Where Did Prot Holo Come From?

[Math.5123]

1 hour ago, Kuma.1503 said:

 

I would like to highlight this. Partially because I think there's value in strong-manning arguments that contradict your own, and partially because there is truth in what this person is saying. 

 

It is possible to achieve greater diversity by removing something from a system. The logic is a tad bit flawed here because he uses the example of B = C = D > A because this scenario describes is three different variables that are effectively the same. A homogenous game where 3 different builds (B C and D) are carbon copies of each other. See the previous posts for an explanation for why this is the case. 

 

The logic behind it is sound however. 

 

Lets look at a real life example. You have an ecosystem that sits in a delicate balance. It contains a wide variety of life, from plants and insects, to mammals and fish. 

 

You introduce an invasive species to the equation. The animals in the ecosystem did not evolve alongside this species, so they have not yet adapted the necessarily defense mechanisms to deal with it. This species exerts strong selective pressures on the organisms within the ecosystem. Selective pressures that they have not evolved to deal with. The ecosystem is thrown into shambles and collapses. 

 

This is why we has humans generally try not to release wild animals into habitats not suited to handle them. (*cough* Emphasis on "try". Looking at you cats).

 

Lets take this to its logical extreme. Humans of the future create a species of godzilla-like monsters. They ravage the landscape, killing off plants, wildlife, people, leveling buildings. The world becomes a wasteland. In this instance, removing the monster as soon as possible is the proper course to preserve the  diversity of life in the world... This is basically the premise of any large monster movie. 

 

The same can occur in Guild Wars 2 where one or more builds are so good and produces such a harsh selective pressure on the meta that any build not able to handle it goes extinct. 

 

If I had to compress it all down to the same variables as before. The argument being made here is that

 

B < C < D < A  is a more diverse system in practice than B < C < D <<<<<<<<<<<<<<<<<<<<<< A

 

A is so far off in it's own planet in terms of power level that you had better either play it or play something that can handle it, or you are an irrelevant side character in any match that A is a part of. 

 

Where the common mistake comes in is that some people ( including Anet ) have taken the approach of "It works in these very specific scenarios, therefore it must work in all circumstances". So they resort to deleting or butchering as the default action whenever something becomes problematic. 

 

This is a problem. 

 

Yeah, I guess people took the = a bit too literal. I know it literally means equal too. But I meant it as close enough to co-exist. Thumbs up. 

[JusticeRetroHunter.7684]

5 hours ago, Math.5123 said:

Yeah, I guess people took the = a bit too literal. I know it literally means equal too. But I meant it as close enough to co-exist. Thumbs up. 

The = sign is not an interpretation. It means exactly what it always means. You describe exactly what you needed to, that "B>C>D being in symbiotic equilibrium" is enough to describe the argument....The distinction between the two is important, because they are very different...expressions or operations that yield the same kind of result.

 

Also, it's unnecessary to make the distinction between viable and non-viable builds. All the builds are contained in the inequality A>B>C>D. What Raz said earlier remains true. Any numerical operation you apply, simply shifts the options around into say D>B>A>C. Again it doesn't matter how strong A is...all that information about A...whether it's Godzilla or a small worm, is contained in the inequality, likewise all the information about the other 3 options, are contained in the inequality. Nothing truly interesting changes in this system.

 

The only thing that is important is the equilibrium/symbiosis.

 

Does A's existence depend on the existence of B, C or D.

 

If the answer is no, then it is an outlier (an invasive species), and its position in the  meta hierarchy, since it's not connected to anything is not relative to anything else, so it's removal "doesn't matter all that much." in terms of global diversity.

 

If the answer is yes, then it will be in some kind of equilibrium, its position in the meta hierarchy matters, and the things related to it will also be effected, and the things that those things were related to, will be effected and this happens in a chain all the way to the end where all things are effected. 

 

It is statistically unlikely for any object, to not be in equilibrium with other objects in a system that has many objects. It's hardly talked about, but some invasive species don't only include the ones that are able to take over the ecosystem...some species that arrive in new ecosystems, simply can't live there at all because they have no defenses from that ecosystem and they immediately die off before they have a chance to develop relationships or invade. This is more likely to happen, you just don't hear about it because well...those species aren't selected for in that environment so why would you actually hear about it.

 

Edited June 4, 2021 by JusticeRetroHunter.7684

[Leonidrex.5649]

retrohuner remind me of my old math teacher, you would ask him a simple question and he would go on a 30minutes rant about unrelated topic and get upset we dont understand what hes talking about

[Math.5123]

2 hours ago, JusticeRetroHunter.7684 said:

The = sign is not an interpretation. It means exactly what it always means. You describe exactly what you needed to, that "B>C>D being in symbiotic equilibrium" is enough to describe the argument....The distinction between the two is important, because they are very different...expressions or operations that yield the same kind of result.

 

Also, it's unnecessary to make the distinction between viable and non-viable builds. All the builds are contained in the inequality A>B>C>D. What Raz said earlier remains true. Any numerical operation you apply, simply shifts the options around into say D>B>A>C. Again it doesn't matter how strong A is...all that information about A...whether it's Godzilla or a small worm, is contained in the inequality, likewise all the information about the other 3 options, are contained in the inequality. Nothing truly interesting changes in this system.

 

The only thing that is important is the equilibrium/symbiosis.

 

Does A's existence depend on the existence of B, C or D.

 

If the answer is no, then it is an outlier (an invasive species), and its position in the  meta hierarchy, since it's not connected to anything is not relative to anything else, so it's removal "doesn't matter all that much." in terms of global diversity.

 

If the answer is yes, then it will be in some kind of equilibrium, its position in the meta hierarchy matters, and the things related to it will also be effected, and the things that those things were related to, will be effected and this happens in a chain all the way to the end where all things are effected. 

 

It is statistically unlikely for any object, to not be in equilibrium with other objects in a system that has many objects. It's hardly talked about, but some invasive species don't only include the ones that are able to take over the ecosystem...some species that arrive in new ecosystems, simply can't live there at all because they have no defenses from that ecosystem and they immediately die off before they have a chance to develop relationships or invade. This is more likely to happen, you just don't hear about it because well...those species aren't selected for in that environment so why would you actually hear about it.

 

Yes, I should've used A ≈ B ≈ C ≈ D in terms of power level. I understood your argument about 17 posts ago, you really don't have to repeat yourself. 

