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

These are exciting days for chess, as the FIDE Candidates Tournament 2024 is currently in full force. Have you ever wondered why the pieces in chess are valued the way they are? I sure have! So I used some maths to compute the true values of all chess pieces - let's see how close they get to what we know!
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Chapters:
00:00 - Intro
00:48 - First attempts
04:12 - Maths ideas
06:58 - Results
09:23 - True values
11:00 - Outro

[@aleksihiltunen7063]

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

[@kurosakiichigo7475]

U traded your rook for your own king?

[@astro_pixel8251]

You mean you traded your king for their rook to gain 1 point

[@aleksihiltunen7063]

@@astro_pixel8251 it’s 1 point up, course I should

[@astro_pixel8251]

@@aleksihiltunen7063 ur original comment isnt phrased correctly btw

[@PROPLAYEN]

​@@astro_pixel8251If you were to think about it you could've figured out what they actually meant

[@GamuSino-dp2pr]

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.

[@finnfinity9711]

In addition to this, the value for every piece depends on the position. Even in the middle game a knight can be worth more than a rook. It's probably too hard/impossible to account for all of these niche scenarios though

[@TheLazyPirateGuy]

You could safely conclude that depending on the position bishops and knights change valuation Fischer argued that a bishop has a value of 3.25 at the start of the game If I recall correctly.

[@athul_c1375]

your 1st and 3rd points can be ignored because piece values do not matter when there are less than 8 pieces on the board, due to table base

[@GamuSino-dp2pr]

@@TheLazyPirateGuy Wow, so he gave a very similar value to Aphacero

[@simonwillover4175]

But the bishop can stop a line of pawns from moving if they are lined up correctly.

[@MorningNapalm]

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"

[@masaltzero]

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

[@Mikeastro]

Thanks for watching! Glad you liked it :]

[@klaus.sfc01official30]

@masaltzero rooky mistake

[@ZDTF]

Idk why but it sounds sus

[@CheckmateSurvivor]

@@Mikeastro Piece value will always lose against numbers. If a Knight equals 3 Pawns, give a player a row of Knights, and the other player 3 rows of Pawns. Knights will lose every single time.

[@zoetje9817]

@@CheckmateSurvivor This is interesting, but not true. Set up a board with King + 9 pawns vs King + Queen for example, and the side with the Queen has a +4.7 advantage at depth 30 (using Stockfjsh 14.1 NNUE). With knights, it’s *mostly* equal, but there’s a slight advantage to the side with more pawns.

[@G102Y5568]

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.

[@jayfeatherblah]

a note on 3, the rook still protects all four friendly pieces, so I would argue that it serves the same value

[@michaelr1661]

Yes but your pieces are controlled by you therefore if you play reasonably well, this efect is negligible bc it's restricted to mostly those cases in which you want to protect the piece in question.

[@G102Y5568]

@jayfeatherblah I considered this same idea when I first thought about it, but that's a fallacy because "protection" doesn't create value, it just prevents you from losing it. You can't count undoing a negative as a positive for yourself. If I went gambling and lost $1000 and then won $1000 back, I didn't make $1000, I broke even. Let's use a more chess-related example. Say there was a position where all of the opponent's pieces were undefended but none of them were being attacked, while all of your pieces are being attacked and protected simultaneously by your pieces. According to your logic you should be winning because all of your pieces are protected and none of your opponent's pieces are. But really it's obvious the pieces are just cancelling out. Then you factor in that the opponent's pieces all have extra squares they can move to because they can still capture your defended pieces for sacrifice plays, it's obvious the opponent has the advantage in activity.

