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Topic: It's all about the E (Read 5207 times) |
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RonWeasley
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Re: It's all about the E
« Reply #30 on: Feb 22nd, 2010, 9:42am » |
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Quidditch usually seems to be all about the golden snitch if you've read Harry's books. If you play keeper or chaser, the game is much more about the quaffle. which the spectators can follow more easily. If you play on a team without talented beaters, the bludgers can be the most important thing.
Like arimaa, quidditch at a high level is all about the coordination of your efforts. And once you have reached a certain age, you realize most of the coordination is centered around the beer.
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Victorwss
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Re: It's all about the E
« Reply #31 on: Feb 24th, 2010, 5:27pm » |
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on Feb 22nd, 2010, 7:31am, Fritzlein wrote:
Victorwss, we all agree that the elephant is very important in Arimaa and that it gets moved often. We don't all agree that that is a problem. Can you explain a little bit more why you think it is a problem? Is it just too easy for you to find the right move? Do you win every time that you move your elephant a lot? Or is the "deficiency" simply a matter of taste, about which there is nothing to say?
Consider this argument: Although Chopin's Raindrop Prelude has many notes besides A-flat, it is the most frequently played note and it is being played most of the time. Some notes don't even get played at all! So, in my opinion, this is a deficiency in Chopin's Raindrop Prelude. Maybe we can experiment with a variation where we only play half of the A-flats that Chopin put in.
To me my argument seems to jump from true facts straight to an unrelated conclusion without anything in the middle to connect the logic.
I'm glad you like it! My guess is that learning more and more about Arimaa makes the game seem less and less boring. This is not true for every game; some games seem interesting at first and only become boring once you know how to play them well. But Arimaa seems fairly boring on the surface and then gets more and more interesting as you understand its depth. That is my opinion, of course, but I think it is the experience of other players as well. We have seldom had a player quit Arimaa because they thought there was "nothing left" to it. |
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Comparing arimaa to Chopin's Raindrop Prelude is non-sense. They are things completely different in nature. Is something like to comparing an orange with a cell phone.
What is one of deficiency in chess? Lets say, the opening follows some few memorizable patterns that can be compiled in an opening book. Why is this a problem? Instead of learn how to play with talent, people and computer just recognize the patterns and use the pre-built solutions.
Other chess deficiency: Due to the rule that white always start, white normally gets a significant advantage over black and is more likely to win than black.
In arimaa the elephant moves frequently. This can be considered a deficiency because suggests to people and computers to think about the elephant first. So, a computer using some sort of alpha-beta search could consider the elephant moves first to get a better pruning in the search tree. This leads to a bit of mechanical playing in opposition of intelligent life playing.
The additional observed pattern that cats and dogs rarely moves because they have few to do in the game leads people and computer to further machanical thinking. A computer may consider dogs and cats moves last, because they are more likely to be pruned in the search. Humans frequently don't care too much in thinking about cats and dogs because they know that will use they in the game just a few.
Arimaa is a very good and intelligent game, but this does not mean that it could be even better with just minor changes in the rules. If the game was not so centered around the elephant and if cats and dogs had more to do, I bet that arimaa would be much better and computers much weaker.
However these deficiencies (or whatever you prefer to call them) are small enough to not lead to significant problems that we see in chess, checkers or tic-tac-toe. However small or not, they are still there.
Another thing I was thinking about is the initial positing of the pieces. Although the distribution may be very random, we see that people and computers follows some patterns that significantly reduces the number of likely-to-see opening positions and so there are just a few that are likely to occur with just some minor variations. For example, if we see both elephants in the same column, gold may start moving his elephant four steps forward blocking silver's elephant. We very frequently see the elephant and the camel in the central rows, the horses in the sides and 6 to 8 rabbits in the first row, specially for gold which does not know how silver will plot his pieces. In the other hand, Silver normally has few non-stupid options to choose how to plot their pieces in response to the way that gold plotted. This sort of commons opening patterns may lead to the development of an arimaa opening book someday.
