[HN Gopher] How some common material imbalances affect your win-...
       ___________________________________________________________________
        
       How some common material imbalances affect your win-rate
        
       Author : Luc
       Score  : 154 points
       Date   : 2023-06-12 21:20 UTC (1 days ago)
        
 (HTM) web link (lichess.org)
 (TXT) w3m dump (lichess.org)
        
       | AdamH12113 wrote:
       | Neat results! I am very much a rank amateur, but it looks like
       | these results roughly align with the traditional 1-3-5-9 point
       | values for the pieces, with a couple exceptions. Advantages of a
       | single pawn, minor piece, or rook have proportionate values
       | similar to what you'd expect. In particular, bishops and knights
       | are very close with bishops being worth a tiny bit more. Winning
       | the exchange is a sizeable advantage. Trading a rook for two
       | minor pieces is an advantage except at the lowest skill levels.
       | 
       | Two rooks turn out to be significantly worse than a queen instead
       | of slightly better. But the most surprising thing to me is that
       | having the two bishops seems to be worth almost nothing -- close
       | to 50% odds across skill level. A single pawn advantage is more
       | valuable! (The article says "4-5% more likely to win" at Elo
       | 1200-1400, but that doesn't match the graph.) These surprising
       | results were also more consistent across skill level, while the
       | well-known advantages are worth significantly more to skilled
       | players.
        
         | rmetzler wrote:
         | > In particular, bishops and knights are very close with
         | bishops being worth a tiny bit more.
         | 
         | I don't think so, and I would like to see the author look in
         | the data for 2 knights vs 2 bishops. My gut feeling is that
         | knights are hard to block and can attack every square, while
         | bishops can only attack their color, so half the board. They
         | can't protect each other and rely on pawns for this.
         | 
         | I'm happy to trade my knights for the pawns, but I would like
         | to see if I'm wrong.
        
           | compiler-guy wrote:
           | Bishops have a great deal more mobility than knights. A
           | knight in the wrong corner takes several moves to catch up to
           | a distant passed pawn. A bishop needs far fewer.
           | 
           | Generally the trade off is a bishop pair, where you do have
           | all the squares covered, vs the slower knights, who are also
           | the only pieces that can threaten a piece without being
           | threatened back.
           | 
           | The advantages are highly position dependent, with closed
           | games largely going to the knights and open games going to
           | the bishops pair.
           | 
           | But most games turn into open games at some point, so
           | statistically the bishops end up better.
        
       | lordnacho wrote:
       | What does "clean" mean in this context? What makes you up a pawn
       | "cleanly"?
        
         | PeterisP wrote:
         | Contrary to the definitions in the other answers, I'd assume
         | that this refers to selecting positions from the database where
         | that's the only difference in material - so "clean up a pawn"
         | means that all the other figures are the same, that there isn't
         | also a bishop vs knight difference, for example.
         | 
         | I really doubt if there was any attempt to check if there's
         | some "strategic compensation" - how would you do that at a
         | large scale? I doubt that even running a solid chess engine
         | evaluation on all these positions is feasible, you need
         | something where you can simply/cheaply filter positions from
         | the database and then just count the winrate.
        
           | [deleted]
        
         | beyondCritics wrote:
         | I agree, this is the first annoying question to ask. As a
         | seasoned club player, i believe to know that "clean Pawn up"
         | means chess wise roughly what you think it means, namely: a)
         | You are in a position, where it really does make sense to count
         | pawns, a so called "quiet" position would be one of those. b)
         | Your opponent has no pawn compensation of any kind, not even in
         | the form of a nuisance. I personally learned this expression
         | from GM Nakamura. Clean pawns were unknown before the "golden
         | era of chess", you could utilize the idiom "ceteris paribus"
         | instead, this was used by the old german vice WM Tarrasch.
         | 
         | Hence it should be clear to anyone that without engine
         | assistance it is totally impossible to determine for 10^6 games
         | if a certain material difference was "clean" or not. Most
         | likely he just checked for a stability over n consecutive
         | moves, at least this is the usual way. Others have noted more
         | problems, that might help put his highly deviating, washed out
         | results in order.
        
           | [deleted]
        
         | iib wrote:
         | I think it means that the opponent does not gain a strategic
         | advantage from an exchange. You both have good enough positions
         | and you are simply a pawn up.
        
