Why Knight and (lone) Bishop are so nearly equal in value

[Vinvin]

I forgot to write about "goal" and "sub-goal" :
In chess, the main goal is obviously to mate, but in different phase there are sub-goal which can affect the use of the pieces ...

In the early opening, the sub-goal is to put your piece on good squares
In wild position, the sub-goal is to win piece (and not lose yours) or mate
...

[Don]

Nice subject, but the brainstorming should be larger and concerns all pieces (rather than only N and B).
In my sens the value of piece is multi-dimensional, the best evidence is the king, he has a very big "value" (as he can't be lost) and he have a small "value" (as he moves slowly).

I made quickly draft list of concepts of piece values :
1) movement, action (be in few moves on a wishing square)
2) "price" (= classical values about 200,9,5,3,3,1)
3) "control" (control squares without moving)
4) fragility (wishing not to be attacked)
5) power (not clear in my head right now )
6) mutual protection, easiness to protect and to be protect (=coordination?)

examples :
a position seen some days ago here.
[d]kb4B1/p7/P6p/1K3P1P/3p3P/3P4/1p1N4/qB6 b - - 0 1
The trapped queen on a1 control squares, cannot moves (loses his power) but hold his value (=potential).

Another one to show mutual protection
[d]N3k2N/8/4q3/8/8/8/8/N2K3N b - - 0 1
The knights have a very low value for mutual protection...
(4 B or 4 N , far for each others vs 1 centralized queen) (or 7 N vs Q)

May be with growing power of computers, the new concepts will be more valuable because actually simple concepts are enough to define value of pieces ...

My best

Wasn't it show here once that a queen cannot beat 3 knights? Your constructed position has the 4 knights on the 4 worst squares of the board and the queen on the most mobile possible square with the king exposed. The only way to make the knights look worse would be to create a position where the queen has mate on the move

So really I could argue that knights have excellent mutual protection ability considering they are the least mobile piece. The common case gives them 8 squares of mobility and even less on the edge, but they can usually protect each other (when they are not on opposite corners) with reasonable placement in just a move or two. Take the opening position for example, how many reasonable moves for the rooks to protect each other? The quickest way is with ridiculous moves like a4, h4 Ra3 Rh3 but the knights can play d3, Nf3 and Nd2 and in fact from most openings positions can find a way to protect each other in 1, 2 or 3 moves (if they want to.)

The rooks can do even better, but only when the position has been "set up" in their favor. i.e. they have already cleared the back rank or there are not many pawns on the board. In highly open positions of course they can almost always protect each other in 1 move. But in general they are just awkward. When you consider all the limitations of the knights it's a surprise that they are consider almost the same as the bishops - but the bishops (in the common case) can never protect each other and of course their mobility is limited to half the board.

The bottom line is that it's hard to gauge mutual protection because it varies highly on the situation and play of the game - but in the general case, at least for the middle game I think the knight holds it's own.

[Vinvin]

Nice subject, but the brainstorming should be larger and concerns all pieces (rather than only N and B).
In my sens the value of piece is multi-dimensional, the best evidence is the king, he has a very big "value" (as he can't be lost) and he have a small "value" (as he moves slowly).

I made quickly draft list of concepts of piece values :
1) movement, action (be in few moves on a wishing square)
2) "price" (= classical values about 200,9,5,3,3,1)
3) "control" (control squares without moving)
4) fragility (wishing not to be attacked)
5) power (not clear in my head right now )
6) mutual protection, easiness to protect and to be protect (=coordination?)

examples :
a position seen some days ago here.
kb4B1/p7/P6p/1K3P1P/3p3P/3P4/1p1N4/qB6 b - - 0 1
The trapped queen on a1 control squares, cannot moves (loses his power) but hold his value (=potential).

Another one to show mutual protection
N3k2N/8/4q3/8/8/8/8/N2K3N b - - 0 1
The knights have a very low value for mutual protection...
(4 B or 4 N , far for each others vs 1 centralized queen) (or 7 N vs Q)

May be with growing power of computers, the new concepts will be more valuable because actually simple concepts are enough to define value of pieces ...

