Go and chess provide clear demonstrations of opportunity cost, the first-mover advantage (esp. go), and the importance of not wasting time on trivial moves.
Go provides proof of, and some understanding of, the power of human intuition. My dad can make moves that I don’t think he knows the reasons for, that turn out to have amazing consequences 10 moves later when I discover eg. that a group of stones of mine is dead because, even though I have more liberties in that group than he has in his attacking group, he can use his liberties while I can’t use mine due to side-effects of those moves. But one is not inclined to view this as mystical intuition; it’s patterns in the stones that his unconscious learned to recognize without his conscious mind knowing why.
Small advantages escalate. In chess, at the start of the game I might focus on trying to force my opponent to take a move back, or to trade a piece I haven’t moved for a piece he has moved. Once that’s done, I take advantage of my increased deployment to try to make an otherwise-even trade that disrupts his pawn structure. Once that’s done, I try to take the isolated pawn. Once I’m a pawn ahead (I know then that I’ll probably win) I force trades to make that advantage larger. I don’t know if this works in real life.
Actually, one thing I enjoy about go is that small advantages don’t escalate, at least not nearly as much as they do in chess. In go, if you make a mistake early that puts you behind by, say, 30-40 points, the place where you made that mistake usually interacts with the rest of the board little enough that you’re not hugely disadvantaged elsewhere, and if you play better in the time and space that is left, you can catch up. But as you say about chess, I’m not sure if this is a very generalizable idea, at least when it comes to rationality.
But as you say about chess, I’m not sure if this is a very generalizable idea, at least when it comes to rationality.
For most practical situations I would suggest that it does generalise. Humans have relatively little ability to compound on success in a drastic manner. Exceptions of course include situations such as if Smily and Clippy were created at the same time on the same planet. Clippy getting the first week wrong could well leave tiling the universe with paperclips instead of molecular smiley faces is completely beyond his grasp.
Playing a move to weaken a group by taking away it’s liberties is something that doesn’t happens unconsciously.
Trying to play move that have a side effect of weakening the weakest group of the enemy is a lot of what the middle game is about.
Then the opponent is either forced to add an additional stone to the group to defend it or you can attack that group later when it becomes to weak.
Reading 10 moves into the future is also something that happens frequently in go.
I’m not talking about taking away liberties. I was thinking of a specific recent example in which I had more liberties throughout a long fight, but in the end my liberties weren’t usable because the group grew out to the edge of the board in such a way that his liberties were still playable, while mine would lead to being captured.
My dad has severe short-term memory loss, and usually can’t remember which color he’s playing. So I don’t think he’s reading that many moves ahead. Also—reading 10 moves ahead? Really? Are you talking about ladders or other sequences of moves that each have only 1 good response? If people could read 10 moves ahead in wide-open situations, why would they need to study joseki?
If he has short-term memory loss it’s of course also possible that such a move is intuitive.
When I say here liberties than I mean the amount of moves that it requires to capture a group.
In a fight it’s useful to have that number in mind.
If you have a fight between two groups than the situation isn’t really wide open like a Joseki.
Reading 10 moves ahead doesn’t mean that you see every possible variation but one variation that includes reasonable moves from both players.
You don’t want to spent time reading out the variance tree of a Joseki every time you play it from scratch. It’s also not always straight forward to know what result is better.
If it possible to start an invasion in the corner? To judge whether it’s possible to invade a corner you might well have to read 20 moves deep.
How much points is the influence really worth?
Bad Joseki moves often only make you lose a single point.
Lastly learning Josekis is a way to learn how stones flow in the beginning of the game. You need a bit of a feeling of how a game flows to be able to read far ahead.
Mastermind is considered a “solved” game, much like Tic Tac Toe, or checkers.
This considered, I was given cause for thought that even though it is “solved” it still presents the ideas of “Learned Rules”, “Intrinsic Rules” and “Trial and Error”.
For learned rules the idea is that the rules are related or taught, how one should act according to circumstance.
Intrinsic rules are those rules that are obviated, that the situation itself causes the desire of a solution.
Trial and error is the process of clarifying the rules, related to Occam’s in the idea of using the simplest rules to solve the game.
The real question is what do we do when a game situation presents us with a flip-flop such as explained in Charles Petzold’s book code? (This is a basic computing concept).
Are games representative of real life or are they viable only as a thought experiment?
Can games be more complicated than physical reality?