[JusticeRetroHunter.7684]

2 hours ago, Math.5123 said:

Yes, I should've used A ≈ B ≈ C ≈ D in terms of power level. I understood your argument about 17 posts ago, you really don't have to repeat yourself. 

I'm just trying to help you, because your response, indicates that your not drawing the right conclusion from your own statement. "≈" means approximately equal too. This is not what it means for something to be in equilibrium. Remember, that A>B>C>D or A=B=C=D is representative for the hierarchy in which these builds are selected for, and this selection describes their relative "strength."

 

Like said above, just saying A>B>C>D is in equilibrium, is a perfectly okay statement, you need not clarify it with a symbol (mostly because you can't, it involves a lot more math) so just saying that they are in equilibrium is fine here, so long as you understand what it means when things are in equilibrium. The objects do not need to be equal in "power level" at all to be in equilibrium...not even remotely close. They merely just need to depend on each other's existence. Again, think abut the sucker fish and the shark.

 

I'm just trying to help you clarify, the information that you are trying to convey.

 

 

Edited June 4, 2021 by JusticeRetroHunter.7684

[Math.5123]

1 hour ago, JusticeRetroHunter.7684 said:

I'm just trying to help you, because your response, indicates that your not drawing the right conclusion from your own statement. "≈" means approximately equal too. This is not what it means for something to be in equilibrium. Remember, that A>B>C>D or A=B=C=D is representative for the hierarchy in which these builds are selected for, and this selection describes their relative "strength."

 

Like said above, just saying A>B>C>D is in equilibrium, is a perfectly okay statement, you need not clarify it with a symbol (mostly because you can't, it involves a lot more math) so just saying that they are in equilibrium is fine here, so long as you understand what it means when things are in equilibrium. The objects do not need to be equal in "power level" at all to be in equilibrium...not even remotely close. They merely just need to depend on each other's existence. Again, think abut the sucker fish and the shark.

 

I'm just trying to help you clarify, the information that you are trying to convey.

 

 

I know what I'm trying to clarify, and I know what you're trying to do. Like I've said 5(?) times now, I understand what you've meant for the past 72 hours. So you can stop trying to sound smart.

[JusticeRetroHunter.7684]

5 hours ago, Math.5123 said:

I know what I'm trying to clarify, and I know what you're trying to do. Like I've said 5(?) times now, I understand what you've meant for the past 72 hours. So you can stop trying to sound smart.

Okay you say you know what you are trying to clarify and you say you understand but then you say this "Yes, I should've used A ≈ B ≈ C ≈ D in terms of power level." and its like... this answer doesn't make sense or line up with what you are trying to say at all... being "approximately equal to" is not the same as something being in equilibrium....not even remotely compatible. 

 

 

Edited June 4, 2021 by JusticeRetroHunter.7684

[apharma.3741]

Looking at Math's responses he didn't state it was in equilibrium, that was you.

 

Given that ANet does balance and system updates and actively changes the state variables of the system over time I don't believe you can claim it is in equilibrium. You could make this claim about Guild Wars 1 however as that game is no longer receiving updates of that nature.

[JusticeRetroHunter.7684]

 

50 minutes ago, apharma.3741 said:

Looking at Math's responses he didn't state it was in equilibrium, that was you.

 

He did. Being in Symbiosis with something, is an equilibrium. There's all kinds of equilibrium...rather there's many ways that a system can be in an equilbirum.

 

50 minutes ago, apharma.3741 said:

Given that ANet does balance and system updates and actively changes the state variables of the system over time I don't believe you can claim it is in equilibrium. You could make this claim about Guild Wars 1 however as that game is no longer receiving updates of that nature.

 

This sentence doesn't make that much sense. A harmonic oscillator is in equilibrium so long as you find it oscillating around it's fixed points. You can analyze the system in discrete time, and even if the system at that time isn't resting on it's fixed point in phase space, it's still obeying the function as it goes through time. Just because you analyzed it discretely where it's not sitting at it's fixed point doesn't mean that it's no longer oscillating lol.

 

In other words, there's probably many equilibriums one could study in gw2 as we speak right now, and if you knew what you were doing, and could map out the behavior of objects in phase space with respect to some function that describes how these objects interact, you can work out the equilibrium those objects are in, and where their fixed points are. It has nothing to do with Gw2 balance patch cadence.

Edited June 4, 2021 by JusticeRetroHunter.7684

[apharma.3741]

32 minutes ago, JusticeRetroHunter.7684 said:

 

 

He did. Being in Symbiosis with something, is an equilibrium. There's all kinds of equilibrium...rather there's many ways that a system can be in an equilbirum.

 

 

This sentence doesn't make that much sense. A harmonic oscillator is in equilibrium so long as you find it oscillating around it's fixed points. You can analyze the system in discrete time, and even if the system at that time isn't resting on it's fixed point in phase space, it's still obeying the function as it goes through time. Just because you analyzed it discretely where it's not sitting at it's fixed point doesn't mean that it's no longer oscillating lol.

 

In other words, there's probably many equilibriums one could study in gw2 as we speak right now, and if you knew what you were doing, and could map out the behavior of objects in phase space with respect to some function that describes how these objects interact, you can work out the equilibrium those objects are in, and where their fixed points are. It has nothing to do with Gw2 balance patch cadence.

 

Yes but we're not talking about a system that experiences a correcting force that is proportional to the displacement, you're now using a mechanical type of equilibrium so can we assume this is the equilibrium type you mean? You were talking about evolution not long ago so can we all be on the same page for which type you're applying?

https://en.wikipedia.org/wiki/List_of_types_of_equilibrium

 

You shouldn't apply a mechanical type equilibrium to game balance, it actually has it's own equilibrium type.

 

[JusticeRetroHunter.7684]

42 minutes ago, apharma.3741 said:

 

Yes but we're not talking about a system that experiences a correcting force that is proportional to the displacement,

Yea, this is what we are talking about.

 

You can think of a simple harmonic oscillator with the example of Rock/Paper/Scissors. Let's say that people just play Rock all the time...Paper is what defeats rock so this increases the growth rate of Paper being played, so Rock goes down in population, and Paper goes up. When Paper goes up, Scissors now becomes popular, and it lowers the population of Paper. Because Scissors population went up, Rock comes back up to lower the population of scissors. The whole system is oscillating with respect to some fixed point in phase space, and the equilibrium solution looks like some sinusoidal wave. This is just one of many possible equilibrium solutions, which depend on the relationship the objects have with each other. In other words, the solution doesn't have to look like a sin wave...it can look like any kind of wave you want to imagine.