[@G102Y5568]

​@@michaelr1661jay is incorrect and I explained why above, but it's for a different reason than the one you explained. There IS actually a small advantage to having protected pieces even if they aren't being immediately attacked, because it gives you more freedom to move your pieces around without concern for getting forked, counterattacked, or losing the initiative. And you may say "but you can calculate ahead and see if the opponent has a tactic", but then you can't play the move you wanted to because of the tactic you spot so you have to settle for a slightly less good move. And you may argue you could calculate THAT and avoid it, but then by avoiding THAT you have to settle for a slightly less good move too. So sooner or later you're going to have to compromise SOMETHING as a consequence of having an undefended piece, it's unavoidable outside of exceptional cases where it can never cause problems.

[@camerson1313]

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?

[@michaelsmith4904]

i don't know the exact value but it's a complex number.

[@magmablaze69]

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

[@B-fq7ff]

my favorite moment was when Mike rounded all the numbers to integers because he thinks his viewers are little babies who don't know how decimals work

[@sebastiangrau8409]

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

[@TreuloseTomate]

That's why the bishop pair is usually so strong compared to knights in the endgame. But a lone bishop can also be weaker than a knight in the endgame if he can't protect crucial squares. It all depends...

[@sebastiangrau8409]

@@TreuloseTomate that is very true, the two bishops have a synergy with each other, which makes them both more powerful. It does make it pretty difficult to assign them a flat value in terms of pawns. There are also various scenarios where the bishop you have either can or cannot block a crucial promotion square which will also affect its circumstantial value.

[@sylenzos6869]

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

[@levistepanian5341]

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.

[@bigsmoke6414]

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

[@mathmagician8191]

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.

[@Houshalter]

It's less efficient, but probably easier to explain hill climbing to a layman. Just randomly modify the values, play a number of games, and select which values did better. Do this a bunch of times and you will reach a point when no further improvement is possible.

[@mathmagician8191]

@@Houshalter That's called SPSA tuning, which is used for tuning search and time management parameters for an engine, but not normally used to tune evaluation parameters

[@elytron6758]

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

[@KenFullman]

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.

[@ecMathGeek]

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.

[@BinaryDragon]

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.

[@Osiris261]

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.

[@Nosthus]

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.

[@64chess]

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

[@64chess]

I should also point out that bishops do gain in value in the endgame, but without their partner they’re still blind to half the squares and either way they have less to attack by definition. So their increase in value is often mitigated. If a knight need only help hold a few points on one side of the board, the enemy bishop being able to zip around the empty side is irrelevant.

[@B-fq7ff]

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

[@Mikeastro]

Rounding can cause such things

[@tomas-wi8dy]

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.

[@12jn00gb]

Now you are thinking with... chess

[@gmdFrame]

Good. Finally not portals

[@exsurgemechprints2671]

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).

[@jaybingham3711]

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.

[@sthenx0r]

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.

[@thatdiamondminer2673]

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.

[@adammartin2431]

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.

[@Mikeastro]

Thanks for watching! It's true, I locked myself a bit into the portal content but tried branching out a bit with this video; maybe I will make more videos in this style :]

[@RikMaxSpeed]

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?

[@robotnik0889]

Nice! This information is useful to improve my chess bot

[@justfive1321]

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.)

[@Mikeastro]

Thank you very much for the kind words :D Interesting suggestion too!

[@brianhiles8164]

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.

[@fanamatakecick97]

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

[@michaelr1661]

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

[@piotrkawaek6640]

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.

[@VibratorDefibrilator]

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.

[@WingedEspeon]

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.

[@FLooper]

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!

[@Pinky149]

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.

[@KaitlynBurnellMath]

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).

[@ethribin4188]

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.

[@DanielLCarrier]

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.

[@georgehornsby2075]

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).

[@kugelblitzingularity304]

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.

[@zakhankimah2226]

According to the formula shouldn't you remove 2 pieces from the probability of a piece being on a certain square, because there is no chance all 32 pieces can come to any square? The players start with bishops that stay on their colours. What about kings? Sure they can always go to 8 squares but never into a check. The likelihood of an enemy piece controlling a square should be higher than the probability of a piece being a square. With this in mind, the King's value should increase as the pieces drop off the board which is consistent with chess principles. However, I really enjoyed the video, so keep it up!

