Please note: I am NOT suggesting that you plug in numbers into this equation to win more games. I am suggesting that you familiarize yourself with this equation because you can intuit it in a way that allows your decisions to be much more rationally informed. This is a mathematical description of winning. Try to understand it intuitively.
So, I was doing a bit of thinking the other day, and I was thinking about how to actually analyze the meta in a more fundamental way. Specifically, I was thinking about how I am always coming up with a new deck idea that tries to address certain fundamental problems, or tries to use different subversive techniques to win more (since I rather like playing around). Then I thought "Well, what deck would a superhuman artificial intelligence play?" I realized that the superhuman A.I. would have figured out a fundamental way of looking at the game that would allow it to win more. Well, of course you would be maximizing your win rate. But how do you conceptualize the output of your winrate in a way that is actually relevant?
I am kind of a fan of math, and there's a rather simple way to sum up (pun fully intended) how to win, based on the meta. In fact, you literally use a summation:
R = ∑ wipi
R is your win rate, wi is the chance that you'll win against a specific deck and pi is the probability that you will encounter that deck. So, here's an example of what that looks like:
Your deck wins against mechmage 50% of the time and encounters it 50% of the time. Your deck encounters fast druid the other 50% of the time, and wins against that 50% of the time. So, intuitively you know that you have a 50% winrate (which is why I picked these numbers, to prove the point).
This is how that general equation works. So, whenver someone asks you "what is the strongest deck?" The answer is the deck that maximizes the output of this equation.
So, what you want to do is do an analysis of what the meta is, and then you want to optimize the output in such a way that you maximize your winrate. Likewise, "the best possible deck" is always changing with the meta, since all of these terms change.
So, you don't have to do an in-depth analysis of this equation for every card, but understanding and conceptualizing this equation (by occasionally plugging numbers in), and roughly guessing at each of its components will help you construct the best deck for the current meta. For instance, maybe mechmage is popular and handlock is as well. But, perhaps you can't beat both of them strongly, but you realize that mechmage is more popular and you can deal with handlock well enough by teching in a few cards.
So, you have to deal with each deck on a case by case basis.
Example:
So, the meta is archetype 1, archetype 2 and archetype 3. So, you run into archetype 1 25% of the time, archetype 2 35% of the time and archetype 3 40% of the time.
Your winrate vs. archetype 1 is 60%. Your winrate vs. archetype 2 is currently 55%. Your winrate vs. archetype 3 is currently 70%. So, you tech out a card and it lowers your winrate against archetype 1 by 10%, raises your winrate against archetype 2 by 10% and lowers your winrate vs. archetype 3 by 5%. Is the change worth it? (If you try to do this before looking at the solution, it'll help you learn this just that much better).
So, you can actually simply calculate the change in overall winrate as the sum of the change in winrates against each deck:
∑∆wipi= ∆R
But, these calculations are only based on estimations. That's where this formula is not a magic bullet, but it's a much more powerful, concise way to conceptualize the game.
In general, yes, your win rate is SUM[P(win|encounter)], but the fact that meta is constantly changing and different ranks have different probability of encounters makes the real analysis an impossible task. Also, sometimes the best way to increase your win rate is to add a tech card and not switch the deck.
In general, yes, your win rate is SUM[P(win|encounter)], but the fact that meta is constantly changing and different ranks have different probability of encounters makes the real analysis an impossible task. Also, sometimes the best way to increase your win rate is to add a tech card and not switch the deck.
Well, the point (which I keep reiterating everywhere I post this) is not that you plug in numbers and do some analysis before you play every day.
The point is a broader point about intuiting how to play this game. Especially pay attention to the ∑∆aibi = ∆R. That is, the change in winrate is the same as the sum of the change in winrates times the prevalence.
That makes those decisions intuitive without plugging in any numbers. (I think I take for granted the fact that equations give intuition because I'm majoring in physics... lol).
While I always support deconstructing any game into math, I'm not convinced what you've posted here is particularly insightful or useful ... the latter because there simply isn't sufficient information to utilize this to select the ideal deck.
While I always support deconstructing any game into math, I'm not convinced what you've posted here is particularly insightful or useful ... the latter because there simply isn't sufficient information to utilize this to select the ideal deck.
