Where Did Prot Holo Come From?

[JusticeRetroHunter.7684]

Just now, Crozame.4098 said:

You are talking about the length of a walk that happens to be a circle?

 

 

Ya. In the math it's explained that once a function gives you the solution that you've already seen before, then it's a fixed point. So if the solution to f(f(f(x))) gives you the same solution f(x) then you've hit a fixed point. It's abstract because it never "settles" on one build, it's settling on 3 builds here. f(x), f(f(x)) and f(f(f(x))).

 

The picture is based on Brouwer fixed point equilibrium, Kakutani is basically the same, but it's the version of brouwers that uses set-valued function instead of real valued function. tbh I still don't truly understand Kakutani as an application of gw2 in an intuitive sense (example, If you can find equilibrium of RPS with a real valued function, then why not gw2 why are they different?)

 

I watched this video to help walk me through the math on Brouwer's.

 

[Crozame.4098]

18 minutes ago, JusticeRetroHunter.7684 said:

Ya. In the math it's explained that once a function gives you the solution that you've already seen before, then it's a fixed point. So if the solution to f(f(f(x))) gives you the same solution f(x) then you've hit a fixed point. It's abstract because it never "settles" on one build, it's settling on 3 builds here. f(x), f(f(x)) and f(f(f(x))).

 

The picture is based on Brouwer fixed point equilibrium, Kakutani is basically the same, but it's the version of brouwers that uses set-valued function instead of real valued function. tbh I still don't truly understand Kakutani as an application of gw2 in an intuitive sense (example, If you can find equilibrium of RPS with a real valued function, then why not gw2 why are they different?)

 

I watched this video to help walk me through the math on Brouwer's.

 

I understand what is a fixed point. But it makes no-sense to me, still, to talk the size of a point. He is using the loop to show the existence of a fixed point. The number of elements in the loop is not the size of the fixed point... At least he did not give the definition of the size of the loop.

 

The only way I can make sense of the size of the fixed point is the fact that, it is a set valued function, and the fixed point's set has larger cardinality.

[JusticeRetroHunter.7684]

47 minutes ago, Crozame.4098 said:

The only way I can make sense of the size of the fixed point is the fact that, it is a set valued function, and the fixed point's set has larger cardinality.

Right that's Kakutani, it's just an extension of Brouwer's, for set valued functions.

[Kolzar.9567]

I think you are not paying attention to the relevant spaces of these mappings. I gather you are more familiar with functions instead of correspondences so I will try to use that, but in the end they hopefully the relation to my earlier posts will be clear. And again all I am suggesting are different notions of diversity, so that when we talk about diversity of meta we have the same thing in mind, as I still think complexity is not appropriate.

 

A function as you might recall, has a domain and a range, and has to take every single element in the domain and has to map it into a single element in the range. Now, lets start from our domain as the set of builds in gw2 (it really doesn't matter how you represent them be it vectors or other things, at the very least you can literally just name them). Say our very simplified space is {scourge, herald, holo, weaver}. If I think of a function on this space it takes every element and spits out a single element. So if we somehow think that this mapping (which we did not make further assumptions on) is representative of how meta evolves (again including all my limitations of the player base) as a counter for a build, if I think of this as a function it restricts me to a single build, although I might be thinking both holo and weaver counters scourge. Furthermore since I am thinking about some collection of builds that are in the meta, and their counters even the domain doesn't do me much good, since the current meta could be sourge and holo, not just singleton objects This part I can fix partially by thinking what counters scourge, holo, individually, and just taking unions, but it doesn't help with the fact that I may have more than one counter. So clearly we need to either change the space or get something more broad then a function. Earlier I directly went with the latter, but this time lets go the former way, however keep in mind I am still trying to describe the same situation and get the same notion. Now I consider the set of all non-empty subsets of the set {scourge, herald, holo, weaver}, this object as you might recall has 2^4 -1 elements, and looks like {{scourge,holo},{scourge,herald,holo},{weaver,holo}{weaver} .....}, all the single, double, triple element subsets and the full set itself. Now a function on this space would take a subset say {scourge,herald, holo} and map into say {weaver, holo}. However we gained some freedom, since the singletons are here, so you can say the counter to a {scourge} meta is {holo, weaver}. Now a fixed point on this function on this space of subsets, is going to be an element of this space, a set. So whatever notion you can have with regards to subsets you can use here. I suggested the size of this set, hence its cardinality. It really has nothing to do with the mapping, but basic properties of the elements. To make an extreme example, suppose we were mapping apples to apples, we could think about the redness or freshness of a fixed point, since it is going to be an apple. 