The real question is what do we do when a game situation presents us with a flip-flop such as explained in Charles Petzold’s book code? (This is a basic computing concept).
I don’t understand the question. By flip-flop, do you mean an electronic circuit with 2 stable states? What did you have in mind, in the game world?
The Fire requires 3 things: Air(A), Heat(H) and a Combustible(C) so that:
F == A+H+C.
We know that there are many true statements about F:
F == H+C+A
F == A+H+C
Etc.
Let’s say that these are also true:
F != A+A+A
F != B+A+A
Etc.
We also, because of trial and error, can enumerate the false statements, starting with:
F != A+H+C.
Etc.
Continuing with:
F == A+A+A
Etc.
Now this is where the flip-flop comes in:
The true and false of the basic circuit have an extraordinary amount of combinations for the purposes of making fire.
I came up with this idea not only because people learn games through both negative and positive reinforcement, but that many times we only have a partial picture of the possible combinations for a win.
This is redoubled when we think of thing in terms of arbitrary meanings such as air, heat and combustible.
Not only that people can learn as much about a game from losing it as they can from winning it, but that they need to loose in order to learn how to win. The flip-flop acts as a helper in the process of trial and error.
The feedback caused by the wiring of two NOR gates of the flip-flop allow this because the switches are controlled by the true and false sets exclusively; one switch is always associated with the true statements and the other with false.
When we start to learn, all possibilities are indeterminate, they can be either true or false; F == A+A+A is just as valid as F != A+H+C.
The flip-flop becomes sort of an ex post facto method of examining the data of the experience depending on win or loss. With a loss there can be mild sorting of possibilities, but the real sorting comes with comparing wins and losses.
Let me know if how I am representing this idea is to brief, it is still in its infancy, and as I have said elsewhere in my posts, I haven’t read everything.
Yes. In go, if a game lasts 300 moves and you win by 10 points (a pretty respectable margin), then on average each of your moves was .06 points [*] better than your opponent’s. (There are some problems with averaging things like this which I probably don’t need to point out to you all, but it still shows that you can win the game by being ever so slightly more efficient than your opponent.)
No, that doesn’t sound right at all. You make it sound like there is linear growth and that all moves are sort of the same. When I hear small advantages escalate, I imagine something more like exponential growth. Small moves, early on, compound throughout chess and can lead to bigger and bigger advantages. From what I understand of go, this is not the same. Small mistakes early on are unlikely to be crippling.
Actually, as your total advantage is growing consistently, even more slowly than linearly, you should be able to say that small advantages accumulate. And similarly, they escalate as long as the growth is accelerating. In calculus terms, small advantages accumulate while the derivative remains positive, and they (also) escalate while both first and second derivatives remain positive. So now I think that my original usage suggestion is too restrictive.
Of course, if you really have a precise mathematical idea about growth, then you could just say that! So don’t read too much into anything that I say.
I think this difference is just a misstatement. One thing pounded into me from Go was how a small difference in skill can produce a dominating effect. The handicap system shows the immense differences in ‘strength’ possible—no other game lets you give up first mover advantage AND several moves and still play on a fair level on a regular basis.
Playing Go feels to me like walking a tightrope, and I’m not even dan-level yet. I would characterize it as ‘small advantages escalate’, but the score only measures a relative difference in play quality. Thus it looks linear.
Small mistakes are unlikely to be crippling for two reasons. First, at a lower level, the other player doesn’t realize how to effectively punish it, so you can get away with your mistake. At a higher level, you don’t make blatant errors (too big of an error and you resign anyhow), so when you do make an error, you have enough skill to play flexibly and partially nullify the relative effect of your opponent’s punishing moves.
As a (poor) Go player, linear growth rather than exponential sounds right to me. In chess, every piece you take is a piece your opponent no longer has—death is permanent. In Go, if you lose a piece, you can hope to make up for it later. You’re down one piece, but it’s not like losing a bishop—it can be replaced*. In Go, poorer play doesn’t necessarily lead to a collapse of a figure and its complete capture, but more usually leads to simply a smaller figure. Big figures, equivalent in value to a queen, say, are almost always alive (either because they’re big enough to have 2 eyes in their own right or because they can connect outwards) and can’t be lost.
* I ignore pawns advancing to the last rank; the promotion rule can matter a lot in chess, but it doesn’t pervasively affect the whole game and rise inexorably out of the game mechanics.
