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Subject: Done to Depth rss

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Russ Williams
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nhjelmberg wrote:
There goes my idea of a Go variant into the trash but thanks for the link. Back to idea of a Chess variant then.

A Chess variant where you only place pieces and capture, but don't move pieces.
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Nicholas Hjelmberg
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russ wrote:
nhjelmberg wrote:
There goes my idea of a Go variant into the trash but thanks for the link. Back to idea of a Chess variant then.

A Chess variant where you only place pieces and capture, but don't move pieces.


Bomber Chess?

No, it's an idea where you start with basic pieces which you may compound during the game. But it's still just an idea.
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Nathan James
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My definition of "the game" was that the tree is a perfect representation of the game. Depth then is a facet of the subjective experience of the game.

I was/am not clear whether the other definition(s) offered were seen as needed corrections or just meandering comments.

It seems necessary to distinguish between the objective aspects of the game, i.e., those that define the tree, and the experiential aspects. Especially because, in my view, the experiential aspect is where conceptual depth resides.
 
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NJames wrote:
My definition of "the game" was that the tree is a perfect representation of the game. Depth then is a facet of the subjective experience of the game.

I was/am not clear whether the other definition(s) offered were seen as needed corrections or just meandering comments.

It seems necessary to distinguish between the objective aspects of the game, i.e., those that define the tree, and the experiential aspects. Especially because, in my view, the experiential aspect is where conceptual depth resides.

Thanks for clearing that up. A tree as 'perfect representation' can still be interpreted differently. A deterministic program searches 'the tree' using alpha-beta pruning. But it has limited depth. At the bottom of its search it evaluates positions using man-made heuristics. But within its analytical depth its evaluation concerns actual positions that are 'a perfect representation' of a small part of the tree.

To have the total tree, purely hypothetical, would allow you to backtrack from the leaves and identify every possible move in every possible position. The truth of every position: win, loss or draw.

But with this information the game in a way ceases to exist. People could still play it of course, but if the truth of every position is known, what's the point.

So my question appears to be (I'm not sure where I'm going) which use you think to make of the tree. Is it just a means to distinguish between analytical depth and conceptual depth?
 
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christianF wrote:
To have the total tree, purely hypothetical, would allow you to backtrack from the leaves and identify every possible move in every possible position. The truth of every position: win, loss or draw.

But with this information the game in a way ceases to exist. People could still play it of course, but if the truth of every position is known, what's the point.


This (often) hypothetical complete game tree is what I'm calling "what a game is, as opposed to what it may seem to be." The complete game tree contains all that is objectively true about the game, without reference to how it is experienced.

christianF wrote:

So my question appears to be (I'm not sure where I'm going) which use you think to make of the tree. Is it just a means to distinguish between analytical depth and conceptual depth?

Well, it isn't so much a means to anything as it is a necessary category distinction. My intention is to point out that "strategic depth" is experiential rather than objective. (Note that this does not mean that depth is not real.) Realizing this is crucial to learning how to design for it.

If you're looking for a mathematical quality of the game tree that would equate to "depth," you're looking objective measure of something exists only subjectively. Doomed to failure.

Conversely, if we recognize that what we are after is a quality that can only be identified by a mind, we can begin to better describe that quality, which will permit better, more intentional design.

After Corey's post (which set off this conversation about depth), Russ asked:
russ wrote:
CoreyClark wrote:
But in Othello the best move doesn't often directly confront the previous move, whereas in Hex the opportunity to tenuki is a rarity. At any rate, I find this definition for depth to be most useful. If a good move often doesn't directly answer the preceding move/s, then you have a mostly strategic game on your hands, if a good move is always a vital response to the preceding move, then your game is mostly tactical.

I like this way of trying to pin down the "tactical vs strategic" distinction, but I wonder how "objective" it is.

I.e. is it possible to objectively define "tenuki" (e.g. does it have any recognizable meaning from a combinatorial game theory point of view?), or is "responding elsewhere vs responding locally" only a subjective human heuristic for grappling with game complexity more easily?

(This is sincere uncertainty; I honestly don't know.)


So we do need to clarify whether the depth we're looking for is in fact objective or subjective. I'm putting my money down on subjective, and attempting to explain why.

To further illustrate the need, I'll mention this
ChristianF wrote:
But sub-goals with some permanency are one of the things that allow Chess players to plan deeply.

