The TRUE values of all chess pieces (according to maths) 

Instructions unclear: I traded my king for a rook and lost the game

Strong chess player here, some caveats: 1- A bishop can only access 50% of all squares, that should be taken in account. It is known that in an end game with all pawns in same side, let's say 3 pawns 1 bishop and king vs 3 pawns knight and king, the knight is stronger than the bishop. Also having the pair of bishops is considered to give an extra value to the bishops, probably due to both toguether being able to access all squares. Also a knight+queen is considereed to be stronger than bishop+queen, due to sinergy in part for same reason. 2-Queen should have a bit of extra value due to being able to strong manuevers in 2+movements. that makes it stronger than Rook+Bishop 3-It is known that in an end game values of pawns actually increase, with little pieces 3 pawns are stronger than a bishop or a knight, with little pieces, many squares are actually not important. 4-Passed pawns increase their value. Also centered pawns in the opening. Doubled pawns decrease it. And other nuances.

i love the part where mike flonged his rook into the exit fizzler to respawn it. very cool video

I love how the Queen is 13, but what is the average value of a bishop in the 8th parallel timeline in a game of 5D Portal Chess with Multiverse Time Travel?

I did a similar thing as you a while back, there are four things that I think need to be taken into consideration in your model: 1. Not all squares are equally valuable to control. Rather, the only squares that really matter are the ones your opponent either has pieces on, or that they control themselves. Typically, this is going to be the top half of the board. As a general rule of thumb, pieces control squares closer to them than ones far away. This is especially true when you take an average over all of your opponent's pieces. And of course, positions may change during the game so that it's not exactly split down the middle, but in objectively equal positions both players should always control around half the board. In unequal positions where one player has a clear lead, there's no real point in calculating piece values since you're already winning. 2. Bishops are the only piece that can only attack half of the squares on the board. Every other piece can eventually reach any other square on the board, including Pawns if you count promotions. This means the opponent can play around a single Bishop avoiding squares of the same color, making it a useless piece as it can't attack anything. Even 8 of the same-colored Bishops can't capture a lone enemy King. 3. When it comes to piece congestion, your pieces interfere with your own pieces more than your opponent's pieces do. This is because you can still technically move to the squares your opponent's pieces are on, AKA capturing them, but you can't capture your own pieces. So if a Rook is surrounded on all four sides by enemy pieces, it has a total of 4 legal moves, but if a Rook is surrounded on all four sides by friendly pieces, it has 0 legal moves. 4. As a consequence of #3, a Queen is actually better than a Rook+Bishop combination because they cause less congestion to your other pieces, since it only takes up 1 square on the board rather than 2. In fact, every piece should have a slight penalty associated to it for the congestion it causes your own pieces, so a Knight + Rook is superior to 3 Pawns + Rook in the opening, for example. This is incredibly easy to prove - for example, open the board editor on lichess, remove any three Pawns from one side and a Knight from the other, and see how the evaluation greatly favors the player with the three missing Pawns.

My favourite moment was when Mike used the portal gun to shoot a portal to e8 transforming his pawn on e2 instantly into a queen and then proceeded to portal all over the black king

Nowhere in this video did you seem to take into account which squares any piece can attack in multiple moves. The knight can reach ANY square on the board, yet the same can't be said for a bishop. The bishop only has domain over half of the squares on the board, yet this limitation doesn't seem to have been taken into account in your evaluations. You seem to only be concerned with what can be achieved in a single move. So in an endgame, the bishop may be thwarted by simply placing a piece on the opposite colour to the bishop. No such strategy is available to escape from a knight.

One thing to consider with the bishops, is that throughout the entire game, they can only ever see half of the board, making them a bit less valuable. The queen on the other hand can move like a bishop, but can also move to different color squares, which is what makes it more valuable than simply a bishop and a rook combination. The pawn is also limited by the number of spaces it can ever reach, but promotion offsets this. One additional thing to consider may be the number of moves it takes a piece to control a particular square, which may close the gap between knights and kings a bit. The value of a king though is usually not as important as it is the entire objective of the game, and is usually set to an arbitrarily high number

I see comments about how bishops can only access half the board. It is a 2 player game after all, so pieces being more or less avoidable by the opponent matters. This could be captured in some measure like the average minimum number of moves necessary to attack each certain square, given the piece is uniformly distributed throughout the rest of the board (not whole board for pawns ofc). Something similar to this would capture pawns moving up, knights having to remaneuver, a queen being better than rook+bishop in general (someone else had a good suggestion for this to do with piece congestion), and bishops only accessing half of all squares. Incorporating piece congestion into this seems hard but not undoable. Separately, pawn promotion is pretty big; it'd be a shame to not include it in some way.