         | WJW wrote:
         | From the other imbalances listed (like being up "the
         | exchange"), I take it that they mean being up a pawn without
         | having to give up anything in return. For pawns that is kinda
         | obvious, but for more valuable pieces like a rook you can win
         | it "cleanly" or maybe you take it with a pawn and then
         | immediately lose the pawn.
         | 
         | You can of course give up material for positional advantages as
         | well, but from the article I don't think the author analyzed
         | that. It would be difficult to accurately measure that anyway,
         | some positions can be up +5 points according to the computer
         | but only if you find 20 perfect moves in a row. Needless to
         | say, most humans would not find those moves especially in the
         | lower ELO brackets that the article analyzed.
        
         | C-x_C-f wrote:
         | Apart from what WJW said, it's not an objective criterion but
         | it essentially means that you don't lose out in terms of
         | initiative or position. A counterexample might be accepting the
         | queen's gambit with black (i.e. 1. d4d5 2. c4cxd4), that way
         | you win a pawn but you arguably give white an advantage in
         | development (an in many cases white will be able to recapture
         | the pawn as a result)
        
         | mtlmtlmtlmtl wrote:
         | "Cleanly" in chess parlance means you're up a pawn, and there
         | is no "compensation" for the other player. I.e there are no
         | other advantages in the opponents position that make up for the
         | fact they're down a pawn(things like an attack on your king,
         | piece activity, pawn structure).
         | 
         | In simple mathematical terms a clean material advantage is:
         | 
         | Overall eval >= material eval
         | 
         | If overall eval < material eval, then compensation(other
         | player) > 0 = "the opponent has some compensation for the pawn"
        
         | EricMausler wrote:
         | I agree with this question. How was clean determined in the
         | dataset?
         | 
         | Knowing how the word is used in chess is not enough to know
         | where the line was drawn in the data here, which is the basis
         | for the entire analysis shown.
        
           | ifdefdebug wrote:
           | I guess you put the game position right after the pawn
           | capture into a good chess engine and check if you're one
           | point up a a result.
        
             | bitshiftfaced wrote:
             | This would mean that, in the case of "up 2 rooks for a
             | queen," the board would evaluate to a player's advantage
             | even though that player would tend to lose at the highest
             | level. I think something weird must be going on.
        
         | nebulous1 wrote:
         | Should have been defined (along with more information about the
         | data set), but I assume it means that the opponent has no
         | tactic that wins them material. For instance if I just take
         | their protected bishop with my bishop then I'm briefly up a
         | bishop. So it would not include that situation or where the
         | trade is more complicated.
        
       | janalsncm wrote:
       | Would be interesting to see material imbalances as a function of
       | plies into the game. As in, we know an extra pawn is more
       | important in the endgame than in the opening, but how much more?
        
       | KingOfCoders wrote:
       | (Please Lichess, this is not ELO)
        
         | nebulous1 wrote:
         | Honestly "Elo" is commonly used to denote most Elo-like
         | systems. I don't think it's confusing here.
         | 
         | Also, it's "Elo" not "ELO".
        
           | evandale wrote:
           | One of my favourite things on the internet is when someone
           | tries to make a pedantic correction and someone else comes
           | along with something that's even more pedantic.
           | 
           | Thanks for giving me a good start to the morning!
        
         | smatija wrote:
         | This is, if I see correctly, not official Lichess blog, but
         | personal blog of marcusbuffett (hosted on lichess). Lichess
         | offers personal blogs for some months now.
        
         | shawabawa3 wrote:
         | What does this mean?
        
           | jackmott42 wrote:
           | OP is making an unimportant pedantic point.
        
           | [deleted]
        
           | czx4f4bd wrote:
           | Quoting Wikipedia because I'm lazy:
           | 
           | > The Elo rating system is a method for calculating the
           | relative skill levels of players in zero-sum games such as
           | chess. It is named after its creator Arpad Elo, a Hungarian-
           | American physics professor.
           | 
           | https://en.m.wikipedia.org/wiki/Elo_rating_system
           | 
           | There are other rating systems that are similar, but people
           | commonly call any similar rating system "Elo". In this case,
           | Lichess uses
           | https://en.m.wikipedia.org/wiki/Glicko_rating_system, so the
           | GP is arguing against calling it "Elo". I think it's kind of
           | a pedantic point given that more people will understand "Elo"
           | than "glicko-2", but I suppose "rating" would be clearer than
           | either term.
           | 
           | https://en.m.wikipedia.org/wiki/Chess_rating_system
        
