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!

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

You should just download 10000 lichess games and average available squares in each position. You can only focus on endgames/openings and middle games if you like but it should be a decent estimate

This is pretty dumb. A computer or a human following these guidelines will lose every game.

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.

The king's value is infinite.

Y is the King worth 3.75?

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

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?

Excellent analysis!

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.

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!

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.

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

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

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

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

Simple rounding error, I win

so we were quite accurate after all

Anyone ignorant enough to suggest that the king has a finite value is not worth your time.

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.

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

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.

This Video is quite similar to my portal 2 journey. I have about 3.7k hours in the game and have been playing it since 2011, and a lot of thinks that happened in this video happened to me. Even getting people like LB and Nock to play my maps (they were both terrible maps, but I was really young and thought they were good) I even somehow got one of my Beemod maps In 2017 on the front page. No idea how, it was a terrible puzzle, and I remember libbybaba saying that it wasn’t a good map (I still really like his maps). And then in 2020 when we were all locked inside, I started making hammer maps, which were my first maps to get really popular. I looked back on when I released those maps and I had been cranking them out once a week! (Probably why I got bored) but now I’ve come back to Portal 2, and I hope that now after 4 years of not making anything, I can actually make a decent puzzle. Thanks for making this video!

could you put this in an exel file so then you can put it in a graph

The best way to describe this test is frightening

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.

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.

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.

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.

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.

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

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.

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.

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.

8:56 interesting mechanic!

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

I love the 7:23 puzzle, the main point of introducing a aspect is no red herriing at all, including portal gun

The obvious next step is to compute the average of these values over a database of actual chess games.

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.

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

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?

Bishop + Rook = Queen + 1

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

My valuation would be knight=3.25, bishop=3.5, rook=5, queen=9

Of course, it's a puzzle game

If the player doesn't know the aspects of the turret, the puzzle can never be solved🥲 thanks for introducing these

11:17 So tricky😮😮😮