As I've been saying, it helps you deconstruct the game conceptually. That is practical.
That's why I say to conceptualize the equation: you can say a lot about how something works by describing it mathematically. Plugging in numbers isn't the point. It just improves your understanding of the game, and can help you ask more relevant questions when building a deck, just by understanding it intuitively.
Beyond that, even if it was absolutely 0% practical, I just find there to be an elegance to understanding the game in this way.
you can not only bump your ai's by swaping out cards, but also by improve your play which will increase your winrate as well... probably the more important thing i think
Well, it's all important. And you can recognize these things through understanding the equation intuitively.
But, these calculations are only based on estimations. That's where this formula is not a magic bullet, but it's a really powerful way to conceptualize the game.
This is probably biggest issue with that formula, of course it works, but you need good data for it to be precise enough. You can estimate winrate against certain decks if you have some win/loss tracker (But you still have to pay attention of distribution of games like 5 mech mages, 1 freeze mage, 2 tempo mages.) Also you need large sample of games against that deck, to have a good enough estimate of your winrate, as you can't say I have 100% wr against secret mages if you only met this deck once. Also you will likely have different chance to win against same deck, that has some different choices, than other deck of the same archetype. For example when you are less likely to win against mech mages who play mad scientist+mirror entity, than if he don't run these cards.
And I don't even have to talk about meta changing and your possibly biased estimations of meta, like if you get against 5 mech mages in a row and you may start to think they are really popular, just to fight 10 other decks after.
Since you suggested not plugging numbers into that equations, rather familiarize with this equation in general, I agree with that and these were just some examples, why trying to directly plug in numbers is not a good idea.
This familiarizing with this equation can mean a lot from a simple decision of adding kezan mysthic if you start facing lot of hunters to completely changing your deck, if you feels it perform better against decks you face the most on ladder. I hope this helped a little and I didn't make any big mistakes or misunderstandings of your text here.
I feel bad for the OP, he explained how it should be used as a concept and yet everyone is crunching numbers. You can never determine exact winrates as decks evolve and change every day, so trying to actually apply the formula is nonsense. The concept is that you understand that your deck should have favorable winrates against prevailing decks in the meta in order to climb. The question you should ask yourself would be: is my deck good enough against the variety of decks I am facing at the moment?
This just about sums it up.
I go a little bit further in implementing the conceptual basis as a means for determining exactly which cards to swap out in your deck. "Will this swap improve/decrease my overall winrate vs. the field?" is the question you want to be asking. Also, "how should I play against deck X, how do my win conditions change?"
That's why it is THE equation, not just an equation. It's the description.
your deck should have favorable winrates against prevailing decks in the meta in order to climb.
Profound... :/ Anyone who didn't understand that already probably won't find mathematical analysis helpful.
While I'm as much in favour of math as the next guy, the equation thing seems a bit superfluous. "Here's a bunch of math, now ignore the math and use gut instinct to figure out the blatantly obvious." Not trying to be critical, just don't really see the point here.
your deck should have favorable winrates against prevailing decks in the meta in order to climb.
Profound... :/ Anyone who didn't understand that already probably won't find mathematical analysis helpful.
While I'm as much in favour of math as the next guy, the equation thing seems a bit superfluous. "Here's a bunch of math, now ignore the math and use gut instinct to figure out the blatantly obvious." Not trying to be critical, just don't really see the point here.
If you think it seems superfluous, I don't think you're totally appreciating it, or, at least, don't fully get it intuitively. I mean, a key difference here is that I used my conceptual understanding of math to write the equation, and you're approaching it like most do, with the plug'n'chug sort of thinking, as opposed to the robust intuitive understanding approach. That's the only way I can see anyone seeing this as superfluous.