           | nebulous1 wrote:
           | lichess uses glicko 2
        
       | ladberg wrote:
       | Am I misunderstanding something or is there a glaring statistical
       | error here? I think this analysis works if the matches are all
       | between two players of _truly_ identical skill that will roughly
       | win 50% of the time against the other, but that 's not really the
       | case here. They attempt to do that by controlling for Elo but I'm
       | decently sure that you could pick any two players of the same Elo
       | on lichess and one will win the majority of games.
       | 
       | Now if we assume there is often a slightly better player in these
       | games, the better player will more likely get into an
       | advantageous position early in the game _and_ win more often, but
       | not entirely because they got into the more advantageous
       | position. What I 'm trying to say is that someone who is up a
       | rook will win partially because they're up a rook and partially
       | because they are likely a better player in the first place and
       | will continue to play better than their opponent.
       | 
       | I think to do this study correctly you'd need to place new
       | players in a random position drawn from the dataset and actually
       | evaluate the win rate without the confounding factor of having
       | gotten into that position in the first place.
        
         | spoonjim wrote:
         | > I'm decently sure that you could pick any two players of the
         | same Elo on lichess and one will win the majority of games.
         | 
         | I think that's true at 2000+ but at my rating (500) it's just a
         | comedy of blunders
        
         | fairity wrote:
         | > I'm decently sure that you could pick any two players of the
         | same Elo on lichess and one will win the majority of games.
         | 
         | Interesting hypothesis, but my intuition is that this effect is
         | small. It should be easy to measure. Just take random matches,
         | and study winrate when a player has beaten their opponent once
         | before.
        
           | yamazakiwi wrote:
           | Players can also swap back and forth between winning a
           | majority of games, so it depends on the amount of games we're
           | talking about.
        
         | queuebert wrote:
         | That's exactly what Elo controls for: all things, being equal,
         | what is the skill component. Now there could be a second-order
         | effect in which a higher Elo player is _even better_ at winning
         | from an advantageous position, but given large enough samples
         | this should be in the mix.
        
           | ladberg wrote:
           | Yeah I guess my claim is that two people of the same Elo on
           | lichess are likely to not be perfectly equal. If the average
           | winrate of any random two players with the same Elo is
           | something like 60/40 (random speculation) then you'd bias the
           | results in the article enough to create the kind of graphs
           | shown.
           | 
           | The problem is that this doesn't really average out with more
           | samples.
        
             | BurningFrog wrote:
             | I accept that players with the same ELO can have unequal
             | strength.
             | 
             | But I don't see how that wouldn't average out over lots of
             | games?
        
               | ladberg wrote:
               | The probability of player A being up a rook over player B
               | is higher when player A is better than player B, even if
               | their Elo is the same. So the author has picked scenarios
               | where player A is already more likely to be better than
               | player B but assumed they were of equal skill, which will
               | cause the results to be biased.
               | 
               | If I have time later I'll try to whip up a small coin-
               | tossing demo in Python that demonstrates what I mean, I'm
               | definitely not the best explainer!
        
       | fairity wrote:
       | I'd love to see the winrate data for Stockfish against Stockfish,
       | for each data set. To get a sense of what the objective winrate
       | is.
        
         | queuebert wrote:
         | Once a computer gets to an advantage of +/-1.0 or so, the win
         | is virtually assured in all positions, barring a few weird
         | theoretical draws and misevaluations of the score of the
         | position (e.g. closed positions with no way to make progress),
         | although the latter problem has gotten a lot better with NN
         | engines.
         | 
         | (I used to play computer correspondence chess.)
         | 
         | Edit: I should add that +/-1.0 is roughly a pawn's worth of
         | material. So a computer up a knight or bishop -- nominally 3
         | points of material -- should virtually always win.
        
       | d_burfoot wrote:
       | This thread makes me want to build a simple chess-practice tool
       | that works like this: 1) Get a big database of positions
       | 
       | 2) Use StockFish to evaluate all the positions
       | 
       | 3) Pick a position at random, show it to the user, and ask for
       | their evaluation
       | 
       | 4) User gets a score based on the difference b/t their eval and
       | StockFish's.
       | 
       | The idea being that this could allow you to rapidly hone your
       | position-evaluation skill. That might be a faster way to improve
       | at chess than just grinding through many games.
        
         | swader999 wrote:
         | Chess engines already do this.
        