 

Now instead of just builds, we could have started from builds and their frequency in population, which would be probability distribution over the set of builds. An example would be {Scourge 1/3, holo 2/3, herald 0, weaver 0} that is currently 1/3 of the population is playing scourge, 2/3s are playing holo, noone plays herald or weaver. Again there is a space of such distributions, usually denoted of the simplex over this set (not to be confused with the simplex algorithm), and usually denoted with delta. So our space would be Delta{Scourge, Holo, Herald, Weaver}. Again we can define a function here, this time from this space of distributions to space of distributions, and again a fixed point will have whatever we can define on an element. Now since these are probability distributions candidates could be standard deviation, mean, kurtosis, etc, or again just the size of the support of the distribution that is the fixed point. The advantage of such a formulation might be a better representation of reality, the disadvantage is that you are dealing with a more complicated object.

 

With that in mind to reiterate why complexity is odd here. Nash equilibrium, or loosely speaking any equilibrium/stability notion can usually be described as a fixed point. The complexity of calculating  very well behaved fixed points such as even Nash equilibria is usually bordering on NP (there are lower complexity ones such as contraction mappings, but those are rather special). And Nash equilibrium is a beast, it assumes way too much about what people know and how people think that is why it is so well behaved. If you think a fixed point of how fotm builds change/evolve is an accurate description of a stable meta you are going to run into the same issues and more. (Again mine is a mere suggestion, you can certainly disagree with all or parts of the descriptions in paragraphs 2/3) The moment you start relaxing some of these assumptions so you get weaker solution concepts than Nash (and you really should given that you don''t know why people play the game), the complexity as defined in the standard way increases, just because now you need to also consider how people think, what people know about the strategic interaction they are in, each other etc. 

Edited June 12, 2021 by Kolzar.9567

[JusticeRetroHunter.7684]

5 hours ago, Kolzar.9567 said:

I think you are not paying attention to the relevant spaces of these mappings. I gather you are more familiar with functions instead of correspondences so I will try to use that, but in the end they hopefully the relation to my earlier posts will be clear. And again all I am suggesting are different notions of diversity, so that when we talk about diversity of meta we have the same thing in mind, as I still think complexity is not appropriate.

 

A function as you might recall, has a domain and a range, and has to take every single element in the domain and has to map it into a single element in the range. Now, lets start from our domain as the set of builds in gw2 (it really doesn't matter how you represent them be it vectors or other things, at the very least you can literally just name them). Say our very simplified space is {scourge, herald, holo, weaver}. If I think of a function on this space it takes every element and spits out a single element. So if we somehow think that this mapping (which we did not make further assumptions on) is representative of how meta evolves (again including all my limitations of the player base) as a counter for a build, if I think of this as a function it restricts me to a single build, although I might be thinking both holo and weaver counters scourge. Furthermore since I am thinking about some collection of builds that are in the meta, and their counters even the domain doesn't do me much good, since the current meta could be sourge and holo, not just singleton objects This part I can fix partially by thinking what counters scourge, holo, individually, and just taking unions, but it doesn't help with the fact that I may have more than one counter. So clearly we need to either change the space or get something more broad then a function. Earlier I directly went with the latter, but this time lets go the former way, however keep in mind I am still trying to describe the same situation and get the same notion. Now I consider the set of all non-empty subsets of the set {scourge, herald, holo, weaver}, this object as you might recall has 2^4 -1 elements, and looks like {{scourge,holo},{scourge,herald,holo},{weaver,holo}{weaver} .....}, all the single, double, triple element subsets and the full set itself. Now a function on this space would take a subset say {scourge,herald, holo} and map into say {weaver, holo}. However we gained some freedom, since the singletons are here, so you can say the counter to a {scourge} meta is {holo, weaver}. Now a fixed point on this function on this space of subsets, is going to be an element of this space, a set. So whatever notion you can have with regards to subsets you can use here. I suggested the size of this set, hence its cardinality. It really has nothing to do with the mapping, but basic properties of the elements. To make an extreme example, suppose we were mapping apples to apples, we could think about the redness or freshness of a fixed point, since it is going to be an apple. 