 

So now, just replace Rock, Paper and Scissors with builds A B and C. If A depends on B, and B depends on C, and C depends on A, then they will have some kind of equilibrium solution, and that solution can look like any kind of wave or CURVE in the phase space, so long as the behavior of the objects are properly described with the function.

 

The link you posted, is not exactly relevant here, because all the different types of equilibrium are mathematically the same, they just apply the same mathematical concept of equilibrium to different kinds of systems. All these equilibrium are described in terms of mapping some function to the position of x on a phase space, where f(x) describes the velocity of x at any given point on the phase space. In discrete time, it just looks like a regular old graph x/y plot, and as you advance in time, each solution at each discrete time, gives you the trajectory of the system, and the sum of all those trajectories describes the equilibrium solution. In the case of a Rock Paper Scissors game, they have a harmonic oscillatory solution.

 

If you want to get an intuition for this specific solution, I suggest watching this video. If you want to see different kinds of solutions, including a solution about the specific thing we've been talking about over the past 2 pages (invasive species), I can link you a video to something much more in depth, that will give you an idea for how complex the different solutions can be.

 

 

Edited June 4, 2021 by JusticeRetroHunter.7684

[apharma.3741]

10 minutes ago, JusticeRetroHunter.7684 said:

Yea, this is what we are talking about.

 

You can think of a simple harmonic oscillator with the example of Rock/Paper/Scissors. Let's say that people just play Rock all the time...Paper is what defeats rock so this increases the growth rate of Paper being played, so Rock goes down in population, and Paper goes up. When Paper goes up, Scissors now becomes popular, and it lowers the population of Paper. Because Scissors population went up, Rock comes back up to lower the population of scissors. The whole system is oscillating with respect to some fixed point in phase space, and the equilibrium solution looks like some sinusoidal wave. This is just one of many possible equilibrium solutions, which depend on the relationship the objects have with each other. In other words, the solution doesn't have to look like a sin wave...it can look like any kind of wave you want to imagine.

 

So now, just replace Rock, Paper and Scissors with builds A B and C. If A depends on B, and B depends on C, and C depends on A, then they will have some kind of equilibrium solution, and that solution can look like any kind of wave or CURVE in the phase space, so long as the behavior of the objects are properly described with the function.

 

The link you posted, is not exactly relevant here, because all the different types of equilibrium are mathematically the same, they just apply the same mathematical concept of equilibrium to different kinds of systems. All these equilibrium are described in terms of mapping some function to the position of x on a phase space, where f(x) describes the velocity of x at any given point on the phase space. In discrete time, it just looks like a regular old graph x/y plot, and as you advance in time, each solution at each discrete time, gives you the trajectory of the system, and the sum of all those trajectories describes the equilibrium solution. In the case of a Rock Paper Scissors game, they have a harmonic oscillatory solution.

 

If you want to get an intuition for this specific solution, I suggest watching this video. If you want to see different kinds of solutions, including a solution about the specific thing we've been talking about over the past 2 pages, I can link you a video to something much more in depth, that will give you an idea for how complex the different solutions can be.

 

 

I know what a harmonic oscillator is, many of us have been to college as well. The point is that you need to be explicit in your definitions so that people are using the correct context and frame of mind for what you're talking about.

 

You're talking about state variables in discrete-time systems (The balance state for this game version only) however most people here may be thinking and talking about a continuous-time system (The balance state of the game across multipl versions). I believe you didn't state this till this page, if you did it was drowned out by the sheer volume of text for a simple answer that needed fewer than 150 words.

[JusticeRetroHunter.7684]

21 minutes ago, apharma.3741 said:

You're talking about state variables in discrete-time systems (The balance state for this game version only) however most people here may be thinking and talking about a continuous-time system (The balance state of the game across multipl versions).

 

There is no difference between continuous time and discrete time. You chop up continuous time into discrete packets to analyze the system at any given moment in time, and analyzing the change of the system at each discrete time gives you the continuous solution. That doesn't need to be clarified...

 

And the balance cadence of Anet doesn't effect your ability to make a calculation on the dynamics of the system. You'll always get a solution.

 

If you are asking whether A>B>C>D is an analysis of a system through time, then yes, it is. This representation is discrete, and you can analyze it continuously using the discrete solution. 

 

Edited June 4, 2021 by JusticeRetroHunter.7684

[razaelll.8324]

28 minutes ago, JusticeRetroHunter.7684 said:

 

There is no difference between continuous time and discrete time. You chop up continuous time into discrete packets to analyze the system at any given moment in time, and analyzing the change of the system at each discrete time gives you the continuous solution. That doesn't need to be clarified...

 

And the balance cadence of Anet doesn't effect your ability to make a calculation on the dynamics of the system. You'll always get a solution.

 

If you are asking whether A>B>C>D is an analysis of a system through time, then yes, it is. This representation is discrete, and you can analyze it continuously using the discrete solution. 

 

I have to disagree here.

There is a significant difference between continuous and discrete domains.

discrete domain  is a set of input values that consists of only certain numbers in an interval.

continues domain is a set of input values that consists of all numbers in an interval.

discrete function is  function with distinct and separate values (meaning the values of the functions are not connected to each other)

continuous function can take any value within a curtain interval 

Continues function always connects its values while discrete function has separations ...

Ofcourse you can transfer continues function in discrete one , but they are not the same thing

Edited June 4, 2021 by razaelll.8324

[Kolzar.9567]

1 hour ago, JusticeRetroHunter.7684 said:

The link you posted, is not exactly relevant here, because all the different types of equilibrium are mathematically the same, they just apply the same mathematical concept of equilibrium to different kinds of systems.

