giving my ideas as to why it might've happened that way. If I remember correctly, I thing the original piece values are derived from how strong each piece is in the endgame, but I could be wrong about that.
A knight deeply entrenched in enemy territory past the threat of pawns and supported by a friendly pawn is seriously dangerous. Understanding some Peaces can easily have a 1.5x multiplier based on board position. In the french defense the black light squared bishop probably sits at a meek 2.3 points for example. The other pieces black have probably increase in value however due to excellent board position.
Seems a convoluted way to calculate piece value. I figured it would be “maximum amount of squares a piece can attack while in the enemy camp divided by 2”
If you talk about static value, bishops are the same. About the pawns, usually the 2 on the edge (first and last column) may be considered weaker because they can take only on one side, but it depends by the position because in endgame they can be a winning condition (any pawn can be a winning condition in the right scenario). Usually the center pawns are considered more important because controlling the center is a strategic advantage... but it all depends by the position. Every chess piece has a variable value, a rook can be considered useless if it's trapped and a pawn can be considered as good as a queen on certain scenarios, so it's impossible to say certantly the value of each piece. For the sake of semplicity we just say all pawns are all 1, knights and bishops 3, and so on. The value of the king is impossible to calculate for me, you must protect it or you lose so it must have the highest value, but if we count value as the strenght of a piece then it's for sure weaker than a rook (and most of the time if not almost always weaker than bishops and probably knights too)
I don't disagree with the idea but I'd like to find a methodology to obtain an objective value to add on (or adjust all other values according to it) rather than one that "feels" right.
You would probably like this then as it goes over giving a pair of bishops a value of 7.5: archive.ph/20120714204040/mywebpages.comcast.net/danheisman/Articles/evaluation_of_material_imbalance.htm.
I like the concept here, but in practice these values are way off. And that is because the board is not empty. With pieces in the way, a lot of those pieces will have far fewer squares they can really go to in practice than if the board were empty. It should also be considered that the total possible influence is not necessarily linear with respect to actual value. As in, sure a queen can move as much as a bishop plus a knight, but even the standard values for a queen give it more value than a bishop plus a knight. There's also another flaw in that pieces are not in all places on the board an equal amount of the time. As in, knights rarely end up in corners so that shouldn't really count against them. But queens can choose to stay near the center of the board and influence a ton of spaces. Anyway, I'd recommend doing an analysis of averaging the influence over like 1000 real game situations. And then also giving a score from those games based on who won and with which piece differences, or based on a comparison with stockfish evaluations to those positions or something.
I love your suggestion but I have no idea how I would code that up. I know I can code something up to get 1000 random positions from the LiChess database but how to turn those positions into influences for each piece (outside of doing so by hand)...might be a worthy summer project or completely beyond me.
The values are dynamic. In a closed position knights fare better than bishops and in open it's the other. Rooks are pretty useless in early and midgame but become extremely valuable in endgame. Especially for the side winning.
Indeed and numerous individuals have developed systems which try to account for this. Additionally, the values of the pawns aren't all identical either (a connected pair is better than two in the same file and so on).
In my view, the standard and chess engine valuations of a piece generally estimate its value in the endgame. In the middle game, position and coordination of pieces matter more for which player is able to deliver a checkmate or decisive attack. If you are checkmated in the middle game, the value of your pieces does not matter, as the Opera game demonstrates - but even without a checkmate, a material advantage can be temporary in the face of a decisive attack. A chess game is won either by a checkmate in the middle game or by reaching an endgame with a material advantage. Hence for individual piece values, it is better to consider them in the endgame.
For what you want to do it
Wrong. The equation is non linear. So the coefficients of a piece are dependent on high order terms. However there is an objective exact average value for each piece. But we aren't anywhere near proving that. But the estimates we use are likely +/-1 of the true value given engine with heavy training are using these values
@@theanimatedchemisthaving watched GothamChess, he mentions the fluidity of the points system depending on the position of individual pieces. For example, a rook wasn't able to move more than a couple of squares, due to it's position, so became, for all intents and purposes, a 3 point rook.
Another method is "How Many is required to force Checkmate." . Three Knights, a Bishop and a Knight, Two Bishops, One Rook, and One Queen suffice.
N + N + N = # and B + N = # thus N + N + N = B + N therefore N + N = B
also
B + N = # and B + B = # this B + N = B + B therefore N = B
so
N ≤ B ≤ 2N
And my calculation puts B (3.0) between N (1.8) and 2N (3.6) 😱
Before anyone thinks I'm being truly serious here I'm just having a bit of fun :)
With a king though
How about pawns 🤔 4? 5?
@@Carboy45 two
Mating potential is a very small component of material value. A piece which moves like a king but cannot be checkmated can force checkmate against a bare king with the help of almost any conceivable royal piece, which is more than can be said for a Rook. But clearly no sane person would consider a non-royal king to be worth more than a rook, and in many situations it wouldn't even be preferable to a knight or bishop.
The Knight gets extra value because it is harder to block in. A single bishop can't cover dark and light squares, a knight can.
You're definitely correct. I did some new calculations giving additional weight to whether the piece can capture or defend and the Knight and Bishop values are more aligned with an ever-so-slight advantage to the Bishop. Probably worth a video update. I've got a chemistry one in the queue then two more to make before I go back to chess.
@@theanimatedchemist We're also only talking about relative average value. The values of pieces change based upon how they are positioned. A Bishop that is hemmed in by its pawns has little value. If you watch a lot of lower-level games you will see people that will leave the a, b & c pawns on their starting positions and never move their queen's side Rook or Bishop. In this situation that Rook was nearly useless because all it did was sit on A1 and do nothing but guard the unmoved Bishop and a-pawn, and it is doubtful that the a-pawn even needed a defender.
@@kingrex1931 That's also true. The other calculation I've done also considers it on an open board for simplicity. I guess AlphaZero's calculation of the values may be the most sensible given that they were derived from actual games. I wonder what the deviation would be if the analysis was done with human games instead.
but a bisop pair vs knights or bishop vs knight is considered better for the person with bishop
@@deepalnanayakkara1887 That is true because a Bishop pair controls both the dark and light squares. But it is still relative. A well-placed Knight + a good Bishop is better than a Bishiop pair where one of the Bishops is completely hemmed in by its pawns.
I think perhaps the knight was so overvalued over the bishop because you should have halved the Bishop’s average influence value. You averaged its squares influenced value over what it could influence on the light squares, but technically it has 0 influence on the dark squares (and the dark square bishop has 0 influence on the light squares). Had you halved the bishops average, it would have been comparable to the knight, though both would have been undervalued.