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| « Last Edit: Feb 24th, 2010, 5:41pm by Victorwss » |
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Fritzlein
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Re: It's all about the E
« Reply #32 on: Feb 24th, 2010, 6:25pm » |
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on Feb 24th, 2010, 5:27pm, Victorwss wrote:| In arimaa the elephant moves frequently. This can be considered a deficiency because suggests to people and computers to think about the elephant first. So, a computer using some sort of alpha-beta search could consider the elephant moves first to get a better pruning in the search tree. This leads to a bit of mechanical playing in opposition of intelligent life playing. |
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Thanks for the further explanation. The power of the elephant might lead players to automatic moves and therefore might take the thinking and creativity out of the game. I can see the concern. I would agree that this was a real problem if there was an absolute rule, i.e. if players always knew that moving the elephant was a good idea.
However, although I can see how theoretically a very powerful piece could make the play of the game mechanical, I don't see that it actually does affect Arimaa that way. For example, look at move 30g in the ongoing postal game TheMob vs. Fritzlein. If the Mob had mechanically thought that they must do something with their elephant, they would have missed the fact that their elephant was already on an excellent square at e5, even though it wasn't directly attacking or defending any trap. Moving the gold elephant in any direction on 30g would have put it on a weaker square. So you can't just think that because the elephant is strong you must always be moving it around. It isn't that easy.
You are quite right that humans and computer alike should pay great attention to the elephant. But fortunately, that doesn't tell us what to do with the elephant. Should I move it here or there or just leave it still? There is no simple rule for it. Therefore, there is still room for truly intelligent play.
Quote:| The additional observed pattern that cats and dogs rarely moves because they have few to do in the game leads people and computer to further machanical thinking. A computer may consider dogs and cats moves last, because they are more likely to be pruned in the search. Humans frequently don't care too much in thinking about cats and dogs because they know that will use they in the game just a few. |
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Certainly cats and dogs are less important than the elephant, but it is not true that they can be ignored, or that it doesn't matter whether it is a cat or a dog on a given square. For example, look at this game that I just played against chessandgo in the World Championship. I purposely set up a dog on the c7 square, not a cat or a rabbit. Chessandgo commented after the game how important this decision was. Look at the position before Gold's move 7g. If I had a cat defending the c6-trap, the Gold dog from c4 could just push it away and I would be completely losing the game. But since I have a dog there, the Gold camel has to come from the opposite side of the board to make a threat, which means I am winning.
Once again, if it did not matter at all what the cats and dogs do, I would have to agree with you that Arimaa is less interesting and requires less thoughtful play. But, just like with the elephant, there is no simple rule to let you avoid thinking what to do with cats and dogs.
Quote:| Arimaa is a very good and intelligent game, but this does not mean that it could be even better with just minor changes in the rules. |
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I agree that we should not assume that Arimaa is perfect just because it is good.
Quote:| If the game was not so centered around the elephant and if cats and dogs had more to do, I bet that arimaa would be much better and computers much weaker. |
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I quite disagree that the changes you propose would make Arimaa better and make computers weaker relative to humans. The fact that an elephant can't be pushed or pulled in any circumstance is a fact that benefits human strategy more than it benefits computer calculation. In order to win, computers need sharp changes to the position, and they need humans to say, "Oh, I didn't see that!" The fact that there are ways for humans to understand Arimaa strategically is what allows them to play Arimaa better than computers.
For example, the backbone of Arimaa strategy is the camel hostage. If a horse and camel together could push an elephant, the camel hostage position would not be stable. When there are no stable positional features, it is hard for a human to see further ahead than a computer can see ahead.
However, even though I think the changes you propose would not make Arimaa a better game or harder for computers, I have to admit that I can't really tell without playing a lot of games. The rules themselves don't make it obvious what strategies will be "in there" at a high level of play. Your proposal would destroy the camel hostage strategy, but maybe it would create a new strategy that is even deeper. It is impossible to tell from the rules alone.
So when I disagree with your rule changes, it is half in ignorance. I know that the Arimaa rules work, but I don't know whether your rules work or not. I don't think there is any problem caused by the powerful Arimaa elephant, but that doesn't mean I think Arimaa is perfect, and I admit the possibility that your suggested rules could have marvelous benefits that I simply can't see without trying them.
Quote:| For example, if we see both elephants in the same column, gold may start moving his elephant four steps forward blocking silver's elephant. This sort of commons opening patterns may lead to the development of an arimaa opening book someday. |
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Yes, standard Arimaa openings may happen some day, and I am curious to see whether they will or not. They haven't happened yet, though. For example, the opening setup I just used against chessandgo in the game I linked above was one I had never used before. Probably the position after chessandgo set up his pieces and I set up mine was a position that had never occurred between any two players ever in 135,490 previous games on the Arimaa server. Such a thing could not happen in chess where after each player makes a move only 400 positions are even possible. So if you don't like memorized openings from chess, I believe that Arimaa will provide you some relief.