This is grossly misleading, you can in principle call any equilibrium a fixed point, but the space it maps to as well as what is endogenous and what is exogenous makes a huge difference. To illustrate the point let me give two rather rough examples with regards to the game and also highlight a few points I disagree with in the preceding discussion. A player in the game has finitely many choices with regards to their talents, amulet, sigils etc. Roughly we can represent this a vector (or a vector of vectors if you will), so roughly we can say a build is something that looks like (x,y,z,r,t,a,...n) where each of the letters is a vector from a finite space. I am going to call again a finite space of such vectors S (that is the finite space of all builds, where a build is a class, amulet, talent choices, utility choices, sigils), just because math notation in this forum is rather difficult. Now we can go in two ways: one we take a collection of such vectors which is a subset of the set S, and call them the current meta, say M subset S. Now, a mapping you might be interested is E, which is a correspondance (a set valued function) over S. You input M, and E(M) is going to be another subset of S. A simple notion of equilibrium will be E(M) = M (the equality in the set sense, and the appropriate fixed point theorem would be the Kakutani one for those that are interested). The dynamics you are interested in can be if you were to put an arbitrary set M_1, what will be E(M_1) which we will denote M_2 and so on and so forth and ask eventually does this operation converge to a fixed point. Notice here the operation E is not well defined here, but loosely we can say that is jointly controlled by the entire player base. (For those of you that have seen a proof of Nash equilibrium, you can loosely think of this as the Best response correspondance of the associated game). Notice this object being endogenous actually causes a lot of issues as opposed to where it is exogenous such as in a mechanical system, there is a rich literature in computer science/econ/math that deals exactly with the complexity of such calculations. One important point we need to realize here is that the computational complexity of such a calculation has nothing to do with the diversity. In particular the diversity in equilibrium of this system would be the cardinality of the set E(M), where M is the fixed point, or notably how many different builds are in this set M. This would be one way of defining equilibria.

 

Another way would be to start from the same set, S, and let Delta(S) denote the set of all distributions of over S, and now define another mapping F which takes an element m of Delta(S) and maps it back into Delta(s). Here we would take it as a function instead of a correspondance (indeed you can actually show this is without loss here due to the different topology) as at any point in time now m corresponds to the a distribution over the builds we are interested in. We can again consider a sequence of m's m_1, m_2, ... and again ask if such a sequence converges to a fixed point m=F(m). Now, you might notice that unlike the previous case this defining a Markov process, and a fixed point is just its ergodic distribution. However again unlike a physical object the Kernel of this Markov process is again jointly controlled and the problem is fairly messy. This becomes a topic on stochastic control/stochastic game theory (stochastic control oddly enough the actually this is the original ``rocket science'' if you will although it is fairly elementary) again you can find a rich literature on this, now it will be more towards operations research/math/econ as opposed to computer science. Unlike the previous example now our notion of diversity is rather hard to define,  it could be the standard deviation of the fixed point m, or it could be the entropy of the distribution (not the thermodynamics part) or just the support of the distribution which would be more inline with our earlier definition. 

 

Now on either case for example you might kill an overperforming build, a,   from a set m_n, so that E(m_n/{a}) is actually a larger set then E(m_n). Or the support of F(m_n/{a}) is larger than F(m_n) and similarly this could be true for the fixed point as well. 

 

On some topics that has come up, treating either of these as continuous or discrete (time) has very little impact since we started from a finite space the continuous time limits are well posed (the markov one needs checking for measurability which will be trivially satisfied here).  Very clearly the set of (pure) choices that are available to a player is discrete, thus creating a continuous domain via randomization is trivial. The second one is if A>B, this defines a binary relation, which is ordinal, treating this as a cardinal object requires you to make very significant assumptions. Lets assume we are trying to jointly define a lunch meta and is an example Burger>Pizza for my lunch choice. It doesnt say anything about how close they are as substitutes for me, and doesn't even say how the whole dynamics of lunch will proceed either. Finally Rock Paper Scissors has a unique equilibrium in the static version, but if you consider a stochastic game extension you will see the set of equilibria(now equilibria will be defined as sequences of actions) and hence the path of play is infinitely rich ( these kind of anything goes results are usually called  ``folk theorems'' if interested, the basic version is for repeated games but there are quite a bit of variations) so I am not sure how you can pinpoint a particular dynamic.

 

 

Edit: The reason for describing our evolution using the distribution is because want to endogenize player proportions in our ``meta-evolution''. This will allow for correcting for how a player evaluates the current meta and take proper expectations. For example suppose there is one grossly overperforming build but only 0.001 percent of the player base plays it because of some reason(say extreme difficulty or needing to purchase the game), then when players are deciding on what to do they take the proportion into account.

The reasoning for the entire post is to try to put a little more structure into the entire discussion, as opposed to randomly quoting random bits of math.

 

Edit 2: Throughout dynamics refers to the characteristics of either the stochastic or deterministic process, M_1, M_2, M_3... or m_1, m_2, m_3 that are determined through our mappings, for whichever definition you like.

Edited June 4, 2021 by Kolzar.9567
Clarification

[JusticeRetroHunter.7684]

1 hour ago, Kolzar.9567 said:

This is grossly misleading, you can in principle call any equilibrium a fixed point, but the space it maps to as well as what is endogenous and what is exogenous makes a huge difference. To illustrate the point let me give two rather rough examples with regards to the game and also highlight a few points I disagree with in the preceding discussion. A player in the game has finitely many choices with regards to their talents, amulet, sigils etc. Roughly we can represent this a vector (or a vector of vectors if you will), so roughly we can say a build is something that looks like (x,y,z,r,t,a,...n) where each of the letters is a vector from a finite space. I am going to call again a finite space of such vectors S (that is the finite space of all builds, where a build is a class, amulet, talent choices, utility choices, sigils), just because math notation in this forum is rather difficult. Now we can go in two ways: one we take a collection of such vectors which is a subset of the set S, and call them the current meta, say M subset S. Now, a mapping you might be interested is E, which is a correspondance (a set valued function) over S. You input M, and E(M) is going to be another subset of S. A simple notion of equilibrium will be E(M) = M (the equality in the set sense, and the appropriate fixed point theorem would be the Kakutani one for those that are interested). The dynamics you are interested in can be if you were to put an arbitrary set M_1, what will be E(M_1) which we will denote M_2 and so on and so forth and ask eventually does this operation converge to a fixed point. Notice here the operation E is not well defined here, but loosely we can say that is jointly controlled by the entire player base. (For those of you that have seen a proof of Nash equilibrium, you can loosely think of this as the Best response correspondance of the associated game). Notice this object being endogenous actually causes a lot of issues as opposed to where it is exogenous such as in a mechanical system, there is a rich literature in computer science/econ/math that deals exactly with the complexity of such calculations. One important point we need to realize here is that the computational complexity of such a calculation has nothing to do with the diversity. In particular the diversity in equilibrium of this system would be the cardinality of the set E(M), where M is the fixed point, or notably how many different builds are in this set M. This would be one way of defining equilibria.

 

Right. This is the much more formal version of the argument that A>B>C>D where A is some attractor. You can imagine a system where if all players in this system are playing A, then the system is homogenous, which is an equilibrium. So I don't see why that's misleading? But I appreciate that you came here to formalize it in good detail, thank you.