Another thing to consider is that some squares, a good player will refuse to place their piece on. The corner: no one will put a night there, preferably not a bishop, queen or king in endgames. Rooks however often thrive on a/h / 1/8 files and ranks respectively. The reason why we don’t put pieces in the corner is because this reduces the number of possible moves/squares controlled. However, in the middle game we put our king right next to the corner for protection due to the checkmate mechanic.

i was honestly surprised by these results. I thought that the video would be about taking a look at tons of recorded chess games and taking the controlled squares of the pieces there, i wasnt expecting this basic approximation to do so well

i would be interested to see an average controlled squares taken not on some uniform distribution of pieces, but from a sample data set of actual chess positions, and see if this improves (or perhaps, degrades?) the quality of this approximations

I think this method of evaluating their values ignores that control doesn't just apply to the next move, but to subsequent moves as well. A knight may only have 8 squares (at most) immediately available to it, but because of the odd nature of its movement, in just a few moves it can access the entirety of the board. While a bishop will always be limited to accessing specific squares. Similarly, it's much easier to completely block off a bishop's range of movement than it is a knight's. Also, control applies to how well a piece controls other pieces. A rook can completely block the opposing king from moving across its controlled rank and file. A rook can also protect an advancing pawn for the entirety of its journey (something a bishop and knight cannot do). A knight can produce a forking threat against multiple pieces.

I could get into the math, but as a 2041 OTB player I’ll use my chess experience to state the following: queen is better than rook and bishop in almost all cases and the reason is simple: flexibility. The queen can do sequences of moves the R+B cannot do together, AND the queen can switch the color complex she is on to influence diagonals the cannot without its counterpart. In fact, there is definitely something to be said for having the bishop pair for that reason, and to some extent the queen mimics this. Side note, I’ve actually read much literature that puts the king at 4 points in strength so your 3.75 makes sense. If I had to give my own subjective values and make them nice round numbers I’d say pawn 2, knight 6, bishop 6, having both bishops bonus 1, rook 9, queen 17, king priceless of course but strength of 7. It covers a lot of scenarios. 2 rooks usually (not always) better than queen 18>17. Knights and bishops equal but bishop pair is a slight edge 13>12, neither of the previous advantages usually worth a full pawn (117) but also not by a full pawn (16 but again not by a full pawn, etc…I feel it’s the closest “nice” numbers you’ll find for easy calculation

2:35 "So, let's change our metric, to counting not the number of squares a piece can move to, but instead, the number of squares a piece can troll"

this was the exact chess lesson i needed to improve my game because i think I understand transitioning to and from midgame significantly better now

Next video: "The TRUE value or all PeTI Testing Elements (according to maths)"

Should we consider the amount of pieces a piece controls at the same time? A rook can control up to 4 pieces at once, unlike the knight, who's limit is 8

5:41 I had a bad feeling about this calculation since it seemed to take the different cases completely independently so I did it myself: 1(x) + 2(1-x)x + 3(1-x)(1-x) x + 2x - 2x^2 + 3 -6x + 3x^2 3 - 3x + x^2 1 + 1-x + 1-2x+x^2 Got the same answer. All good!

Its cool that the values made using math was very similar to the typical values most of the time

Cool presentation. One significant misstep...pawn values are far from being static throughout a game. Pawn values spike going into the endgame. Passed pawns are particularly elevated. Better yet, connected passed pawns hold massive value.

0:09 huh? I mean king being the only losing condition should at least make him worth 16

An observation I have remembered, and deem reasonable, is that a Bishop´s value is 0.5 units greater when both are on the board.