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omar
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Re: It's all about the E
« Reply #33 on: Feb 25th, 2010, 9:30pm » |
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Victorwss, your variations do sound interesting and I never thought to try those while play testing Arimaa. The closest I came to reducing the power of the elephant was allowing the rabbits to push/pull/freeze the elephant. That version didn't seem to work. See my posting of Jan 15, 2010:
http://arimaa.com/arimaa/forum/cgi/YaBB.cgi?board=talk;action=display;nu m=1263419748
Replacing one rabbit with a mouse has been suggested before and even played tested by IdahoEv and ocmiente. However, I would not have even considered that when I was play testing Arimaa since it would not have fit one of my original goals for Arimaa being that it should be playable with a standard chess set.
You are most likely correct that within the space of games possible with the Arimaa mechanics and using only a standard chess set the variant I picked is probably not the optimal in terms of being the most difficult for computers. But when you have multiple criteria it is difficult to optimize all of them and there will have to be some trade-offs. In addition to being playable with a standard chess set and being difficult for computers I was also trying to make it easy to learn and fun and interesting to play. The easy to learn criteria always made me not want to add any more rules than what was absolutely necessary. And of course the fun and interesting criteria is very subjective. Although one somewhat objective aspect of it is that a game should not take too long to finish. I strongly suspect that variants which are more difficult for computers would also take longer to finish. So the particular version of Arimaa we play does an amazingly good job of trying to achieve these four goals in a well balanced way.
When I released Arimaa I think I was more confident of how well it met my goals than I should have been. Only later did I come to realize how lucky I got. There was no way my play testing could tell me if the game was deeply flawed. I could only look for obvious flaws. Only after the game has been played many, many times at very strong levels of play can we begin to say that it is not deeply flawed. It could have turned out that a camel hostage gave the hostaging player such an advantage that they always win and if say the first player could alway take a camel hostage then the game would be seriously flawed. It is only after intense playing that we come to know about the deeper characteristics of the game.
I've come to think of the rules of a game to be analogous to the genes of a specimen. You can't tell by looking at the genes what the final organism is going to be like. Those genes have to be expressed in the slow and meticulous genotype to phenotype mapping process. Only then will you know it's characteristics and whether or not it is a mutant and can't survive. Likewise it is hard to tell the nature of the game and whether it is flawed or not by looking at the rules. We have to play out the rules, create a collection of games and analyse them to begin expressing it. Only then can we begin to see if it might be deeply flawed. I guess in this analogy that makes us players the ribosomes 
With the current Arimaa rules we are beginning to build some confidence that it is not deeply flawed after having played thousands of games. However even slight changes to the rules can completely alter the game and we have to start over. So even being the creator of the game I don't think I can easily alter the game anymore. The genie is out of the bottle and the only thing that can stop this variant of Arimaa from continuing to build it's dominance is if we find that it is flawed.
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Victorwss
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Re: It's all about the E
« Reply #34 on: Feb 26th, 2010, 1:07am » |
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Thanks for the further explanation. The power of the elephant might lead players to automatic moves and therefore might take the thinking and creativity out of the game. I can see the concern. I would agree that this was a real problem if there was an absolute rule, i.e. if players always knew that moving the elephant was a good idea.
However, although I can see how theoretically a very powerful piece could make the play of the game mechanical, I don't see that it actually does affect Arimaa that way. For example, look at move 30g in the ongoing postal game TheMob vs. Fritzlein. If the Mob had mechanically thought that they must do something with their elephant, they would have missed the fact that their elephant was already on an excellent square at e5, even though it wasn't directly attacking or defending any trap. Moving the gold elephant in any direction on 30g would have put it on a weaker square. So you can't just think that because the elephant is strong you must always be moving it around. It isn't that easy.
You are quite right that humans and computer alike should pay great attention to the elephant. But fortunately, that doesn't tell us what to do with the elephant. Should I move it here or there or just leave it still? There is no simple rule for it. Therefore, there is still room for truly intelligent play. |
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Well, I said that it made the thinking a bit more mechanical. Players (both humans and computers) first consider moving the elephant because it is likely that it should be moved, but this does not means that they always should move the elephant nor to where. So, yes there is still a very large room for intelligent play.