 

Now instead of just builds, we could have started from builds and their frequency in population, which would be probability distribution over the set of builds. An example would be {Scourge 1/3, holo 2/3, herald 0, weaver 0} that is currently 1/3 of the population is playing scourge, 2/3s are playing holo, noone plays herald or weaver. Again there is a space of such distributions, usually denoted of the simplex over this set (not to be confused with the simplex algorithm), and usually denoted with delta. So our space would be Delta{Scourge, Holo, Herald, Weaver}. Again we can define a function here, this time from this space of distributions to space of distributions, and again a fixed point will have whatever we can define on an element. Now since these are probability distributions candidates could be standard deviation, mean, kurtosis, etc, or again just the size of the support of the distribution that is the fixed point. The advantage of such a formulation might be a better representation of reality, the disadvantage is that you are dealing with a more complicated object.

 

With that in mind to reiterate why complexity is odd here. Nash equilibrium, or loosely speaking any equilibrium/stability notion can usually be described as a fixed point. The complexity of calculating  very well behaved fixed points such as even Nash equilibria is usually bordering on NP (there are lower complexity ones such as contraction mappings, but those are rather special). And Nash equilibrium is a beast, it assumes way too much about what people know and how people think that is why it is so well behaved. If you think a fixed point of how fotm builds change/evolve is an accurate description of a stable meta you are going to run into the same issues and more. (Again mine is a mere suggestion, you can certainly disagree with all or parts of the descriptions in paragraphs 2/3) The moment you start relaxing some of these assumptions so you get weaker solution concepts than Nash (and you really should given that you don''t know why people play the game), the complexity as defined in the standard way increases, just because now you need to also consider how people think, what people know about the strategic interaction they are in, each other etc. 

 

Ya... you know I don't think I disagree with the model (Most of my posts have merely been to try to understand it more).

 

The only thing I disagree with seems to be this applicability of complexity... The first thing to note is the time it takes for a system to approach equilibrium. There needs to be a notion of time in the system in order to talk about how that system changes with time...so even if your system is an appropriate set up, it's not encompassing those notions.

 

This is why classification of NP exists, If your system gets lucky, in which the players just happens to play the best build in the game the very first moment the game was released, you have to then ask how long does it take to verify the answer. If the problem is NP hard, there is no algorithm, you must brute force your way to verifying it by checking every configuration, and this defines the complexity of the thing you want to talk about. The more elements there are, the longer time it will take (because you're using brute force)

 

 

So ya...I don't think I disagree with your model at all, just the notion of how your treating computational complexity for it. I know you've explained the set up to me quiet a few times now, and each explanation I appreciate because its not the easiest setup for me to follow.

Edited June 12, 2021 by JusticeRetroHunter.7684

[Kolzar.9567]

1 hour ago, JusticeRetroHunter.7684 said:

This is why classification of NP exists, If your system gets lucky, in which the players just happens to play the best build in the game the very first moment the game was released, you have to then ask how long does it take to verify the answer.

This is the nuance here I think, the system getting lucky part. Complexity is a better tailored notion for when how your system transitions from one state/meta (whatever you want to call) follows some known rules even if there is some stochasticity. I gave this example before but if my tea is cooling, the water molecules just follow the rules of physics/thermodynamics. They do not try to deliberately burn my tongue or do some sort of optimization on their own to decide who shall be sacrificed for my tea drinking pleasure. People picking builds try to do such an optimization which causes the issues you face for whichever notion of equilibria you would need to impose on. I mean consider a case where you assumed that people changed their builds while disregarding others and their own goals and instead were using some fixed rules (suppose I am currently playing weaver, tomorrow I am either going to play weaver or herald with some known odds no matter what happened). In that case there are still some transition of builds, and we can think of some mapping of builds to each other and ask about stability/fixed points. Then I would agree that complexity would  make sense as a notion capturing how long it will take to reach stability ( I would still think this is slightly different than diversity at the stable point, but it is still a meaningful notion and you can argue either way). But the moment people try to their best, we add a strategic interaction, and that just inflates the complexity of the problem just like various notions of equilibria in games, in the sense that at every instance in time I am going to do my best and you are going to do your best given what you think I am doing and your evaluation of your options over your beliefs about what I am doing.