You're not wrong. I have another calculation which brings the two pieces more in line without simply doubling one or halving one because it seems sensible.
I agree. The difficulty with this approach is defining "influence" as "average number of squares a piece can move to in principle". It overvalues pawns, for example, because they cannot take forwards and they cannot move diagonally without taking. It also undervalues knights because they cannot be blocked by intervening pieces.
Although bishops are great and fast weapons (I favor them) they are quite predictable pieces, especially compared to a knight giving you fits if it has spots in your house it can jump around in. A knight is more feared, unlike the other pieces. There is something in that the fork power is devastating, but how do you factor it? Can't really, it lives on a pipe dream, that has vanished by the time it arrives on the scene. I think they're only good for some immediate tactic or just defense, not great for planning.
This is not correct. The reason is because in the centre of the board, the bishop controls corner and edge squares, which are less important, but he valued them the same as controlling of the central squares. You can't just take half of the squares away just because it's a different color complex. It's also controlling MORE of the color it does control. If you take half of its influence, you're not considering that aspect.
@@pauls5745 bishops can pin and skewer opponent pieces, knights can't.
That was an interesting analysis. Prior to neural networks, tables would provide values for each piece depending on where it was on the board, thus a white pawn on the 7th rank is worth a lot more than on it starting 2nd rank, and a knight on the rim is dim etc. Furthermore these values would be adjusted for the end-game. Ultimately, assigning static values to pieces is not going to work well in practice.
It was interesting to read about systems where value was adjusted based on the phase of the game or the position (rank/file) of the piece.
I think if you watch analysis by top players, they are usually thinking of piece value in terms of the endgame: thus for example, doubled pawns or isolated pawns are worth less, but advanced, protected and/or passed pawns are worth more. In the middle game individual piece values are not enough, even taking position into account, because the coordination of the pieces is also vital to determine which side has the stronger attack and is more likely to deliver checkmate or reach the endgame with a material advantage.
Assigning static values to pieces works well in 99% of chess positions. I'm saying this as a master-level chess player.
Knight vs bishop: the Knight's influence can never be blocked, it always influences that number of squares. The Bishop's influence (like Rook and Queen) stops at the next piece.
Knights are better in closed positions and bishops are better in open positions
Your comment is why I specifically did this with respect to an open board. It gets hilariously complicated to even think about otherwise.
But your statement is not fully correct: a knight cannot land on a square occupied by a piece of the same color.
But even my counter-statement is not fully correct: a knight can *protect* that piece, of course…but that piece does not necessarily always require protection!
I am nuts enough that I might consider doing another version of this while attempting to take your observation into account. It would be more complicated like I said but that should make it all the more more fun to work on.
@theanimatedchemist Good point about pieces of the same colour. The same goes for bishops.
This sums up the notion that on open boards, bishops are preferred.
I did want to take issue with two points about the worth of a pawn - one an ignorably pedantic view of your model : on an open board, it can't take anything for there isn't anything to take; and secondly if on the seventh rank, its "influence" is to move forward or take sideways, but the effect is promotion. I wonder if that should be added in as a "pawn-on-eighth-rank" case, or if its worth should be upgraded on the 7th rank for the potential it has to promote?
@@jeremywilliams5107So you mean the value of a pawn increases depending on what rank its on? Thats getting silly. The whole point is to try to assign pieces objective values outside of any given position. Not "a pawn is worth 1 point...
unless you are in a game where you have a pawn on the seventh rank and the opponent must suffer material loss or checkmate in order to stop the pawn from promoting thereby increasing its value".
😂😂😂
Your assumption is incorrect. The number of squares a knight can move to is not always the same.
Knight attacks
16 squares of 8 attacks 128
16 squares of 6 attacks 96
20 squares of 4 attacks 80
8 squares of 3 attacks 24
4 squares of 2 attacks 4
242 attacks from 64 squares. 3.78125 average
Bishop attacks
2 squares of 13 attacks 26
6 squares of 11 attacks 66
6 squarrs of 9 attacks . 54
4 squares of 8 attacks. 32
14 squares of 7 attacks. 98
276 attacks from 32 squares 8.625 average
Yes, knights can't be blocked. However, they can not pin opponent pieces either.
Great video, but as a mathematician, I think the main challenge is define what you mean by "influence" and "value". As others have noted here, there are difficulties in comparing the influence of pawns versus knights versus bishops, rooks and queens. Pawns cannot take forwards, and cannot move diagonally with taking; knights can jump over pieces and fork, but bishops, rooks and queens can pin and skewer.
In my view the material value of individual pieces and their positions is most critical in the endgame. In the middle game, coordination of pieces is also very important, whereas in the endgame, it is only the local coordination of pawns that is significant (doubled, isolated, protected, passed), so estimating material value here is crucial. The goal in the middle game is to develop a stronger attack than the opponent in order either to deliver checkmate during the middle game, or reach the endgame with a material advantage. Since engines rarely fall for checkmate in the middle game, they are estimating a combination of activity (attacking potential) and endgame material value.
The pawn's influence should be adjusted to account for the fact that it is the only piece which cannot capture on 1 (and occasionally 2) of its influenced squares. This would explain why normalizing to the pawn's influence left the queen, rook, and knight undervalued compared to the standard model, as well as all four (non-pawn) pieces relative to alpha-zero. Basically, I believe that the 2.916 value is too high considering that the pawn's influence is not the measuring the same thing.
You raise a good point. Here is my version of a 'back of the envelope' calculation.
Capture = 1, Move = 0.75 results in an average for the pawn of 2.64.
P = 1, N = 1.99, B = 3.3, R = 5.3, Q = 8.6, K = 2.5
Capture = 1, Move = 0.5 results in an average for the pawn of 2.33.
P = 1, N = 2.3, B = 3.8, R = 6.0, Q = 9.8, K = 2.8
Given the Q crosses the 9pt value between the two calculations, perhaps a more appropriate "influence" of the pawn is between 2.64 and 2.33. Certainly taking into account @acebharath's point below about the knight's fork would up the knight's value in these calculations too.
However, what would a promotion move's value be? Then what about a capture that results in a promotion?
@@theanimatedchemist I don't think this back-of-the-envelope calculation is the right adjustment, because a pawn cannot move diagonally unless it can capture. Thus all its moves are limited by context, not just the forward ones. However also I don't think you can do a useful a priori calculation without being clear what is meant by "influence".
A good next step might be to compare the pieces occupancy rates on particular squares during / after common openings instead of treating all squares as having an equal chance. I also feel certain pieces lose their value in the late game especially the knights, they’re more useful for capturing rooks and queens early game. Definitely a very interesting topic!