Really, you ought to have some series that accounts for not only the squares attacked on the next move by a piece but also squares that could be attacked in two moves or three moves. For example, a queen or rook can reach any square on an open board in two moves while a knight could take many moves to reach a given square. Bishops can only reach half the squares (but can do so in two moves). For human play, it's not really the actual values that matter but the total order on the various sums. That is, it matters whether rook and bishop are worth more or less than a queen, not whether the rook is worth 5.1 or 4.9. It would be really interesting but much more difficult to use a game database to obtain values of various factors using statistics from game results and to incorporate that into the piece value table. For example, with a standard set of piece values, having the bishop pair while the opponent doesn't is typically considered to be worth half a pawn. Could you similarly say that having an isolated pawn is worth negative .25? Are center pawns worth more than edge pawns? Is it true that three tempi are worth a pawn in the opening? Is a passed pawn inherently worth more than an unpassed pawn? What is the cost of a backward pawn? Positional factors could be assigned numerical scores by looking at performance rating, finding out how being up one pawn affects game outcomes, and then seeing how other factors affect outcomes. For example, if being up a pawn against an equal player leads to a 2% higher score over 1000 games, could we say that a feature that also leads to a 2% difference in performance is also equal to one pawn?

Now you are thinking with... chess

First: Very cool analysis. Thankyou. Second: The idea that a 'blocking' piece eliminates the influence of bishops, rooks and queens on squares behind that piece is not strictly correct due to tactics like pins and skewers. If taken into account, that would increase the expected value of those pieces (along with the complexity of the analysis).

I think both the standard values but your model even more so underestimates how powerful the queen is. a queen can often be as good or better than 2 rooks.

Something that I think should be considered is that the king isn't allowed to move to a space where it would be in check, and therefore can never control such a space. I don't know if you factored this into your calculation, but it would make the king slightly less powerful.

7:40 this looks like a mathematical basis for the "positional rook sacrifice": trade a rook for the opponent's good minor piece in closed/cramped positions (prevent the opponent from uncramping the position though because you're still a rook down).

the piece value increase or decrease in function of key square contacts; for example the pawn on 7th rank is moooore valuable the pawn on 3th rank. Also the value depend the relation/contact with other piece.

I was thinking you'd use some chess algorithm to calculate out how well each player is doing in a bunch of random games and look at how many of each piece they have, and then find a linear approximation to calculate about how well they'd be doing based only on number of pieces.

If you calculate the value for all pieces for each possible count of pieces on the board you could then provide an always-accurate piece value chart that updates after every capture.

What about doing the same analysis you did with bishops, rooks, and queens, but with the pawn that can only move 2 squares on rank 2 if nothing is blocking it? That might affect the analysis a bit, since you are normalizing the pawn. But if the pawn's ability to move can also change over time, then you might need to look at that. That being said, great video!

value depends on the piece activity in the game.. constantly changes for each piece with every move of you and you opponent make. you can have games where a queen is worth 1 point and a pawn is worth 5.

Nice! This information is useful to improve my chess bot

I really enjoyed watching this video, so underated, you should make more of this content. What about to make a video about chess bots? (stockfish, torch etc.)

I like the simply explainable results, but I’d prefer or more statistical approach: approximate these piece values against a billion board positions and the known game outcome for example. Maybe prune the positions where the king is in check or some other heuristics, only use grandmaster games, or top computer games etc. Would that approach generate a more real-life usable set of piece values that humans can simply add up?

In short, the order is still the same in all but the Knight and Bishop. They arent equal, but the knight is worth more early (more than 26 pieces on tge board), but the Bishop becomes more valuable afterwards (once less that 26 pieves are on the board). The rest is effectivly the same order wise.

King is 2.5 pawns in the endgame. Chess engines have found the optimal piece values by playing millions of games

I am not into chess but during first turns rooks fills completely usless but bishop are op and its flips around at the late game.

Enjoyed the note for math nerds. I was thinking that, so thanks for clarifying. I think you covered it well. Sad that theres not more similar videos on your channel, all portal lol.