Quote:| Certainly cats and dogs are less important than the elephant, but it is not true that they can be ignored, or that it doesn't matter whether it is a cat or a dog on a given square |
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I did not said that they can be ignored. If you just ignore they you will probably lose miserably. I said that they have few to do. I did not said that they have nothing to do. Normally, cats and dogs are very shy in the game and they function tends to be to obstruct the way of enemies pieces, protect traps and unfreeze allies pieces. Frequently their moves are just to keep they alive and protected from stronger enemies pieces or to support stronger allies pieces. It is very unlikely to see they hunting or directly attacking and it is very common to see they staying quiet in their places for numerous turns.
Quote:| Yes, standard Arimaa openings may happen some day, and I am curious to see whether they will or not. They haven't happened yet, though. For example, the opening setup I just used against chessandgo in the game I linked above was one I had never used before. Probably the position after chessandgo set up his pieces and I set up mine was a position that had never occurred between any two players ever in 135,490 previous games on the Arimaa server. Such a thing could not happen in chess where after each player makes a move only 400 positions are even possible. So if you don't like memorized openings from chess, I believe that Arimaa will provide you some relief. |
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Yes, an opening book in arimaa whould be much more larger and shallow than a chess opening book. But it is still possible to have some standardized well-known frequently used openings. The start positions does not need to be identical, just similar. Get some frequently used start position, switch a dog with a cat or a cat with a rabbit and probably few things would be different in the opening. It is true that that diffence may be anywhere between decisive and irrelevant in the middle game, but in the opening it is much more likely irrelevant, and so, the same opening tatics would be applicable.
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Replacing one rabbit with a mouse has been suggested before and even played tested by IdahoEv and ocmiente. However, I would not have even considered that when I was play testing Arimaa since it would not have fit one of my original goals for Arimaa being that it should be playable with a standard chess set. |
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Their tests sounds very interessing. There seems to be some stone-scissors-paper pattern which casts the game strategies very complex. And I think that this is something harder for computers, because this jams and twists the calculation of the relative value of the pieces, except when the elephant or the mouse dies.
Quote:| With the current Arimaa rules we are beginning to build some confidence that it is not deeply flawed after having played thousands of games. However even slight changes to the rules can completely alter the game and we have to start over. So even being the creator of the game I don't think I can easily alter the game anymore. The genie is out of the bottle and the only thing that can stop this variant of Arimaa from continuing to build it's dominance is if we find that it is flawed. |
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I don't think that Arimaa is flawed and I don't see any evidence of this. It is much more solid than chess and checkers. Chess has a lot of defficiencies and has some evidences that it is possibly flawed, however it is still a great game. Arimaa has much fewer defficiencies than chess, and so is a very good game.
These variants I suggested earlier where thought when I was posting just to give an example of how minor changes to the rule could change the game to something better or worse. I did not tested that rules nor thought about them deeply, was just the three firsts variations that went out of my mind. However, if you think that would be interessing to test them, it would be nice.
Arimaa is great the way it is, does not needs to be fixed. However, Arimaa's defficiencies must be studied and that was the objective of my posts: to point out and analyze some of such defficiencies. I am not asking for changes in the rules!
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| « Last Edit: Feb 26th, 2010, 1:25am by Victorwss » |
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novacat
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Re: It's all about the E
« Reply #35 on: Feb 26th, 2010, 5:56am » |
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on Feb 26th, 2010, 1:07am, Victorwss wrote:
Normally, cats and dogs are very shy in the game and they function tends to be to obstruct the way of enemies pieces, protect traps and unfreeze allies pieces. Frequently their moves are just to keep they alive and protected from stronger enemies pieces or to support stronger allies pieces. It is very unlikely to see they hunting or directly attacking and it is very common to see they staying quiet in their places for numerous turns. |
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All pieces in Arimaa obstruct the way of enemy pieces (includes framing), protect traps (including taking control of enemy traps), unfreeze allied pieces, and attack weaker enemy pieces (except rabbits). All pieces except the elephant support stronger allied pieces.
Dogs and cats often support by defending the home traps when other allied pieces are attacking, thus they often do not move much but have an important function. They can be used as bait as well to offer a kill while gaining momentum in a race elsewhere on the board. They can also attack, but most people don't use the dogs or cats extensively for this because it takes understanding and foresight that the people haven't developed. If you look at http://arimaa.com/arimaa/games/jsShowGame.cgi?gid=135050&s=w you will see chessandgo uses his dogs very effectively.