 

For another angle, you can ask how long an individual takes to find their best solution to a given meta. That would be just me trying to solve an optimization problem given my understanding and you can ask the complexity of that problem (I mean at the very least something greedy should work, given that there are only finitely many builds) and it will be well defined and won't suffer the issues of fixed point problems. 

[Grand Marshal.4098]

🥱

 

Fr, you guys put way too much thinking in a game where the entire team of developers that has worked on, hasn't.

Edited June 13, 2021 by Grand Marshal.4098

[JusticeRetroHunter.7684]

16 hours ago, Kolzar.9567 said:

This is the nuance here I think, the system getting lucky part. Complexity is a better tailored notion for when how your system transitions from one state/meta (whatever you want to call) follows some known rules even if there is some stochasticity. I gave this example before but if my tea is cooling, the water molecules just follow the rules of physics/thermodynamics. They do not try to deliberately burn my tongue or do some sort of optimization on their own to decide who shall be sacrificed for my tea drinking pleasure. People picking builds try to do such an optimization which causes the issues you face for whichever notion of equilibria you would need to impose on. I mean consider a case where you assumed that people changed their builds while disregarding others and their own goals and instead were using some fixed rules (suppose I am currently playing weaver, tomorrow I am either going to play weaver or herald with some known odds no matter what happened). In that case there are still some transition of builds, and we can think of some mapping of builds to each other and ask about stability/fixed points. Then I would agree that complexity would  make sense as a notion capturing how long it will take to reach stability ( I would still think this is slightly different than diversity at the stable point, but it is still a meaningful notion and you can argue either way). But the moment people try to their best, we add a strategic interaction, and that just inflates the complexity of the problem just like various notions of equilibria in games, in the sense that at every instance in time I am going to do my best and you are going to do your best given what you think I am doing and your evaluation of your options over your beliefs about what I am doing.

 

For another angle, you can ask how long an individual takes to find their best solution to a given meta. That would be just me trying to solve an optimization problem given my understanding and you can ask the complexity of that problem (I mean at the very least something greedy should work, given that there are only finitely many builds) and it will be well defined and won't suffer the issues of fixed point problems. 

 

I don't agree with this. If you stand by this conclusion, then you can justify saying that an RPS game has infinite complexity because there are configurations where players just play configurations of P and S only. There's a reason they set up the class of NP and other complexity classes like they do, it's to avoid conclusions like this. It's to bound the systems capacity for complex computation, and therefor, give it the lowest bound for how complex a problem is.

 

For example, someone can make an algorithm which I'll just call "the dum dum" where the dum dum algorithm, is a brute force algorithm that only looks at the first 3 bits of data and just cycles between those 3 in an endless loop. You might have a program that is a 1000 bits of data, and you run the dum dum on it, and it will never accomplish the task of solving the program you gave it. This does not mean that the program is overly complex, it's just because your algorithm is dumb.

 

One thing that you know for sure about a problem that's NP hard is that the fastest way to solve your 1000 bit program, is an algorithm that has to brute force solve it in 1000 steps. This gives it a lower bound in terms of complex complexity time, and the obvious scaling with the size of the program.

 

 

 

 

Edited June 13, 2021 by JusticeRetroHunter.7684

[Crozame.4098]

On 6/12/2021 at 5:28 PM, JusticeRetroHunter.7684 said:

Right that's Kakutani, it's just an extension of Brouwer's, for set valued functions.

No, its not right. There is no size of the fixed point. And what mentioned was the size of the set in the domain and range, which are again single unit inputs since the function is set valued, in this case, the fixed point is still a point...

[JusticeRetroHunter.7684]

16 hours ago, Crozame.4098 said:

No, its not right. There is no size of the fixed point. And what mentioned was the size of the set in the domain and range, which are again single unit inputs since the function is set valued, in this case, the fixed point is still a point...

 

I mean what are you arguing with me for? It's his model not mine, All i did was to just try to draw it so that other people can understand what's being said. 