Because of the vagueness of the word "deep," I'm not entirely sure whether you meant analytical or conceptual depth, and I wonder if you knew when you wrote it! The ease of describing both as "depth" leads easily to equivocation, which can hinder not only discussion but even contemplation.

To return to the why, note the agreement between these:
NJames wrote:
The depth we care about is the experience of exploring a game through a series of evolving concepts or heuristics. Conceptual depth is entirely a matter of playing games in the absence of a game tree that permits our direct inspection. It is an artifact of how a finite mind interacts with an overwhelmingly large game tree.

ChristianF wrote:
But with [the complete game tree available for inspection] the game in a way ceases to exist. People could still play it of course, but if the truth of every position is known, what's the point.

Yes, what's the point, and furthermore, there is no room for strategy or conceptual depth to function. Instead you will be using "brute force" analysis to definitively value every move as win, lose or draw.
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Joe Joyce
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NJames wrote:
It seems we need to define "the game," although, I think I defined my own use of it a few posts back.

milomilo122 wrote:
To rephrase, the game is the engagement of human perceptual faculties, with the tree, as the tree is represented visually

I suppose this means that no two people have ever played the same game, since their faculties certainly do not engage the, uh, game in the same way.

Does the game not exist unless someone is currently playing it? May we not usefully speak of playing the same game on an iPhone as we play on a table?

I agree that the representation is important, but I'm not sure about the need for the redefinition.

joejoyce wrote:
The game tree is not the game. The game is the particular path through the game tree you and your opponent take.

This is what I would normally refer to as "a game of Chess" as opposed to "the game of Chess."

I wonder if there is a technical term for a single instance of playing the game. We could invent one if we need it badly enough.

Semantics again... To me, 'the game of Chess' is the area of game conceptual space that encompasses *all* the chesslike games past, present, future, and possible, all their complete game trees woven into a complex ecology of potential chess forms, with a tiny bit of the tree known through our 5000 - 10000 published variants. I see what you mean as the game as 'merely' the rules, with the implied associated game tree. I think it's correct to say a game only exists when it is being played. And the play involves the combined experiences of both players in the game.

But is any of this relevant to the situation at hand? I'm B.C. - before computers, but isn't it the case that for any code, the patterns that emerge(d) from the original situation will by necessity/math/logic would create patterns in the code? Cryptographers crack codes by knowing that there is information there, knowing what (language) it translates to, and finding and analyzing the patterns? Each 'message' here is a single game. No matter how you encode it, a pattern in the games must be reflected by a pattern in the coded message. So presentation comes down only to whether or not the presentation language and topic are known to the audience or not.

NJames wrote:
My definition of "the game" was that the tree is a perfect representation of the game. Depth then is a facet of the subjective experience of the game.

I was/am not clear whether the other definition(s) offered were seen as needed corrections or just meandering comments.

It seems necessary to distinguish between the objective aspects of the game, i.e., those that define the tree, and the experiential aspects. Especially because, in my view, the experiential aspect is where conceptual depth resides.

Grin, the way I see it, your first paragraph contradicts your 3rd. The game tree is the rules, expanded to show every eventuality. It's not at all a perfect representation of the game. The tree is static. The game is a moving viewpoint within the tree. No agreement whatsoever. And then you write exactly what I think and agree with, namely your paragraph 3! The game is the experience, and the depth of the game is seen within the minds of the players (and maybe some audience members.)
 
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NJames wrote:
I note that there is always the possibility of discovering an insight that breaks the game. (from World Chess Championship thread)

Why? Why do you believe this? Does it hold true for all abstracts? If not, what separates the two types?

I know some (many/most?) games are breakable, but I do not believe all games, or all abstract games, are breakable in the ruined for play sense of the word. I think I could safely point to games by a number of people involved in our current conversations which are unbreakable abstracts. I know it's a bit of a sidetrack, but we're famous for that here, and new as I am, I have contributed mightily! So tell me what I'm not understanding, please.

 
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(I am speaking only of abstract games)

For any non-trivial game there will be strategies and tricks we have already thought of and others we have not. We couldn't possibly know the effect of the undreamed of strategies. One of them may break (weakly solve) the game.

In order to know whether the human mind is capable of breaking a game, you must either (1)break the game or (2)know all the capabilities of the human mind. I'm confident that it is not possible to know all the capabilities of the human mind. Therefore it is impossible to prove that a game is unbreakable.