There are more uses for dogs and cats, but the main thing I see is that people don't use them for active positions because they are easier to use for jobs that require little movement. If cats and dogs were replaced with horses, most games would likely have a few horses that rarely moved.
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Fritzlein
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Re: It's all about the E
« Reply #36 on: Feb 26th, 2010, 7:29am » |
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J. Mark Thompson has an excellent essay about what qualities an abstract strategy game should have. He points out that there is a kind of contradiction between the need for depth and the need for clarity.
Quote:Depth means that human beings are capable of playing at many different levels of expertise. For most board positions, until the last stages of the endgame, the puzzle of finding the best move should not be completely solvable.
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But in addition to depth a good game must have clarity. Clarity means that an ordinary human being, without devoting his career to it, can form a judgment about what is the best move in a given situation. [...] In a game that lacks clarity, the player simply has no instincts.
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if a usable algorithm is known which will always reveal the best move in any situation of a game, then the game's clarity is perfect, but it has lost all its depth. |
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To enjoy a game, we need it to be deep. We need to have levels of thought. There needs to be a ladder to climb. If we had Elo ratings and everyone was about the same rating, outside observers would wonder what kind of game doesn't have any experts.
Also we need clarity. If you can't write a strategy guide with rules of thumb like "rabbits are more valuable in the endgame", then the game feels inhumane. People would get frustrated at the lack of guiding principles, and think you have to be an alien to understand how to play.
But there is a tension between depth and clarity. Whenever something can be clearly expressed about a game, one should fear that it will make the depth of the game dry up. For example, when I wrote that a key strategy of Arimaa was to overload the enemy elephant, i.e. tie the enemy elephant down to defense of one threat and then create a second threat elsewhere, IdahoEv said that sounded like it made Arimaa "too easy".
Superficially, the criticism is valid. The more I can say about Arimaa strategy, the more it gives the appearance that we are near to figuring it out, or the appearance that Arimaa strategy can be put into an algorithm. But in practice it turns out that as long as the strategies are still fuzzy enough, the clarity doesn't take away from the depth.
In terms of abstract game design, it would be a shame to think poorly of a game as soon as a strategy guide is published. If reading any principle (e.g. "use strong pieces to make a frame, then rotate them out for weak pieces") spurs the thought that the game is too simple and therefore the rules need to be changed to make the strategy unreliable, then we could slip into strategy prevention mode. Look, there is a player trying to execute Strategy X in every game, and it is helping him win. Quick, we must changes the rules to make Strategy X no longer effective! The ultimate goal of a game designer: stop a strategy guide from being written. 
But in reality we don't need to prevent a game from having clarity as long as the depth is still there. The key thing to observe is whether everyone has to start playing Strategy X in order to compete, and if so, whether it makes the resulting games boring when everyone has caught on. If the play is still varied, unsettled, and intense after everyone knows about Strategy X, i.e. if there are ways to counter it or variety within it, then all is well.
To bring it back to the forum topic, we all know the elephant is vastly powerful, and we're not bored yet! Arimaa seems to be one of the few games that can have a lot of clarity and a lot of depth at the same time.
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JoeHead
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Re: It's all about the E
« Reply #37 on: Feb 26th, 2010, 8:35am » |
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Quick look at amount of elephant steps
JoeHead
Introduction
There is a live discussion wheather moving elephant too often is bad for arimaa as a game. How much does the elephant move? The claimes were made without any statistical methodology. So I performed simple yet very effective statistical analysis. Now everybody will have starting point.
Method
World Championship 2010 games were used as a starting population. There were 48 games played up to this moment. From these 5 games were chosen randomly using excel function "randbetween( 1, 48 )". Number of plies of each chosen game was calculated. Final ply is: rabbit reaching the goal, resignation, time. Number of steps was calculated. Piece removal from the board was not recognised as a step. One ply can consist of 1 to 4 steps. Number of elephant steps during the game was counted. Then the ratio "Elephant Steps/Total steps in game". Next average of ratios in 5 games was calculated and finally 95% confidence intervals about the average using t-distribution with 4 degrees of freedom.