As others have rightly pointed out, the values are certainly dynamic. What you suggest first might be within my coding capability. I can definitely go through a database and get board positions for games of a certain length. What exactly to do from there would take a bit of thought.
Good analys!! But i have another varient, if we will count squares, where pieces cant go (pawns and bishops) we get this result:
Pawn - 140 (1)
Knight - 336 (2.4)
Bishop - 280 (2)
Rook - 896 (6.4)
Queen - 1456 (10.4)
King - 420 (3)
Less similar to standard, but also interesting
The knight has a super power. It can attack all other pieces on the board without being in danger from the piece it's attacking ─ which makes it the only piece that can attack the queen.
Other than another knight, of course :)
@@theanimatedchemistIt's like knight in the ruts.
Knights are the best
Do you not consider a skewer an attack? Other pieces can attack the queen through revealed attacks that result from skewers.
@@exisfohdr3904Ewww, you thought of a semantic exception. Good for you.
There’s a fundamental flaw in your calculations: the pawns can’t go backwards once they’ve moved, but they can turn into different pieces, and a pieces value is also determined by its compatibility with other pieces.
5:36 for the bishop’s value, I disagree, it should be 1/2 of 8.75 = 4,375. I think we should considerate the full 64 squares and say that he has a 0 influence value on half of them.
The value of pawns is way more fluid than anything. You must check if the pawn is passed, if it is connected and protected, if it is on the 7th rank (only 1 square away from promotion) instead of say 3rd rank, if it is on the edge of the board and the opposing team has only the wrong bishop left, how solid is the base of the pawn island, etc etc. I think these considerations must skew a bit the value of the pawn and therefore the relative value of all other pieces
One thing I find interesting is comparing the value of pawns. Obviously the value is 1 by default, but by looking at the other values, we can see that influence values pawns more than standard because the other numbers are lower and vice versa for AlphaZero.
The reason why your bishop and knight values are so off, is because you're valuing influence on every square equal. In the centre of the board, the knight attacks 8 squares, and the bishop attacks 13. However, 4 of those squares the bishop attacks are on the edge/corner of the board. You could weight influence, giving greater weighting to squares controlled in the centre.
That would be an interesting approach. I think it's generally agreed upon that the center is more valuable to control. I could keep the influences as I defined them and adjust what it means to be a particular square.
The only issue I see is with the pawn. A pawn on a7 isn't really influencing anything but it's also eyeing promotion. I do love the idea though. It's making me think a lot.
@@theanimatedchemist Threat of promotion is a separate factor to consider, for sure.
@@theanimatedchemist Threat of promotion is the threat of having the value increase to that of a queen on the enemy's home row. I'd imagine you'd just have to know the probability of that happening to assign a value to that possibility.
Looking at AI games and dividing the number of promotions by the number of pawns that make it to the 7th row should give a pretty accurate value for that probability.
But then, the value of a pawn on the 6th row is influenced by the fact that it's eyeing becoming a 7th row pawn...
[edit:] also, the pawn still has to take up one move for that promotion, so the act of expending that has to be assigned some value and it has to be subtracted from the points increase due to the promotion (before their result is then multiplied by the probability of it happening).
@@theanimatedchemist maybe weight the # of squares influenced by the % of the time the piece stays on that square by extracting the info from a chess database?
The first bishop is worth more than the second.
The knight can ALWAYS reach EVERY square he can move to, but a bishop is almost always restricted by intervening pieces of either color.
I get what you're saying, and other's have said similar before, but it's not that simple. You are correct a bishop can be blocked by either color but a knight's movements are restricted by like-color pieces occupying the "landing" square. For example, all knight's have 3 potential squares to move from the starting position but 1 of those 3 is blocked by a pawn.
I think that really the only thing that needs to be added to the “classical” picture is that having the bishop pair is worth about 0.5-1 pawns. It explains why trading one bishop for a knight is bad (you lose the bishop pair), and why trading your second bishop for a second night is almost always dubious (you lose the ability to force a trade between your “twin-less” bishop and one of your opponent’s pair).
I would question the approach of only dividing the bishop's value by the total number of squares it can influence (32) rather than the total number of squares on the board (64), given that it is the only piece on the board that is limited to half of the squares (and given that this means that its value is -- I would argue -- more than halved by not acting as a pair).
That type of deduction is flawed, as the number of squares it can influence inherently already determined what squares it could attack/defend. There is no need to divide it's value in half.
One could argue that the knight has a similar limitation. Although the knight can access all squares, what can it access per move? While a knight is on a light square, it can ONLY access dark squares and vise versa. Same limitation. If you want the knight to be able to attack dark squares while it is on a dark square, it costs you an extra move.
@@exisfohdr3904 _"If you want the knight to be able to attack dark squares while it is on a dark square, it costs you an extra move"_ And if you want the bishop to attack the other color of square, you're SOL. A true analysis of piece value of course should include that bishops only control half the squares and therefore half this influence calculation. And, when you do this, you can also establish why chess players don't actually favor the bishop per se, but the bishop _pair_ . That is to say: two bishops of opposite color squares are better than a bishop and a knight, or two knights. The reason being that these two bishops cannot hinder each other's movement, they are exactly twice as good as one bishop. And the same is not true of the other combinations.
Here is the intrinsic (note 1) piece value system I use and have taught: P = 3, N = 10, B = 10 or 11 (note 2), R = 15, Q = 29, K = 13. As you can see, if you exchange a N & B for 1 R and 1 P, you give up 21 to get 18. Not good! But if you get 1 R and 2 Ps (21), now we're talking! Also we can see that 2 R = 30, a bit better than 1 Q.
Note 2: 2 Bs = 21. Which B is worth 11? The first one to leave the board. This may seem cumbersome, but with a bit of practice it's very easy. Primarily use this when making exchange decisions.
Note 1: the actual value of any piece changes with its role or position. A white N placed on an out[post on e5 for example, is usually considered to be as strong as a R.
The K is a fighting piece in the endgame so use it as such!
That's an interesting system. How did you develop it? It's not exactly the same but it seems like it has the same feel as Sarratt's (1828) where it's just not as simple as a piece having a constant value. It's on page 2 of his book and I think this link will bring you directly there (books.google.com/books?id=_0wCAAAAYAAJ&pg=PA2#v=onepage&q&f=false).
@@theanimatedchemist It's not my system but one that I heard about a long time ago. I'm sorry, but I can't find the originator of it anymore. In any case, it's the best one I have ever seen! I say it's the intrinsic value by which I mean their basic value. But once the game progresses, the actual value will change due to all the positional aspects of their locations. For example, a white N on e5 is quite a dangerous piece so its actual value is greater than 10. Put it on d6, then it's worth a R.