 

The size of the fixed point is just the cardinality of the set and you said this already earlier yourself. The more elements in the set the larger the fixed point...it's still a point but it's "larger" then a set with one element inside.

 

That's all I know about the model in my limited understanding...again if you don't agree with the model talk to him about it not me lol

 

 

[Crozame.4098]

2 hours ago, JusticeRetroHunter.7684 said:

 

I mean what are you arguing with me for? It's his model not mine, All i did was to just try to draw it so that other people can understand what's being said. 

 

The size of the fixed point is just the cardinality of the set and you said this already earlier yourself. The more elements in the set the larger the fixed point...it's still a point but it's "larger" then a set with one element inside.

 

That's all I know about the model in my limited understanding...again if you don't agree with the model talk to him about it not me lol

 

 

The reason is that if you want to discuss something complicated rigorously, you should define the basics very clear. And if you understood what I wrote, you will notice that what I meant was the for the size of the fixed point.

 

This is related to all of your posts regarding balance. Everything is defined vaguely and when people question you, you use other vague answers or hour long vids to cover that. Thats why I dont even bother argue with you on the broad picture anymore. I will focus on the basic if have time.

[JusticeRetroHunter.7684]

42 minutes ago, Crozame.4098 said:

The reason is that if you want to discuss something complicated rigorously, you should define the basics very clear. And if you understood what I wrote, you will notice that what I meant was the for the size of the fixed point.

 

I don't understand what else you can possibly mean by the size of the fixed point.

 

Anyway, I think we've already moved on from discussing the model...it's a good and it works good enough to at least paint a general picture for gw2 balance in a formal way. Again if you have issues with the model talk to him not me. 

Edited June 15, 2021 by JusticeRetroHunter.7684

[Crozame.4098]

57 minutes ago, JusticeRetroHunter.7684 said:

 

I don't understand what else you can possibly mean by the size of the fixed point.

 

Anyway, I think we've already moved on from discussing the model...it's a good and it works good enough to at least paint a general picture for gw2 balance in a formal way. Again if you have issues with the model talk to him not me. 

point is a point, there is no size of a point in terms of a fix point in a graph.

[JusticeRetroHunter.7684]

1 hour ago, Crozame.4098 said:

point is a point, there is no size of a point in terms of a fix point in a graph.

 

Did I ever say the point physically grows bigger on a graph? Stop projecting pedantics...it's annoying.

 

That's why I drew the picture... to illustrate how the points cardinality defines the size of the fixed point... You don't like the picture go draw your own. 

Edited June 15, 2021 by JusticeRetroHunter.7684

[Crozame.4098]

1 hour ago, JusticeRetroHunter.7684 said:

 

Did I ever say the point physically grows bigger on a graph? Stop projecting pedantics...it's annoying.

 

That's why I drew the picture... to illustrate how the points cardinality defines the size of the fixed point... You don't like the picture go draw your own. 

You are saying the size of the fixed point. The freaking size of a fixed point... 

[JusticeRetroHunter.7684]

1 hour ago, Crozame.4098 said:

You are saying the size of the fixed point. The freaking size of a fixed point... 

 

Take a 3 element RPS game, then take a 10 element RPS game and put a dot in the center (the center of gravity aka a fixed point). Both dots are the same size when you plot it here...but one has less elements then the other...

 

We aren't talking about the actual dot's size on a graph when you plot it...we are talking about the number of elements in a set that are in equilibrium...that is the "size" of the fixed point.

Edited June 15, 2021 by JusticeRetroHunter.7684

[Crozame.4098]

23 hours ago, JusticeRetroHunter.7684 said:

 

Take a 3 element RPS game, then take a 10 element RPS game and put a dot in the center (the center of gravity aka a fixed point). Both dots are the same size when you plot it here...but one has less elements then the other...

 

We aren't talking about the actual dot's size on a graph when you plot it...we are talking about the number of elements in a set that are in equilibrium...that is the "size" of the fixed point.

 

Take a 3 element RPS game, then take a 10 element RPS 

What do you even mean. Why there are 10 elements in the RPS?...

 

put a dot in the center (the center of gravity aka a fixed point). 

Well, again centre of gravity have different meanings than a fixed point.