Results
Average number of plies in a sample: 66,2
95% confidence interval for number plies: 38,1 - 94,2
Average ratio of E steps: 0,249
95% confidence interval for ratio of E steps: 0,18 - 0,31
Discussion
Wide spread is caused by small sample size. Nevertheless it is still best possible estimate for true value to be captured 95% of times.
The game length is little lower then experience tells us because one randomly chosen game lasted only 27 plies.
From the data we can see that almost every fourth step is made by elephant. Also one should realise that majority of steps is done during opening. The shortest game in a sample had ratio 0,34. Other games had ratio around 0,22. As the game gets longer non-elephant pieces start their significant contribution.
These are data. How much enjoyment comes from the arimaa playing is subjective. This is answer to how much elephant actualy moves in the game.
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lightvector
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Re: It's all about the E
« Reply #38 on: Feb 27th, 2010, 1:53am » |
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on Feb 24th, 2010, 5:27pm, Victorwss wrote:
In arimaa the elephant moves frequently. This can be considered a deficiency because suggests to people and computers to think about the elephant first. So, a computer using some sort of alpha-beta search could consider the elephant moves first to get a better pruning in the search tree. This leads to a bit of mechanical playing in opposition of intelligent life playing.
The additional observed pattern that cats and dogs rarely moves because they have few to do in the game leads people and computer to further machanical thinking. A computer may consider dogs and cats moves last, because they are more likely to be pruned in the search. Humans frequently don't care too much in thinking about cats and dogs because they know that will use they in the game just a few.
Arimaa is a very good and intelligent game, but this does not mean that it could be even better with just minor changes in the rules. If the game was not so centered around the elephant and if cats and dogs had more to do, I bet that arimaa would be much better and computers much weaker.
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In regards to the computer vs. human strength, actually, my experience in coding for Arimaa and working on my bot seems to indicate that the opposite is true. One of the major advantages for Arimaa strategy, in regards to human strength against bots, is exactly the domination of the elephant. This allows for more stable positions, such as hostages, frames, elephant deadlocks, etc. Getting a bot to play with more than most primitive understanding of these features is a enormous obstacle, precisely because their value and their implications depend very precisely on complex and subtle variations in the position, and yet their stability makes them persist for potentially many moves, many orders of magnitude beyond resolving by standard search methods. The fact that the elephant can't be displaced by weaker pieces seems to be one of the key reasons that such long-term positional features can occur.
Anyways, I also don't believe that human vs. computer strength for a game necessarily has anything to do with how interesting the game is for people to play. But in the particular case of Arimaa, I actually do feel the same way here as well. It's extremely hard to tell how a game would play with alternate rules, as Fritzlein mentioned. But at least my own intuition says that if the elephant could be too easily threatened or overwhelmed by weaker pieces, and if weaker pieces could play more significant roles than they already do in attacking, then sure, more possibilities would now be allowed, but the human-understandable depth may not increase, or could even decrease, because one might have fewer stable positional patterns, and more tactical positions where strategic insight is less important than exhaustively reading out all the combinatorial possibilities.
I would agree with many others that have posted here that the quality of a game like Arimaa should not be judged just on a statistic showing that moving one piece is best several times more often than another. It should be on judged on whether a great deal of dynamics and possibilities for deep strategy remain and develop from such an observation, or whether such an insight threatens to make the game uninteresting or stagnant. For Arimaa, nothing could be further from the case right now, so in my opinion, there's no urgent need to fix anything.
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| « Last Edit: Feb 27th, 2010, 1:57am by lightvector » |
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jdb
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Re: It's all about the E
« Reply #39 on: Feb 27th, 2010, 7:03am » |
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on Feb 27th, 2010, 1:53am, lightvector wrote:
In regards to the computer vs. human strength, actually, my experience in coding for Arimaa and working on my bot seems to indicate that the opposite is true. One of the major advantages for Arimaa strategy, in regards to human strength against bots, is exactly the domination of the elephant. This allows for more stable positions, such as hostages, frames, elephant deadlocks, etc. Getting a bot to play with more than most primitive understanding of these features is a enormous obstacle, precisely because their value and their implications depend very precisely on complex and subtle variations in the position, and yet their stability makes them persist for potentially many moves, many orders of magnitude beyond resolving by standard search methods. The fact that the elephant can't be displaced by weaker pieces seems to be one of the key reasons that such long-term positional features can occur.
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I agree with this 100%. Well said.
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