This equation helps inform you on what adding or subtracting a card from a deck will do to your winrate, since you can see it as a combination of deck popularity and your winrate vs. a specific deck. So, you can ask yourself when you add a card "Well, I see facehunter about 30% of my games, mechmage 25% of my games, demonlock 25 % of my games, control warrior 10% of my games (with "other" as the remaining 10% that you don't worry about since on average it changes less, so you just consider that ~∆0%). I think that adding this card would add 2% vs. mechmage and facehunter, 3% vs. demonlock and maybe -4% vs. control warrior." For this, a is approximately the same as b and c, and they're all larger than d. So, then, you note that mechmage, fachunter and demonlock are a lot more prevalent than control warrior, and you then realize that you're making a card change on much more sound reasoning than "oh, just counter the meta." Well, is it a good change? The answer is +1.45% overall difference. This 1-2% could play out in a way such that you get a couple additional winstreaks over this minor card adjustment.
See, it makes deck construction multidimensional in an intuitive way. See, this deck could be fast druid and the card could be antique healbot, and those numbers wouldn't necessarily be a bad approximation.
Beyond that, how many people do you see, even at high ranks, playing very greedy decks? See, this games a number as your winrate, so your greed becomes "winrate" and "the meta," not just "I want to play a powerful deck and I'll put in these tech cards for the meta." See how uninformed that actually is?
See, I can't even imagine summing that type of decision-making up as "just counter the meta." "Counter the meta" is a murky, vague statement in comparison to an equation.
Scroll to the bottom of that where you see class percentages by rank. Now how much more useful is the equation? You could calculate your expected winrate and test it against your actual winrate, and analyze your previous deck decisions based on this by using game tracking software.
This is actually very analogous to the definition of an asset in finance: a sequence of future cash flows. The present value of an asset is the sum of those future cash flows at different points in time (divided by the interest rate of each term at each term's time in the future). See? You can't actually do finance intelligently without that, but you might see it as intuitive, since you know future dollars are worth less than dollars now because of interest. And, getting the right numbers in valuation is hairy business -- people make mistakes all the time. It's a lot like Hearthstone: the equation is absolutely true and it is actually pretty intuitive. But, you can't make as informed of decisions without it.
You can't even do that approximate conceptual analysis as well without the equation there. It's a lot like saying "well, if you drop a ball from a plane, it falls faster and faster until air resistance makes the acceleration slow down to zero." Of course you know this, but you don't know by how much without differential equations (which still give an approximation! But it's still a useful approximation!). And, if you track your games, you get a decent range of popularity for a deck. And, though you can't get the popularity of the day, you can get the average popularity over a week or month.
differential equations (which still give an approximation! But it's still a useful approximation!).
Well, thats not true. Differential equations are not approximations. You just just dont have the exact equation for a problem and/or not all needed initial conditions. Edit: I think i missread your sentence, my bad.
And i wouldnt be so focused around your formula. You will most likely just do the exact same thing others do while building a deck for the meta. Switch cards, watch your winrate drop or increase and then adjust again. But since the meta has always an uncertainty its more or less guessing which way the meta went, isnt it?
Edit2: But at least its a nice orientation for new players, ill give you that one.
"Physics is the fine art of approximation." - My first physics professor. That generally applies to any mathematical description of something with outcomes in reality. For instance, you wouldn't analyze rolling a dice in terms of its inertial tensor. You're just going to give each side 1/6, and some number might have a minute probability of being higher or lower. See? You have to choose the correct level of mathematics to describe something. You can actually be way too precise.
You don't use general relativity to engineer rockets -- you still use Newtonian physics with post-Newtonian mathematical methods. See, Newton isn't "wrong" -- he's just an approximation of a more robust description of reality, a robust description that likes to act a lot like Newton's laws at low gravities and low velocities.
There's nothing exact in physics, and there's rarely anything exact in applied mathematics. Hell, a major part of math is the art of understanding just how "wrong" you are.
This equation works -- it's just not very precise. And, as a "nice orientation for new players?" Yes, I kind of intended it as that. However, experienced players still get emotional in the form of greed, building decks with too many tech cards or too much late-game. And you can see it, where they think they're just maximizing their overall win-rate by accounting for everything, as opposed to understanding this optimization. That's where it's more subtle at the higher level of play.
If you understand this simple equation, your "greed" becomes the biggest win-rate that you can get.
In MTG this is called having good matchups against the field, I guess this is helpful since many here seem to miss basic CCG concepts, as seen by the contraversies over gang up and the warrior mech.