We could be more or less confident that a game is likely to be unbreakable, but certainty is impossible.
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nhjelmberg wrote:
An interesting and important angle. Would Go become a deeper game if you added rules for moving stones? Its decision tree would expand but would anyone equal that to depth? Permanency (or is inertia a better term?) should definitely be included in a discussion about depth.

Is the game deeper if you can act in 2 areas of the board rather than one? Not just influence 2 areas with one action, but 2 separate actions each turn? That might allow you to influence 4 areas. It is certainly wider; it requires not only pattern recognition but also modified pattern recognition. Yes, some turns you don't move, but in (1,1) GOmove, players did move a stone about 75% of the time. So for 100 stones dropped, there were 175 actions. How does this *not* translate into a noticeable effect on the game. Right now freact and I are playing (4,4) GOmove on the chessvariants site, and there are some 'finished' games there, and one broken one. I put (6,6) on a 31x31 board, and that is way too small for hextuple GOmove. The 31x31 board is even a little small for the current game of (4,4) playing on it.
 
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NJames wrote:
(I am speaking only of abstract games)

For any non-trivial game there will be strategies and tricks we have already thought of and others we have not. We couldn't possibly know the effect of the undreamed of strategies. One of them may break (weakly solve) the game.

In order to know whether the human mind is capable of breaking a game, you must either (1)break the game or (2)know all the capabilities of the human mind. I'm confident that it is not possible to know all the capabilities of the human mind. Therefore it is impossible to prove that a game is unbreakable.

We could be more or less confident that a game is likely to be unbreakable, but certainty is impossible.

Okay, I see where you are coming from. I would argue there is a third option, to know all the capabilities of the game. This might involve knowing the entire game tree in detail so you could determine *all* the legal paths through it, and see if there is a breakdown somewhere.

I gotta ask about the 'weakly solved' part. Macysburg is a purely combinatorial wargame using chess pieces and activators to simulate an early gunpowder era combat. I played a game with a military history buff and occasional wargame designer, as well as a wargame historian of sorts. He beat me using an oblique approach attack, a successful tactic of of centuries gone by. Does his knowledge of military tactics mean Macysburg is weakly solved and thus broken? ;)

 
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Weakly solved would mean that you not only know which side will win before the game begins, but how to do it. For Chess it would mean showing a forced sequence leading to checkmate (or more likely to a draw) from the starting position.

I think the Wikipedia page on solved games is helpful. https://en.wikipedia.org/wiki/Solved_game

It doesn't sound like Macysburg is broken, no!
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NJames wrote:

To further illustrate the need, I'll mention this
ChristianF wrote:
But sub-goals with some permanency are one of the things that allow Chess players to plan deeply.

Because of the vagueness of the word "deep," I'm not entirely sure whether you meant analytical or conceptual depth, and I wonder if you knew when you wrote it! The ease of describing both as "depth" leads easily to equivocation, which can hinder not only discussion but even contemplation.

Be assured I realise the difference and that I meant conceptual depth. Actually I'm not all that interested in analytical depth. Grandmasters use both of course, but the conceptual part is the more intetersting. Of course their ability to literally read out tactical sequences is important too, but these are always encountered within the framework of conceptual strategies. I presume you mean to say that if a game offers no means to frame long term strategies (has no or little conceptual depth) its 'depth' is questionable, however deep, wide or dense its tree. Right?
 
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Right. Every abstract game has analytical depth equal to the size of it's tree, that is not what we're interested in. (Although I'm sure that players enjoy calculating, to some extent.)

What does this tell us about "clarity" as a desirable trait? At the least we can specify whether we're aiming for a clarity that supports conceptual manipulation directly or a clarity that aids calculation.

It also brings to mind the use of randomized setups / boards in online abstracts, like Catchup, for example. These techniques are supplying analytical variety, and only indirectly increasing conceptual depth, if at all.
 
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NJames wrote:
Right. Every abstract game has analytical depth equal to the size of it's tree, that is not what we're interested in. (Although I'm sure that players enjoy calculating, to some extent.)

What does this tell us about "clarity" as a desirable trait? At the least we can specify whether we're aiming for a clarity that supports conceptual manipulation directly or a clarity that aids calculation.