 

We aren't talking about the actual dot's size on a graph when you plot it..

But when you say size of a fixed point, its confusing.

 

You are making your own definition. All I was trying to do is to make you be more rigorous.

Edited June 16, 2021 by Crozame.4098

[JusticeRetroHunter.7684]

4 hours ago, Crozame.4098 said:

 

Take a 3 element RPS game, then take a 10 element RPS 

What do you even mean. Why there are 10 elements in the RPS?...

 

put a dot in the center (the center of gravity aka a fixed point). 

Well, again centre of gravity have different meanings than a fixed point.

 

We aren't talking about the actual dot's size on a graph when you plot it..

But when you say size of a fixed point, its confusing.

 

You are making your own definition. All I was trying to do is to make you be more rigorous.

 

 

I'm not making my own definitions...I'm trying to explain why we are using this language...talking about sizes of fixed points. It's just understood that the actual point on the graph does not change in any physical sense when plotting it on a graph. We are describing it's "size" based on how many elements in the system are in equilibrium...honestly I don't know how much more simple I can explain that to you... we aren't drawing dots that are big dots and small dots on the graphs.

 

Also there are many fixed point theorems... which are just more interesting ways to find the equilibrium of more complex objects...but they are all fundementally the same and all stem from one of the first instances of equilibrium : center of mass. Thought you would maybe understand better using the most intuitive version of equilibrium.

 

[Crozame.4098]

7 hours ago, JusticeRetroHunter.7684 said:

It's just understood that the actual point on the graph does not change in any physical sense when plotting it on a graph

I did not understand. It just confuses people.

7 hours ago, JusticeRetroHunter.7684 said:

..but they are all fundementally the same and all stem from one of the first instances of equilibrium : center of mass

You are making vague statements again.

1) What do you mean by first instances of equilibrium?

2) We are talking fix point of a function, how is it related to a centre of mass? And why centre of mass is a fixed point? When you say centre of mass, the first thing comes to me is imagining a square and then the point in the middle. Since there is no domain and range of that square, how you think it is related to a fixed point?

Edited June 17, 2021 by Crozame.4098

[JusticeRetroHunter.7684]

4 hours ago, Crozame.4098 said:

We are talking fix point of a function, how is it related to a centre of mass?  And why centre of mass is a fixed point? When you say centre of mass, the first thing comes to me is imagining a square and then the point in the middle....how you think it is related to a fixed point?

 

You just said it right now yourself...the point in the middle of an objects mass is a fixed point. 

 

It's the most basic way to think about what a fixed point is, and when you think about more abstract fixed points in systems like a RPS game, they use fixed point theorems, that are extensions of the above concept of mechanical equilibrium...That's not trying to be vague or more confusing...it's helping to simplify the conversation to things people actually know and talk about in every day experience, like gravity...which we experience all the time.

 

Just watch this video on fixed points...Fixed points are very generic and very universal things that apply to many kinds of systems. This video should help you understand why what I'm saying isn't vague but trying to give a more accurate definition for what a fixed point is.

 

 

[Crozame.4098]

5 hours ago, JusticeRetroHunter.7684 said:

 

You just said it right now yourself...the point in the middle of an objects mass is a fixed point. 

 

It's the most basic way to think about what a fixed point is, and when you think about more abstract fixed points in systems like a RPS game, they use fixed point theorems, that are extensions of the above concept of mechanical equilibrium...That's not trying to be vague or more confusing...it's helping to simplify the conversation to things people actually know and talk about in every day experience, like gravity...which we experience all the time.

 

Just watch this video on fixed points...Fixed points are very generic and very universal things that apply to many kinds of systems. This video should help you understand why what I'm saying isn't vague but trying to give a more accurate definition for what a fixed point is.

 

 

You just said it right now yourself...the point in the middle of an objects mass is a fixed point. 

 

I did not say this, I what meant was: the middle of an object exists without functions etc, therefore not directly related to fixed point. And you think your simplification will make others understand. Most people dont even care. Why would they? You cannot even give specific suggestions.

 

[JusticeRetroHunter.7684]

1 hour ago, Crozame.4098 said:

I did not say this, I what meant was: the middle of an object exists without functions etc, therefore not directly related to fixed point. 