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This is cross posed in battle.net -- http://us.battle.net/hearthstone/en/forum/topic/16858806377#1
Please note: I am NOT suggesting that you plug in numbers into this equation to win more games. I am suggesting that you familiarize yourself with this equation because you can intuit it in a way that allows your decisions to be much more rationally informed. This is a mathematical description of winning. Try to understand it intuitively.
So, I was doing a bit of thinking the other day, and I was thinking about how to actually analyze the meta in a more fundamental way. Specifically, I was thinking about how I am always coming up with a new deck idea that tries to address certain fundamental problems, or tries to use different subversive techniques to win more (since I rather like playing around). Then I thought "Well, what deck would a superhuman artificial intelligence play?" I realized that the superhuman A.I. would have figured out a fundamental way of looking at the game that would allow it to win more. Well, of course you would be maximizing your win rate. But how do you conceptualize the output of your winrate in a way that is actually relevant?
I am kind of a fan of math, and there's a rather simple way to sum up (pun fully intended) how to win, based on the meta. In fact, you literally use a summation:
R = ∑ wipi
R is your win rate, wi is the chance that you'll win against a specific deck and pi is the probability that you will encounter that deck. So, here's an example of what that looks like:
Your deck wins against mechmage 50% of the time and encounters it 50% of the time. Your deck encounters fast druid the other 50% of the time, and wins against that 50% of the time. So, intuitively you know that you have a 50% winrate (which is why I picked these numbers, to prove the point).
So, lets plug those numbers in:
∑ wipi= w1p1 + w2p2 = (0.5)(0.5) + (0.5)(0.5) = .25 + .25 = 0.5
This is how that general equation works. So, whenver someone asks you "what is the strongest deck?" The answer is the deck that maximizes the output of this equation.
So, what you want to do is do an analysis of what the meta is, and then you want to optimize the output in such a way that you maximize your winrate. Likewise, "the best possible deck" is always changing with the meta, since all of these terms change.
So, you don't have to do an in-depth analysis of this equation for every card, but understanding and conceptualizing this equation (by occasionally plugging numbers in), and roughly guessing at each of its components will help you construct the best deck for the current meta. For instance, maybe mechmage is popular and handlock is as well. But, perhaps you can't beat both of them strongly, but you realize that mechmage is more popular and you can deal with handlock well enough by teching in a few cards.
So, you have to deal with each deck on a case by case basis.
Example:
So, the meta is archetype 1, archetype 2 and archetype 3. So, you run into archetype 1 25% of the time, archetype 2 35% of the time and archetype 3 40% of the time.
Your winrate vs. archetype 1 is 60%. Your winrate vs. archetype 2 is currently 55%. Your winrate vs. archetype 3 is currently 70%. So, you tech out a card and it lowers your winrate against archetype 1 by 10%, raises your winrate against archetype 2 by 10% and lowers your winrate vs. archetype 3 by 5%. Is the change worth it? (If you try to do this before looking at the solution, it'll help you learn this just that much better).
So, original:
.25*.6+.35*.55+.4*.7= .15+.1925+.28= .6225=62.25% winrate
And
.25*(.6-.1)+.35(.55+.1)+.4*(.7-.05)=.125+.2275+.26=.6125 = 62.25%
So, it lowers your overall winrate by 1%. Bummer.
But, there's an easier way to do this.
.25(-.1)+.35*(.1)+.4(-.05)=-.025+.035-.02= -.01 = -1%
It checks out!
So, you can actually simply calculate the change in overall winrate as the sum of the change in winrates against each deck:
∑∆wipi= ∆R
But, these calculations are only based on estimations. That's where this formula is not a magic bullet, but it's a much more powerful, concise way to conceptualize the game.
Week 11 Design Submission - Bronze Drake
The Fundamental Equation of Hearthstone - The Fundamental Description of Maximizing Winrate
I must say that I support the posting of this thread in it's entirety.
Math 2spooky4me.
I bring life and pffffffffffffff!
In general, yes, your win rate is SUM[P(win|encounter)], but the fact that meta is constantly changing and different ranks have different probability of encounters makes the real analysis an impossible task. Also, sometimes the best way to increase your win rate is to add a tech card and not switch the deck.