My best

Wasn't it show here once that a queen cannot beat 3 knights? Your constructed position has the 4 knights on the 4 worst squares of the board and the queen on the most mobile possible square with the king exposed. The only way to make the knights look worse would be to create a position where the queen has mate on the move

So really I could argue that knights have excellent mutual protection ability considering they are the least mobile piece. The common case gives them 8 squares of mobility and even less on the edge, but they can usually protect each other (when they are not on opposite corners) with reasonable placement in just a move or two. Take the opening position for example, how many reasonable moves for the rooks to protect each other? The quickest way is with ridiculous moves like a4, h4 Ra3 Rh3 but the knights can play d3, Nf3 and Nd2 and in fact from most openings positions can find a way to protect each other in 1, 2 or 3 moves (if they want to.)

The rooks can do even better, but only when the position has been "set up" in their favor. i.e. they have already cleared the back rank or there are not many pawns on the board. In highly open positions of course they can almost always protect each other in 1 move. But in general they are just awkward. When you consider all the limitations of the knights it's a surprise that they are consider almost the same as the bishops - but the bishops (in the common case) can never protect each other and of course their mobility is limited to half the board.

The bottom line is that it's hard to gauge mutual protection because it varies highly on the situation and play of the game - but in the general case, at least for the middle game I think the knight holds it's own.

I totally agree. With the diagram, I mean : "The knights _in this position_ have a very low value for mutual protection... ".
Certain values for pieces depends on the relative positions of others pieces.

[hgm]

Indeed. And I would not call these 'piece value', but 'positional value'. So in the given position it is not that you have inferior material. It is just that you have the material poorly positioned. (In fact this position is not tactically quiet. So it is a bit pointless to talk about any form of evaluation for it.)

I remember there is a position in KQKNNNN that is a full-point mutual zugzwang: he who has the move will lose. (Not really relevant, but an interesting trivia.)

In many end-games the outcome depends on whether the pieces are scattered, or well connected. In the latter case they can then salvage a (fortress) draw, in the former case they are hopelessly lost. The KQKNNNN position reminds me of that. It is difficult to determine the drawishness of such positions based on material only.

One interesting example is the non-royal piece that moves as King (Commoner, C). KQKC is very easily won if K and C are far apart: the Q can almost always fork them in one or two moves, and then grabs the unprotected C. But when they protect each other, they form an unbeatable fortress (if not already trapped on the edge). In this case the Commoner, hardly worth more than a Knight in normal play, outperforms the Rook as defender, as KQKR is virtually always won. The point is that, unlike the Rook, the Commoner cannot be approached by the King. So you can never attack it twice with KQ, and keeping it singly proteted ith King is always enough to keep it 100% safe.

I guess the best treatment for such end-games is to apply a multiplier like it is a fortress draw. Then, when it is not, the search will discover quickly enough that you can gobble up the Commoner, and score as KQK.

[Sven]

Another one to show mutual protection
[d]N3k2N/8/4q3/8/8/8/8/N2K3N b - - 0 1
The knights have a very low value for mutual protection...

Wasn't it show here once that a queen cannot beat 3 knights? Your constructed position has the 4 knights on the 4 worst squares of the board and the queen on the most mobile possible square with the king exposed. The only way to make the knights look worse would be to create a position where the queen has mate on the move

Moreover, black is to move so 1...Qd5+ 2.Kc1 2...Qxh1 and two other knights are hanging (similar with other king moves) so at least one more knight is lost, leaving a KNNKQ endgame after 5 plies

But the intention was, of course, to show exactly that: these knights can't defend each other.

Sven

[Don]

Another one to show mutual protection
[d]N3k2N/8/4q3/8/8/8/8/N2K3N b - - 0 1
The knights have a very low value for mutual protection...

Wasn't it show here once that a queen cannot beat 3 knights? Your constructed position has the 4 knights on the 4 worst squares of the board and the queen on the most mobile possible square with the king exposed. The only way to make the knights look worse would be to create a position where the queen has mate on the move

Moreover, black is to move so 1...Qd5+ 2.Kc1 2...Qxh1 and two other knights are hanging (similar with other king moves) so at least one more knight is lost, leaving a KNNKQ endgame after 5 plies

Actually, I count that 2 knights are immediately lost. After check on d5, the capture of the knight on a8 attacks 2 more knights simultaneously so another knight is lost on the next move.

But the intention was, of course, to show exactly that: these knights can't defend each other.

Sven

[lucasart]

Nice subject, but the brainstorming should be larger and concerns all pieces (rather than only N and B).
In my sens the value of piece is multi-dimensional, the best evidence is the king, he has a very big "value" (as he can't be lost) and he have a small "value" (as he moves slowly).