         | JHonaker wrote:
         | The problem I see with this is that it's not very useful for
         | people to know if something is a +3 or +4.1 position. The
         | qualitative descriptions that underlie that rating is what's
         | important. How does my pawn structure compare to my opponents?
         | What is the balance of minor pieces (knight and knight vs
         | bishop knight, BB vs KK, etc.)? That's how you get to
         | understand chess.
        
           | EchoAce wrote:
           | Yeah, this is probably better for a game like Go where
           | evaluating raw point differentials is a common thing that
           | professionals (and even strong amateurs) do.
        
         | ginnungagap wrote:
         | A guy on reddit made a website that does exactly that, but I
         | cannot find anymore at the moment
        
         | rmetzler wrote:
         | I just play often 10min games on lichess, then look at all the
         | mistakes I made and try to memorize simple rules of thumb for
         | better moves.
        
         | POiNTx wrote:
         | Sort of like this (but without guessing the best move):
         | 
         | https://makotoe.github.io/guess-the-eval/
        
         | disambiguation wrote:
         | like this? https://lichess.org/training
        
         | frakt0x90 wrote:
         | Not to rain on your fun, but this already exists. I watched
         | Hikaru Nakamura play it online once. It just gives a random
         | position and you have to guess the centipawn loss of the
         | position. Kinda like Geoguessr for chess.
        
           | whoisthis12 wrote:
           | Do you know what the site is?
        
         | ebiester wrote:
         | Stockfish eval is often misleading because some positions are
         | very sharp - if you see mate in 15 you're ahead, but otherwise
         | you're far behind at the most pathological case.
        
           | shawabawa3 wrote:
           | You could do some basic filtering on e.g. top 3 moves are all
           | within 0.5 or something
        
             | 6D794163636F756 wrote:
             | I think the problem then is a position where there is only
             | one winning move that is 8+moves away. If one is mate in 10
             | and the others are mate in 3 for your opponent how do you
             | reconcile those into one evaluation?
        
         | light_hue_1 wrote:
         | There are endless tactics trainers out there.
         | 
         | But it takes far more to build something useful for humans than
         | just asking stockfish to give you the best move.
         | 
         | For one thing you don't want to learn to play like machines.
         | Often the moves that machines make are terrible human moves.
         | Machines walk into extremely dangerous positions all the time;
         | positions where you need to make a series of only moves to gain
         | a small advantage. Humans shouldn't be doing this.
         | 
         | Machines also rely on tactics heavily. Unless you understand
         | the tactics in play it will make your play much worse. It might
         | appear to you take a piece can't be taken, but stockfish says
         | it can. Ok? Unless you understand the reason why, you aren't
         | going to gain anything from this exercise.
        
           | TechBro8615 wrote:
           | I wonder how true this will be over the next decade or so
           | when the top humans are people who have trained with machines
           | their whole career.
        
       | billfruit wrote:
       | "1200 Elo players don't know how to take advantage of having the
       | bishop pair."
       | 
       | How exactly does one take advantage of having a bishop pair? Why
       | is that not obvious to players under 1200?
        
         | wonnor wrote:
         | Generally the advantage of a bishop over a knight is its
         | increased mobility. The bishop pair is more mobile than N+N or
         | B+N and can control both color complexes. To exploit the
         | mobility advantage, good players tend to try to "open up" the
         | position more to increase the scope of the bishops. A classic
         | example of a strategic goal if you want to play for an open
         | position is to keep the pawn structure fluid (as opposed to
         | locked up), which can be too subtle for weaker players.
        
       | red_admiral wrote:
       | I'm surprised that in the higher ELO ranges, being up a clean
       | pawn has only a comparatively minor impact - I thought that one
       | clean pawn ahead and no other major positional or structural
       | disadvantages was more or less "GG" among top players.
        
         | tux1968 wrote:
         | Just a friendly pedantic point, it's "Elo", not "ELO". Rather
         | than an acronym, it's a person's name; that being the rating
         | system creator, Arpad Elo.
        
           | xxs wrote:
           | For years the guy calculated the FIDE ratings by hand...
        
           | Olipro wrote:
           | If you want to be fully pedantic, it's Elo, not Elo - the way
           | in which you pronounce the former is dramatically different
           | from the latter.
        
             | neaden wrote:
             | if you want to be fully fully pedantic its Glicko-2 rating
             | system on Lichess, while FIDE uses Elo.
        
               | digging wrote:
               | Rather, players have a FIDE rating within FIDE, and a
               | unique rating in each system, because no organization
               | follows the exact, original Elo system.
        