[@Crazy-dlwah]

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

[@apbuggygaming]

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.

[@cmilkau]

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.

[@Pystro]

In addition to giving black pieces a different value than white pieces, you could also consider positional information: like whether a pawn is on rank 2 or 7, how close to the border or center a knight/ bishop is, if you have bishops on both colors and so on. Except that usually you'd be going for maximum simplicity in these schemes, so you'd want a set values for the 6 pieces, and then maybe only the 3-5 most important modifiers. But those could probably be identified by looking at the results of the "full complexity" values. But then it would just be a matter of finding data for realistic/real positions, and doing a mountain of least squares fitting.

[@CatSurfer]

Great video! Chess has always interested me.

[@tom_curtis]

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).

[@GlacialScion]

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.

[@Five-Star-General]

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 😂

[@Mikeastro]

We've all been there :'D

[@tom-kz9pb]

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.

[@JackBond1234]

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.

[@frostiesr345]

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 :)

[@Dunebug1985]

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

[@knivetsil]

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.

[@belaji]

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.

[@hlibprishchepov322]

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.

[@SmartsellerGaming]

so we were quite accurate after all

[@daboffey]

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.

[@TheMazarineIsReal]

ah cool, like this new style of content!

[@ramonsobrevalsarce.7634]

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.

[@shreyjain3197]

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

[@mynamemywish8563]

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

[@mrdraw2087]

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.

[@VincentHondius]

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

[@guillaumeparadis6331]

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.

[@Mikeastro]

That is actually a great point; the king still puts indirect pressure on squares that are also controlled by your opponent, but not directly anymore - one could consider factoring in this effect for the piece strength though the maths might be a bit tricky... interesting!

[@paristhewabbit]

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.

[@arashaboora891]

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.

[@punch7793]

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.

[@perplexedon9834]

For the knight square access issues, you could calculate the number of squares that can be reached within two, three, four, etc. moves and then take a weighted average with the one move case. The weights would be a subjective value judgement, but I'd say the ability to get to a square in two moves is worth roughly half the ability to get there in 1, and so on.

[@ChibiRuah]

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.

[@ckq]

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

[@polemopoleno7104]

AFAIK he's worth 4

[@drdca8263]

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?

[@klaus.sfc01official30]

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

[@LendriMujina]

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.

[@The-Anathema]

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.

[@chrisdavis2161]

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.

[@rubensilvera1164]

incredible ! do you have the game reference ? I have also a game where a full knight is traded for basically nothing than space et slight development and whites won the game against 3300 elo Computer.

[@billmcgahey1926]

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.

[@aliensconfirmed3498]

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.

[@briansinger5258]

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

[@noob_jr_2sjrkc]

Now this is some quality chess power scaling.

[@thermitty_qxr5276]

Did bro slove chess? Btw cool video, dont stop

[@Trizzer89]

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

[@DownDance]

Interesting video! The bass (beginning at 0:23 and stopping at 6:45) makes me go crazy (sry to be that guy) 😅

[@Mikeastro]

Thanks for watching! You're not the only one to hear it - sadly for me it is completely inaudible so I had no idea :[ I'll make sure to not let it happen again though!

[@BorisGamingChannel]

Why doesn't the pawn promoting account for its power? It totally should, and it would make evaluations more accurate. Not accounting for it must be due to no mathimatical way to calculate that I am assuming.