 

So just to elaborate some more here, if all players play build A, then the system is homogenous, and in contrast if the system has no attraction to a fixed attractor A, then the system is maximally heterogenous where all configurations of possible builds are being played...which would be complete diversity. We can get a little philosophical here and talk about what maximal heterogeneity is and whether it's even possible to attain (unlikely), but mathematically speaking in this formal context, it would be a system with no fixed attractor.

 

Quote

"One important point we need to realize here is that the computational complexity of such a calculation has nothing to do with the diversity"

 

So I like your post, and how you outlined everything very nicely. But there is one thing I disagree with, which is the above statement. The diversity of a system definitely scales with the computational complexity of the system. I had a good talk on previous thread with @razaelll.8324  on this very concept, and the concept was derived by a well known top physicist. The best way to think about it is that you have elements in a system, each with dimensionality. Each element therefor has a set of different configurations it can be in on the order of m^k where k is the dimensionality, and m is the number of elements. As the elements increase, the number of possible configurations increases on the order of K. So adding more things into a system increases the complexity exponentially with k. Example: A system with 4 things, with 2 possible parameters will have 16 possible configurations. In a system with 4 things with10 parameters will have a million configurations. So if you add another element with a dimensionality of 10, your increasing it by a factor of 10. In other words, the system is highly sensitive to k, so introducing more elements m, or increasing k, increases the computation complexity by a lot.

 

So in terms of diversity, when you notice how the computational complexity increases as the number of possible configurations increases, then as the system evolves toward maximal complexity, the longer the computation takes to go from a maximally heterogenous state, to a maximally homogenous state. So for example, you have a computer with 4 objects each with 2 parameters...it will take 16 computations to reach maximal complexity, to answer a problem like "Which object is the best object to play". In a system with 1000 objects with 10 parameters, it will take...10^30 computations to find the answer to that question. So if you have a game, where the goal of the game is to play the best build, and you have a highly complex game, then it is an NP hard problem and it will take you exponential time to figure it out. So even though you are trying to find out what A is, you don't actually know what it is until you go through 10^30 computations.

 

I think you know how computational complexity works, but if it's not too much to ask, I would encourage you to watch the derivation made by Susskind, and it will give you a bit more insight into why the above makes sense, and the reason for why i talk about complexity theory alot. Now if you disagree, I'm totally down for discussing it

 

 

 

Edited June 4, 2021 by JusticeRetroHunter.7684

[Kolzar.9567]

4 hours ago, JusticeRetroHunter.7684 said:

The diversity of a system definitely scales with the computational complexity of the system.

This depends on your definition of diversity, you can certainly define complexity itself to be diversity, but I believe this doesn't serve any purpose for the discussion at hand. The notion of complexity of finding a fixed point (or an equilibrium if you will) is fairly clear and has a rather rich literature. Notably if you start from some set m_1 what is the worst case scenario to reach a fixed point, that is how many times must you use our evolution whether it is E or F to eventually reach a fixed point. Now clearly it has a correlation with the dimensions and the size of the objects in S, whether it is computable in polynomial time or whatnot. You might use this as a proxy for how long it will take to reach a stable meta. This however does not tell you how many different classes are in the meta, which I believe is the appropriate notion of diversity here. Notably, you can have a fixed point that requires you to have say N^298179181981 iterations where N is the cardinality of S, but every time E(M_n) is a singleton, that is everyone plays a single build of scourge, followed by a single build of engi so and so forth. Also computational complexity is not particularly useful when people are involved either. A simple example you can think of is the Walrasian equilibrium, (you know where supply equals demand) with at least 3 goods is a well known NP hard problem. However, unlike computers humans can calculate the fixed points much faster albeit not algorithmically, so that you can easily ask college students to do it. Furthermore again the number equilibria (in this case prices and amounts sold) could be 1 or infinitely many. Now that number of equilibria, that is how many triplets of prices and demands you can get to me is the appropriate notion of diversity here, that is the cardinality of the fixed point. If you want to take it as a game as in game theory, how many equilibria exists (that is a tuples of strategies that are best responses to each other) is a distinctly separate question compared to how long it will take to reach one equilibria and how it scales with the parameters of the problem.. 

 

Edit: I realize my answer is very long winded, but the gist is, even if you add arbitrary dimensions to every object in S, it does not necessitate increasing the cardinality of the set E(M) =M. And that cardinality at least in my description is the number of meta vectors if you will. A trivial albeit unrelated example is thinking of a menu from a restaurant that has fantastic steak and mediocre everything else. Now no matter how many more vegan options you have, you can think that every week you will have some sort of steak dinner at the said restaurant, maybe changing the sides/type of steak etc. Now if you remove steak from the menu, then there is possibility that sometimes you can eat pasta, sometimes you eat salad and all the more variations, because there is no more superlative option.

Edited June 4, 2021 by Kolzar.9567

[JusticeRetroHunter.7684]

15 hours ago, Kolzar.9567 said:

This depends on your definition of diversity, you can certainly define complexity itself to be diversity, but I believe this doesn't serve any purpose for the discussion at hand. The notion of complexity of finding a fixed point (or an equilibrium if you will) is fairly clear and has a rather rich literature. Notably if you start from some set m_1 what is the worst case scenario to reach a fixed point, that is how many times must you use our evolution whether it is E or F to eventually reach a fixed point. Now clearly it has a correlation with the dimensions and the size of the objects in S, whether it is computable in polynomial time or whatnot. You might use this as a proxy for how long it will take to reach a stable meta. This however does not tell you how many different classes are in the meta, which I believe is the appropriate notion of diversity here. Notably, you can have a fixed point that requires you to have say N^298179181981 iterations where N is the cardinality of S, but every time E(M_n) is a singleton, that is everyone plays a single build of scourge, followed by a single build of engi so and so forth. Also computational complexity is not particularly useful when people are involved either. A simple example you can think of is the Walrasian equilibrium, (you know where supply equals demand) with at least 3 goods is a well known NP hard problem. However, unlike computers humans can calculate the fixed points much faster albeit not algorithmically, so that you can easily ask college students to do it. Furthermore again the number equilibria (in this case prices and amounts sold) could be 1 or infinitely many. Now that number of equilibria, that is how many triplets of prices and demands you can get to me is the appropriate notion of diversity here, that is the cardinality of the fixed point. If you want to take it as a game as in game theory, how many equilibria exists (that is a tuples of strategies that are best responses to each other) is a distinctly separate question compared to how long it will take to reach one equilibria and how it scales with the parameters of the problem.. 