@@belaji Dynamic values are probably the way to go. Finding one that is sensible and teachable may be difficult though it seems like you were lucky enough to find one!
How can a king be assigned a value? Anyway regarding rooks, my experience says, practically queen is actually worth more than two rooks in most circumstances. Only if your opponent has some major permanent weakness like a shelterless king or destroyed pawn structure, may be in those cases two rooks may offer you some hope. Even the great Carlsen couldn't win with two rooks versus queen in an otherwise equal position.
Plain Q vs 2R is difficult for either side to win. But if you have an extra pawn the Rooks will find it easier to push.
Certainly 3 minor pieces are a bit better than a Q, but they have to be coordinated, so how do you score that?
Great job Animated Chemist! Very interesting on lots of levels and always great to see new analysis of the game!
Appreciate the kind words! I do mostly chemistry and science here but there will be a couple for chess videos in the future…if my laptop can manage all the PGN analysis that is!
While the knight lacks in board influence, it gains a powerful ability that when a knight puts your king in a check, you either have to take that knight or move your king because you cant block the knight's attack with other peices. This jumping mechanic makes one of the most unique checkmate pattern in my opinion, the smothered mate.
I think the fact that the knights are unobstructed by your own pieces, or your opponents pieces, needs to be considered. The board is rarely fully clear. I would have liked to see this but considering the number of pieces you would typically see on the board and the influence of the pieces in those positions. I.e., knight is more powerful at the bigging of the game and less so at the end, assuming that the board is open.
Woah, the influence section gave me a huge realization: Knights are known as being the powerhouses in a closed position (a position with many pawns that aren’t able to move on the board. The knight cant be obstructed except by same color peices, so it can still make of it’s moves and captures. However the bishops lose their advantage worse than the knight.
My hypothesis is that if you had an application running that calculates turn-by-turn influence of each peice, the knight’s avg influence would be higher for closed games, or games where you put it on a “good square.”
I also believe that the number of pawns on the board, and their ability to move, devaluese the bishop when the pawns are on it's color. This also increases the value of the knight. Closed positions favor knights, open positions favor bishops.
The main issue with this analysis is that it assumes the chess board is empty, negating the main advantage the knight has over all the other pieces. If we counted the average available squares in positions from real games, you'd get a much higher number for knights as, unlike bishops, rooks and queens, they don't get stuck between other pieces.
everything is slightly under which can be explained by the fact the pawn is off. it considers all pawn moves as equal but it usually doesnt have 3 moves since its rare for a pawn to have both captures and forward be available. a pawn wont always have a capture and can easily be blocked meaning it can even have 0 possible moves sometimes
and the knight has special powers such as attacking a queen while not being attacked itselt and jumping over pieces which is not considered
I think the most crucial aspect in your underestimate of Knights (or overestimate of long range pieces) is the fact the long range pieces can be blocked in by enemy or even your own pieces. As a crude example, at the starting position, none of your long range pieces actually have any available moves, while Knights have two possible moves.
Maybe split the evaluation into max mobility count and max attack count. E.g. the bishop can attack max 4 pieces, while the knight can attack max 8 pieces. This would bring them closer together. Also, the pawn is currently overvalued as it should be much worse when ignoring its promotion ability.
This is an interesting idea and "max attack count" can be adjusted for position. On a8, for example, a knight attacks 2 squares and the bishop only 1. In terms of percentage, the knight can attack 100% of it's total influence while a bishop only 14.3%.
I think the knight got shortchanged because the influence value doesn't take into account its ability to jump.
I think its notable that the pawns value is a lot more than its sake covering, having the highly nonstandard ability of promotion
In terms of power, the king has classically been determined as 2.5. Knights can access 64 squares. Bishops only 32.
I agree thats a big oversight
Why 2.5? I thought the King was worth 4 because it can attack up to 8 squares, which is 4 times more than a pawn.
I still think bishop > knight tho
Only going off of legal moves on any given square is not the best way to evaluate piece value. You need more factors than that. For example, the number of squares until promotion is a significant factor in determining the value of a pawn. A couple of connected pawns on the 7th is usually more valuable than a rook.
A few things:
1. The pawn's influence is extremely overestimated. When we say that a pawn can influence 3 squares, it cannot actually capture or move to all of them. Pieces that move and capture differently are difficult to analyze.
2. Leaping pieces are far less likely to be blocked than sliding ones. In reality, a rook on one side of the board on the middlegame has basically no chance to make those cross-board 7 square moves. This can be shown by the fact that a jumping rook is pretty much a fair alternative to the queen.
3. Movement direction and the ability to jump over pieces or defended squares have quite a bit of influence on value. One reason that a knight is more useful in the early game than a non-royal king is that a knight has 4 ways to make forward progress and two of them completely bypass a row of pieces. One reason a pawn is useful at all is that all of its moves point toward the enemy and the promotion squares. A pawn with no forward bias in its movement pattern would be absolute hot garbage.
4. Pieces are more likely to be on good squares than bad squares even after moving randomly. For example, if you jump a knight around the board at random, it's much more likely to end up in the middle of the board than in the corner, because there are many more squares that a knight can move from to reach the middle of the board than the corner, and this applies recursively too. Low mobility due to board edge effect is way less important than placing pieces on random squares suggests, as in reality everything but a rook is more likely to be in the center even if moved by a monkey, and human or engine play tends to compound this effect, not negate it, since random play assumes you're equally likely to open with Nf3 or Nh3, while grandmaster level classical and rapid games played Nf3 as their first move literally one thousand times more often than Nh3. Real humans do not like moving pieces to low mobility squares even when the option is available, and even a random move agent will avoid those squares most of the time because low mobility means for pieces that move symmetrically they are also difficult to get to. The "correct" average influence of a knight is much higher than 5.25. Likely over 6. With a random move agent you could argue that knights are more likely to end up sitting still and not moving on the rim than in the middle, but that's a rather moot point unless you actually plan on using a random move agent.
5. A piece that doesn't develop anywhere can't really do much other than sit there and defend. This is part of the reason why the Ferz (1,1 leaper) is stronger than the Wazir (1,0 leaper) despite being more influenced by the board edges and also colorbound. The reality is that if you have a Wazir, marching it to the middle of the board is almost never gonna be a good use of 5 moves in a row. But marching a Ferz to the middle of the board in 3 is a different story, and bringing out a knight attacks the middle immediately. Being able to quickly and efficiently bring a piece to attack or defend an important square is very much needed for attacking. It is one reason that the King does not feel stronger than the Knight in practice outside of the endgame even if you make it not royal. The king moves significantly slower.