In go, I don’t think of mistakes as costing me stones; I think of them as costing me chunks of territory. A mistake that puts you one stone behind can turn a large group of stones from alive to dead.
A strong group of stones can’t move across the board like pieces can in chess, so winning is localized in go. Winning one corner of the board doesn’t have a huge effect elsewhere on the board; losing a rook in chess has a huge effect everywhere.
If you lose a stone in go (as opposed to sacrificing it), you aren’t only losing territory but the group that captures your stone gets an eye.
That eye gives the group strength that can be used to attack elsewhere.
If you capture a stone and don’t get an additional eye you probably not gaining a small advantage through that move but are doing an even exchange.
I think you are being too general. But discussions such as this should happen about concrete positions; it’s too easy to talk past each other when speaking in the abstract.
You don’t really need concrete positions to discuss what gets considered as general go theory.
To take the relevant proverb, ponnuki is supposed to be worth 30 points. Of course you can find examples where ponnuki isn’t worth 30 points, I however wouldn’t consider those relevant enough to drop the proverb.
My objection to your original statement was the specificity about gaining eyes. Yes, a ponnuki is strong, but it’s not necessarily a guaranteed eye. There’s more to strength than eyes. That’s what I was trying to say and apparently failed miserably at.
I am 1d AGA FWIW. Just for fun, I feel like guessing your level based off this conversation. :) I’m guessing you’re probably between 5-10k, but 10% chance you’re weaker than that, 20% chance you’re 1-5k, and 10% chance you’re same level/stronger than me. What level are you?
Okay, I accept that point. However the main point I wanted to make is that a mistake usually not only leads you to lose points locally but also leads you to lose strength.
If the mistake would only lead to the local loss of points than I would speak about linear development. The fact that you however also get strength when you are making points (especially through actions such as capturing stones) suggests to me that the effect is larger than linear.
What level are you?
As written above I’m 1 kyu in Germany. At least that was my ranking when I played regularly two years ago.
I didn’t realize “escalate” implied exponential growth. I am now torn as to whether advantages scale linearly or exponentially in go. It may depend on how strong the players are. (i.e., do you actually know how to punish that?) It can easily scale exponentially if the player with the slight disadvantage tries something crazy to catch up.
I don’t think early mistakes in go are less severe in an absolute sense than mistakes in chess—but go gives you more time to recover (and more time for your opponent to screw up), so relatively speaking they might be.
9x9 go is more similar to chess in that a single mistake is most likely game ending.
EDIT: having thought about this further, I think advantage in go scales linearly. Having a small advantage does not make you more likely to gain additional advantages. Assuming correct play from opponent, etc..
I don’t think that’s an improvement. As I said in another comment just now, I think that in go having a small advantage does not make you more likely to gain additional advantages.
Then why does handicapping work? Giving someone 3 stones on star points at the start of a game will have a much larger impact than giving them 3 stones on star points at the end of the game.
I finally saw your point—moves are more valuable at the beginning of the game, mistakes come at a more or less constant rate, therefore the margin of victory shouldn’t be divided up evenly into every move of the game. Yes.
I tried to put a blanket disclaimer in my post that started this thread (“There are some problems with averaging things like this which I probably don’t need to point out to you all...”) in the interest of brevity but perhaps that was a mistake.
There are problems with my calculation that yours does not solve. Namely, mistakes do not tend to be small and come at a constant rate. If I lose by 10 points it’s entirely possible that I made a single 20 point mistake and my opponent made 10 single point mistakes. (well, for example only. In reality amateurs make a lot more mistakes than that)
That said, now that I understand why you suggested it, your calculation does represent the situation more accurately.
The escalate/accumulate/linear/exponential discussion threw me off, as did the fact that I was looking for an answer expressed in points (it’s easier to visualize what that means), and the fact that I have seen this calculation done by stronger players than I am. Obviously an answer expressed in points can’t be constant throughout the game, and I should have seen that.
Go and chess provide clear demonstrations of opportunity cost, the first-mover advantage (esp. go), and the importance of not wasting time on trivial moves.
Go provides proof of, and some understanding of, the power of human intuition. My dad can make moves that I don’t think he knows the reasons for, that turn out to have amazing consequences 10 moves later when I discover eg. that a group of stones of mine is dead because, even though I have more liberties in that group than he has in his attacking group, he can use his liberties while I can’t use mine due to side-effects of those moves. But one is not inclined to view this as mystical intuition; it’s patterns in the stones that his unconscious learned to recognize without his conscious mind knowing why.
