Chess math - knight vs bishop value

So we all take for granted that the value of the pieces is as follows...

P=1, N=3, B=3, R=5, Q=9

Different people over the years have attributed slightly different values to the pieces - but whilst the logic of the evaluation of the pieces makes sense, can anyone justify mathematically these values. The logic suggests that a knight can cover all squares whereas a bishop can only cover half of the squares on the board. A white squared bishop is useless against black squared pieces. A knight that is attacked may have to retreat from covering a vital square. Logically it makes sense, but not mathematically.

If you place the knight on all of the squares on the board, and add up the total number of moves it can make, you get a possible 336 moves. Averaged over 64 squares, the knight can move to 5.25 squares per move (336 / 64 = 5.25).

If you do the same with the bishop, it can move to a total of 560 squares. Averaged over 64 squares, it can move to 8.75 squares per move (560 / 64 = 8.75).

So, mathematically speaking, that would suggest that the bishop is vastly superior to the knight - more than 1 and a half times it's value.

But the bishop is limited to only half of the available squares, so that would give it an average of half the original value = 4.375. 

Yet the knight is almost always considered to be the weaker of the two pieces where their value is concerned.

So, mathematically speaking (not logically), how do we justify their values of approx 3?

I haven't halved the value twice. The second picture shows the total number of possibilities for a bishop from all of the squares on the board. This is the combined values of both bishops as it covers all squares and totals 560 possible moves, an average of 8.75 moves per square. Halving this value (to account for the fact that the bishop can only access one colour) gives 4.375 squares on average.

I'm asking for a mathematical explanation, not a logical one. Yes the knight can jump, but that's already taken into account in picture 1. The total number of squares a knight can jump to, from any square on the board, is 336 - an average of 5.25 squares per move).

So, can you justify the knight and bishops value mathematically?

Some people may not understand the true value of Pieces in chess. The order from highest to lowest

is Q=9 R=5 B=3 K=3 P=1. I think that the bishop is more superior to the knight so I think the bishop is worth 4 pawns. Does anyone agree?

I'm not asking for an opinion, or a justification based on positional benefits. I'm asking for a mathematical justification of the values that we accept for the pieces.

MyRatingis1523 - you're not looking at it correctly. The total of 560 possible moves for the bishop is for all of the squares on the board, therefore BOTH bishops. The average value is also calculated on this.

Any mathematicians out there?

Sorry, you're absolutely right - I'm just very tired and not thinking straight. It's 5am here and I've been up all night. Please accept my apologies. 

OP, You ask a question that can only be answered subjectively based on position and experience because how you subjectively weight the relative move properties of the pieces is important, but perhaps statistics is the branch of mathematics that will give an approximate answer if you analyze a large number of Knight vs Bishop GM games. Also just a few things wrong with your analysis about Knight superiority:

1) Bishops can cover more squares on the next turn in an open position if properly placed, what happens in the near term in chess is important.

2) Bishops can move a lot further, so they move faster in general

3) while true that Bishops only cover the same colored squares as they are, the opponents pawns will eventually have to move through that color several times before queening which is usually necessary in the endgame.

4) I assume you are just talking when each side has only one Knight or one Bishop...it is known that two Bishops are a lot better than two Knights.

This question can be answered objectively, by the means of statistics: https://www.chess.com/article/view/the-evaluation-of-material-imbalances-by-im-larry-kaufman

GM Larry Kaufman states that Bishop and Knight are equal, but Bishop's pair is worth additional 0.5 pawn, whereas Knight's value gets modified by something like 0.06 pawn * (number of pawns for the same player - 4). So in the opening position Bishop and Knight are equal, later on Knights gradually lose their value, whereas Bishops lose it abruptly when they get unpaired.

knights can jump over pieces, not just over other squares.

I think this not perfect mathematics. If you saying bishop can move on half of the squares than why did yo divide 560/64. You can divide half of square i.e. 280/32=8.75

But mathematically night is 5.25 and bishop is 8.75.

Now considering two advantages of night (1. it can cover all squares 2. It can jump over).

I just want to say it is not possible to compare these pieces mathematically. 

In one video Garry Kasparov said that he thinks that the bishop is worth closer to 3.25

Now do the same, but put a knight/bishop in the center, and count how many moves it takes to reach each square.

The bishop's board will have a lots of N/A

The knight's will be mostly 3s.

Here are the values I calculated.

Pawn: 1.00

Knight: 3.23

Bishop: 3.81

Knight: 5.28

Queen: 9.09

You are not taking into account that the knight can move over pieces, while the bishop gets blocked by white and black pieces that limits it’s movement. Also you wouldn’t divide by 2 because you are already accounting for the squares the bishop cannot move to. 

I started by looking at every possible position of each piece. From those positions, I calculated the number of squares the piece could move to, attack, or defend. 

To do this I had to calculate the average probability that a given square was empty, black controlled, or white controlled. 

To get the square occupancy percentages, I had to calculate the average scenario giving every combination of remaining black/white pieces equal weight and having them randomly located on the remaining 63 squares.

I also valued 1 out of every 40 pawns as a Queen to account for promotions (0.4 total promotions per game). This was admittedly a guess. 

Finally I adjusted the values so that a pawn is worth 1 by definition while keeping the ratios the same. 

Some additional findings include B through G pawns being worth 18% more than A and H pawns. Also, every other piece gradually gets more valuable the closer to the center of the board they are located. Center vs corner positions are 1.5X, 2X, 3X, 4X as valuable for Rooks, Queens, Bishops, Knights respectively.

Piece values are always math based, but chess is too complex for anyone to have developed a proof in favor of their scheme. We should ask AlphaZero. 

I think the Bishops "penalty" has to do mostly with the fact that there are a lot of squares it can NEVER go to, making it's value fluctuate wildly depending on the position.

On an empty board, a Bishop has more squares to go to than a kNight; but the kNight can reach ANY square in at most 4 moves, the B only half of them.

Ironically their specialized influence is an extra reason NOT to trade them off against anything but their counterparts without due consideration; even if it translates into a decrease in their objective value..

@14

"We should ask AlphaZero."
P = 1
N = 3.05
B = 3.33
R = 5.63
Q = 9.5

Table 6 https://arxiv.org/pdf/2009.04374.pdf

Any pcs is more valuable than u urself

White’s 13th move, exchanging bishop for knight, was guided by the understanding that the position of the board would more than likely favor knights over bishops. Later in the game, White’s knights controlled the dark squares in the center of the board. At one point, one of the knights pulled back from an outpost because it was threatened by a bishop. That exchange would have favored Black.

https://www.chess.com/game/live/95868818049