6. Colorbound pieces usually suffer somewhat if there are some parities of them which aren't available.
7. Move synergies often have important effects on a piece's strength. A queen has a power that neither the rook nor the bishop has. That is, the ability to move like one and attack like the other. A king can reach a square a knight's hop away in 2 moves, while a Wazir takes 3, and a Ferz can't do so at all. Another advantage that the Knight has over the king is that it has substantially more synergistic movement than the king. That is, a king can attack 25 different squares including its initial square should a knight can attack 41 (of which 8 require it to remain in place, a knight cannot "triangulate," unlike a king). Meanwhile, a pawn can only attack 4 squares unless it captures, and 2 of them require it to stay in place. It cannot attack its own square either. This is significant because making a non-capturing move to threaten a square is probably the vast majority of moves made in chess, and even if they are not made, they force your opponent to consider them. The fact that a rook or queen can instantly decide to attack any square on the board in a sufficiently late endgame is quite valuable. And the fact that the queen can potentially fork or pin most of the combinations of squares on the board is hugely valuable as well.
Jumping rook sounds like infinite check nightmare. I would trade this crap at the start of every game 😂
Pin this already
Is there a way to take those those tables and that influence value and create the Piece Value Tables the chess engines use to determine where pieces are best positioned?
I took a very naive approach to finding piece values but I think at best some "principles" of chess are revealed by them.
For example, centralize the queen, put the bishop on the longest diagonal, "a knight on the rim is grim", and place rooks on open files/ranks.
As for engines, my knowledge of how they work is about as deep as my opening theory so here are a couple of papers regarding AlphaZero which may give you a sense of how they work (arxiv.org/pdf/1712.01815.pdf and arxiv.org/pdf/2009.04374.pdf).
I have somewhat of an understanding as to how the engines work. They use Piece Value Tables as a way of determining if a piece is on a good square or not. That then gets included in the calculation of the position to determine what the next best course of action is. I was just wondering if there was a way to take the tables you showed with how many spaces each piece can move from certain squares and derive those Piece Value Tables. But thanks for the info I'll check it out.
@@Michael-sb1tt ok, now I get where you're coming from. The second of the two links describes how AlphaZero's piece values were derived (Section 3.8 - Material).
I like the general idea, but I feel there are a few flaws with the concept. First of all, I guess it's obvious that pieces can only influence all these squares if the board is empty. Which isn't the case. The fact that the knight can jump over pieces and therefore ALWAYS influences the numbers written on screen whereas a bishop's will nearly always be reduced is something that should be accounted for. Secondly, by only averaging over the squares the piece can land on, you are overvaluating certain pieces, i.e. the pawns and bishops. Overall, they can influence less squares, but since we are only averaging over the ones that they can access, this disadvantage is not taken into account. It seems a bit weird to use a system like that and then don't account for these things at all. Which gives the feeling that you searched for a way to get close enough to the standard values and then worked your way backwards from there. Because if you included the squares the pieces can not control, the bishop would already be weaker than the knight. And we haven't even taken into account that the knight can jump at this point. Which I feel already shows that the system isn't really telling you much about the pieces' values.
Idk if u r gonna update this video ever with the feedback. But the value of the queen is bigger because it can cause "draws" from losing positions via perpetual checks. And its the only piece that can do this by itself
You also forgot the fact that you can castle making 2 extra moves for the rook and king, and en passant.
En Passant doesn't affect the number of squares a pawn can move to.
I was always taught that the Knight was sligtly higher value then a Bishop (3.1 to 3) due to its superior attacking and area influencing potential. The Knight retains its threat on all squares it can reach irregardless of the pressense of pieces which would otherwise block every other piece type, while the bishop can only each reach half the board. I think you would find that statistically knights are moved more then Bishops are and are only the pieces most likely to put the king in Check.
You were taught wrong.
Who taught you this BS? Every school kid knows that a bishop is worth slightly more than a knight on average. Ever heard of the bishop pair?
Why knight’s ability to fork the king and the queen is overlooked?
I mean , its the only piece that can fork the two most important piece on the board and eat up the most valuable piece (queen) without being in the danger of being captured.
Sure , the bishop , the rook and the pawn can fork the king and the queen but they cant do it on their own. They need a supporting piece to do the royal fork. The knight OTOH dont need any piece. The royal fork by the knight is considered as game over for 99.9% of the players.
Its also really really tough to understand what the knight’s intention are when he is on F5. A protected knight on F5 is definitely worth a queen as it has great chances to fork
A, The Rook and the Queen
B , The rook and the king
C, The queen and the king
D, The Queen and an unprotectable Bishop
These forking adds lots of value to the knight.
I purposely "overlooked" plenty and I state as much. You are certainly correct: a knight does have a devastating move that it can pull of on its own that no other piece can. This absolutely would increase its value over the simplistic calculation I did but the question would be by how much? You say "lots of value" but it's not as if every move by the knight results in a fork let alone a royal fork. So how on earth would I go about quantifying that? If I ever do a follow up, and I might, I would absolutely take it into consideration. Probably still wouldn't satisfy everyone though!
Also, a knight can get to any square, a bishop, only half the squares.
The knight is a great piece ofcourse, with all the abilities you have described, but its negated to some extent by some of its shortcomings. All the other pieces can continue attacking from different positions, but once a knight is dislodged it needs two or more moves to attack the same point. Also if you are playing on one side of the board, and your opponent switches the attack to the other side, your knight will need 1 or two more moves to reach that side, while rooks and bishops can travel instantly.
check mate is usually worked by surprise! In my opinion, the Knight has the great advantage of stealth, valued in the same way as the `assist` to a goal scorer. The value of a football `striker` is huge in comparison to a great winger of midfielder, BUT without that pass made to the striker, the striker would be pretty useless! This is why I personally value the knight highly because he can creep up all of a sudden from out of nowhere, to afford the glorious queen her checkmate move.
Someone wise once said, "Knights are tricky b*st*rds!"
And it was in this position, at 9:59 , that a shout out to the great Agadmator was made and there is nothing more to be done.
His videos popped up on my feed in the summer of 2017. I'm quite glad they did!
"Hello everyone, and it is as of this move that we have a completely new greeting"
What's missing from your assessment method is the fact that the knight, as you said, moves uniquely, any piece it attacks cannot take it.
For undefended pieces, the Queen can take any of them except for the knight, to give an example. Also a king can never capture a cheking knight, altough a king might be able to capture any other checking piece. This is one of the reasons your knight was ranked to low relatively to the gameplay and experience based assessment that's 'default' today.