Small advantages escalate. In chess, at the start of the game I might focus on trying to force my opponent to take a move back, or to trade a piece I haven’t moved for a piece he has moved. Once that’s done, I take advantage of my increased deployment to try to make an otherwise-even trade that disrupts his pawn structure. Once that’s done, I try to take the isolated pawn. Once I’m a pawn ahead (I know then that I’ll probably win) I force trades to make that advantage larger. I don’t know if this works in real life.
Actually, one thing I enjoy about go is that small advantages don’t escalate, at least not nearly as much as they do in chess. In go, if you make a mistake early that puts you behind by, say, 30-40 points, the place where you made that mistake usually interacts with the rest of the board little enough that you’re not hugely disadvantaged elsewhere, and if you play better in the time and space that is left, you can catch up. But as you say about chess, I’m not sure if this is a very generalizable idea, at least when it comes to rationality.
For most practical situations I would suggest that it does generalise. Humans have relatively little ability to compound on success in a drastic manner. Exceptions of course include situations such as if Smily and Clippy were created at the same time on the same planet. Clippy getting the first week wrong could well leave tiling the universe with paperclips instead of molecular smiley faces is completely beyond his grasp.
It generalizes to real-time strategy games, at least.
Playing a move to weaken a group by taking away it’s liberties is something that doesn’t happens unconsciously. Trying to play move that have a side effect of weakening the weakest group of the enemy is a lot of what the middle game is about. Then the opponent is either forced to add an additional stone to the group to defend it or you can attack that group later when it becomes to weak.
Reading 10 moves into the future is also something that happens frequently in go.
I’m not talking about taking away liberties. I was thinking of a specific recent example in which I had more liberties throughout a long fight, but in the end my liberties weren’t usable because the group grew out to the edge of the board in such a way that his liberties were still playable, while mine would lead to being captured.
My dad has severe short-term memory loss, and usually can’t remember which color he’s playing. So I don’t think he’s reading that many moves ahead. Also—reading 10 moves ahead? Really? Are you talking about ladders or other sequences of moves that each have only 1 good response? If people could read 10 moves ahead in wide-open situations, why would they need to study joseki?
If he has short-term memory loss it’s of course also possible that such a move is intuitive.
When I say here liberties than I mean the amount of moves that it requires to capture a group. In a fight it’s useful to have that number in mind. If you have a fight between two groups than the situation isn’t really wide open like a Joseki.
Reading 10 moves ahead doesn’t mean that you see every possible variation but one variation that includes reasonable moves from both players.
You don’t want to spent time reading out the variance tree of a Joseki every time you play it from scratch. It’s also not always straight forward to know what result is better. If it possible to start an invasion in the corner? To judge whether it’s possible to invade a corner you might well have to read 20 moves deep. How much points is the influence really worth? Bad Joseki moves often only make you lose a single point.
Lastly learning Josekis is a way to learn how stones flow in the beginning of the game. You need a bit of a feeling of how a game flows to be able to read far ahead.
This reminds me of the general rules of games...
I was recently playing a game of mastermind with a friend
http://en.wikipedia.org/wiki/Mastermind_(board_game)
Mastermind is considered a “solved” game, much like Tic Tac Toe, or checkers.
This considered, I was given cause for thought that even though it is “solved” it still presents the ideas of “Learned Rules”, “Intrinsic Rules” and “Trial and Error”.
For learned rules the idea is that the rules are related or taught, how one should act according to circumstance.
Intrinsic rules are those rules that are obviated, that the situation itself causes the desire of a solution.
Trial and error is the process of clarifying the rules, related to Occam’s in the idea of using the simplest rules to solve the game.
The real question is what do we do when a game situation presents us with a flip-flop such as explained in Charles Petzold’s book code? (This is a basic computing concept).
Are games representative of real life or are they viable only as a thought experiment?
Can games be more complicated than physical reality?
I don’t understand the question. By flip-flop, do you mean an electronic circuit with 2 stable states? What did you have in mind, in the game world?
Sorry for the delay.
Let’s start a Fire.
The Fire requires 3 things: Air(A), Heat(H) and a Combustible(C) so that:
F == A+H+C.
We know that there are many true statements about F:
F == H+C+A
F == A+H+C
Etc.
Let’s say that these are also true:
F != A+A+A
F != B+A+A
Etc.