If we start talking about tablebases, I think it is useful to somehow extrapolate values from them too. F.e. if winning by a queen leads to a victory most of the time then it should somehow influence the value of it, since that information should influence how we trade think about positions and trades before 7-men endgame. This could very easily lead to an AlphaZero-like evaluation where you compute the average game result for each piece but it would be more problematic to create a traditional point-based system. One approach is that if we normalize pawns at 1 point in all endgames and expect [value_of_white_pieces]/[value_of_black_pieces] equal to the expected outcome (1 for a win, 0.5 for draw and 0 for a loss), we could get the endgame values of all pieces and factor that into the equation by simply counting the number of positions certain piece wins and 1/2 of all positions where it draws and dividing by the number of positions in total. It would be interesting to see the results of such analysis and how it would affect the overal ratings

I didn't even know that chess pieces have values assigned to them! TIL! Ah, at the end was the table I was looking for! Maybe I'll plot them tomorrow to visualize them! Nice, informative video!

I like how you say "the queen's value will always the the sum of the rook and the bishop" and then 30 seconds later have 2.75 + 3.5 = 6.2

Great video! Chess has always interested me.

Is there any chance you could try this same evaluation with the King as the baseline? With the Pawn as the base, it's impossible to tell to what degree the other pieces are becoming more valuable or the pawns are becoming less.

Something this model lacks (tho that isn’t exactly on you) is relative piece value. Rather, this model lacks it for the knight and the pawn. When your knight is seeing squares in enemy territory, it has a higher value. The more of its squares are in enemy territory, the higher value it becomes. In the ever so iconic Octopus Knight, Antanoli Karpov leading with e4, and Garry responding with c5; the Sicilian defense. At one point in this game, Kasparov had a knight with a relative value of 9, and Karpov traded his queen, with a relative value of 6, to get rid of it. Pawns also become worth more based on two factors: advancement and locking a chain. If a pawn is locking a chain, it becomes tantamount for it to maintain that role until the position evolved otherwise. The closer it gets to enemy territory, the higher value it has by threat of promotion Amazing video, otherwise

One place where this analysis, which is otherwise quite good, is deficient in my opinion, is that a piece's value is not only the number of squares it can control, but also how many squares it could control after a single move (either by the piece itself or via discovered attack after the movement of another piece).This gives the bishop, rook, and queen some extra value that the other pieces lack, and this value is extremely pronounced in games where these pieces will commonly slide in from an uncontested area of the board to give check, protect a threatened piece, or sit in front of a pawn on the way to promotion. It also helps explain why the traditional valuations of 5 and 3 respectively for rooks and bishops are more disparate than the values you were calculating - both pieces are able to control close to the same number of squares on average, but rooks can threaten basically twice as many "1-move" squares as bishops can. I wouldn't suggest that these 1-move squares should count the same as the squares being directly controlled, but rather I think a piece's value is a function of both. Making value a two-variable function also has the fun effect of creating a table that hypothetical new piece types could inhabit. For example, (using some rough, but easy to work with rounded numbers) if we say a knight controls 6 squares on average and 12 1-move squares, while a rook controls 12 squares plus an additional 30 1-move squares on average, these would sum to 18 squares controlled plus 42 1-move squares (ignoring for the moment that the sum is not the same as the union here). We could then see what value the f(18, 42) is to estimate the value of a piece that could move like either rook or knight. Of course, all of this is assuming that trying to use mobility as a metric is ideal to begin with. Given the results in the video, it's clear it's not a terrible metric, but I'd posit that the TRUE definition of relative piece value is that two sets of pieces would be equal value if games played between similarly skilled players where each player had one set of pieces resulted in neither side winning more than the other. The fact that endgame strategies dominate over piece value when there is little material on the board that you pointed out means that one couldn't simply see if a knight is worth three times the value of a pawn by measuring the average outcome of KN vs KPPP games, but you should be able to, with enough different combinations of pieces, set up a system of equations that would allow you to solve for each piece's value as a function of a pawn's. One could look at a database of chess games, index each position by the combination of pieces each player has at that point, and then measure the tendency for one side to win over the other across all games with the same index. All indices with an even (or close enough to even) win balance would thus represent equal value sets of pieces, those sets being the ones that would be used in the system of equations to solve for final value.

I would like to see the fact that the bishop can only reach half the board and pawns can only reach a relatively small number of squares, whereas the other pieces can reach the full board included in the analysis.

The absolute value of pieces is just one aspect of the overall evaluation of any particular position. There are also their mobility, perspectives, cooperation with other pieces, combinational, strategic and theoretical motives. You cannot compute all this with satisfactory precision. So, it's preferable to keep things simple and use 1:3:3:5:9 scale throughout the entire game for the exchanging purposes only. Beginners have to learn just this: pawn < knight = bishop < rook < queen. Later on, the experience will make the evaluation ability of the player much more detailed and precise.