             | shawabawa3 wrote:
             | To be correctly pedantic, it's named after Elo but it is
             | called the Elo rating system
        
         | zeteo wrote:
         | You are correct, but this uses Lichess ratings, which are a bit
         | inflated, and probably a variety of time controls including
         | blitz. At levels of play below super-grandmaster with full time
         | control the accuracy of play can be rather low, and picking a
         | suboptimal move will often cost you half a pawn (or more) in
         | engine evaluation. Add this up over an entire game and a one
         | pawn advantage is easily lost in the variability of both
         | players making compounding errors.
        
         | [deleted]
        
         | erostrate wrote:
         | I suspect the author only counts raw material. So players who
         | lose a pawn but gain compensation reduce the perceived impact
         | of losing a pawn. Including many openings where you temporarily
         | give up a pawn (queen's gambit, benko, etc).
        
           | nomilk wrote:
           | I was thinking the same. It could be true for higher ELO
           | players being down a pawn is a good thing, since your
           | opponent took the bait.
        
         | yuvadam wrote:
         | Many pawn-up endgames are still technically a draw (king and
         | pawn vs. king [1] in the most extreme example)
         | 
         | [1] -
         | https://en.wikipedia.org/wiki/King_and_pawn_versus_king_endg...
        
           | xxs wrote:
           | That depends both on the position of the pawn and the king...
        
           | smatija wrote:
           | Take care that not all of king and pawn versus king are
           | drawn. Their result depends on relative position of kings.
           | 
           | Nice example of more extreme imbalance is that sometimes
           | King+Queen vs King+Pawn is drawn - that is when pawn is on
           | seventh rank of bishop or rook file with defending king in
           | front of it, while attacking king is not close enough to
           | force checkmate.
           | 
           | See: https://en.wikipedia.org/wiki/Queen_versus_pawn_endgame#
           | Quee...
        
           | mtlmtlmtlmtl wrote:
           | Yeah. More commonly rook endings are notoriously hard to win.
           | 
           | I gained a lot of rating once I started really studying the
           | ins and outs of all the different rook and pawn vs rook
           | endings. Because no matter what the tablebase might say, you
           | have to be very precise with or without the pawn.
           | 
           | So I started getting a lot of wins out of equal positions,
           | and got a lot better at saving a draw in bad positions.
           | 
           | Pawn and minor piece vs minor piece is also fraught with
           | drawing chances, especially with bishops. Because you can
           | just sac the piece for the pawn and there's insufficient
           | material for mate.
           | 
           | Queen and pawn vs queen is basically a 3 result game at the
           | club level. So many ways to blunder your queen. And at GM
           | level it tends towards a draw I think. Though it's so
           | stupidly complicated I never even tried to properly study it.
        
             | littlestymaar wrote:
             | > Yeah. More commonly rook endings are notoriously hard to
             | win.
             | 
             | Oh, so it's not just me.
        
               | mtlmtlmtlmtl wrote:
               | There's a famous saying in chess: "all rook endings are
               | drawn".
               | 
               | Of course, this isn't actually true. But it's a useful
               | rule of thumb to just assume it's a draw unless you're
               | sure it's a win, because it probably is, fucking somehow.
        
       | wmsiler wrote:
       | I wonder how this would change if you only looked at highly rated
       | chess engines. For example, is a rook pair really not as good as
       | a queen or is it just that _humans_ aren't as good at using the
       | rooks effectively?
        
       | KingOfCoders wrote:
       | Really nice insights.
       | 
       | Though
       | 
       | "Even at lower Elo ranges like 1200-1400, you're 4-5% more likely
       | to win if you have the only bishop pair. 1200 Elo players don't
       | know how to take advantage of having the bishop pair, and yet it
       | helps them win all the same [...] A 1000-1200 Elo player is only
       | about 8% more likely to win when up two minors for a rook."
        
       | tech_ken wrote:
       | Maybe I'm just a choke artist but if I win the exchange I'm
       | probably losing the game. Opponent always seems to "turn on" and
       | I just get blown away
        
       | rauljara wrote:
       | Honestly, two rooks for a queen being a disadvantage is kind of a
       | relief.
       | 
       | I've always been taught the two rooks are better and who can
       | argue with 5 + 5 > 9? But also, I've also lost almost every game
       | where I've had the rooks. I always thought that I just needed to
       | be a better player to take advantage of it. Glad to know it
       | wasn't just me.
       | 
       | Just goes to show that these shortcuts, like the point system,
       | are only heuristics, and pretty shallow ones at that. Knowing
       | that being up a bishop gives you slightly more of an advantage
       | than a knight is better than nothing. But learning in which sorts
       | of positions a knight is actually better than a bishop will give
       | you a much deeper understanding (and correspondingly more wins).
        