[@Mikeastro]

That, and also somehow by disregarding it the values I computed were quite close to the standard values so I figured maybe it was the way to go, as making pawns even stronger would bring all other values further down

[@skyhistory6602]

I definitelyly can say even how tis calculation consider more factor than just calculator, however it's isn't perfect, it still a lot of thing to consider such as move and attack isn't equal, piece with more powerful attacking is more value than piece that did thing in reverse as can be seen in variant called Hoppel-Poppel that knight move like knight but capture as bishop and bishop move like bishop but capture as knight found that knight are near 2 times stronger than bishop in this variants, or another such as volourboundness that piece with coulourbound piece tend to be less value that non-colourbound a bit (87% value of non-colourbound) and it can be compensate by having another colour bound piece on different colour

[@gatoleyo7739]

I wonder why the king is worth more than the knight if they always control the same amount of squares (unless they are on an edge). Also I see a lot of ppl that claim the bishop only control one color of squares but knights sort of work the same, the only difference is that knights switch of color whenever they move

[@Mikeastro]

I think it's just the edge that makes the king control more squares; its power is only reduced when on the edge and typically still controls more than a knight, whose power is also weaker one square away from the edge

[@LuisAldamiz]

The king has infinite value, also the knight is slightly less valuable than the bishop (2.5 vs 3). All values should be adjusted by position anyhow.

[@JimCullen]

Gonna be honest, I found this video significantly less interesting than I was hoping for from the title. My disappointment is exemplified by the basic premise: "normalised in such a way the value of a pawn is 1". When I saw the title, I thought it might be doing something interesting with the relative value of different positions. The D and E pawns are intuitively probably worth a bit more than the A and H pawns, in terms of how important the space they control is. Relatedly, maybe the queenside bishop will be worth more than the kingside, or vice versa. I'm not sure how you would calculate that to be honest. It's obviously a lot more difficult. But that was what I was hoping to see from the title, so even though the analysis itself is actually really good, I ended up disappointed.

[@ethribin4188]

The mathematical point values seem to be quite close to the base game point values. Except that the Bishop is slightly more valuable then the Knight. And the Rooks value is inflated.

[@enderyu]

1:24 - 3:00 How did the values of the other pieces increase relative to the pawn if the pawn is now considered stronger?

[@Mikeastro]

That's a good question actually, I realise that I never double checked Gik's calculations. Somehow they arrive at a pawn having on average 2.1875 squares of mobility. Perhaps they combined normal pawn moves (average 7/6 = 1.167 squares) with a boost for maybe being able to take a piece (say 16/64 = 1/4 of the time, yielding another 7/16 = 0.4375) and then maybe some kind of additional boost for promoting? I'm not sure :[

[@carlospena98]

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

[@brianmulholland2467]

Nice. But I think you can do better by borrowing the concept of WAR from baseball and other sports analytics. What needs to be done is a database crawl through high level games to explore not how many squares can be moved to or threatened, but how well having particular pieces correlates with winning. As you noted, the value of a piece is likely not static throughout the game, but an excessive level of detail here defeats the purpose of the analysis, which is to create something usable. I would invite someone with access to the necessary data to try to analyze the value of pieces related to helping to create wins.

[@totallynotsj9005]

as if chess wasn't confusing enough already

[@scottpope7835]

Since pieces attack/defend different numbers of squares depending on the square they occupy, their value should be the average of their values on each of the 64 squares.

[@Mikeastro]

I agree, that's why I computed it that way in the video!

[@Anonymityfan]

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

[@ulrikebeckmann4986]

What about the bishop on z 294 ?

[@llamallama1509]

Is the value of a queen just rook + bishop? Wouldn't a rook and bishop be more powerful, because there would have to be two captures to reduce their power to zero, compared to one capture to reduce a queen to zero. Plus having two pieces increases flexibility, and if you go off to defend one square with your bishop, you're still defending with your rook.

[@Medz-bl9lw]

When you move a queen you change more of the controlled squares and it has the ability to fork a bishop and rook and it’s also harder to capture because it has more squares it can move too

[@nain3517]

Great video, bro, wow...

[@ethribin4188]

I always felt weird with Bishop and Knight at the same value. Bishop is stronger. Period. The Knight simply is a wild card piece, the only one aside of a pawn reaching the opposit side. Mainly due to the strange move pattern and its ability to ignore enemy pieces in its path. Powerfull yes, but unreliable.