Or the ez way out is to yoink what chess nerds say (theres a wiki page on a whole host of values). Or what engines say generally, people have used some methods to extract it not sure how exactly.

Realative piece values have some meaning as broad generalities, but are very position-dependent. Getting a "bad bishop" that is hopelessly hemmed in by your own gridlocked pawns can be as bad as being down a piece. If you can trade it for an opponent's knight, it is a good swap, even though on average a bishop has more mobility and slightly higher value than a knight. In an endgame where the opponent has only a single bishop, if you can keep all your pawns and pieces on the opposite colored squares, it can be like an "invisibility shield" that can effectively put you up a piece. In the endgame, the King becomes a key fighting piece. Many a kingdom won or lost by a single tempo of King moves.

as if chess wasn't confusing enough already

ah cool, like this new style of content!

The value of particular pieces or pawns often depends on the position.

so we were quite accurate after all

The approach usually used to determine evaluation parameters in a chess engine is called texel tuning, it takes a few million positions from games of the engine against itself and uses gradient descent to find the parameter values that best predict the final result of the positions. Engines also usually interpolate between an evaluation for the middlegame and an evaluation for the endgame based on the material currently on the board, which accounts for factors like pawns being much more valuable in the endgame and the king being best in the corner in the middlegame and in the center in the endgame.

I disagree that the queen is a sum of rook + bishop. Each bishop controls exactly half of the spaces on the board, but the queen (by spending an extra turn on a rook-like move) can move like either of the bishops to control any space on the board.

Lol ‘are you a Chess computer’ ive always been suspicious about the relative value. Bro I am about 1531 on Chess tempo; took a risk and played a 1888 level player and got absolutely Annihilated, his combinations, intermetzo moves, cunning knight maneuvers, and absolutely impeccable removals of defenders got me creased 😂

I see other people mentioning bishops not being able to switch colour, but I do want to emphasize this cause there are historical pieces that make this stand out. First off, the camel, which is a knight variant, but instead of doing a 2,1 jump it does a 3,1 jump, which always lands it on the same colour. Due to being "colourblind" the camel is generally only assigned 2 points instead of 3 points. Second, there are historical versions of bishops that can switch colours, sometimes just given the ability to step one to the left or right. These are typically assigned point values very close to a rook, like around 4.5. Third, there are hybrid pieces. Knight+Rook, and Knight+Bishop. These end up very close in value, like 7.7 to 8.3, again, closer than Bishop/Rook, because the Knight movement makes the Bishop no longer stuck to one colour and ups its value. (And then obviously this has implications for the queen as well. The estimated value of a queen used by engines like komodo is actually a bit higher than 9, more like 9.7. More than the value of a bishop plus a rook, closer to the value of two rooks, because it's a bishop that can switch from white to black squares).

I never heard the king is worth 3.5 peice or the value of it. I always learn the king value was infinite as the game is over and there is no way either side will not have 1 king, so computing it is not needed. Maybe its about having an active king, but active king early is negative value and end game is very context dependent.

i dont think its accurate to evaluate value of a queen to be rook+bishop, because in most cases queen is better than rook and bishop because its usually much easier to coordinate 1 piece than 2

The king may be a week piece, but it can't be captured. A game over is forced before that happens. That makes the weak king infinitely more valuable than all the other pieces.

Very nice video! However, if you want to rly assess the strength of the piecess you should take games from database of GM players and do similar calculation (and the more frequent the position the more impact it should have on the final result). Random positions do not aproximate well the positions you can realisticly get in a chess game.

The original concept for values was actually roughly based on how many pawns a piece is worth. So, the idea was that a minor (bishop/knight) piece is worth roughly 3-3.5 pawns (depending on the scholar's opinion). Therefore, it's generally seen as slightly unfavorable to trade two minors for a rook and pawn, which is surprisingly not an uncommon position. Also, your queen is generally seen a slightly stronger than a minor+ a rook, and move valuable than even all 8 pawns on the board.

Hello Mike, great vid. For calculating piece strength, you might want to concider the fact that the bishop can only move on 1 color.

