A Mathematical Dilemma

Date:- 29.02.2012

Time:- 1.00 a.m. IST (GMT+05.30)

Status:- Sleepless, Hungry, Thoughtful

Result:- Formation of a mathematical dilemma

So I don't know why this just came up an hour past midnight. I'll get this straight. You are given a certain situation. 

Situation:- A square is occupied by a Black Rook (henceforth called R1), being attacked by a White Queen and a White Bishop. Another Black Rook (henceforth called R2) supports the first Rook. It is White to play, the Bishop captures the Rook. The other Rook captures the Bishop and the Queen finally captures the last Rook. 

Analysis 1:- When R1 is captured by B, Black suffers a loss of 5 piece points. After R2 takes B, the net effect is 5-3=2 points loss. And finally, Q takes R2 for a net effect of 5-3+5=7 points loss. Since, a loss of 5 is better than a loss of 7, Black should have never taken the Bishop. 
This was an additive-subtractive type of calculation. 

Analysis 2:- When B captures R1, the total loss ratio of Black to White is 5:0. If taken as a fraction, it turns out that White is infinitely times better than Black. Or in other words, Black's loss is infinite times more than White's loss. When R2 captures B, the ratio becomes 5:3. And finally, after R2 is also captured, the ratio of losses is 10:3. If this is taken in a fractional form, Black's loss is only 1.67 (approx.) times that of White. Since 1.67 times loss is better than infinite times loss, it is better that Black captures the Bishop. 
This was a multiplicative-divisive method of calculation. 

 Either of the analyses was right in its own way. Both bequeath their justifications in their very own unique manner. Yet the first is the most preferred and recommended to use by players. But again, isn't the second one too, a point to be noted? 

What?

What application does this have to practical chess?

All you need to know in chess terms is that the bishop is en prise and can be safely captured, unless there is some other reason in the position why capturing it would be bad.

EDIT: Sorry I meant the rook is en prise

What "What?" ?

Sorry, but I don't see any advantage in applying a multiplicative scale over an additive one in deciding who comes out better materially. There aren't any situations where your Analysis 2 would give a different answer (in terms of better/same/worse) to Analysis 1, and Analysis 1 has the advantage of giving an answer that is in the same units (pawns=1) as the pieces. Or am I missing something??

@OP - To answer your question with a rhetorical question: is a 1:0 exchange better than a 30:1 exchange as suggested by your analysis?  How about a 2:1 versus a 30:16? (the 2:1 favored in your analysis)

Run away from your ratio method as fast as you can.

Just because you assign numbers to something doesn't mean there's any intrinsic meaning to your analysis. This is a fallacy of those who understand how to manipulate arithmetical symbols but fail to understand if and when their mathematical model has any ties to the larger problem they are solving. In that sense, your second model is just fine in and of itself, except that you won't win any chess games if you use it to make decisions. I also have a model where pawns are worth more than rooks, it's just that I lose a lot when I use it :-)

Ultimately what matters is the position on the board. Remember, checkmate ends all debate. 

Because the "right" answer is specific to the actual game in question and not an abstract rendition of it, there isn't a correct answer because it is an incorrect question. 

Now, if you add a qualifier of "assuming no additional positional considerations, initiative, or other forms of comp at all, which is the better situations," then the answer is that you want to be materially down the lowest point count differential possible. 

But that general answer is in practical terms meaningless as there is always the position to consider.  

"Look at the pieces left on the board, not the ones taken off"

Analysis #3 -- You are gazing at your navel.

Consider reading Morris Kline, Mathematics: The Loss of Certainty (1982).

And buy some decent chess books, to get yourself started on this crazy game.

Points don't matter, play only for mate. Numerical numbers are meaningless without a perspective reference point.

[Ref: Chaos Theory]

The fallacy is that you can't divide by zero. This is not allowed by the laws of mathematics. It doesn't give the result of "infinite;" it gives you the result of "meaningless." If you allow the expression 1/0 in algebra, it is easy to prove that 1=2. So your "ratio" of 5:0 yields no fraction and is null and void.

The queen was never captured paul. It is 7 points net.

Is this a hoax or do you not know how to read nor write, leaves me to wonder why no one caught on this?

I guess a BA in math isn't that great.  Paul should have coughed up a little more dough and gotten himself a BS.

And for those worried about dividing by zero, remember, for large values of ONE, One approaches Two, ..that is... for small values of TWO.  Extrapolate as needed. 

Interesting, don't know why so many decided to dump on this.

As said though, take the ratio of the pieces remaining. 6:16   or  1:13 (I give the black king a value of 1 due to heavy pieces on an open board and the white king a value of 4 due to safe king on an open board).

To those that said run away from the ratios... this is how I've always considered piece trades... I actually never thought that by adding and subtracting I could get the same result lol :)

 This thread is its own worst dilemma, mathematics notwithstanding.

Wow, I think you missed something. The queen is not taken; the rook is. The rook is worth 5, and 5-3+5=7 Net Points. You should get your mathematics straight.

Also, as a side-note. The point value of pieces is given as an additive value, not an absolute one. This means that they must be revalued if we wish for a ratio-based understanding.

"The Queen is never captured", FYI. (BA in Mathematics, OK, I see). Thanks for pointing out a hoax mistake. I top my class in Math every year. 

How's that 1=2 thing work?

@StrategicPlay: What type of math are you studying right now?

The problem here is the you are taking the material loses of the players in the ratio. You should take their absolute material instead.

Here's what you're saying:

Position 1 : 0:0

Position 2 : 5:0 = infinite

Position 3 : 5:3

Position 4 : 10:3

Heres what you should say:

Position 1 : (Bishop+Queen):(Rook+Rook) = 12:10

Position 2 : (Bishop+Queen):(Rook) = 12:5

Position 3 : Queen:Rook = 9:5

Position 4 : Queen:null = 9:0 = infinite

Hence black shouldnt recapture with the rook as it loses his advantange from 5/12 to 0. This also matched with additive/substractive evaluations.