I made quickly draft list of concepts of piece values :
1) movement, action (be in few moves on a wishing square)
2) "price" (= classical values about 200,9,5,3,3,1)
3) "control" (control squares without moving)
4) fragility (wishing not to be attacked)
5) power (not clear in my head right now )
6) mutual protection, easiness to protect and to be protect (=coordination?)

examples :
a position seen some days ago here.
[d]kb4B1/p7/P6p/1K3P1P/3p3P/3P4/1p1N4/qB6 b - - 0 1
The trapped queen on a1 control squares, cannot moves (loses his power) but hold his value (=potential).

Another one to show mutual protection
[d]N3k2N/8/4q3/8/8/8/8/N2K3N b - - 0 1
The knights have a very low value for mutual protection...
(4 B or 4 N , far for each others vs 1 centralized queen) (or 7 N vs Q)

May be with growing power of computers, the new concepts will be more valuable because actually simple concepts are enough to define value of pieces ...

My best

Interesting but you have to translate that into a 1-dimensional eval at the end, so you can call it multidemensional or simply eval feeatures it's all the same thing

[Vinvin]

Nice subject, but the brainstorming should be larger and concerns all pieces (rather than only N and B).
In my sens the value of piece is multi-dimensional, the best evidence is the king, he has a very big "value" (as he can't be lost) and he have a small "value" (as he moves slowly).

I made quickly draft list of concepts of piece values :
1) movement, action (be in few moves on a wishing square)
2) "price" (= classical values about 200,9,5,3,3,1)
3) "control" (control squares without moving)
4) fragility (wishing not to be attacked)
5) power (not clear in my head right now )
6) mutual protection, easiness to protect and to be protect (=coordination?)

examples :
a position seen some days ago here.
kb4B1/p7/P6p/1K3P1P/3p3P/3P4/1p1N4/qB6 b - - 0 1
The trapped queen on a1 control squares, cannot moves (loses his power) but hold his value (=potential).

Another one to show mutual protection
N3k2N/8/4q3/8/8/8/8/N2K3N b - - 0 1
The knights have a very low value for mutual protection...
(4 B or 4 N , far for each others vs 1 centralized queen) (or 7 N vs Q)

May be with growing power of computers, the new concepts will be more valuable because actually simple concepts are enough to define value of pieces ...

My best

Interesting but you have to translate that into a 1-dimensional eval at the end, so you can call it multidemensional or simply eval feeatures it's all the same thing

The idea is to detect bad placed pieces without seeing too late by depth and have a better evaluation. But I understand well that little bean counters dominate now ...

[lucasart]

The idea is to detect bad placed pieces without seeing too late by depth and have a better evaluation.

Yes, that is what an eval (beyond simple material) is for. You have the value of a piece and a ton of corrective terms, or you decide to call the result the value of a piece and make it a function of the paremeters, it's really about how you name things, but it's the same thing.
I think what's really interesting here is HGM's idea of making the piece value depend on the material of the opponent: the more lower pieces the opponent has the less the value of our big piece. In principle it makes a lot of sense. In practice however, the entire problem is how to do it numerically and opti,izing the parameters. That is the real difficulty...

But I understand well that little bean counters dominate now ...

Well this is simply because all good chess programs use the minmax algorithm (very much refined and improved but still the logic is to maximize your score under the constraint that your opponent does the same and so on recursively).
And I've never heard of a multi-dimensional minmax algorithm. I really don't see what you could do with a multi-dimensional score. But if you have an idea of such an algorithm, please let us know, as it would be quite a breakthrough in game theory.

[smrf]

... And I've never heard of a multi-dimensional minmax algorithm. I really don't see what you could do with a multi-dimensional score. But if you have an idea of such an algorithm, please let us know, as it would be quite a breakthrough in game theory.

if you would reinterprete a good evaluation function's purpose to well PREDICT an opponents move instead of calculating the OPTIMAL move in the view of your evaluation that would have some consequences. It seems to be more useful to have a good simulation of the opponents behavior. This establishes a need to KNOW the opponent's identity or more precise its evaluation function. It then would be a new task for an engine to approximate the opponent's behavior. If this would be working to a relevant quality, it will make sense to exchange the evaluation routine (the own one, the simulated from opponent) whenever a ply changes. Then the most efficient way would be to supply two evaluations in parallel for any position, and selecting the current relevant.

The traditional negamax approach will no longer be usable without change, because level optima will probably depend on the view of the active side.