Best are these intrinsic values: P = 3, N= 10, B1 = 10, B2 = 11 (the first B to leave the board is the 11 point one), R = 15, Q = 29, K = 13. This makes sense as 2Rs = 30 and a Q =29 (2Rs are better than a Q). Or how about this one: N + B = 21 vs. R + 2Ps = 21. It has always been said that B + N are better than a R + P (but with 2Ps, it's equal). Then obviously in a game, the values of the pieces change depending on the position. With a little practice, it's easy to use.

You've stumbled into the same kind of thinking a computer does when calculating positions. They always try to calculate the value of their pieces as it relates to the position on the board, and what squares they control has a big effect on how the positions are valued. Your math doesn't even need to be very precise for the computer to be able to beat any human alive since they can calculate so many positions and simply choose the best move that they find. The raw crunching power will usually result in a computer outclassing the human by sheer volume of calculated potential positions. I would recommend setting the King's value to maximum, however, so that your computer doesn't try to trade its king for a rook.

Just pointing out some objections without providing any fix: Chess squares are not equally likely to be occupied by a piece. In the beginning four ranks are always occupied whereas the other four are always empty. Progressing from that state backranks tend to empty over time and so on. A fully occupied chess board still lets any chess piece other than the knight see their immediate neighbours. A bishop would be worth at least twice as much as a pawn in that situation. There are some more things missing that are hard to put into a calculation. The ease with which pawns are blocked from moving as compared to the other pieces, the bishops both being unique pieces, the fact that king and another minor piece aren't able to checkmate the opposing king alone whereas rook or queen can, and probably more I cannot think of right now. Still the average comes surprisingly close to what we are used to. Maybe whoever invented the point system went a similar route? Of course none of this is able to truly give us an accurate evaluation of the value of a piece in action. This is the chess players job, to correctly asses true values as the game unfolds. Pawns are worth more the further they are advanced in ranks for example, due to the potential threat of promotion. Doubled past pawns especially can be very terrifying.

Thanks for your excellent video, I love the intersection of chess and math, and any effort to explore this is welcome! :) In addition to all you considered, however, it is important to consider some intangibles (unquantifiable qualities)... One is the raw checkmate power of the pieces, especially in the basic endgames where they are alone with the two kings. The major pieces have the most checkmate power, because they can attack both color squares at the same time. The minor pieces have less, because they can only attack one color at a time. The pawn has even less, in this regard. The king has none, other than it can help others. Another is that the differences - intangible strengths and weaknesses of certain pieces. The knight can jump, making it greater usefulness in specific positions. It can also attack/defend both colors, though not at the same time. The bishop cannot attack/defend half the chessboard at all, ever. The queen is worth more than R+B because it is both pieces in one piece, it is worth more than the simple sum of its parts. The rook is a notoriously awkward defender, but an awesome attacker -- how can this be quantified? And how do you measure the intangible aspect of a pawn -- the potential for promotion? Its value must be augmented by its prospects to promote, in addition to its attack/defense ability. I think what you're doing is indeed a noble effort, and I loved your video for its earnestness and attention to detail. But the intangible/unmeasurable qualities have a place, and I believe your project is doomed from the start. It is still lots of fun to try, though! :) Peace, Nathan G.

Knight is a 3 and bishop is a 3.25 because with less pieces it can dominate knight and more than often games will go to endgames rather than being decisive in middle game. Rook is 5 mainly because it's the only non queen piece that can escort pawn sitting at one square and also deliver checkmate with king. Queen should be 10 cause if rook pair isn't well coordinated, queen dominates it. Finally there is no point in talking about King cause both sides are always going to have one.

Each piece has a space-time specific value. The averages are useful though, obviously.

Hm, what if we took a big dataset of chess games, and did like, sigmoid(difference in score) as if it were probability of winning, and tried to fit the point value of each piece, to make this probability have the lowest loss , when averaged over all moves of all games in the dataset?

fascinating stuff, but properly factoring in the king requires integration (calculus). he is worth the game. His 'value' is infinite. he graphs as a line toward which the value of other players can approach asymptotically along a hyperbolic curve. with those parameters in the algorithm, a chess engine could give the value of any piece in any position during any game. it is fascinating - albeit pretty worthless if you are actually playing chess.