It also brings to mind the use of randomized setups / boards in online abstracts, like Ketchup, for example. These techniques are supplying analytical variety, and only indirectly increasing conceptual depth, if at all.

Catchup starts on an empty board as do loads of games. Maybe Chess960 is a better example. Analytical variety for sure but whether it 'increases' conceptual depth is questionable. Part of Chess is the theory regarding conceptual depth, that is, the concepts specific to Chess with its current rules. You can't play say a Sicilian Defence in Chess960, nor add anything to its theory.

To reiterate a good metaphor, Chess is a book, Chess960 is a collection of tabloids. And here's the thing: a book eventually has an ending. Tabloids go on forever.
 
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christianF wrote:
NJames wrote:
Right. Every abstract game has analytical depth equal to the size of it's tree, that is not what we're interested in. (Although I'm sure that players enjoy calculating, to some extent.)

What does this tell us about "clarity" as a desirable trait? At the least we can specify whether we're aiming for a clarity that supports conceptual manipulation directly or a clarity that aids calculation.

It also brings to mind the use of randomized setups / boards in online abstracts, like Ketchup, for example. These techniques are supplying analytical variety, and only indirectly increasing conceptual depth, if at all.

Catchup starts on an empty board as do loads of games. Maybe Chess960 is a better example. Analytical variety for sure but whether it 'increases' conceptual depth is questionable. Part of Chess is the theory regarding conceptual depth, that is, the concepts specific to Chess with its current rules. You can't play say a Sicilian Defence in Chess960, nor add anything to its theory.

To reiterate a good metaphor, Chess is a book, Chess960 is a collection of tabloids. And here's the thing: a book eventually has an ending. Tabloids go on forever.


Couldn't you argue that Chess960 has 960 Sicilian Defences depending on setup? (I don't, I think Chess960 is more like 960 different chess variants, but I play the devil's advocate here.)
 
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christianF wrote:
To reiterate a good metaphor, Chess is a book, Chess960 is a collection of tabloids. And here's the thing: a book eventually has an ending. Tabloids go on forever.

Hmm, that strange tabloid seems to not go on forever, but to have a limited print run of 960 issues... one of which is the famous book Chess!
 
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russ wrote:
christianF wrote:
To reiterate a good metaphor, Chess is a book, Chess960 is a collection of tabloids. And here's the thing: a book eventually has an ending. Tabloids go on forever.

Hmm, that strange tabloid seems to not go on forever, but to have a limited print run of 960 issues... one of which is the famous book Chess!

Shades of "The Library of Babel". Borges is so good (and, incidentally, a chess fan, I believe).

My view is that Chess was an unsuitable object for fischerisation. The differentiation of piece-powers is likely to make for a certain number of the permutations being in some sense of the term less 'sensible' than Chess itself. (Though this doesn't seem to hold to anything like the same extent for Arimaa - possibly not at all - and I can't for the moment put my finger on why.)

Speaking as a firm believer in variable setups as the way to a new future so long as the game is right (perfect in Morelli, for example, though Richard Malaschitz's principle of diametrical opposition is a crucial element), I'll beg leave to disagree with Nathan's view that

Quote:
These techniques are supplying analytical variety, and only indirectly increasing conceptual depth, if at all.
.

The thing is, because there are then far too many lines ever to be learnable by rote, you're thrown back on your understanding of principles. So every encounter has the potential to be like those exciting (but increasingly rare) Chess games where the players find themselves on uncharted ground.




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mocko wrote:
Shades of "The Library of Babel". Borges is so good (and, incidentally, a chess fan, I believe).

Ha, indeed! One of the relatively few short stories which I've read more than just a couple times in my life, at various ages. Hmm, perhaps the time is approaching to read it again!
 
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mocko wrote:


The thing is, because there are then far too many lines ever to be learnable by rote, you're thrown back on your understanding of principles. So every encounter has the potential to be like those exciting (but increasingly rare) Chess games where the players find themselves on uncharted ground.

Mmm.

I put to you this question: is the uncharted ground you refer to analytically uncharted, or conceptually uncharted?

Does it force the players to use new concepts, or does it permit them to use old concepts that were made unnecessary by thorough analysis?
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nhjelmberg wrote:
christianF wrote:
NJames wrote:
Right. Every abstract game has analytical depth equal to the size of it's tree, that is not what we're interested in. (Although I'm sure that players enjoy calculating, to some extent.)