We also, because of trial and error, can enumerate the false statements, starting with:
F != A+H+C.
Etc.
Continuing with:
F == A+A+A
Etc.
Now this is where the flip-flop comes in:
The true and false of the basic circuit have an extraordinary amount of combinations for the purposes of making fire.
I came up with this idea not only because people learn games through both negative and positive reinforcement, but that many times we only have a partial picture of the possible combinations for a win.
This is redoubled when we think of thing in terms of arbitrary meanings such as air, heat and combustible.
I still don’t understand what the idea is.
The idea is this:
Not only that people can learn as much about a game from losing it as they can from winning it, but that they need to loose in order to learn how to win. The flip-flop acts as a helper in the process of trial and error.
The feedback caused by the wiring of two NOR gates of the flip-flop allow this because the switches are controlled by the true and false sets exclusively; one switch is always associated with the true statements and the other with false.
When we start to learn, all possibilities are indeterminate, they can be either true or false; F == A+A+A is just as valid as F != A+H+C.
The flip-flop becomes sort of an ex post facto method of examining the data of the experience depending on win or loss. With a loss there can be mild sorting of possibilities, but the real sorting comes with comparing wins and losses.
Let me know if how I am representing this idea is to brief, it is still in its infancy, and as I have said elsewhere in my posts, I haven’t read everything.
http://en.wikipedia.org/wiki/Arthur_Samuel
Yes. In go, if a game lasts 300 moves and you win by 10 points (a pretty respectable margin), then on average each of your moves was .06 points [*] better than your opponent’s. (There are some problems with averaging things like this which I probably don’t need to point out to you all, but it still shows that you can win the game by being ever so slightly more efficient than your opponent.)
[*] 10 points / 150 moves
No, that doesn’t sound right at all. You make it sound like there is linear growth and that all moves are sort of the same. When I hear small advantages escalate, I imagine something more like exponential growth. Small moves, early on, compound throughout chess and can lead to bigger and bigger advantages. From what I understand of go, this is not the same. Small mistakes early on are unlikely to be crippling.
Suggested usage:
Exponential growth: small advantages escalate.
Linear growth: small advantages accumulate.
That makes sense to me. Upvoted.
Does it make sense to talk about chaotic growth?
Small advantages bounce around?
Actually, as your total advantage is growing consistently, even more slowly than linearly, you should be able to say that small advantages accumulate. And similarly, they escalate as long as the growth is accelerating. In calculus terms, small advantages accumulate while the derivative remains positive, and they (also) escalate while both first and second derivatives remain positive. So now I think that my original usage suggestion is too restrictive.
Of course, if you really have a precise mathematical idea about growth, then you could just say that! So don’t read too much into anything that I say.
I think this difference is just a misstatement. One thing pounded into me from Go was how a small difference in skill can produce a dominating effect. The handicap system shows the immense differences in ‘strength’ possible—no other game lets you give up first mover advantage AND several moves and still play on a fair level on a regular basis.
Playing Go feels to me like walking a tightrope, and I’m not even dan-level yet. I would characterize it as ‘small advantages escalate’, but the score only measures a relative difference in play quality. Thus it looks linear.
Small mistakes are unlikely to be crippling for two reasons. First, at a lower level, the other player doesn’t realize how to effectively punish it, so you can get away with your mistake. At a higher level, you don’t make blatant errors (too big of an error and you resign anyhow), so when you do make an error, you have enough skill to play flexibly and partially nullify the relative effect of your opponent’s punishing moves.
As a (poor) Go player, linear growth rather than exponential sounds right to me. In chess, every piece you take is a piece your opponent no longer has—death is permanent. In Go, if you lose a piece, you can hope to make up for it later. You’re down one piece, but it’s not like losing a bishop—it can be replaced*. In Go, poorer play doesn’t necessarily lead to a collapse of a figure and its complete capture, but more usually leads to simply a smaller figure. Big figures, equivalent in value to a queen, say, are almost always alive (either because they’re big enough to have 2 eyes in their own right or because they can connect outwards) and can’t be lost.
* I ignore pawns advancing to the last rank; the promotion rule can matter a lot in chess, but it doesn’t pervasively affect the whole game and rise inexorably out of the game mechanics.
In go, I don’t think of mistakes as costing me stones; I think of them as costing me chunks of territory. A mistake that puts you one stone behind can turn a large group of stones from alive to dead.