 

wow...you're just being argumentative just for the sake of it. How do you think you find the middle of an objects mass? You plot it on a graph like any kind of function. Honestly stop wasting thread space and time and just read the links and watch the video's and look it up yourself so we don't have to have this conversation...do your own diligence.

-----

You can also just read the Fixed Point Wikipedia page if you want an exact notion of the picture I drew on the last page...I didn't know it had a special name, but it's called a "periodic point" where the function returns the same value after a certain number of steps and it just repeats in that loop. The more iterations, the larger the loop, the "larger" the fixed point is. 

 

just the quote directly from the wiki:

Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one.

 

 

Edited June 17, 2021 by JusticeRetroHunter.7684

[Crozame.4098]

26 minutes ago, JusticeRetroHunter.7684 said:

 

wow...you're just being argumentative just for the sake of it. How do you think you find the middle of an objects mass? You plot it on a graph like any kind of function. Honestly stop wasting thread space and time and just read the links and watch the video's and look it up yourself so we don't have to have this conversation...do your own diligence.

Well, I dont care about how to find the middle of an object masses. I am only thinking if you wanna talk about equilibruim / fixed point, you need to first define the strategies etc. Simply saying the middle of an object means nothing.

 

Btw, that finding middle of object thing, might only work for squares. What about circles? triangles? or other irregular objects?

[JusticeRetroHunter.7684]

 

1 hour ago, Crozame.4098 said:

Well, I dont care about how to find the middle of an object masses. I am only thinking if you wanna talk about equilibruim / fixed point, you need to first define the strategies etc. Simply saying the middle of an object means nothing.

 

That's your issue here dude...the center of mass of an object or a system of objects is a fixed point... If you want to find the equilibrium of a system of celestial bodies, you count up all the mass, count up their vectors and the middle of those gives you the fixed point of the system (the central gravitational attraction of the system).

 

You're thinking about game theory equilibrium only and using it in a very very rigid sense...which is not fine...equilibrium does not just apply to game theory...it is a universal construct and the concept of equilibrium is much older then game theory. This is probably why you are having trouble visualizing how a game like RPS can have a very obvious fixed point without requiring any math (the center of 3 object RPS game and the center of 10 object RPS game have the same fixed point.).

 

https://i.imgur.com/Tiwzrba.png

 

Continuity solutions aside, the above is the physical version of an RPS game, where the center of mass is the fixed point, and it's vectors indicate the direction these bodies are heading towards. Both games have the same fixed point, but the size, which is what me and Kolzar have been talking about is merely a way of expressing the number of elements in the system that are in equilibrium. Kolzar's model is just a bit more complicated...so it's not as simple as the above picture...but this is a good enough example for what we are talking about when we talk about the size of fixed points.

 

 

Edited June 17, 2021 by JusticeRetroHunter.7684

[Crozame.4098]

12 hours ago, JusticeRetroHunter.7684 said:

 

 

That's your issue here dude...the center of mass of an object or a system of objects is a fixed point... If you want to find the equilibrium of a system of celestial bodies, you count up all the mass, count up their vectors and the middle of those gives you the fixed point of the system (the central gravitational attraction of the system).

 

You're thinking about game theory equilibrium only and using it in a very very rigid sense...which is not fine...equilibrium does not just apply to game theory...it is a universal construct and the concept of equilibrium is much older then game theory. This is probably why you are having trouble visualizing how a game like RPS can have a very obvious fixed point without requiring any math (the center of 3 object RPS game and the center of 10 object RPS game have the same fixed point.).

 

https://i.imgur.com/Tiwzrba.png

 

Continuity solutions aside, the above is the physical version of an RPS game, where the center of mass is the fixed point, and it's vectors indicate the direction these bodies are heading towards. Both games have the same fixed point, but the size, which is what me and Kolzar have been talking about is merely a way of expressing the number of elements in the system that are in equilibrium. Kolzar's model is just a bit more complicated...so it's not as simple as the above picture...but this is a good enough example for what we are talking about when we talk about the size of fixed points.

 

 

I mean, seriously, think of some other games: publics good game, Cournot completion, common value auctions etc. For these games, the  equilibrium is the centre of the mass? LMAO.

 

Seriously dude, all your statements are meaningless if you are not rigorous.