 

Edit: I realize my answer is very long winded, but the gist is, even if you add arbitrary dimensions to every object in S, it does not necessitate increasing the cardinality of the set E(M) =M. And that cardinality at least in my description is the number of meta vectors if you will. A trivial albeit unrelated example is thinking of a menu from a restaurant that has fantastic steak and mediocre everything else. Now no matter how many more vegan options you have, you can think that every week you will have some sort of steak dinner at the said restaurant, maybe changing the sides/type of steak etc. Now if you remove steak from the menu, then there is possibility that sometimes you can eat pasta, sometimes you eat salad and all the more variations, because there is no more superlative option.

 

Appreciate another comment from you thank you! There's a lot to say here so I hope you don't mind me parsing your comment down into sizeable chunks for me to manage.

 

Quote

This depends on your definition of diversity, you can certainly define complexity itself to be diversity, but I believe this doesn't serve any purpose for the discussion at hand .Notably if you start from some set m_1 what is the worst case scenario to reach a fixed point, that is how many times must you use our evolution whether it is E or F to eventually reach a fixed point. Now clearly it has a correlation with the dimensions and the size of the objects in S, whether it is computable in polynomial time or whatnot. You might use this as a proxy for how long it will take to reach a stable meta. This however does not tell you how many different classes are in the meta, which I believe is the appropriate notion of diversity here.

 

So diversity here is defined strictly as the opposite of homogeneity (sameness) as you would a mixture. If you have a bunch of things, and they are all the same, that's homogenous. If you have a bunch of things, and they are all different, then that's heterogenous (differentiation). You can imagine that if you have a system of 2 coins, and each coin can exist in either heads or tails, then the system can be in 1 of 4 possible states : HH, HT, TH, TT. The state of HH and TT are homogenous, while the state of HT and TH are heterogenous. You can extend this example to system in which there are 2 3sided coins, with Heads, Tails and Snouts, and what you get are three homogenous states (HHH, TTT, SSS) and 27 heterogenous states (7 unique). HTS, HTT, HHT, SHH, STT, SST, SSH. Looking at the heterogenous state, you can see that it is gradient, where HTS is the most heterogenous and the other 6 states are heterogenous to some lesser degree.

 

It's best to visualize diversity as the above, so that you can get a hold of the the notion that homogeneity and heterogeneity are not symmetric. You are more likely to find a system in a heterogenous state, then you are to find it in a homogenous state, and as you add more elements with dimension into a system, there are statistically more heterogenous states then there are homogenous states, and you also get the notion that systems are really just degrees of heterogeneity with homogeneity being outliers. In essence finding a system being in the configuration of all H's or all T's is statistically unlikely rather than finding the system in a state being just a semi-random slew of H's and T's.

 

Again we can get philosophical here, in that if you were to look at a system that consists of just completely random H's, T's, you notice that a system in perfect heterogeneity (HTS) is basically the definition of a system in thermodynamic equilibrium...and that becomes obvious when you realize that the solutions HHH, TTT, SSS and HTS are all equilibrium solutions to the above system. Given this, you find that you are statistically more likely, to find the system in one of the configurations that are somewhere in between the equilibria and in a nutshell this points to a nice explanation as to why we live in such a world that isn't in complete equilibrium or in perfect equity...we are somewhere in between, because we are statistically more likely to be there. 

 

My arguments here on the forum is basically this idea in extension. I sort of do away with the idea of equilibrium, and just focus on complexity side of it, because most things like this which are statistical will have regression to the mean, scale invariant behavior. You analyze part of a system locally, you will find some parts of it are homogenous, other parts will be highly diverse, other parts will be in equilibrium somewhere in between and on average, will reflect the system as a whole...so what matters to me more then things finding equilibrium, is just the complexity and how that has a global effect on the capacity of a system, finding it in heterogeneity. Luckily for this argument, it only varies as a component of time. More complexity = more time spent in heterogenous evolution. That to me seems rather simple and beautiful way to orient a solution to the diversity/balance problem.

 

Quote

. Also computational complexity is not particularly useful when people are involved either. A simple example you can think of is the Walrasian equilibrium, (you know where supply equals demand) with at least 3 goods is a well known NP hard problem. However, unlike computers humans can calculate the fixed points much faster albeit not algorithmically, so that you can easily ask college students to do it. Furthermore again the number equilibria (in this case prices and amounts sold) could be 1 or infinitely many.

 

So I have to look up many of the things you've told me so far, like Walrasian equilibrium, but I'm just gonna leave this comment here first. I wanted to mention that the game of gw2 and it's complexity is basically a NP hard problem, but its being run in parallel process like an algorithm, much like how ants optimize paths to find food, the fact that we are all parallel processing, and sharing information turns an NP hard problem into a problem that is solvable in polynomial time, and I think this is why the seemingly complex game of gw2 can find meta's so fast, within a few days or weeks rather then years. I guess it would be fun to imagine, what gw2 would be like if the game existed with only 2 players playing it, and asking how long would it take to find the optimal strategy and the best build.

 

Quote

This however does not tell you how many different classes are in the meta, which I believe is the appropriate notion of diversity here. Notably, you can have a fixed point that requires you to have say N^298179181981 iterations where N is the cardinality of S, but every time E(M_n) is a singleton, that is everyone plays a single build of scourge, followed by a single build of engi so and so forth....

 

Edit: I realize my answer is very long winded, but the gist is, even if you add arbitrary dimensions to every object in S, it does not necessitate increasing the cardinality of the set E(M) =M.

 

Okay so it took me some time to get this because you are computer science guy and I'm a physics guy and I was having a hard time visualizing the phase space of this problem, but allow me to simplify it for my own benefit and you tell me if I'm misunderstanding. E(Mₙ) in this problem, is that in every instance, all players are just changing from one build to a new build to a new build etc at every time interval, and all the points in the phase space is just the space of all the different possible builds that exist, where each build has a coordinate. As x moves along in the phase space, the vector changes (the amount or value of this vector is cardinal), with respect to some attractor(s) in the space, which creates the ordinality of the system. To simplify this, the system is in some non ordinal state of ABCD where at time=1 all players are playing D, at t=2 all players are playing C, at t=3 all players are playing B, and t=4 all players are playing A, and it creates the ordinance of A>B>C>D. 