Meta changes the moment you switch your deck.
Well, the point (which I keep reiterating everywhere I post this) is not that you plug in numbers and do some analysis before you play every day.
The point is a broader point about intuiting how to play this game. Especially pay attention to the ∑∆aibi = ∆R. That is, the change in winrate is the same as the sum of the change in winrates times the prevalence.
That makes those decisions intuitive without plugging in any numbers. (I think I take for granted the fact that equations give intuition because I'm majoring in physics... lol).
Week 11 Design Submission - Bronze Drake
The Fundamental Equation of Hearthstone - The Fundamental Description of Maximizing Winrate
While I always support deconstructing any game into math, I'm not convinced what you've posted here is particularly insightful or useful ... the latter because there simply isn't sufficient information to utilize this to select the ideal deck.
As I've been saying, it helps you deconstruct the game conceptually. That is practical.
That's why I say to conceptualize the equation: you can say a lot about how something works by describing it mathematically. Plugging in numbers isn't the point. It just improves your understanding of the game, and can help you ask more relevant questions when building a deck, just by understanding it intuitively.
Beyond that, even if it was absolutely 0% practical, I just find there to be an elegance to understanding the game in this way.
Week 11 Design Submission - Bronze Drake
The Fundamental Equation of Hearthstone - The Fundamental Description of Maximizing Winrate
Well, it's all important. And you can recognize these things through understanding the equation intuitively.
Week 11 Design Submission - Bronze Drake
The Fundamental Equation of Hearthstone - The Fundamental Description of Maximizing Winrate
This is probably biggest issue with that formula, of course it works, but you need good data for it to be precise enough. You can estimate winrate against certain decks if you have some win/loss tracker (But you still have to pay attention of distribution of games like 5 mech mages, 1 freeze mage, 2 tempo mages.) Also you need large sample of games against that deck, to have a good enough estimate of your winrate, as you can't say I have 100% wr against secret mages if you only met this deck once. Also you will likely have different chance to win against same deck, that has some different choices, than other deck of the same archetype. For example when you are less likely to win against mech mages who play mad scientist+mirror entity, than if he don't run these cards.
And I don't even have to talk about meta changing and your possibly biased estimations of meta, like if you get against 5 mech mages in a row and you may start to think they are really popular, just to fight 10 other decks after.
Since you suggested not plugging numbers into that equations, rather familiarize with this equation in general, I agree with that and these were just some examples, why trying to directly plug in numbers is not a good idea.
This familiarizing with this equation can mean a lot from a simple decision of adding kezan mysthic if you start facing lot of hunters to completely changing your deck, if you feels it perform better against decks you face the most on ladder. I hope this helped a little and I didn't make any big mistakes or misunderstandings of your text here.
This just about sums it up.
I go a little bit further in implementing the conceptual basis as a means for determining exactly which cards to swap out in your deck. "Will this swap improve/decrease my overall winrate vs. the field?" is the question you want to be asking. Also, "how should I play against deck X, how do my win conditions change?"
That's why it is THE equation, not just an equation. It's the description.
Week 11 Design Submission - Bronze Drake
The Fundamental Equation of Hearthstone - The Fundamental Description of Maximizing Winrate
Profound... :/ Anyone who didn't understand that already probably won't find mathematical analysis helpful.
While I'm as much in favour of math as the next guy, the equation thing seems a bit superfluous. "Here's a bunch of math, now ignore the math and use gut instinct to figure out the blatantly obvious." Not trying to be critical, just don't really see the point here.
If you think it seems superfluous, I don't think you're totally appreciating it, or, at least, don't fully get it intuitively. I mean, a key difference here is that I used my conceptual understanding of math to write the equation, and you're approaching it like most do, with the plug'n'chug sort of thinking, as opposed to the robust intuitive understanding approach. That's the only way I can see anyone seeing this as superfluous.