I came across this issue when trying to create my own heuristic for a chess bot. We often hear that a rook is valued at pieces in a open board but what about a closed position? I tried providing two heuristics: A control "Frontier" that explores the squares it directly controls, and a hard value. The intuition is that the hard value is a long term evaluation of how desirable it is to preserve this piece and acts as a value "floor" when computing trades. This way, a trapped queen doesn't immediately get traded for a knight. Unsurprisingly enough, the rook had the largest hard value, reflecting the discrepancy between the early game and the late game.
I would argue that a piece only "influences" the squares it attacks, so for the pawn only the diagonals should be counted giving an average of 1.75
A couple rational steps to improve the influence rating could be to factor in how likely it is in a real game for a piece to be on that square and be able to move to the squares it can see with rational play. Another step could be to factor in how many moves it would take for a piece to get to each square on the chess board
Assuming that reach (or influence, as you call it) is a good measure for the value of a piece… the number of surviving pieces on the board limits reach.; Your own pieces even more so than opponent pieces, since you cannot take your own pieces. So if the pieces were randomly distributed (which they aren‘t), you‘d need to adjust each piece‘s value by how many white and black pieces are left at any given moment. The knights would be the least negatively affected by this analysis, which explains why a) your analysis undervalued them and b) why they are relatively more powerful on crowded boards (which correlates with closed positions).
There are additional complexities stemming from the caveat that pieces indeed aren’t randomly distributed, particularly at the beginning of the game. However, the conclusion that the value of pieces changes over the course of a game as pieces get eliminated - even before considering their position on the board - will be familiar to every chess player.
The main flaw with just looking at number of squares is that being unable to switch colour seriously hurts its point value. In an endgame you can just move every pawn and your king off of that colour.
This obviously applies to the bishop, but this principle holds true for a lot of alternate and historical chess pieces too.
An alternate version of the bishop that is just given the ability to step one to the left or right to switch colours goes up almost to the value of a rook (like 4.5 points or something).
The "Camel" which is a variant on the knight that can jump 3-1 (instead of the Knight's 2-1 jumps) ends up stuck to one colour and therefore ends up with a value of around 2 instead of the value of 3 for the knight.
The same way the queen is a hybrid bishop/knight, there are historical hybrid knight/rooks and bishop/rooks, and they actually end up with very similar piece values to each other because once again the bishop gets a lot stronger when it can switch colours--which knight movement allows. (These compound pieces are worth like 8 and 8.5 respectively).
With just standard pieces, there is often a point bonus that is given for owning the bishop pair, because both bishops together can cover every square (the bonus is roughly half a pawn).
I don't think I had ever heard of the camel or other variants/hybrids you mentioned. Always nice to learn something new.
All pieces except the pawn can attack on all counted squares; the pawn can either capture or occupy an empty space without fighting. It seems to me that the pawn was overrated based on the squares.
But a pawn can become a piece, this is not taken into account at all, there is a slight underestimation here.
The knight is also noticeably undervalued, since its moves, unlike all other pieces, cannot be blocked by enemy pieces. Your own or someone else's piece on the bishop diagonal blocks all the squares behind it. Same with other figures. This does not happen with a knight; blocking each of its cells only blocks that one.
Your "averages" are calculated based on the total number of squares any piece *could* move to *IF* there are no obstacles (such as its allies in the way; enemies in the way; or tempo issues such as checks, pins, and opponent's attacks on higher-value pieces). But the true value of a piece is intrinsic to what can do in the current position, extending to its realistic potential. I would submit that the value of a piece is a very dynamic number. We've all seen games where a single pawn becomes more valuable than everything else on the board, simply because its next move results in checkmate, and the opponent cannot prevent it.
Even if the true valuations are wrong due to a variety of factors, just the "influence" grid is interesting to see, and probably useful for all beginning chess players to try to maximize their coverage/utility of pieces.
bishops are usually considered having more value than knights but a piece has a different value depending on the scenario of the game. At times, a knight pair can be stronger than a bishop pair but it is rare. Knights are not long range pieces,which is one of the reasons they get weaker in end games or open games. but some times a bishop can be stronger than a rook, depending on the game. I have been playing chess for a long time and I am pretty good at it so I sort of think about this practically.
What if influence was calculated not by what they can move to, but on a given square, how many ways could a piece move to that square? What about captures? Considering that pawns can only capture in certain directions but they can move in a different directions, would it be better to evaluate the influence based off of their threat or defense to a square if they were in every board position? Idk.
Knights extra value compared to this comes of course from the fact it's the only piece that can attack any other piece without being threatened, for its movement.
And bishops value gets demoted once we consider only one bishop.
You again forgot that pawns can promote at the end, allowing them to turn into any of the pieces
p sure he knows that
He addressed this, when a party reaches the end it is no longer a pawn so it's a different piece with a new value separate
a knight can attack other pieces without supports (like you can chase the queen with a single knight while other pieces need at least one cover). It can also move better in close game.
To the table makes sense to add price of a move. How much one tempo worth.
The influence of all pieces but the knight can't go through other pieces except via x-ray attacks and fossilizations. That's probably the biggest thing that this video didn't cover. because if you factor in that most of the pieces trajectories will be blocked in some way, the knight then regains its status as being very similar in value to the bishop
I expected that stockfish evaluates a bishop distinctly above 3, but I was most surprised when the rook had a value above 5.6. Shows that endgame really is everything, huh.
Many things not accounted for, like the value of a pawn changing into a queen, or how near it is to changing to a queen, or how a bischop en pawn can auto/mutual defend itself, these are also very valuable properties.
I definitely thought about these things but there are so many permutations, never mind trying to assign values to each, that trying to properly account for them all would just get in the way.
I think considering the number of squares covered by two moves into would be fairer on the knight, it doesn’t reach the bishops level, but it does double its influence for most squares it can land on, whereas the bishop at most adds 3/4s of its own and lowest doesn’t add anything
adding an addendum, the second move should only consider the average of each potential second move, that wasn’t very clear, so for example, a bishop in a corner would move to each square on the diagonal, covering those 7 squares + (2 * (2 + 4 + 6) + 1) / 7 squares for the average number of squares covered by each of those moves and not considering the 7 squares along the diagonal or the corner square the bishop started on, because otherwise the queen and rook cover the whole board each, and ask any chess player, a queen and rook are not equal
Chess960 drastically changes the values of the pieces. Theoretically, they can be vaued the same, but its starting position can change the outcome & strategy of a game.
Rook much less than Bishop + Knight...even Rook+Pawn for N+B is a bad idea, barely 2 pawns compensate
I have also believed that the opposing bishops of opposite colors are friends. They can never capture each other.
We should also give points to the Queen for being the only piece that is able to put the King in perpetual checks...
counting a pawn positioned one step from promotion as the same influence as being one step away from starting position is wild.