Could you create a value system that crosses this with position on the board? Ie a pawn on the 7th tank could be worth more than a knight on the 1st. I guess I’m less asking if it’s possible and more saying I would love to see that video :)

I have no idea how you would go about doing it but using both a piece's movement potential and the number of squares it currently controls in conjunction with each other with some sort of discounting for number of moves in the future would be interesting. Eg a knight controls (I assume) an average of 7.x squares (most of the time 8 but if it is on the edge of the board then fewer). But it has the potential to control y number of square 2 ply in the future and z number of squares 4 ply in the future. Obviously controlling a square currently is more valuable than theoretical control after 2 ply, 4 ply etc but those also have some value. Agree that controlling squares is a more useful metric than mobility but it is not perfect. A rook and queen would be able to control every square on an empty board at 2 ply so could be an interesting starting point for estimating how much you should discount 2 ply vs 0 ply (13 for the queen and 8 for the rook at 0 ply).

I love math and Chess This is the best video I've seen in a while. Made my day :)

Chess pieces cannot be given any "Constant Values"; as in they have variable values, the values keep changing depending on the positions that come forth on the board, in some instances two bishops can be more valuable than a rook or queen in a scenario to deliver BODEN'S checkmate pattern and same for other like the ARABIAN Mate. The piece values keep changing and sometimes a king can rise in value during endgames as for opposition and moving towards pawns and such. Chess is a really interesting game, requiring constant attention to what is happening on the board.

How about if you got the evaluation of lots of positions, removed each piece one by one, and see the difference in evaluation to determine their value.

the one main problem with the analysis on the bishop vs the knight is that the bishop will only ever be able to see one type of square (white or black) while the knight will be able to control both types of squares or will control the different color squares when it moves. Now i'm not sure on how you'd implement this but it's something that should be taken into account in my opinion. Another problem is that the queen in this analysis is rook+bishop, but a queen is usually way better since having a rook and a bishop doesn't grand you the same mobility or pressure compared to a queen (the bishop will only be able to do one type of square) though if you find a way to like "punish" the bishop for not having this the queens value will be more accurate in comparison to the bishop and the rook, since the queen isn't affected by the same disability the bishop has.

LIked it but did your calculations include the fact that a king unlike other pieces, can't go to a case that are curently controled by an enemy piece? Thus it reduce the king value a lot when there is many pieces on board.

One big factor with pawns is that they have the potential to become a queen

The king's value is obviously infinity since losing the king loses the game on the spot, no matter what other conditions are at play.

Very interesting video and a very interesting take. Aside from the recorrent comment that your analisys didn't look into future (mainly impacting the balance between knights and bishops) and other relevant actual chess based comments, for me, one comparison that it lacked (in the video, and even more so in the comments) was against the values in which strong AI (AlphaZero or LeelaZero, for instance) evaluate them

in practice a Queens is most often better than a bishop and a Rock. Probalby because when you move a Queen you make more changes to the controled squears. That is Queen is more Dynamic.

personally: 7 bishop pair 9 queen rook anywhere between 4 and 5 miner anywhere between 3 and 4

If we actually would want to re-evaluate pieces, rather than making up formulae, I'd consider a statistical fit of "frequency of white wins with pieces ABC vs. DEF" to "sum of (unknown) values of pieces ABCDEF", then rescale for white pawn = 1. Note that black pieces have negative values here, which can differ from their white counterparts.

I think an important point is that a bishop can only visit 32 squares while a knight can visit them all

IMO, I'd cut the King's value in half at least. His power to attack and defend is very limited by the fact that you can't put yourself in Check. All other pieces are able to take much more risks and deal some serious damage while moving into harm's way. - Vs. Pawn: 75% coverage. - Vs. Bishop: 50% coverage. - Vs. Rook: In theory 50% coverage, but in practice less since a Rook can so easily counterattack while a King tries to approach. - Vs. Knight: Difficult to predict. - Vs. Queen: Absolutely powerless. The only scenario where a King can attack a Queen at all is on her own terms, _forcing_ him to by moving to an adjacent square, in which case attacking her is one of the only legal moves. - Vs. King: _Almost_ powerless. Not as much as against a Queen since a King can _indirectly_ restrict the other King's movement from _two_ spaces away, but still not a very hopeful matchup.

Now this is some quality chess power scaling.