 

So, maybe this is just me, but it's doesn't seem like a perfect model, maybe because x representing all players as instances of builds seems strange, rather then x representing a single person playing a single build, and just representing the whole evolution of x as a phase flow? You can imagine that if you dropped 100 players onto the coordinate plane where each coordinate is a different player playing the build at that coordinate, you'd get the freedom to use many instances at the same time interval, and then you just represent the whole thing as phase flow to see the attraction.

 

So ya, to put what I said in more simple terms, is that the ordinance (the attraction to certain builds) , is a consequence of the cardinality (the computation occurring to create a vector at each time interval ) of the system. Such a model means that at each coordinate, one has to make a computation, that computation gives that coordinate cardinality in the form of a vector, and it moves on to the next coordinate as a function of that vector which gives the system ordinance. This is probably hard to see if you have all players being instances of one build rather then representing it as each player playing a build at that coordinate and just using a phase flow.

 

If I got anything you said wrong, please let me know.

Edited June 5, 2021 by JusticeRetroHunter.7684

[Kuma.1503]

You could listen to the physics guy or the computer science guy. 

 

I draw pretty pictures for a living. I like the meta more when there are more symbols on screen. It gives the game a nice aesthetic. I remember a time when there was balance in PvP, you had a wibbly wobbly diamond things. Pointy red diamond things. There were as many symbols as there were grains of sand. 

 

Now it's all circles. I'm tired of circles. They're boring! smooth (Lol just like the brains of the people who play them),  and they have this garish shade of green. 

 

This isn't to throw shade at the circles. It's an unavoidable side effect when you assimilate into circle society. I was able to document this phenomenon myself. I will never get those 5 IQ points back. 

 

...

 

Thank you for your time. 

Edited June 5, 2021 by Kuma.1503

[JusticeRetroHunter.7684]

1 hour ago, Kuma.1503 said:

You could listen to the physics guy or the computer science guy. 

 

I draw pretty pictures for a living. I like the meta more when there are more symbols on screen. It gives the game a nice aesthetic. I remember a time when there was balance in PvP, you had a wibbly wobbly diamond things. Pointy red diamond things. There were as many symbols as there were grains of sand. 

 

Now it's all circles. I'm tired of circles. They're boring! smooth (Lol just like the brains of the people who play them),  and they have this garish shade of green. 

 

This isn't to throw shade at the circles. It's an unavoidable side effect when you assimilate into circle society. I was able to document this phenomenon myself. I will never get those 5 IQ points back. 

 

...

 

Thank you for your time. 

 

I'm a physics guy, but also just a regular guy, at first I just wanted to ask questions or give insight into the diversity/balance problem that's all I wanted and I thought it would be easy. I just didn't realize how hard the question was going to be and I had to spend years (it's seriously been about 2 or 3 years) attempting to understand all the components to the problem, forget the solution. You can see from the above that the topic spans 3 subject areas at least if not more (Chaos Theory-Physics-Computer Science) It's  a lot and I am no where near smart enough to formalize it, let alone simplify it to ape brain level where I can actually understand how it describes gw2.

 

But to clarify, complexity just means that right now, it's too easy to find the optimal strategy. It's too easy for players  to look at skills, and to find out which ones are the best, even if that notion is approximate. It's an optimization problem, and the problem is too easily solved. I believe that, by making things more heterogenous, rather then homogenizing them, it makes that optimization harder to quantify...

 

Like what Kolzar says about Pizza > Burgers. Pizza and burgers are so different from each other, that quantifying which one is better is hard to do. You can turn this into an optimization problem, and quantify whether pizza actually is better then a burger...but this involves a lot of complex computation. You can think of that computation in terms of asking questions, and providing answers to all possible scenarios like :

 

Pizza will give me 150 calories, Burgers will give me 450 calories. Eating a burger will make me use the bathroom 1.4 times as often. Cheese is poorly digested, so I get more nutrition from the burger then the pizza. Fat is a component of calorie intake so you will gain 5 more pounds eating a burger then you would pizza, 75% of my taste buds activate eating pizza while 40% of my taste buds activate eating burgers...

 

And this goes on and on into an infinite array of possible situations to quantify whether a Pizza slice is objectively better then a burger...in theory you can compute that but it is an NP Hard problem. 

 

In the case of a comparison of skills, think about how easy it is to quantify whether one skill is better then another skill. If you look at one skill that says "+10% damage" and another skill that says "+20% damage" then how long does such a computation take...it's probably a split second decision. In other words the game is too easy. because the skills here are too similar...too homogenous and not different enough where the problem is hard to solve.

 

So this is what is meant by complexity, and how differentiation makes the optimization problem more computationally complex, and that this is a good thing. If we look at the skills available to us, and we immediately know which skills are basically the best skills...then when we go to fight other players (who've done the same calculation for themselves) and our build can't beat theirs...then what else is there to play? Well you play their build on their class, and you can see how this  process is collapsing a system towards a meta game. 

 

Edited June 5, 2021 by JusticeRetroHunter.7684

[Kolzar.9567]

15 hours ago, JusticeRetroHunter.7684 said:

I'm a physics guy and I was having a hard time visualizing the phase space of this problem, but allow me to simplify it for my own benefit and you tell me if I'm misunderstanding. E(Mₙ) in this problem, is that in every instance, all players are just changing from one build to a new build to a new build etc at every time interval, and all the points in the phase space is just the space of all the different possible builds that exist, where each build has a coordinate.