He took influence only as the number of squares controlled. You talk about the dynamic value, but he talks about the static value. There is no point in calculating dynamic value since it changes with every position and only a computer would be able to evaluate all the pieces according to the position on the board for most of the positions, and still it would not be 100% correct since chess isn't solved to this day (with more than 7 pieces on the board)
If I am playing a Benoni type I am killing my bishops for their knights. Same with closed Dutch. Typically - I like more open games and with that I will throw my knights away to kill their bishops.
One thing with your analysis.
Take any board position on any move. Calculate the influence for that piece, on that move.
Very early in the game, the pawns and knights have more spaces. Bishops, queens, and rooks are useless. So you can make an argument that early in a game - pawns are important at impeding influence of competing pieces. And since knights can jump, this helps their mobility.
Meaning…
The time of the game can have an influence on value.
The type of game (open/closed) can create a skewed value for knights over bishops/rooks.
So time of game is a big deal. An end game wide open- queens and rooks rule, with bishops the next important and knights the last.
well one thing is that most pieces are blocked by other pieces. So the bishop can only hit the first piece in every direction, and it's rarely covering behind it, whereas the knight always covers all squares it can reach and is good at covering backwards.
If you counter the max number of square a knight control it's 8, but for both a bishop and a rook it's 4 if you account for being stopped by other pieces of course that's not a good way to look at it either, just a way to point out that the board coverage without other pieces on board isn't always a good metric.
Fair comment. This is actually a simple way I could have "fixed" the low value of knight after seeing I got 1.8. Of course, simply doubling it to 3.6 doesn't seem right either.
Should also be noted that AlphaZero's point values for pieces are conditional on its strategy. The points are not applicable for any random player
I think the flaw in the Influence scores is that they do not take into account the fact that the knights can jump. At any one given moment your opponent can block the full range of moves open to any of your pieces. Not so the knight. So this adds to its value - though not sure how one could use maths to factor this in ...
Yeah that's beyond me as well. I would attempt something like this: given the position of a knight, how likely is it that it one of its potential moves is blocked by a piece of the same color? For example, say white is down to a knight and king. The knight can move to b6 or c7 as long as the king does not occupy one of those squares. Since the king has the potential to be on 63 other squares, it has a 2 in 63 chance of blocking one of the knight's moves. Now if white has a bishop on a8 and a king, then the king has a 7 in 63 chance of blocking one of the bishop's moves. Obviously more pieces would get absurdly complicated but maybe it's worth a go for just 1 piece and 1 potential blocker. Maybe it'll give some type of "factor" which would increase the knight's value compared to the others.
Every endgame up to 7 pieces has been solved. The results of the endgame table bases confirms that in most endgames, the bishop outperforms the knight by far, especially when there are two of them. Two bishops can even outperform 3 knights. There’s an article that has all of the data called “Bishop vs Knight”. While the knight is very handy in the early game, I believe trading it for a bishop is typically a good idea.
I've heard of the endgame table bases before but haven't looked into them much. I wonder if there will be a number of pieces where the knights are superior or if it will always tend towards the bishops.
@@theanimatedchemist I forgot to address your question at the end of your comment! Bishops will always be better than rooks in higher numbers. A common trend in chess programming is to actually modify the value of pieces depending on their position and other factors. Pawns, for example, loose half a point of doubled or isolated. The double bishop is often hard-coded to be worth significantly more than 6 material because they can make an incredibly strong mating net.
The most unintuitive thing to me as a chess player is that Stockfish values rook+pawn higher over a knight and bishop or two knights and approxilately as much as two bishops.
I'm a little confuser. What is the point of the video?
I came up with my own value system for the pieces, but everyone says it is wrong, and actually so do I?
The reason the horse is more valuable is because it can hit every square, not just half of them like a bishop.
It can also smother mate.
It is the best piece for attacking a queen.
It cannot be blocked.
It can get in and out of high traffic areas by jumping over them, where rooks, bishops, and queens need higher clearance to pass through.
If it is harder to anticipate, so many people get tricked by simple forks that they would easily see from another piece.
It is also the only piece, othe than a pawn, that can start a game, giving it high stretegic importance.
It is harder to master, but it can be extremely useful if honed carefully.
The point? I think I made it clear around 9:45 when I said "So will this analysis, flaws and all, add much to the discussion or end up in the hallowed digital halls of Wikipedia? Probably not - *but I enjoyed giving it a go anyway*."
Nuh uh. you're clearly a beginner. I assumed this because first of all you said the knight is a "horse" lol.
Knight is decent in the middlegame and gets almost completely dominated by the bishop in the endgame.
Smother mate is very rare you'll very likely not get it in most of your games.
It can't be blocked that's true but it's easier to trap it. knights get trapped more than bishops.
Those jams are mostly temporary after trading few pieces it clears up.
It's not hard to anticipate knights movement in fact most high rated people keep an eye on knights movement so they usually know what's going with knight.
Bishops > knights in most cases
Knights > bishops in closed position
Bishop's long range makes it effective. even tho it can only control half of the board it's a very good attacking piece
Bishop pair is nightmare against 2 knights
The value of pieces depend upon its position and forking ability of the Knight gives an upper hand to Bishop.
Philidor's estimations are impressively accurate
I can sac a king *for a bishop! Yay!
This was an extremely interesting video! I'd like to echo what some other comments have said in that I believe the reason bishops and knights are rated as equal in the standard system is their value compared to each other is determined largely by the position as a whole. They're equal by standard, but the player elevates them or hinders them with their handling of the position. Did you put all your pawns on light squares? Well your dark squared bishop DEFINITELY is now worth more than your knight, but your light squared bishop may be worth less than one. Excellent video though, seeing piece value expressed as influence is something I need to consider more in my own games.
I'm glad you found it interesting. You and the other similar comments aren't wrong, of course. I haven't included any consideration to actual positions which is an obvious flaw but it did keep things simple. We have all seen or played games where a bishop is reduced to a pawn with a pointy hat and the knight is getting all the glory so there is certainly more to the story.
My understanding of Alpha Zero is it determines values off pieces, moves, and positions on the probability of winning.
Knights have an ability to jump so it can't be restricted. Its influence always stays the same. If you put other pieces on the board, knight still can move the same however bishop movement can be restricted highly. So, your measurement has a big flaw.
Knights can still be restricted by other pieces of the same color but a knight would need 8 pieces on 8 particular squares (or whatever number in other situations) to lose access to all of them while a bishop can lose access to 7 squares (say from a8) with a single piece on b7.
What is the bonus for having 2 bishops according to alphazero? 3.33 x 2.1 ?