This equation helps inform you on what adding or subtracting a card from a deck will do to your winrate, since you can see it as a combination of deck popularity and your winrate vs. a specific deck. So, you can ask yourself when you add a card "Well, I see facehunter about 30% of my games, mechmage 25% of my games, demonlock 25 % of my games, control warrior 10% of my games (with "other" as the remaining 10% that you don't worry about since on average it changes less, so you just consider that ~∆0%). I think that adding this card would add 2% vs. mechmage and facehunter, 3% vs. demonlock and maybe -4% vs. control warrior." For this, a is approximately the same as b and c, and they're all larger than d. So, then, you note that mechmage, fachunter and demonlock are a lot more prevalent than control warrior, and you then realize that you're making a card change on much more sound reasoning than "oh, just counter the meta." Well, is it a good change? The answer is +1.45% overall difference. This 1-2% could play out in a way such that you get a couple additional winstreaks over this minor card adjustment.
See, it makes deck construction multidimensional in an intuitive way. See, this deck could be fast druid and the card could be antique healbot, and those numbers wouldn't necessarily be a bad approximation.
Beyond that, how many people do you see, even at high ranks, playing very greedy decks? See, this games a number as your winrate, so your greed becomes "winrate" and "the meta," not just "I want to play a powerful deck and I'll put in these tech cards for the meta." See how uninformed that actually is?
See, I can't even imagine summing that type of decision-making up as "just counter the meta." "Counter the meta" is a murky, vague statement in comparison to an equation.
https://hearthstats.net/
Scroll to the bottom of that where you see class percentages by rank. Now how much more useful is the equation? You could calculate your expected winrate and test it against your actual winrate, and analyze your previous deck decisions based on this by using game tracking software.
This is actually very analogous to the definition of an asset in finance: a sequence of future cash flows. The present value of an asset is the sum of those future cash flows at different points in time (divided by the interest rate of each term at each term's time in the future). See? You can't actually do finance intelligently without that, but you might see it as intuitive, since you know future dollars are worth less than dollars now because of interest. And, getting the right numbers in valuation is hairy business -- people make mistakes all the time. It's a lot like Hearthstone: the equation is absolutely true and it is actually pretty intuitive. But, you can't make as informed of decisions without it.
You can't even do that approximate conceptual analysis as well without the equation there. It's a lot like saying "well, if you drop a ball from a plane, it falls faster and faster until air resistance makes the acceleration slow down to zero." Of course you know this, but you don't know by how much without differential equations (which still give an approximation! But it's still a useful approximation!). And, if you track your games, you get a decent range of popularity for a deck. And, though you can't get the popularity of the day, you can get the average popularity over a week or month.
Week 11 Design Submission - Bronze Drake
The Fundamental Equation of Hearthstone - The Fundamental Description of Maximizing Winrate
"Physics is the fine art of approximation." - My first physics professor. That generally applies to any mathematical description of something with outcomes in reality. For instance, you wouldn't analyze rolling a dice in terms of its inertial tensor. You're just going to give each side 1/6, and some number might have a minute probability of being higher or lower. See? You have to choose the correct level of mathematics to describe something. You can actually be way too precise.
You don't use general relativity to engineer rockets -- you still use Newtonian physics with post-Newtonian mathematical methods. See, Newton isn't "wrong" -- he's just an approximation of a more robust description of reality, a robust description that likes to act a lot like Newton's laws at low gravities and low velocities.
There's nothing exact in physics, and there's rarely anything exact in applied mathematics. Hell, a major part of math is the art of understanding just how "wrong" you are.
This equation works -- it's just not very precise. And, as a "nice orientation for new players?" Yes, I kind of intended it as that. However, experienced players still get emotional in the form of greed, building decks with too many tech cards or too much late-game. And you can see it, where they think they're just maximizing their overall win-rate by accounting for everything, as opposed to understanding this optimization. That's where it's more subtle at the higher level of play.
If you understand this simple equation, your "greed" becomes the biggest win-rate that you can get.
So, it can definitely have impact.
Week 11 Design Submission - Bronze Drake
The Fundamental Equation of Hearthstone - The Fundamental Description of Maximizing Winrate
In MTG this is called having good matchups against the field, I guess this is helpful since many here seem to miss basic CCG concepts, as seen by the contraversies over gang up and the warrior mech.