You are almost right, but I think this is the source of confusion. Your space is over the subsets of all builds, you are not mapping builds to builds, but sets of builds to builds. Most of mechanical behavior is a real valued function, you take a point in space and map into another point in space. However, what I meant even in the most basic form was a correspondence, or a set valued function if you will. A very simplified example is the following: suppose there are only two classes and each class only has two selections. Class A with build x or y, and class B with build 1 or 2. Then the space S that I have defined would be the set of tuples S= { (A,x), (A,y), (B,1),(B,2)}. (As a reminder sets, unlike sequences do not have repetition,  for example in the real numbers set there is only one number 1. If you want to have proportions you need to define a measure over this set, which was the intention of putting a distribution over there.) Then a set valued mapping would take a subset of S, say {(A,y),(B,2)}, let's call this M_1, the initial meta and maps it into another subset of S, say {(A,x), (A,y),(B,2)}. So formally we have E({(A,y),(B,2)}) = {(A,x), (A,y),(B,2)}. Now the cardinality of the set  E({(A,y),(B,2)})=  {(A,x), (A,y),(B,2)}. Now we will call a set meta/viable set of builds if E( set) = set itself (that is both sides must be sub and supersets of each other). For example if we have  E({(A,x),(B,1)}) = {(A,x),(B,1)}, then you have reached a meta, or equilibrium, or formally a fixed point. Computational complexity asks if I start with some arbitrary set S_1 how many iterations would it take to reach some n such that E(S_n)=S_n. Clearly in most mechanical objects if we add more dimensions you would expect this n to increase, that is how long will it take a meta to settle down, it doesnt say how many builds are viable, which would be the cardinality of the set S_n, in our elementary example of E({(A,x),(B,1)}) = {(A,x),(B,1)} it was two. Now clearly before you reach your fixed point all of S_k, and S_l k,l<n these sets are different, but this is just the speed that meta evolves. That is why I tried to give the example that you can have singletons, say you start with the full set M(S) which is mapped to a single object say (A,x), which is mapped to E((A,x))=(B,y), where everyone plays a single build, it just changes what the flavor of the month is.  If in addition to having a rich meta, you also want to have a rich transition to meta you would need E(set) for any set to have a large cardinality, but the number of iterations before reaching a fixed point is irrelevant. Now if you wanted to incorporate the proportion of players playing the builds {(A,x),(B,1)} then you would move on to my second example, where you would put a distribution over this set, say there are twice as many players (A,x), (B,1), then you can define a simple measure over this set, which would give you a Markov process, A set along with a probability measure that only relies on the previous set.  A fixed point in this operation is a set and a measure which maps into the same set and measure now, which is formally known as the ergodic distribution of your markov process.  Now the number of iterations, that is the number n is at least in my opinion and experience is mostly irrelevant, as people move much faster and fill in the gaps of whatever algorithm you can come up with via intuition. Furthermore since the objects E or F are endogenous, references to complexity that is directly tied to how a fixed transition is a little odd, to examplify, you can try to look up ``a guessing half of the average game''  which has a unique equilibria (in my experience undergraduate students usually converge in 6-7 repetitions, graduate students usually do it in 1 or 2, if you do it algorithmically via elimination it depends on the number of elements in your inital set, say whether it is 10 or 1000) or for a more famous example a Cournot game. Notably although it is a little off topic a rather basic Cournot game (a quantity duopoly) can be solved by iterated elimination of dominated strategies but it takes infinitely many steps, yet you can just ask a student to solve it via intersecting best responses which hopefully they should be able to do in less than half a page. (something that is essentially very complex by complexity theory,  like the Walrasian equilibrium as well). The main gist of this difference is any kind meta/equilibrium is a fixed point, and unless that is a rather real valued function that you are interested in fixed point calculations are very complex, yet humans do it very efficiently.

 

Also as a I side note my intention with Walrasian equilibrium etc was not to confuse but to try to come up with the simplest examples I could although given the poor medium of an internet forum and unknown math background it is proving much more difficult than I imagined. 

Edited June 5, 2021 by Kolzar.9567

[JusticeRetroHunter.7684]

4 hours ago, Kolzar.9567 said:

Computational complexity asks if I start with some arbitrary set S_1 how many iterations would it take to reach some n such that E(S_n)=S_n. Clearly in most mechanical objects if we add more dimensions you would expect this n to increase, that is how long will it take a meta to settle down, it doesnt say how many builds are viable, which would be the cardinality of the set S_n,

 

I'm just not seeing where this idea about viability comes from when what's describing the evolution of the system for what is the meta is the ordinance, which is equal to the cardinality.

 

Adding dimensions increases n, which increases the cardinality of the set S which can be mapped equally to the systems ordinance. If it was a singleton evolution like stated before, then n should be proportional to the number of time intervals needed to reach equilibrium.

 

Quote

Also as a I side note my intention with Walrasian equilibrium etc was not to confuse but to try to come up with the simplest examples I could although given the poor medium of an internet forum and unknown math background it is proving much more difficult than I imagined. 

I can follow along with your comments, but you got to level with me a bit here because this is the precipice of my knowledge on the subject in this detail. Thanks

Edited June 5, 2021 by JusticeRetroHunter.7684

[Kolzar.9567]

26 minutes ago, JusticeRetroHunter.7684 said:

I'm just not seeing where this idea about viability comes from when what's describing the evolution of the system for what is the meta is the ordinance, which is equal to the cardinality.

That is the issue, the operators E or F is jointly controlled by your players as opposed to being an exogenous object like say heat dispersion where you have a nice heat equation. I have not attempted to describe how these E and F are determined other than saying they are endogenous. That is I have described to  you the space S, the objects in this space, and an implicit operator. Clearly designing the game so that players can form E and F is dev's job. However, regardless of how E or F is determined the appropriate notion of diversity, at least in my opinion is related to the size of the fixed point, as opposed to the complexity of the fixed point, which is how many iterations it would take you to reach.

 

As a side note complexity can increase richness of play, when the set of available strategies to a player is way too large to compute. A good example is chess, by Zermelo's theorem it is actually a fairly simple game, however a strategy, which is a complete contingent plan of action is rather hard to compute hence people use approximations, hence we keep playing it still. Guild wars on the other hand is relatively simple as far as build choices go.

[JusticeRetroHunter.7684]

5 hours ago, Kolzar.9567 said:

However, regardless of how E or F is determined the appropriate notion of diversity, at least in my opinion is related to the size of the fixed point, as opposed to the complexity of the fixed point, which is how many iterations it would take you to reach.

Honestly, you aren't wrong, and I agree with you on that, but I just think you're model is missing the component of computation time and how that time scales with the size and dimensionality of the system.

 

The video i linked earlier in the derivation basically does the same thing you did, but instead of mapping things via a set valued function, Susskind maps it like you would a real valued function, where all possible states are mapped 1 for 1 to a coordinate space, where each coordinate is just one configuration in a space of all possible configurations. So rather then their being vectors within vectors, all that dimensionality is just mapped out to it's own coordinate. In this way he can give the system ordinance and treat it like a geometry.  Frankly I don't see why you can't do that here too.

 

Is there a reason why you are using a set valued function rather then a real valued function? I'm not exactly knowledgeable on set theory so hopefully that's not a dumb question. Bubblegum and tape learning here.

Edited June 6, 2021 by JusticeRetroHunter.7684

[Pimsley.3681]

The last page of this thread is something you'd expect from PvPers that play Engie. Tech company recruiters will have a field day on here. This made my day, such a different tone from mapchat.