What does this tell us about "clarity" as a desirable trait? At the least we can specify whether we're aiming for a clarity that supports conceptual manipulation directly or a clarity that aids calculation.

It also brings to mind the use of randomized setups / boards in online abstracts, like Ketchup, for example. These techniques are supplying analytical variety, and only indirectly increasing conceptual depth, if at all.

Catchup starts on an empty board as do loads of games. Maybe Chess960 is a better example. Analytical variety for sure but whether it 'increases' conceptual depth is questionable. Part of Chess is the theory regarding conceptual depth, that is, the concepts specific to Chess with its current rules. You can't play say a Sicilian Defence in Chess960, nor add anything to its theory.

To reiterate a good metaphor, Chess is a book, Chess960 is a collection of tabloids. And here's the thing: a book eventually has an ending. Tabloids go on forever.


Couldn't you argue that Chess960 has 960 Sicilian Defences depending on setup? (I don't, I think Chess960 is more like 960 different chess variants, but I play the devil's advocate here.)

If you argue that there may be very interesting set-ups I agree. But their interest is embedded in emerging opening concepts and their interest lies in further exploration. If there's little further exploration, the interest can only lie in the particular tactics of a particular line.
Chess960 may be useful for practice purposes, keeping you sharp so to say, but it seems far less suitable for deep exploration of the kind that makes Chess so fascinating.
 
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russ wrote:
christianF wrote:
To reiterate a good metaphor, Chess is a book, Chess960 is a collection of tabloids. And here's the thing: a book eventually has an ending. Tabloids go on forever.

Hmm, that strange tabloid seems to not go on forever, but to have a limited print run of 960 issues... one of which is the famous book Chess!

Maybe we read the metaphor differently. I'm not in the Library of Babel but in the 'real world' together with humans that may eventually become all too human or superhuman, but not in my lifetime.
So yes, in the library all 960 trees are there. Among them is one we hold dear because we know so much about it. We'd like to explore it ad infinitum but, alas, it's finite so we have to content ourselves with a, humanly speaking, real long time.
And we've come a real long way. That's quite different from playing random selections of 960 where you're climbing up a trunk and branch and go to the next tree. You can do that forever. We can't forever play Chess the way it is played now. But at least we can, eventually, close a book instead of opening a new tabloid every day, holding the same 'Trunk & Branch' news every day.
 
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christianF wrote:
russ wrote:
christianF wrote:
To reiterate a good metaphor, Chess is a book, Chess960 is a collection of tabloids. And here's the thing: a book eventually has an ending. Tabloids go on forever.

Hmm, that strange tabloid seems to not go on forever, but to have a limited print run of 960 issues... one of which is the famous book Chess!

Maybe we read the metaphor differently. I'm not in the Library of Babel but in the 'real world' together with humans that may eventually become all too human or superhuman, but not in my lifetime.
So yes, in the library all 960 trees are there. Among them is one we hold dear because we know so much about it. We'd like to explore it ad infinitum but, alas, it's finite so we have to content ourselves with a, humanly speaking, real long time.
And we've come a real long way. That's quite different from playing random selections of 960 where you're climbing up a trunk and branch and go to the next tree. You can do that forever. We can't forever play Chess the way it is played now. But at least we can, eventually, close a book instead of opening a new tabloid every day, holding the same 'Trunk & Branch' news every day.

Have to just check here. (The problem with trying to engage with metaphorical exchanges.)

In the UK, tabloid newspapers are just sheets of smaller area - to those who read them, that is.

To everyone else they're crap newspapers.

In what sense are we using the word "tabloid"?
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PS: I should clarify that although the Guardian not long ago went tabloid in format, obviously it's still really a broadsheet.



("A million people can't be right.")

 
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christianF wrote:
russ wrote:
christianF wrote:
To reiterate a good metaphor, Chess is a book, Chess960 is a collection of tabloids. And here's the thing: a book eventually has an ending. Tabloids go on forever.

Hmm, that strange tabloid seems to not go on forever, but to have a limited print run of 960 issues... one of which is the famous book Chess!