A strong group of stones can’t move across the board like pieces can in chess, so winning is localized in go. Winning one corner of the board doesn’t have a huge effect elsewhere on the board; losing a rook in chess has a huge effect everywhere.
If you lose a stone in go (as opposed to sacrificing it), you aren’t only losing territory but the group that captures your stone gets an eye. That eye gives the group strength that can be used to attack elsewhere.
captured stone != eye (not always!)
eye != additional strength (not always, anyway—only weak groups need eyes, and they only need two, a third one doesn’t make them stronger)
If you capture a stone and don’t get an additional eye you probably not gaining a small advantage through that move but are doing an even exchange.
In the end game you are right that additional strength through more eyes doesn’t really exist. In the middle game it however often does.
Beginner games are a bit different because beginners often overconcentrate their stones and then an added eye won’t do any good.
I think you are being too general. But discussions such as this should happen about concrete positions; it’s too easy to talk past each other when speaking in the abstract.
You don’t really need concrete positions to discuss what gets considered as general go theory.
To take the relevant proverb, ponnuki is supposed to be worth 30 points. Of course you can find examples where ponnuki isn’t worth 30 points, I however wouldn’t consider those relevant enough to drop the proverb.
By the way, what your Go ranking?
My objection to your original statement was the specificity about gaining eyes. Yes, a ponnuki is strong, but it’s not necessarily a guaranteed eye. There’s more to strength than eyes. That’s what I was trying to say and apparently failed miserably at.
I am 1d AGA FWIW. Just for fun, I feel like guessing your level based off this conversation. :) I’m guessing you’re probably between 5-10k, but 10% chance you’re weaker than that, 20% chance you’re 1-5k, and 10% chance you’re same level/stronger than me. What level are you?
Okay, I accept that point. However the main point I wanted to make is that a mistake usually not only leads you to lose points locally but also leads you to lose strength. If the mistake would only lead to the local loss of points than I would speak about linear development. The fact that you however also get strength when you are making points (especially through actions such as capturing stones) suggests to me that the effect is larger than linear.
As written above I’m 1 kyu in Germany. At least that was my ranking when I played regularly two years ago.
I didn’t realize “escalate” implied exponential growth. I am now torn as to whether advantages scale linearly or exponentially in go. It may depend on how strong the players are. (i.e., do you actually know how to punish that?) It can easily scale exponentially if the player with the slight disadvantage tries something crazy to catch up.
I don’t think early mistakes in go are less severe in an absolute sense than mistakes in chess—but go gives you more time to recover (and more time for your opponent to screw up), so relatively speaking they might be.
9x9 go is more similar to chess in that a single mistake is most likely game ending.
EDIT: having thought about this further, I think advantage in go scales linearly. Having a small advantage does not make you more likely to gain additional advantages. Assuming correct play from opponent, etc..
Try redoing the calculation with geometric averaging: 300 moves, 150 of which are yours, suppose the final score is 80 to 70:
x^150 = 70, x = (exp (/ (log 70) 150)) = 1.028728
y^150 = 80, y= 1.029644
y / x = 1.00089
I don’t think that’s an improvement. As I said in another comment just now, I think that in go having a small advantage does not make you more likely to gain additional advantages.
Then why does handicapping work? Giving someone 3 stones on star points at the start of a game will have a much larger impact than giving them 3 stones on star points at the end of the game.
I finally saw your point—moves are more valuable at the beginning of the game, mistakes come at a more or less constant rate, therefore the margin of victory shouldn’t be divided up evenly into every move of the game. Yes.
I tried to put a blanket disclaimer in my post that started this thread (“There are some problems with averaging things like this which I probably don’t need to point out to you all...”) in the interest of brevity but perhaps that was a mistake.
There are problems with my calculation that yours does not solve. Namely, mistakes do not tend to be small and come at a constant rate. If I lose by 10 points it’s entirely possible that I made a single 20 point mistake and my opponent made 10 single point mistakes. (well, for example only. In reality amateurs make a lot more mistakes than that)
That said, now that I understand why you suggested it, your calculation does represent the situation more accurately.
The escalate/accumulate/linear/exponential discussion threw me off, as did the fact that I was looking for an answer expressed in points (it’s easier to visualize what that means), and the fact that I have seen this calculation done by stronger players than I am. Obviously an answer expressed in points can’t be constant throughout the game, and I should have seen that.