         | mtlmtlmtlmtl wrote:
         | Yeah, agreed. 2 rooks vs queen is much easier to play for the
         | queen in most cases, I've found.
         | 
         | The queen can be good as a single piece whereas the rooks have
         | to coordinate, which is hard when there's a queen on the board
         | ready to fork. Often the rooks end up having to defend
         | eachother, and if this happens suboptimally they can become
         | very immobilised on a useless file or rank whereas the queen
         | can fly around the board attacking stuff at will.
         | 
         | Like you said, material imbalance is very difficult to evaluate
         | and understand. I generally recommend not to incorporate them
         | into one's play until ~1700 FIDE elo. Though sometimes you're
         | forced into it of course.
         | 
         | It's also a matter of style. I personally love materially
         | imbalanced positions, and I'm pretty good at them, so I
         | incorporate a fair bit of exchange sacrifices into my play.
         | Because then I often get positions I understand better than my
         | opponent and that makes up for the material on its own, I find.
         | 
         | But other players just prefer a different style of play and
         | maybe shouldn't go for it even if objectively it's the best
         | move, because they'll end up misplaying the position.
        
           | Enginerrrd wrote:
           | As someone low in rating, there are few things in chess more
           | satisfying to me than when the engine confirms I was correct
           | in sacrificing an exchange or a full piece for some sort of
           | positional advantage. The opportunity itself comes up very
           | rarely for me, so recognizing it just makes me giddy.
        
             | mtlmtlmtlmtl wrote:
             | Yeah, there's nothing that feels better in chess than that.
             | I still remember my first win in the norwegian
             | championships(in the noob category, not bragging, they let
             | anyone play in it) almost 20 years ago(yikes); found,
             | correctly calculated and played a rook sacrifice leading to
             | mate(it was like a 3 move combination, all checks, nothing
             | too fancy but I did consciously threaten it the move
             | before). I also remember walking into mate in 1 in another
             | game that tournament, the rest is a blur.
             | 
             | And in fact most of the wins I remember the most clearly
             | involved some kind of well-found sacrifice. The others tend
             | to blend together too much in my mind. And most of the
             | losses I remember most clearly involved horrible blunders.
        
           | superhuzza wrote:
           | edit: my comment was silly, removed.
        
             | jpmoral wrote:
             | I don't think being up a full piece counts as material
             | imbalance.
        
               | superhuzza wrote:
               | You are absolutely right, I deleted my comment as it was
               | misleading.
        
         | theonemind wrote:
         | The pieces definitely vary in value by position. Chess engines
         | that people have no chances of beating will occasionally
         | sacrifice a rook to take an enemy knight on a good outpost
         | square (past ability to get attacked by enemy pawns and
         | attacking enemy space). Hard to argue they don't know what
         | they're doing when they can beat all humans.
        
         | im3w1l wrote:
         | The author doesn't list which time control he studied (maybe a
         | mishmash of bullet, blitz and rapid?), but I'm thinking that
         | shorter time controls favor the queen.
        
       | alexmolas wrote:
       | Nice article!!
       | 
       | But this is not what you're testing. You're not determining the
       | effect of imbalance alone, but imbalance + particular position of
       | pieces. To determine the effect of imbalance you should study how
       | does not having a certain piece since the beginning affect the
       | result of the game.
       | 
       | Also, being a pawn up in the opening is much more different than
       | being a pawn up in the end game. It would be interesting to redo
       | this analysis but splitting by opening, middle game, and end
       | game.
        
         | moi2388 wrote:
         | If you take into account all variables aren't you just writing
         | a chess engine at that point?
        
         | jackmott42 wrote:
         | I am amazed that you think the author doesn't already
         | understand that.
        
           | alexmolas wrote:
           | Well, I don't know if the author understands that or not, but
           | I guess they do. I'm just pointing to the fact that the
           | analysis can be misleading, and I just shared a proposal that
           | I think it would make it more interesting :)
        
       ___________________________________________________________________
       (page generated 2023-06-13 23:02 UTC)