Maybe we read the metaphor differently. I'm not in the Library of Babel but in the 'real world' together with humans that may eventually become all too human or superhuman, but not in my lifetime.
So yes, in the library all 960 trees are there. Among them is one we hold dear because we know so much about it. We'd like to explore it ad infinitum but, alas, it's finite so we have to content ourselves with a, humanly speaking, real long time.
And we've come a real long way. That's quite different from playing random selections of 960 where you're climbing up a trunk and branch and go to the next tree. You can do that forever. We can't forever play Chess the way it is played now. But at least we can, eventually, close a book instead of opening a new tabloid every day, holding the same 'Trunk & Branch' news every day.

I was mostly just making a joke since the metaphor seemed to not quite work for me, but to be serious: I saw & see your point, and agree to some extent, but on the other hand, it seems to assign too much importance to knowledge of the opening lines, as if exploring the opening is the whole essence of the game, and that no real interesting improvement is possible if one plays Chess960.

But surely part of the important intellectual challenge & skill (as well as enjoyment) of both Chess or Chess960 is the general strategy and tactics throughout the game (midgame and endgame too, not just opening), and in that skill one can study and improve and explore progressively further, not just in some light unsatisfying shallow "tabloid-esque" way, even if one starts from a different setup everytime in Chess960. Right?

(Just as e.g. one can progress in Go even without studying opening joseki.)
 
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russ wrote:
I was mostly just making a joke since the metaphor seemed to not quite work for me, but to be serious: I saw & see your point, and agree to some extent, but on the other hand, it seems to assign too much importance to knowledge of the opening lines, as if exploring the opening is the whole essence of the game, and that no real interesting improvement is possible if one plays Chess960.

I think there's a difference here between Chess as a game and Chess as a sport. Everyone can learn and become better and the particularly gifted and determined players can make it to grandmaster level. I'm not sure how many there are, but it must be an awful number. They're all supporting and expanding the Book of Chess. Some have been so important that they have openings named after them.
A measure of the Book of Chess's expansion are the world championships and in particular the matches. These show the current state of its expansion to a larger and larger audience, over many platforms, minute by minute. Not the whole of it of course, but flashes of it. Meanwhile the theory around those games expands too.


The golden pathways
Hundreds may at any time expand the Book in some significant way, thousands write it and millions support it. It has become an autonomous process that keeps grinding on and digging for new lines to explore. But what exactly are these efforts aimed at? They try to estabish the golden pathways, the small ridges between the abyss to the one side and the abyss to the other side.
Of course stepping of can be very interesting because suddenly you are in a winning/losing line. It is possibly advantageous for the one who discovers the line, particularly if it is in otherwise well charted territory. After its disclosure it becomes part of the ever growing Book.

To avoid misunderstandig, much of this process may be confirmed by the tree, and certainly human/AI generated analytical lines are clear and can be depended on, but the final evaluation is still human: "this line leads to an advantageous position because of this and that and such and such".
We're not at the end of course. We've mined well into the middle game, and certain lines have progressed even beyond that. That's due to the historical and 'accidental' size of the game. Who could have imagined 500 years ago that the Book of Chess would ever be finished? I'm not talking about the whole world enjoying a great game, but about the world championship. There's a deep irony embedded in these efforts. Theory, and not only opening theory because we're beyond that, has evolved so far that the games as a rule, once they come into uncharted territory, will offer little chances of winning. The beginning of uncharted territory lies by definition at the end of a golden patway and thus in a balanced position. You can see it happen. Another irony is that a dramatically grown audience has on average less understanding. A win is a win and a draw is a draw and how interesting a draw was doesn't matter much to the majority of the audience. So we get more exposure and more draws, let's see how that works out.

Chess960 of course doesn't have a similar issue to cope with, and it's available and convenient if one wants to sharpen one's wits in basically the same game but in a different setting. I have nothing against it except its architecture.

russ wrote:
But surely part of the important intellectual challenge & skill (as well as enjoyment) of both Chess or Chess960 is the general strategy and tactics throughout the game (midgame and endgame too, not just opening), and in that skill one can study and improve and explore progressively further, not just in some light unsatisfying shallow "tabloid-esque" way, even if one starts from a different setup everytime in Chess960. Right?

(Just as e.g. one can progress in Go even without studying opening joseki.)

I agree but I hope you've noticed I'm talking about the Book of Chess as opposed to the game.


P.S. I aplogise if this seems a bit off topic, but my actual angle is why Chess offers such a degree of conceptual depth, especially since a totally different game like Go offers it to a similar degree. Is there a general characteristic of games with this feature?
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