The authors of the A0 paper do note "...there is usually an additive adjustment while in possession of a bishop pair" but I could not find a specific bonus. Perhaps I missed it though.
the bishop and knight is same value because i think knight does some movement and attacks surprise locations and bishop can only move on one color
It should be a weighted average; the influence of a piece in a position of the board should be weighted by the probability that it will be there in the first place - which naturally changes across time, and the probability that it's influence will be neutralized by surrounding pieces. This is a calculation only the sophisticated neural net of an AlphaZero is able to do.
You just don't put your knight on the edge of the board and rotate them wisely and it's around 3 and bishop is likely 3.3 or around that
My guy, the reason the knight's valuation comes out as lower is because this system doesn't consider that other pieces are usually locked in or prevented from actually having access to all of their squares they could theoretically reach because of the pieces on their respective side.
Props for addressing me as "My guy" 🤣
Yes I did not make such a consideration. All pieces can be restricted to some degree and trying to account for all possibilities was beyond what I wanted to do. I may take the idea of move restriction into account in a follow up as many people have posted ideas or criticisms, some of which I never would have thought of on my own.
Right but it is rather relevant since Knights can't be blocked by their own team while bishops can. assuming the same number of pieces on each side (usually) for a fair evaluation, this is important to consider, MY GUY@@theanimatedchemist
Here from agadmator 😊 nice video 😀😊 really nice calculations 🙂
Thanks for watching! I’m glad you enjoyed it. I’m a long ways off on another chess video but my computer is plugging away at millions of PGNs as I type.
@@theanimatedchemist imma be watching your other stuff too ☺️ seems like you care about the vudeos yiu produce and keep the good work going friend, best regards 🙂
@@burimsaliji23 Thank you! I will do my best but other than the 2 chemistry videos I've got in the queue it's going to be some time before I've got others ready to add. Since my comment 11 days ago, I've analyzed every Lichess PGN (rated, classical) from Jan-2013 to Dec-2017...I'll leave it a mystery as to why...and I'm downloading 2018s as I type!
The Anitmated Chessist.
I think the conditional moveset of the pawn and the jumping of the knight was overlooked in the final calculation
Without a doubt something with respect to the knight is overlooked.
I think that the reason your influence analysis was a little off was because it didn't consider some more strategical elements. Influence isn't the whole story in a Chess game, rather it's the Quality of that influence. Knights may control fewer squares than most other pieces, but their ability to jump pieces, the fact that they can see both colour complexes, and the fact that no piece other than another Knight can easily challenge their influence I think increases their value.
If you saw me play, you would understand why I didn't include any "strategical elements" :)
As for the knights, definitely something is missing in what I did to result in such a low value. I am still curious if the three things you mentioned, or something you didn't, could somehow be evaluated and added on to it to some degree.
@@theanimatedchemistI'm by no means an expert I was just giving my ideas as to why it might've happened that way.
If I remember correctly, I thing the original piece values are derived from how strong each piece is in the endgame, but I could be wrong about that.
A knight deeply entrenched in enemy territory past the threat of pawns and supported by a friendly pawn is seriously dangerous. Understanding some Peaces can easily have a 1.5x multiplier based on board position.
In the french defense the black light squared bishop probably sits at a meek 2.3 points for example. The other pieces black have probably increase in value however due to excellent board position.
Seems a convoluted way to calculate piece value. I figured it would be “maximum amount of squares a piece can attack while in the enemy camp divided by 2”
How would you define "enemy camp"? Is it based on the position of the king or something like half the board (for simplicity)?
@@theanimatedchemist just by splitting the board in half by rank. rank 1-4 = white, rank 5-8 = black
@@adawg853 ok, that's certainly something I could try to incorporate into a follow up. I am curious what differences there may be.
"Hello Everyone"
Agadmator is that you?
I would argue that a pawn’s value slightly increases the further it goes, and if it is paired with additional pawns.
Absolutely. The wiki page linked in the description has some examples.
Could the seconf spring have moore value then the first, wich pawn is worth the most? Is the white bishop moore worth then the black bishop etc!
If you talk about static value, bishops are the same. About the pawns, usually the 2 on the edge (first and last column) may be considered weaker because they can take only on one side, but it depends by the position because in endgame they can be a winning condition (any pawn can be a winning condition in the right scenario). Usually the center pawns are considered more important because controlling the center is a strategic advantage... but it all depends by the position. Every chess piece has a variable value, a rook can be considered useless if it's trapped and a pawn can be considered as good as a queen on certain scenarios, so it's impossible to say certantly the value of each piece. For the sake of semplicity we just say all pawns are all 1, knights and bishops 3, and so on. The value of the king is impossible to calculate for me, you must protect it or you lose so it must have the highest value, but if we count value as the strenght of a piece then it's for sure weaker than a rook (and most of the time if not almost always weaker than bishops and probably knights too)
The knight is the only piece that retains those numbers regardless of the position.
So I'd give it a plus point for that
I don't disagree with the idea but I'd like to find a methodology to obtain an objective value to add on (or adjust all other values according to it) rather than one that "feels" right.
I like the Kaufman values (can be found on the Wikipedia page the video mentioned near the end
Love the phrase upper lower middle value ELO player
Two bishops should be considered as 7 points of material.
You would probably like this then as it goes over giving a pair of bishops a value of 7.5: archive.ph/20120714204040/mywebpages.comcast.net/danheisman/Articles/evaluation_of_material_imbalance.htm.
I like the concept here, but in practice these values are way off. And that is because the board is not empty. With pieces in the way, a lot of those pieces will have far fewer squares they can really go to in practice than if the board were empty. It should also be considered that the total possible influence is not necessarily linear with respect to actual value. As in, sure a queen can move as much as a bishop plus a knight, but even the standard values for a queen give it more value than a bishop plus a knight. There's also another flaw in that pieces are not in all places on the board an equal amount of the time. As in, knights rarely end up in corners so that shouldn't really count against them. But queens can choose to stay near the center of the board and influence a ton of spaces.
Anyway, I'd recommend doing an analysis of averaging the influence over like 1000 real game situations. And then also giving a score from those games based on who won and with which piece differences, or based on a comparison with stockfish evaluations to those positions or something.
I love your suggestion but I have no idea how I would code that up. I know I can code something up to get 1000 random positions from the LiChess database but how to turn those positions into influences for each piece (outside of doing so by hand)...might be a worthy summer project or completely beyond me.
Counting the squares of pawns who can just move without taking OR just take the same way like the other pieces who can do both, must be incorrect.
Personally, I agree with the fact that Knights can be as useful as the Bishop but I also think that Bishops are slightly better.