Value of a piece= probability of losing for not having that piece * probability of winning for having that piece

0:23 that is why i am here mr Daas. (I also feel called out lol)

I think that two of the top chess engines in the world have solved piece valuation (Lc0 and SF). I think that they've both concluded that piece value is extremely dynamic from move to move and that piece placement adds/subtracted from value more than is comprehendible by humans. I've watched Leela force Stockfish into positions where the rook value not only became worthless but it ultimately became a liability (in my mind that means it has a negative value) because The rook could do nothing else but stay on that square or the game was lost. The rook was under attack and it could not be traded because it would result in a worst position. The game looked like it would have been better if the rook wasn't on the board.

While rudimentary, this assessment is fairly accurate. Other suggestions may aim to refine the calculations, but what would be the reason for improvement? "Official values" are not better than yours. After all, both you and the official value calculation generalize the average value of the pieces and, inevitably, lose some nuance. In actual chess gameplay, the value of pieces can change dramatically depending on their positions on the board and their stability, with additional advantages if they are on the opponent's side. Pawn structure also plays a crucial role, not only in the value of the pieces but in the assignment of pawn value. A single passed pawn can make the difference between winning and losing in a chess game. Your calculation already assigns a value of 1, which is the same as the "official" value, but at high levels of play, this falls short. Additionally, the pairing of pieces is significant; this is why chess masters emphasize the importance of having "the pair of bishops." (for example). Anyway, it was both fun and interesting to explore this from mathematical perspectives.

Your mic has a low rumble that messes with the audio. Filter the lowest frequencies out of the signal so my subwoofer doesn't hum constantly

I think the reason your methodology values bishops slightly higher than most existing piece valuation systems is because it doesn’t factor in color complexes. In a game, if your opponent only has one bishop, that bishop can only ever control squares of the same color that it’s on. So you can try to manipulate the pawn structure to clog up the squares of that color in order to hem the bishop in, more so than a random placement of an equal number of pieces would do. This is also the reason why having both bishops is valued by players more so than the sum of the values of each bishop alone. Perhaps a way to account for this is to add a (downweighted) term in your calculation for the number of squares a piece can move to or control in TWO moves. I suspect this term would be significantly larger for knights than for bishops in most cases.

where did you get the numbers in 9:55

Cool video. I've never seen a mobility value for the king. Its importance value is infinite of course. The values of other pieces are not to dissimilar from the values found by Kaufman in 1999, based on game results, or by AlphaZero. Interestingly, these point systems seem to value pieces slightly more compared to pawns, even though this analysis ignores the pawn's promotion ability. The problem with pawns is that they can easily be immobilized, after which both sides play around them. Bishops have a similar disadvantage, as they can only cover half of the board. Although this usually doesn't matter too much, it makes the piece more predictable and as such an opponent can counter it, for example by putting his pawns on the same color. Having the bishop pair is therefore usually an advantage, although Kaufman explains it in terms of redundancy; two bishops never interfere with each other. You could argue this is a disadvantage as well, as they can't protect each other or attack the same target, but overall it's an advantage. I would say rooks may still be overrated compared to knights in this analysis. Knights are usually centralized, so they control a lot of useful squares early in the game, while the rooks are usually staring at their own pawns for most of the time. The problem with rooks is that they cannot easily be activated in the opening, so until the endgame they are usually not that strong. The queen is far easier to activate than a rook and isn't bound to one color like the bishop, and is therefore significantly stronger than a rook and a bishop. In fact, I'd say the queen is about as strong as two rooks. Similar synergy advantages can be found with the princess (a piece that can move like a knight and a bishop), which is a truly powerful piece.

But the chance of a particular piece being on a particular square is not n/63. For one thing, there are four to eight times as many pawns per other piece type. For another, the pieces do not start out on random squares. Next, the pieces tend to go to certain squares by the structure and tactics of the game. And also, given a bishop of one color, the chance (even under the arbitrary and over-simplified rule you gave), the other bishop will be in one or two states: it will have a chance of 0/x of being on the same color as the other bishop, and (n-2)/[32 or 31 or 30] of being on the other color; or else, if it is a promoted pawn, it's square color is indeterminate, making calculations even less meaningful.

My favourite moment was when He became a 🤓 and 🤓☝️ed all over the place